Abstract:
A method and system for increasing the detection, location, identification or classification of objects hidden on the surface or buried below the surface of the ground is disclosed. The method acquires image data in separate IR and/or visible spectral regions simultaneously and converts the data into intensity value arrays for each spectral region. These intensity value arrays are transformed into two-dimensional discrete wavelet transform arrays for each spectral region. The background clutter from the two-dimensional discrete wavelet transform arrays is removed; forming clutter reduced two-dimensional discrete wavelet transform arrays. The inverse two-dimensional discrete wavelet transform is performed on the clutter reduced two-dimensional discrete wavelet transform arrays to form clutter removed intensity value arrays. These arrays are subtracted in a pair-wise mariner to obtain chemical-specific spectral signatures. The processed images are correlated with 3-dimensional matched filters of known emissive signatures of objects to detect the presence of the object.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application claims the benefit of U.S. Provisional Patent Application No. 61/112,245 titled “Methods of Using Spatial/Spectral/Temporal Imaging for Hidden or Buried Explosive Detection,” filed Nov. 6, 2008, the contents of which are hereby incorporated by reference in its entirety. 
     
    
     TECHNICAL FIELD 
       [0002]    The present disclosure is in the field of electronic imaging. More particularly in the area of multispectral surveillance imaging systems for identifying explosives hidden on the surface of or buried under the ground. 
       BACKGROUND OF THE INVENTION 
       [0003]    The detection and identification of explosives hidden on the surface or buried under the ground has a multitude of commercial applications. After Sep. 11, 2000 the United States proclaimed war on the terrorists responsible for the attack on the Twin Towers in New York City. This included those directly involved as well as those who actively aided in the attack. Military intelligence from a number of sources identified specific individuals including the then ruler of Iraq. In previous United Nations investigations of Iraq sanctions were instituted against the country and its ruler to force compliance with international nuclear weapons reduction treaties. When attempts failed the United States invaded Iraq. During the attack and during the Iraq occupation non-conventional weapons were used by the insurgence to retaliate against United States forces. These devices were called improvised explosive devices or “IED”s. Because of their non-conventional appearance they were difficult to identify and consequently caused a number of casualties. Currently a number of methods exist for the detection of IED&#39;s including (1) Ground Penetrating Radar, which does not meet performance targets established by the military, (2) Electrical Impedance Tomography, which requires physical contact with the ground and which cannot be used in very dry soil, (3) X-Ray Backscatter, which is physically large, consumes large power and which requires high X-ray power to operate, (4) Acoustic-Seismic device, which is not effective when surface vegetation is present, (5) Chemical Vapor detection system, which is not sensitive to very low chemical density, (6) Biological method where mammals like dogs do not perform well under fatigue, (7) Nuclear Quadrupole Resonance, which is effective only at high signal-to-noise ratios, (8) Prodders and Probes, which are risky due to the possibility of detonation, (9) Thermal Signature Detection using analytical models, which requires soil-specific parameters such as heat capacity, soil density and thermal conductivity and which requires intensive iterative computations that may be unstable. Most of these existing methods either make physical contact with the ground or are not reliable. 
         [0004]    Consequently there is a need for a multispectral imaging system that can detect, identify, locate and classify explosives such as improvised explosive devices (IEDs) both hidden on the surface or buried below the surface of the ground that is more reliable than current methods, does not require substantial amounts of energy to operate and does not require physical contact with the soil. 
       SUMMARY OF THE INVENTION 
       [0005]    The following presents a simplified summary in order to provide a basic understanding of some aspects of the claimed subject matter. This summary is not an extensive overview, and is not intended to identify key/critical elements or to delineate the scope of the claimed subject matter. Its purpose is to present some concepts in a simplified form as a prelude to the more detailed description that is presented later. 
         [0006]    In one aspect of the present disclosure, a method for detecting, identifying or classifying of objects hidden on the surface or buried below the surface of the ground is provided, comprising: acquiring image data in separate infrared (IR) and/or visible spectral regions simultaneously; converting the acquired image data into intensity value arrays for each spectral region; applying a transformation on the intensity value arrays to form two-dimensional discrete wavelet transform arrays for each spectral region; removing background clutter from the two-dimensional discrete wavelet transform arrays to form clutter reduced two-dimensional discrete wavelet transform arrays; performing an inverse two-dimensional discrete wavelet transform on the clutter reduced two-dimensional discrete wavelet transform arrays to form clutter removed intensity value arrays; subtracting the clutter removed intensity value arrays in a pair-wise manner to obtain chemical-specific spectral signatures; and correlating the chemical-specific spectral signatures with 3-dimensional matched filters of known emissive signatures of objects to detect a presence of the object. 
         [0007]    In another aspect of the present disclosure, the above method is provided, further comprising: applying at least one of a 3- and 4-dimensional feature space to compare a thermal spectral signature of an inspected soil with a thermal spectral signature of surrounding soil, to determine a disturbed soil condition. 
         [0008]    To the accomplishment of the foregoing and related ends, certain illustrative aspects are described herein in connection with the following description and the annexed drawings. These aspects are indicative, however, of but a few of the various ways in which the principles of the claimed subject matter may be employed and the claimed subject matter is intended to include all such aspects and their equivalents. 
         [0009]    Other advantages and novel features may become apparent from the following detailed description when considered in conjunction with the drawings. As such, other aspects of the disclosure are found throughout the specification. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0010]      FIG. 1  is an illustration of capturing spatial/spectral/temporal signature of explosive chemical; 
           [0011]      FIG. 2  is an illustration of image acquisition and processing; and 
           [0012]      FIG. 3  is an illustration of clusters of soil constituents in a 3-dimensional feature space. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0013]    The present disclosure provides methods for detecting explosive hidden above or buried under the ground, such as IEDs, exploiting the thermal emissivity of chemical explosives. Each chemical explosive has a distinct emissivity characteristic or signature over a particular spectral region. This spectral signature falls in the long wave infrared (LWIR) region. Spectral signatures of explosives also change with time. The present disclosure detects and classifies buried IEDs using this thermal detection. An infrared detector array captures the thermal emissivity of the soil and its constituents. In order to distinguish the chemical signature from background clutter, we use infrared detector array as it is more suited to the spectral range of interest. This disclosure is unique in that it utilizes spatial, temporal and two spectral bands—a total of four dimensions to detect as much information about the chemical and background signatures as possible to maximize the probability of detection and minimize false alarm rates. 
         [0014]      FIG. 1  depicts one embodiment of the present disclosure. The system captures the spectral signatures of the land surface area being imaged in two distinct bands of the electromagnetic spectrum. The land surface is scanned over a certain area at a fixed rate. The cylindrical lens focuses the scene on to the detector array of size M pixels by N rows. The focused thermal image is split by a beam splitter. Each beam then passes through a prism, which decomposes the corresponding thermal signal into its spectral components and the spectral components are captured by the detector array. Thus, each column of the detector array represents the emissive spectral signature of the land surface area that is being imaged on to a pixel. This process of capturing emissive signatures is carried out in time sequence thereby acquiring spatial-spectral-temporal information of the land surface area of interest. 
         [0015]      FIG. 2  is a schematic diagram showing signal acquisition and processing. Each pixel value is subtracted from the background value and the difference signal is digitized by an analog-to-digital converter (ADC). This scheme of analog-to-digital conversion improves the accuracy of representing a pixel value in digital form with a fixed number of bits of representation. The same procedure is used for the pixels in the second spectral band, as shown. After the signal is acquired, the images from the two spatial-spectral-temporal dimensions are processed either by a special purpose signal processing hardware or by a computer (PC or mainframe). The processing consists of optimal weighted differencing whose purpose is to eliminate background clutter and extract chemical-specific spectral signatures. After signature extraction, detection and classification tasks are performed. 
         [0016]    Detection is achieved by correlating the processed images with 3-dimensional matched filters. These matched filters are known a priori as they correspond to the emissive signatures of known chemical explosives. If the form of these filters are known analytically (for example if they are treated as Gaussian functions), then the filter parameters such as the mean and standard deviation are known or may be estimated from the acquired thermal images. The processed images are correlated with all the matched filters from a database and corresponding test statistics are computed. These test statistics are used in a maximum a posteriori (MAP) sense to test the hypotheses to decide whether an IED is present or not. 
         [0017]    Classification of the detected chemical explosives is conducted using a suitable clustering procedure. In the 3- or 4-dimensional feature space, each soil constituent forms a cluster that is distinguished from the others, as shown in  FIG. 3 . In order to classify the detected chemical explosives, we compute the feature vectors of the emissive signatures and use the maximum likelihood (ML) method to identify the corresponding classes. 
         [0018]    Our current disclosure exploits the following features of chemical explosives: (1) Disturbance of soil with respect to the surrounding—digging and placing explosive chemicals in the soil changes the appearance and is captured by both visible and infrared detectors. (2) Altered thermal signature—changes the thermal signature of soil by amounts exceeding 0.1° K. (3) Thermal emissive spectral signature—soil disturbance enhances spectral contrast. Because of these informative features used for detection and classification, the present disclosure is more robust and reliable as compared to the currently used methods. Moreover, the present disclosure can be mast-mounted on a vehicle or mounted on a helicopter and has little risk of detonation, covers a large area and can be carried out covertly. Additionally, the present disclosure uses (I) custom sensor in the visible spectrum to give a strong signal of the land surface, (II) focal plane arrays for the near IR spectrum at room temperature, (III) cantilever technology for the sensor in the LWIR spectrum. It should be pointed out that the infrared sensor using cantilever technology is readily available in the market and that it requires no cooling as opposed to other systems which require cooling by liquid nitrogen. 
         [0019]    Camera 
         [0020]    The multispectral surveillance system (MLSS) is a system for the detection of emissivity characteristics or signatures of explosives on or below the surface of the ground in the long wave infrared (LWIR) region through the elimination of most of the surface reflected thermal emission clutter components from multispectral images. The digitized outputs of the multispectral sensor are first captured and stored in the memory of a frame grabber card. Two of the three available spectral bands, X and Y, are selected for the processing that follows. 
         [0021]    Sensors must exhibit a linear response to the input thermal emission; i.e., the output signal doubles for a doubling of the intensity of IR signal on the sensor. Deviation from this behavior is a non-linear response, which is measured for each pixel. A set of curve fitting coefficients is calculated for each pixel or class of pixels. These coefficients are then used to calculate the correction to be applied to each pixel during frame readout, and are stored in a ROM and table lookup. 
         [0022]    The algorithms, operating system, and system control files are stored in an operational file folder that is downloaded to the appropriate units upon power turn on of the system. The algorithms and system control functions are then executed by the appropriate CPU, DSP, FPGA, or ASIC. 
         [0023]    Use 
         [0024]    The present disclosure is a processing method used for the detection of explosives hidden on the surface or buried in the ground using spectral images. These processing methods work with either two or three component images. In either case, the processing methods can be classified as optimal or suboptimal. Optimal processing methods use the statistics of the component images while the suboptimal ones use ad hoc parameters. FIG. 8 (former FIG. 1, patent app. 2) shows the classification of the various pre-detection processing methods described below. These processing methods could be performed purely in the spatial domain, frequency domain or the time and spatial domain. 
         [0025]    In optimal processing, the camera sensors produce intensity images with differing intensity values in the three IR/visible spectral regions depending on the emissivity characteristics of the explosive, the presence of other infrared emitting sources such as fires or geothermal areas and reflections from the ground in the area being screened. The main idea behind explosive detection, therefore, is to subtract one component image from another so that the difference image retains only the explosive&#39;s signature. Preferably the signal from the surface reflection has been completely removed. It should be pointed out that the contrast due to the target between the component images is very small and depends on the emissivity of the explosives in the spectral regions. Thus, the process methods discriminate a very small contrast from image components. The ability to detect such a miniscule contrast depends on 1) the sensor dynamic range, 2) effectiveness of the preprocessing methods in eliminating background clutter, and 3) detection algorithm itself. Our processing methods address the third element primarily because the first is a limitation of the sensor and the second can be eliminated with standard methods known to those skilled in the art. 
         [0026]    The processing methods of the present disclosure may be applied to two-channel and three-channel imaging. The processing methods for two-channel imaging may be used to minimize the mean square error, error variance, correlation and/or covariance. To minimize the mean square error a scaled image component is subtracted from another to enhance the low contrast of the explosive&#39;s signature. Consider the channels spanning three contiguous IR spectral bands. For the sake of argument, let us denote the three such contiguous channel pixels at location (m,n) by R(m,n), G(m,n) and B(m,n) and name them as red, green and blue respectively. Then, define the difference pixel e R (m,n) at the same pixel location as Equation 1 
         [0000]        e   R ( m,n )= R ( m,n )−α R   G ( m,n ) 
         [0000]    We want to determine the scale factor α R , so as to minimize the mean square error 
         [0000]    
       
         
           
             
               ɛ 
               R 
             
             = 
             
               
                 1 
                 MN 
               
                
               
                 
                   ∑ 
                   
                     m 
                     = 
                     0 
                   
                   
                     M 
                     - 
                     1 
                   
                 
                  
                 
                   
                     ∑ 
                     
                       n 
                       = 
                       0 
                     
                     
                       N 
                       - 
                       1 
                     
                   
                    
                   
                     
                       
                         e 
                         R 
                         2 
                       
                        
                       
                         ( 
                         
                           m 
                           , 
                           n 
                         
                         ) 
                       
                     
                     . 
                   
                 
               
             
           
         
       
     
         [0000]    The optimal coefficient α* R  is obtained by setting the derivative of ε R  with respect to α R , to zero. This results in the optimal coefficient value as given by Equation 2 
         [0000]    
       
         
           
             
               α 
               R 
               * 
             
             = 
             
               
                 
                   ∑ 
                   
                     m 
                     = 
                     0 
                   
                   
                     M 
                     - 
                     1 
                   
                 
                  
                 
                   
                     ∑ 
                     
                       n 
                       = 
                       0 
                     
                     
                       N 
                       - 
                       1 
                     
                   
                    
                   
                     
                       R 
                        
                       
                         ( 
                         
                           m 
                           , 
                           n 
                         
                         ) 
                       
                     
                      
                     
                       G 
                        
                       
                         ( 
                         
                           m 
                           , 
                           n 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ∑ 
                   
                     m 
                     = 
                     0 
                   
                   
                     M 
                     - 
                     1 
                   
                 
                  
                 
                   
                     ∑ 
                     
                       n 
                       = 
                       0 
                     
                     
                       N 
                       - 
                       1 
                     
                   
                    
                   
                     
                       G 
                       2 
                     
                      
                     
                       ( 
                       
                         m 
                         , 
                         n 
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
         [0000]    Note that both the numerator and denominator have the same normalizing factor 1/MN and can, therefore, be omitted. The numerator of Equation 2 is a measure of the correlation between the two spectral bands R(m,n) and G(m,n) under consideration. 
         [0027]    In a similar manner we can obtain the optimal scaling coefficients for the green-blue and blue-red channels and are given, respectively, by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       α 
                       
                         G 
                          
                         
                             
                         
                       
                       * 
                     
                     = 
                     
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               G 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               B 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           
                             ∑ 
                             
                               m 
                               = 
                               0 
                             
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               B 
                               2 
                             
                              
                             
                               ( 
                               
                                 m 
                                 , 
                                 n 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       α 
                       B 
                       * 
                     
                     = 
                     
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               B 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               R 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 0 
                               
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               R 
                               2 
                             
                              
                             
                               ( 
                               
                                 m 
                                 , 
                                 n 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   3 
                 
               
             
           
         
       
     
         [0028]    Minimization of the error variance ε R  instead of minimizing the mean square error, is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           ɛ 
                           R 
                         
                         = 
                           
                          
                         
                           
                             1 
                             MN 
                           
                            
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         e 
                                         R 
                                       
                                        
                                       
                                         ( 
                                         
                                           m 
                                           , 
                                           n 
                                         
                                         ) 
                                       
                                     
                                     - 
                                     
                                       μ 
                                       
                                         cR 
                                          
                                         
                                             
                                         
                                       
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             1 
                             MN 
                           
                            
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         
                                           
                                             ( 
                                             
                                               
                                                 R 
                                                  
                                                 
                                                   ( 
                                                   
                                                     m 
                                                     , 
                                                     n 
                                                   
                                                   ) 
                                                 
                                               
                                               - 
                                               
                                                 μ 
                                                 
                                                   R 
                                                    
                                                   
                                                       
                                                   
                                                 
                                               
                                             
                                             ) 
                                           
                                           - 
                                         
                                       
                                     
                                     
                                       
                                         
                                           
                                             α 
                                             R 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 G 
                                                  
                                                 
                                                   ( 
                                                   
                                                     m 
                                                     , 
                                                     n 
                                                   
                                                   ) 
                                                 
                                               
                                               - 
                                               
                                                 μ 
                                                 G 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   4 
                 
               
             
           
         
       
     
         [0000]    where μ R  is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     μ 
                     cR 
                   
                   = 
                   
                     
                       
                         1 
                         MN 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             R 
                              
                             
                               ( 
                               
                                 m 
                                 , 
                                 n 
                               
                               ) 
                             
                           
                         
                       
                     
                     - 
                     
                       
                         
                           α 
                           R 
                         
                         MN 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             G 
                              
                             
                               ( 
                               
                                 m 
                                 , 
                                 n 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   5 
                 
               
             
           
         
       
     
         [0000]    This results in the optimal coefficient, which is expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     α 
                     R 
                     * 
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           m 
                           = 
                           0 
                         
                         
                           M 
                           - 
                           1 
                         
                       
                        
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ( 
                             
                               
                                 R 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 μ 
                                 R 
                               
                             
                             ) 
                           
                            
                           
                             ( 
                             
                               
                                 G 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 μ 
                                 G 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         ∑ 
                         
                           m 
                           = 
                           0 
                         
                         
                           M 
                           - 
                           1 
                         
                       
                        
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ( 
                             
                               
                                 G 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 μ 
                                 G 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   6 
                 
               
             
           
         
       
     
         [0029]    The numerator of 
         [0030]    Equation 6 is a measure of the covariance rather than correlation and the denominator is the variance rather than sum of squares. Following the same argument for the red-green channel, we obtain the optimal coefficients for the green-blue and blue-red channels as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       α 
                       G 
                       * 
                     
                     = 
                     
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               ( 
                               
                                 
                                   G 
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   μ 
                                   G 
                                 
                               
                               ) 
                             
                              
                             
                               ( 
                               
                                 
                                   B 
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   μ 
                                   B 
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               ( 
                               
                                 
                                   B 
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   μ 
                                   B 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       α 
                       B 
                       * 
                     
                     = 
                     
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               ( 
                               
                                 
                                   B 
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   μ 
                                   B 
                                 
                               
                               ) 
                             
                              
                             
                               ( 
                               
                                 
                                   R 
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   μ 
                                   R 
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               ( 
                               
                                 
                                   R 
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   μ 
                                   R 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   7 
                 
               
             
           
         
       
     
         [0031]    While the above procedures determine the optimal coefficients by minimizing the mean square error or the error variance, there exists an alternative minimization criterion to compute the optimal weights. This is the minimum correlation between the difference channels. Consider the two error images corresponding to the red and green channels, as given by: 
         [0000]        e   R ( m,n )= R ( m,n )−α R1   G ( m,n ),  e   G ( m,n )= G ( m,n )−α G1   B ( m,n )  Equation 8 
         [0000]    Determination of the two coefficients, α R1 , and α G1 , by minimizing the cross correlation is expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           ɛ 
                           RG 
                         
                         = 
                           
                          
                         
                           
                             1 
                             MN 
                           
                            
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   
                                     e 
                                     R 
                                   
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     e 
                                     G 
                                   
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             1 
                             MN 
                           
                            
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     R 
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                   - 
                                   
                                     
                                       α 
                                       
                                         R 
                                          
                                         
                                             
                                         
                                          
                                         1 
                                       
                                     
                                      
                                     
                                       G 
                                        
                                       
                                         ( 
                                         
                                           m 
                                           , 
                                           n 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           ( 
                           
                             
                               G 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                             - 
                             
                               
                                 α 
                                 
                                   G 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                                
                               B 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   9 
                 
               
             
           
         
       
     
         [0000]    Using the standard procedure for the minimization, the following normal equations are obtained: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         α 
                         
                           G 
                            
                           
                               
                           
                            
                           1 
                         
                         * 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               G 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               B 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     - 
                     
                       
                         ∑ 
                         
                           m 
                           = 
                           0 
                         
                         
                           M 
                           - 
                           1 
                         
                       
                        
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                          
                         
                           
                             G 
                             2 
                           
                            
                           
                             ( 
                             
                               m 
                               , 
                               n 
                             
                             ) 
                           
                         
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   10 
                 
               
             
             
               
                 
                   
                     
                       
                         α 
                         
                           R 
                            
                           
                               
                           
                            
                           1 
                         
                         * 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               G 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               B 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     - 
                     
                       
                         ∑ 
                         
                           m 
                           = 
                           0 
                         
                         
                           M 
                           - 
                           1 
                         
                       
                        
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                          
                         
                           
                             B 
                              
                             
                               ( 
                               
                                 m 
                                 , 
                                 n 
                               
                               ) 
                             
                           
                            
                           
                             R 
                              
                             
                               ( 
                               
                                 m 
                                 , 
                                 n 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   11 
                 
               
             
             
               
                 
                   
                     
                       
                         α 
                         
                           B 
                            
                           
                               
                           
                            
                           1 
                         
                         * 
                       
                        
                       
                         
                           
                             ∑ 
                             
                               m 
                               = 
                               0 
                             
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               B 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               R 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     - 
                     
                       
                         ∑ 
                         
                           m 
                           = 
                           0 
                         
                         
                           M 
                           - 
                           1 
                         
                       
                        
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                          
                         
                           
                             B 
                             2 
                           
                            
                           
                             ( 
                             
                               m 
                               , 
                               n 
                             
                             ) 
                           
                         
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   12 
                 
               
             
             
               
                 
                   
                     
                       
                         α 
                         
                           G 
                            
                           
                               
                           
                            
                           1 
                         
                         * 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               B 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               R 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     - 
                     
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                         
                         
                           M 
                           - 
                           1 
                         
                       
                        
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                          
                         
                           
                             R 
                              
                             
                               ( 
                               
                                 m 
                                 , 
                                 n 
                               
                               ) 
                             
                           
                            
                           
                             G 
                              
                             
                               ( 
                               
                                 m 
                                 , 
                                 n 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   13 
                 
               
             
             
               
                 
                   
                     
                       
                         α 
                         
                           R 
                            
                           
                               
                           
                            
                           1 
                         
                         * 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               R 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               G 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     - 
                     
                       
                         ∑ 
                         
                           m 
                           = 
                           0 
                         
                         
                           M 
                           - 
                           1 
                         
                       
                        
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                          
                         
                           
                             R 
                             2 
                           
                            
                           
                             ( 
                             
                               m 
                               , 
                               n 
                             
                             ) 
                           
                         
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   14 
                 
               
             
             
               
                 
                   
                     
                       
                         α 
                         
                           B 
                            
                           
                               
                           
                            
                           1 
                         
                         * 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               R 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               G 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     - 
                     
                       
                         ∑ 
                         
                           m 
                           = 
                           0 
                         
                         
                           M 
                           - 
                           1 
                         
                       
                        
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                          
                         
                           
                             G 
                              
                             
                               ( 
                               
                                 m 
                                 , 
                                 n 
                               
                               ) 
                             
                           
                            
                           
                             B 
                              
                             
                               ( 
                               
                                 m 
                                 , 
                                 n 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   15 
                 
               
             
           
         
       
     
       Since 
     Equation 11 and 
       [0032]    Equation 14 are distinct, as are the other equations, by adding them α* R1 , can be resolved by: 
         [0000]    
       
         
           
             
               
                 
                   
                     α 
                     
                       R 
                        
                       
                           
                       
                        
                       1 
                     
                     * 
                   
                   = 
                   
                     
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               R 
                               2 
                             
                              
                             
                               ( 
                               
                                 m 
                                 , 
                                 n 
                               
                               ) 
                             
                           
                         
                       
                       + 
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               B 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               R 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               R 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               G 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       + 
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               G 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               B 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   16 
                 
               
             
           
         
       
     
         [0000]    In a like manner, the optimal coefficients for the green and blue channels are given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       α 
                       
                         G 
                          
                         
                             
                         
                          
                         1 
                       
                       * 
                     
                     = 
                     
                       
                         
                           
                             ∑ 
                             
                               m 
                               = 
                               0 
                             
                             
                               M 
                               - 
                               1 
                             
                           
                            
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 G 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                         + 
                         
                           
                             ∑ 
                             
                               m 
                               = 
                               0 
                             
                             
                               M 
                               - 
                               1 
                             
                           
                            
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 R 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                   ) 
                                 
                               
                                
                               
                                 G 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             ∑ 
                             
                               m 
                               = 
                               0 
                             
                             
                               M 
                               - 
                               1 
                             
                           
                            
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 G 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                   ) 
                                 
                               
                                
                               
                                 B 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         + 
                         
                           
                             ∑ 
                             
                               m 
                               = 
                               0 
                             
                             
                               M 
                               - 
                               1 
                             
                           
                            
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 B 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                   ) 
                                 
                               
                                
                               
                                 R 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   17 
                 
               
             
             
               
                 
                   
                     
                       α 
                       
                         B 
                          
                         
                             
                         
                          
                         1 
                       
                       * 
                     
                     = 
                     
                       
                         
                           
                             ∑ 
                             
                               m 
                               = 
                               0 
                             
                             
                               M 
                               - 
                               1 
                             
                           
                            
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 B 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                         + 
                         
                           
                             ∑ 
                             
                               m 
                               = 
                               0 
                             
                             
                               M 
                               - 
                               1 
                             
                           
                            
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 G 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                   ) 
                                 
                               
                                
                               
                                 B 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             ∑ 
                             
                               m 
                               = 
                               0 
                             
                             
                               M 
                               - 
                               1 
                             
                           
                            
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 B 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                   ) 
                                 
                               
                                
                               
                                 R 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         + 
                         
                           
                             ∑ 
                             
                               m 
                               = 
                               0 
                             
                             
                               M 
                               - 
                               1 
                             
                           
                            
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 R 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                   ) 
                                 
                               
                                
                               
                                 G 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   18 
                 
               
             
           
         
       
     
         [0033]    These are listed in Table 1. 
         [0034]    By minimizing the covariance instead of the correlation between the channels, we obtain another set of optimal coefficients. These are identical in form to the coefficients in 
         [0035]    Equation 11 and 
         [0036]    Equation 14 except that each variable is replaced by its mean-removed value. These are listed in Table 1. 
         [0037]    The processing methods for three-channel imaging may also be used to minimize the mean square error, error variance, correlation and/or covariance. In the three-channel imaging, all the three channels are considered to generate the difference image. Consequently, Equation 1 is rewritten as: 
         [0000]        e   R ( m,n )= R ( m,n )−α R   G ( m,n )−β R   B ( m,n )  Equation 19 
         [0000]    Similarly, the other two difference channels are given by: 
         [0000]        e   G ( m,n )= G ( m,n )−α G   B ( m,n )−β G   R ( m,n )  Equation 20 
         [0000]        e   B ( m,n )= B ( m,n )−α B   R ( m,n )−β B   G ( m,n )  Equation 21 
         [0000]    Similar to the two-channel case, the minimization of the mean square error gives us the normal equations for the optimal coefficients: 
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   G 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   G 
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   B 
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   G 
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   B 
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   B 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                    
                   
                       
                     
                       
                         [ 
                         
                           
                             
                               
                                 α 
                                 R 
                                 * 
                               
                             
                           
                           
                             
                               
                                 α 
                                 R 
                                 * 
                               
                             
                           
                         
                         ] 
                       
                       = 
                       
                         [ 
                         
                           
                             
                               
                                 
                                   ∑ 
                                   
                                     m 
                                     = 
                                     0 
                                   
                                   
                                     M 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       n 
                                       = 
                                       0 
                                     
                                     
                                       N 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                     
                                       R 
                                        
                                       
                                         ( 
                                         
                                           m 
                                           , 
                                           n 
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       G 
                                        
                                       
                                         ( 
                                         
                                           m 
                                           , 
                                           n 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ∑ 
                                   
                                     m 
                                     = 
                                     0 
                                   
                                   
                                     M 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       n 
                                       = 
                                       0 
                                     
                                     
                                       N 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                     
                                       B 
                                        
                                       
                                         ( 
                                         
                                           m 
                                           , 
                                           n 
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       R 
                                        
                                       
                                         ( 
                                         
                                           m 
                                           , 
                                           n 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   22 
                 
               
             
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   B 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   B 
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   R 
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   B 
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   R 
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   R 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                    
                   
                       
                     
                       
                         [ 
                         
                           
                             
                               
                                 α 
                                 G 
                                 * 
                               
                             
                           
                           
                             
                               
                                 β 
                                 G 
                                 * 
                               
                             
                           
                         
                         ] 
                       
                       = 
                       
                         [ 
                         
                           
                             
                               
                                 
                                   ∑ 
                                   
                                     m 
                                     = 
                                     0 
                                   
                                   
                                     M 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       n 
                                       = 
                                       0 
                                     
                                     
                                       N 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                     
                                       G 
                                        
                                       
                                         ( 
                                         
                                           m 
                                           , 
                                           n 
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       B 
                                        
                                       
                                         ( 
                                         
                                           m 
                                           , 
                                           n 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ∑ 
                                   
                                     m 
                                     = 
                                     0 
                                   
                                   
                                     M 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       n 
                                       = 
                                       0 
                                     
                                     
                                       N 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                     
                                       R 
                                        
                                       
                                         ( 
                                         
                                           m 
                                           , 
                                           n 
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       G 
                                        
                                       
                                         ( 
                                         
                                           m 
                                           , 
                                           n 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   23 
                 
               
             
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                   
                                     R 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                   
                                     R 
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     G 
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                   
                                     R 
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     G 
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                   
                                     G 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               α 
                               B 
                               * 
                             
                           
                         
                         
                           
                             
                               β 
                               B 
                               * 
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                         
                           
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                   
                                     B 
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     R 
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                   
                                     G 
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     B 
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   24 
                 
               
             
           
         
       
     
         [0038]    The optimal coefficients for the minimum error variance case will be identical to those in 
         [0039]    Equation 22 through 
         [0040]    Equation 24, for the three-channel case, except that each term must be replaced by its mean removed value. These are listed in Table 2. 
         [0041]    The difference channels are as defined in 
         [0042]    Equation 19 to 
         [0043]    Equation 21. Minimization of the pair-wise correlations is as follows. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ɛ 
                       R 
                     
                     = 
                     
                       
                         1 
                         MN 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               
                                 e 
                                 R 
                               
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               
                                 e 
                                 
                                   G 
                                    
                                   
                                       
                                   
                                 
                               
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       ɛ 
                       G 
                     
                     = 
                     
                       
                         1 
                         MN 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 0 
                               
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               
                                 e 
                                 G 
                               
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               
                                 e 
                                 B 
                               
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       ɛ 
                       B 
                     
                     = 
                     
                       
                         1 
                         MN 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               
                                 e 
                                 B 
                               
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               
                                 e 
                                 R 
                               
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   25 
                 
               
             
           
         
       
     
         [0000]    By setting the derivatives of the three terms in
 
Equation 25 with respect to the three coefficients the following equations are obtained:
 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         α 
                         G 
                       
                        
                       
                         
                           
                             ∑ 
                             
                               m 
                               = 
                               0 
                             
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               G 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               B 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     + 
                     
                       
                         β 
                         G 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               R 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               G 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         m 
                         = 
                         0 
                       
                       
                         M 
                         - 
                         1 
                       
                     
                      
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                        
                       
                         
                           G 
                           2 
                         
                          
                         
                           ( 
                           
                             m 
                             , 
                             n 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   26 
                 
               
             
             
               
                 
                   
                     
                       
                         α 
                         R 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               G 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               B 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     + 
                     
                       
                         β 
                         R 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               B 
                               2 
                             
                              
                             
                               ( 
                               
                                 m 
                                 , 
                                 n 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         m 
                         = 
                         0 
                       
                       
                         M 
                         - 
                         1 
                       
                     
                      
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                        
                       
                         
                           B 
                            
                           
                             ( 
                             
                               m 
                               , 
                               n 
                             
                             ) 
                           
                         
                          
                         
                           R 
                            
                           
                             ( 
                             
                               m 
                               , 
                               n 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   27 
                 
               
             
             
               
                 
                   
                     
                       
                         α 
                         G 
                       
                        
                       
                         
                           
                             ∑ 
                             
                               m 
                               = 
                               0 
                             
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               B 
                               2 
                             
                              
                             
                               ( 
                               
                                 m 
                                 , 
                                 n 
                               
                               ) 
                             
                           
                         
                       
                     
                     + 
                     
                       
                         β 
                         G 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               B 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               R 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         m 
                         = 
                         0 
                       
                       
                         M 
                         - 
                         1 
                       
                     
                      
                     
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             0 
                           
                         
                         
                           N 
                           - 
                           1 
                         
                       
                        
                       
                         
                           G 
                            
                           
                             ( 
                             
                               m 
                               , 
                               n 
                             
                             ) 
                           
                         
                          
                         
                           B 
                            
                           
                             ( 
                             
                               m 
                               , 
                               n 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   28 
                 
               
             
             
               
                 
                   
                     
                       
                         α 
                         R 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               R 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               G 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     + 
                     
                       
                         β 
                         R 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               B 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               R 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         m 
                         = 
                         0 
                       
                       
                         M 
                         - 
                         1 
                       
                     
                      
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                        
                       
                         
                           R 
                           2 
                         
                          
                         
                           ( 
                           
                             m 
                             , 
                             n 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   29 
                 
               
             
             
               
                 
                   
                     
                       
                         α 
                         B 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               B 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               R 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     + 
                     
                       
                         β 
                         B 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               G 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               B 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         m 
                         = 
                         0 
                       
                       
                         M 
                         - 
                         1 
                       
                     
                      
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                        
                       
                         
                           B 
                           2 
                         
                          
                         
                           ( 
                           
                             m 
                             , 
                             n 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   30 
                 
               
             
             
               
                 
                   
                     
                       
                         α 
                         G 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 0 
                               
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               B 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               R 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     + 
                     
                       
                         β 
                         G 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               R 
                               2 
                             
                              
                             
                               ( 
                               
                                 m 
                                 , 
                                 n 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         m 
                         = 
                         0 
                       
                       
                         M 
                         - 
                         1 
                       
                     
                      
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                        
                       
                         
                           R 
                            
                           
                             ( 
                             
                               m 
                               , 
                               n 
                             
                             ) 
                           
                         
                          
                         
                           G 
                            
                           
                             ( 
                             
                               m 
                               , 
                               n 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   31 
                 
               
             
             
               
                 
                   
                     
                       
                         α 
                         B 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               R 
                               2 
                             
                              
                             
                               ( 
                               
                                 m 
                                 , 
                                 n 
                               
                               ) 
                             
                           
                         
                       
                     
                     + 
                     
                       
                         β 
                         B 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               R 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               G 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         m 
                         = 
                         0 
                       
                       
                         M 
                         - 
                         1 
                       
                     
                      
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                        
                       
                         
                           B 
                            
                           
                             ( 
                             
                               m 
                               , 
                               n 
                             
                             ) 
                           
                         
                          
                         
                           R 
                            
                           
                             ( 
                             
                               m 
                               , 
                               n 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   32 
                 
               
             
             
               
                 
                   
                     
                       
                         α 
                         G 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               G 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               B 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     + 
                     
                       
                         β 
                         G 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               R 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               G 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         m 
                         = 
                         0 
                       
                       
                         M 
                         - 
                         1 
                       
                     
                      
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                        
                       
                         
                           G 
                           2 
                         
                          
                         
                           ( 
                           
                             m 
                             , 
                             n 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   33 
                 
               
             
             
               
                 
                   
                     
                       
                         α 
                         R 
                       
                        
                       
                         
                           
                             ∑ 
                             
                               m 
                               = 
                               0 
                             
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               R 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               G 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     + 
                     
                       
                         β 
                         R 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               B 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               R 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         m 
                         = 
                         0 
                       
                       
                         M 
                         - 
                         1 
                       
                     
                      
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                        
                       
                         
                           R 
                           2 
                         
                          
                         
                           ( 
                           
                             m 
                             , 
                             n 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   34 
                 
               
             
             
               
                 
                   
                     
                       
                         α 
                         R 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               R 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               G 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     + 
                     
                       
                         β 
                         B 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               G 
                               2 
                             
                              
                             
                               ( 
                               
                                 m 
                                 , 
                                 n 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         m 
                         = 
                         0 
                       
                       
                         M 
                         - 
                         1 
                       
                     
                      
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                        
                       
                         
                           G 
                            
                           
                             ( 
                             
                               m 
                               , 
                               n 
                             
                             ) 
                           
                         
                          
                         
                           B 
                            
                           
                             ( 
                             
                               m 
                               , 
                               n 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   35 
                 
               
             
             
               
                 
                   
                     
                       
                         α 
                         R 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 0 
                               
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               G 
                               2 
                             
                              
                             
                               ( 
                               
                                 m 
                                 , 
                                 n 
                               
                               ) 
                             
                           
                         
                       
                     
                     + 
                     
                       
                         β 
                         R 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               G 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               B 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         m 
                         = 
                         0 
                       
                       
                         M 
                         - 
                         1 
                       
                     
                      
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                        
                       
                         
                           R 
                            
                           
                             ( 
                             
                               m 
                               , 
                               n 
                             
                             ) 
                           
                         
                          
                         
                           G 
                            
                           
                             ( 
                             
                               m 
                               , 
                               n 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   36 
                 
               
             
             
               
                 
                   
                     
                       
                         α 
                         B 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               B 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               R 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     + 
                     
                       
                         β 
                         B 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               G 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               B 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         m 
                         = 
                         0 
                       
                       
                         M 
                         - 
                         1 
                       
                     
                      
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                        
                       
                         
                           B 
                           2 
                         
                          
                         
                           ( 
                           
                             m 
                             , 
                             n 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   37 
                 
               
             
           
         
       
     
         [0000]    Since equations (29) and (34) are identical we retain one of them, Equations (27) and (36) involve the same set of variables but are distinct. Therefore, we form a new equation by subtracting (27) from (36) to get: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         α 
                         R 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             [ 
                             
                               
                                 
                                   G 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   G 
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   B 
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                     + 
                     
                       
                         β 
                         R 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             [ 
                             
                               
                                 
                                   G 
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   B 
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                               
                               - 
                               
                                 
                                   B 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         m 
                         = 
                         0 
                       
                       
                         M 
                         - 
                         1 
                       
                     
                      
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               R 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               G 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                           - 
                           
                             
                               B 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                              
                             
                               R 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   38 
                 
               
             
           
         
       
     
         [0000]    Now equations (29) and (38) can be consolidated as: 
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               RG 
                             
                           
                         
                         
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               BR 
                             
                           
                         
                       
                       
                         
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 [ 
                                 
                                   
                                     G 
                                     2 
                                   
                                   - 
                                   GB 
                                 
                                 ] 
                               
                             
                           
                         
                         
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 [ 
                                 
                                   GB 
                                   - 
                                   
                                     B 
                                     2 
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                    
                   
                       
                     
                       
                         [ 
                         
                           
                             
                               
                                 α 
                                 R 
                               
                             
                           
                           
                             
                               
                                 β 
                                 R 
                               
                             
                           
                         
                         ] 
                       
                       = 
                       
                         [ 
                         
                           
                             
                               
                                 
                                   ∑ 
                                   
                                     m 
                                     = 
                                     0 
                                   
                                   
                                     M 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       n 
                                       = 
                                       0 
                                     
                                     
                                       N 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                     R 
                                     2 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ∑ 
                                   
                                     m 
                                     = 
                                     0 
                                   
                                   
                                     M 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       n 
                                       = 
                                       0 
                                     
                                     
                                       N 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                     [ 
                                     
                                       RG 
                                       - 
                                       BR 
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   39 
                 
               
             
           
         
       
     
         [0000]    In Equation (39), the double subscripts are omitted for want of space. In a similar manner, we arrive at the rest of the equations: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 GB 
                               
                             
                           
                           
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 RG 
                               
                             
                           
                         
                         
                           
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                   [ 
                                   
                                     
                                       B 
                                       2 
                                     
                                     - 
                                     BR 
                                   
                                   ] 
                                 
                               
                             
                           
                           
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                   [ 
                                   
                                     BR 
                                     - 
                                     
                                       R 
                                       2 
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               α 
                               
                                 G 
                                  
                                 
                                     
                                 
                               
                             
                           
                         
                         
                           
                             
                               β 
                               G 
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                         
                           
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   
                                     ∑ 
                                     
                                       n 
                                       = 
                                       0 
                                     
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                   G 
                                   2 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                   [ 
                                   
                                     GB 
                                     - 
                                     RG 
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   40 
                 
               
             
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               BR 
                             
                           
                         
                         
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               GB 
                             
                           
                         
                       
                       
                         
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 [ 
                                 
                                   
                                     R 
                                     2 
                                   
                                   - 
                                   RG 
                                 
                                 ] 
                               
                             
                           
                         
                         
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 [ 
                                 
                                   RG 
                                   - 
                                   
                                     G 
                                     2 
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                    
                   
                       
                     
                       
                         [ 
                         
                           
                             
                               
                                 α 
                                 B 
                               
                             
                           
                           
                             
                               
                                 β 
                                 B 
                               
                             
                           
                         
                         ] 
                       
                       = 
                       
                         [ 
                         
                           
                             
                               
                                 
                                   ∑ 
                                   
                                     m 
                                     = 
                                     0 
                                   
                                   
                                     M 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       n 
                                       = 
                                       0 
                                     
                                     
                                       N 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                     B 
                                     2 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ∑ 
                                   
                                     m 
                                     = 
                                     0 
                                   
                                   
                                     M 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       n 
                                       = 
                                       0 
                                     
                                     
                                       N 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                     [ 
                                     
                                       BR 
                                       - 
                                       GB 
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   41 
                 
               
             
           
         
       
     
         [0000]    The optimal coefficients for the three-channel minimum correlation case are obtained by solving the normal equations (39) through (41). 
         [0044]    For the minimum covariance case the optimal coefficients will be identical to those corresponding to the minimum correlation case, except that each entry in the 
         [0045]    Equation 39 through 
         [0046]    Equation 41 must be replaced by its mean-removed value. These are listed in Table 2. 
         [0000]    
       
         
               
             
               
               
               
             
               
               
               
             
           
               
                 TABLE 1 
               
               
                   
               
               
                 Optimal Coefficients: Two-channel case 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 Case 
                 Optimal Coefficients 
                 Minimum MSE or Variance Value 
               
               
                   
               
               
                 Minimum MSE 
                 
                   
                     
                       
                         
                           α 
                           R 
                           * 
                         
                         = 
                         
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   R 
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   G 
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   G 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             ∑ 
                             
                               m 
                               = 
                               0 
                             
                             
                               M 
                               - 
                               1 
                             
                           
                            
                           
                               
                           
                            
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 R 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                         - 
                         
                           
                             
                               { 
                               
                                 
                                   ∑ 
                                   
                                     m 
                                     = 
                                     0 
                                   
                                   
                                     M 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       n 
                                       = 
                                       0 
                                     
                                     
                                       N 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       R 
                                        
                                       
                                         ( 
                                         
                                           m 
                                           , 
                                           n 
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       G 
                                        
                                       
                                         ( 
                                         
                                           m 
                                           , 
                                           n 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                               } 
                             
                             2 
                           
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   G 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                   
                 
                   
                     
                       
                         
                           α 
                           G 
                           * 
                         
                         = 
                         
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   G 
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   B 
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   B 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             ∑ 
                             
                               m 
                               = 
                               0 
                             
                             
                               M 
                               - 
                               1 
                             
                           
                            
                           
                               
                           
                            
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 G 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                         - 
                         
                           
                             
                               { 
                               
                                 
                                   ∑ 
                                   
                                     m 
                                     = 
                                     0 
                                   
                                   
                                     M 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       n 
                                       = 
                                       0 
                                     
                                     
                                       N 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       G 
                                        
                                       
                                         ( 
                                         
                                           m 
                                           , 
                                           n 
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       B 
                                        
                                       
                                         ( 
                                         
                                           m 
                                           , 
                                           n 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                               } 
                             
                             2 
                           
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   B 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                   
                 
                   
                     
                       
                         
                           α 
                           B 
                           * 
                         
                         = 
                         
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   B 
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   R 
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   R 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             ∑ 
                             
                               m 
                               = 
                               0 
                             
                             
                               M 
                               - 
                               1 
                             
                           
                            
                           
                               
                           
                            
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 B 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                           
                         
                         - 
                         
                           
                             
                               { 
                               
                                 
                                   ∑ 
                                   
                                     m 
                                     = 
                                     0 
                                   
                                   
                                     M 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       n 
                                       = 
                                       0 
                                     
                                     
                                       N 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       B 
                                        
                                       
                                         ( 
                                         
                                           m 
                                           , 
                                           n 
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       R 
                                        
                                       
                                         ( 
                                         
                                           m 
                                           , 
                                           n 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                               } 
                             
                             2 
                           
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   R 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 Minimum Variance 
                 
                   
                     
                       
                         
                           α 
                           R 
                           * 
                         
                         = 
                         
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   
                                     R 
                                     ~ 
                                   
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     G 
                                     ~ 
                                   
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   
                                     G 
                                     ~ 
                                   
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           σ 
                           R 
                           2 
                         
                         - 
                         
                           
                             
                               { 
                               
                                 
                                   ∑ 
                                   
                                     m 
                                     = 
                                     0 
                                   
                                   
                                     M 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       n 
                                       = 
                                       0 
                                     
                                     
                                       N 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       
                                         R 
                                         ~ 
                                       
                                        
                                       
                                         ( 
                                         
                                           m 
                                           , 
                                           n 
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       
                                         G 
                                         ~ 
                                       
                                        
                                       
                                         ( 
                                         
                                           m 
                                           , 
                                           n 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                               } 
                             
                             2 
                           
                           
                             σ 
                             G 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                   
                 
                   
                     
                       
                         
                           α 
                           G 
                           * 
                         
                         = 
                         
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   
                                     G 
                                     ~ 
                                   
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     B 
                                     ~ 
                                   
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   
                                     B 
                                     ~ 
                                   
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           σ 
                           G 
                           2 
                         
                         - 
                         
                           
                             
                               { 
                               
                                 
                                   ∑ 
                                   
                                     m 
                                     = 
                                     0 
                                   
                                   
                                     M 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       n 
                                       = 
                                       0 
                                     
                                     
                                       N 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       
                                         G 
                                         ~ 
                                       
                                        
                                       
                                         ( 
                                         
                                           m 
                                           , 
                                           n 
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       
                                         B 
                                         ~ 
                                       
                                        
                                       
                                         ( 
                                         
                                           m 
                                           , 
                                           n 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                               } 
                             
                             2 
                           
                           
                             σ 
                             B 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                   
                 
                   
                     
                       
                         
                           α 
                           G 
                           * 
                         
                         = 
                         
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   
                                     G 
                                     ~ 
                                   
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     B 
                                     ~ 
                                   
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   
                                     B 
                                     ~ 
                                   
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           σ 
                           B 
                           2 
                         
                         - 
                         
                           
                             
                               { 
                               
                                 
                                   ∑ 
                                   
                                     m 
                                     = 
                                     0 
                                   
                                   
                                     M 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       n 
                                       = 
                                       0 
                                     
                                     
                                       N 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       
                                         B 
                                         ~ 
                                       
                                        
                                       
                                         ( 
                                         
                                           m 
                                           , 
                                           n 
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       
                                         R 
                                         ~ 
                                       
                                        
                                       
                                         ( 
                                         
                                           m 
                                           , 
                                           n 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                               } 
                             
                             2 
                           
                           
                             σ 
                             R 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                   
                 {tilde over (R)}(m, n) = R(m, n) − μ R , 
               
               
                   
                 {tilde over (G)}(m, n) = G(m, n) − μ G   
               
               
                   
                 {tilde over (B)}(m, n) = B(m, n) − μ B   
               
               
                   
               
             
          
           
               
                 Case 
                 Optimal Coefficients: 1 st  set 
                 Remark 
               
               
                   
               
               
                 Minimum Correlation 
                 
                   
                     
                       
                         
                           α 
                           R 
                           * 
                         
                         = 
                         
                           
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     R 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                               
                             
                             + 
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     B 
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     R 
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     R 
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     G 
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             + 
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     G 
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     B 
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                   
                 
                   
                     
                       
                         
                           α 
                           G 
                           * 
                         
                         = 
                         
                           
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     G 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                               
                             
                             + 
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     R 
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     G 
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     G 
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     B 
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             + 
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     B 
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     R 
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                   
                 
                   
                     
                       
                         
                           α 
                           B 
                           * 
                         
                         = 
                         
                           
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     B 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                               
                             
                             + 
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     G 
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     B 
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     B 
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     R 
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             + 
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     R 
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     G 
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 Minimum Covariance 
                 
                   
                     
                       
                         
                           α 
                           R 
                           * 
                         
                         = 
                         
                           
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     
                                       R 
                                       ~ 
                                     
                                     2 
                                   
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                               
                             
                             + 
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     
                                       B 
                                       ~ 
                                     
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     
                                       R 
                                       ~ 
                                     
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     
                                       R 
                                       ~ 
                                     
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     
                                       G 
                                       ~ 
                                     
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             + 
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     
                                       G 
                                       ~ 
                                     
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     
                                       B 
                                       ~ 
                                     
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                 {tilde over (R)}(m, n) = R(m, n) − μ R   
               
               
                   
               
               
                   
                 
                   
                     
                       
                         
                           α 
                           G 
                           * 
                         
                         = 
                         
                           
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     
                                       G 
                                       ~ 
                                     
                                     2 
                                   
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                               
                             
                             + 
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     
                                       R 
                                       ~ 
                                     
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     
                                       G 
                                       ~ 
                                     
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     
                                       G 
                                       ~ 
                                     
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     
                                       B 
                                       ~ 
                                     
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             + 
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     
                                       B 
                                       ~ 
                                     
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     
                                       R 
                                       ~ 
                                     
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                 {tilde over (G)}(m, n) = G(m, n) − μ G   
               
               
                   
               
               
                   
                 
                   
                     
                       
                         
                           α 
                           B 
                           * 
                         
                         = 
                         
                           
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     
                                       B 
                                       ~ 
                                     
                                     2 
                                   
                                    
                                   
                                     ( 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                               
                             
                             + 
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     
                                       G 
                                       ~ 
                                     
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     
                                       B 
                                       ~ 
                                     
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     
                                       B 
                                       ~ 
                                     
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     
                                       R 
                                       ~ 
                                     
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             + 
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     
                                       R 
                                       ~ 
                                     
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     
                                       G 
                                       ~ 
                                     
                                      
                                     
                                       ( 
                                       
                                         m 
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                 {tilde over (B)}(m, n) = B(m, n) − μ B   
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
             
           
               
                 TABLE 2 
               
             
             
               
                   
               
               
                 Optimal Coefficients: Three-channel case 
               
             
          
           
               
                 Case 
                 Optimal Coefficients 
               
               
                   
               
               
                 Minimum MSE 
                 
                   
                     
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           G 
                                           2 
                                         
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           G 
                                            
                                           
                                             ( 
                                             
                                               m 
                                               , 
                                               n 
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                           B 
                                            
                                           
                                             ( 
                                             
                                               m 
                                               , 
                                               n 
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           G 
                                            
                                           
                                             ( 
                                             
                                               m 
                                               , 
                                               n 
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                           B 
                                            
                                           
                                             ( 
                                             
                                               m 
                                               , 
                                               n 
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           B 
                                           2 
                                         
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             ] 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     α 
                                     R 
                                   
                                 
                               
                               
                                 
                                   
                                     β 
                                     R 
                                   
                                 
                               
                             
                             ] 
                           
                         
                         = 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     ∑ 
                                     
                                       m 
                                       = 
                                       0 
                                     
                                     
                                       M 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       ∑ 
                                       
                                         n 
                                         = 
                                         0 
                                       
                                       
                                         N 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         R 
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         G 
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     ∑ 
                                     
                                       m 
                                       = 
                                       0 
                                     
                                     
                                       M 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       ∑ 
                                       
                                         n 
                                         = 
                                         0 
                                       
                                       
                                         N 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         B 
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         R 
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                   
               
               
                   
                 
                   
                     
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           B 
                                           2 
                                         
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           B 
                                            
                                           
                                             ( 
                                             
                                               m 
                                               , 
                                               n 
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                           R 
                                            
                                           
                                             ( 
                                             
                                               m 
                                               , 
                                               n 
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           B 
                                            
                                           
                                             ( 
                                             
                                               m 
                                               , 
                                               n 
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                           R 
                                            
                                           
                                             ( 
                                             
                                               m 
                                               , 
                                               n 
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           R 
                                           2 
                                         
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             ] 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     α 
                                     G 
                                   
                                 
                               
                               
                                 
                                   
                                     β 
                                     G 
                                   
                                 
                               
                             
                             ] 
                           
                         
                         = 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     ∑ 
                                     
                                       m 
                                       = 
                                       0 
                                     
                                     
                                       M 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       ∑ 
                                       
                                         n 
                                         = 
                                         0 
                                       
                                       
                                         N 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         G 
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         B 
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     ∑ 
                                     
                                       m 
                                       = 
                                       0 
                                     
                                     
                                       M 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       ∑ 
                                       
                                         n 
                                         = 
                                         0 
                                       
                                       
                                         N 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         R 
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         G 
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                   
               
               
                   
                 
                   
                     
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           R 
                                           2 
                                         
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           R 
                                            
                                           
                                             ( 
                                             
                                               m 
                                               , 
                                               n 
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                           G 
                                            
                                           
                                             ( 
                                             
                                               m 
                                               , 
                                               n 
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           R 
                                            
                                           
                                             ( 
                                             
                                               m 
                                               , 
                                               n 
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                           G 
                                            
                                           
                                             ( 
                                             
                                               m 
                                               , 
                                               n 
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           G 
                                           2 
                                         
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             ] 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     α 
                                     B 
                                   
                                 
                               
                               
                                 
                                   
                                     β 
                                     B 
                                   
                                 
                               
                             
                             ] 
                           
                         
                         = 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     ∑ 
                                     
                                       m 
                                       = 
                                       0 
                                     
                                     
                                       M 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       ∑ 
                                       
                                         n 
                                         = 
                                         0 
                                       
                                       
                                         N 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         B 
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         R 
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     ∑ 
                                     
                                       m 
                                       = 
                                       0 
                                     
                                     
                                       M 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       ∑ 
                                       
                                         n 
                                         = 
                                         0 
                                       
                                       
                                         N 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         G 
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         B 
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                   
               
               
                 Minimum Variance 
                 
                   
                     
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           
                                             G 
                                             ~ 
                                           
                                           2 
                                         
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           
                                             G 
                                             ~ 
                                           
                                            
                                           
                                             ( 
                                             
                                               m 
                                               , 
                                               n 
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                           
                                             B 
                                             ~ 
                                           
                                            
                                           
                                             ( 
                                             
                                               m 
                                               , 
                                               n 
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           
                                             G 
                                             ~ 
                                           
                                            
                                           
                                             ( 
                                             
                                               m 
                                               , 
                                               n 
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                           
                                             B 
                                             ~ 
                                           
                                            
                                           
                                             ( 
                                             
                                               m 
                                               , 
                                               n 
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           
                                             B 
                                             ~ 
                                           
                                           2 
                                         
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             ] 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     α 
                                     R 
                                   
                                 
                               
                               
                                 
                                   
                                     β 
                                     R 
                                   
                                 
                               
                             
                             ] 
                           
                         
                         = 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     ∑ 
                                     
                                       m 
                                       = 
                                       0 
                                     
                                     
                                       M 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       ∑ 
                                       
                                         n 
                                         = 
                                         0 
                                       
                                       
                                         N 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         
                                           R 
                                           ~ 
                                         
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         
                                           G 
                                           ~ 
                                         
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     ∑ 
                                     
                                       m 
                                       = 
                                       0 
                                     
                                     
                                       M 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       ∑ 
                                       
                                         n 
                                         = 
                                         0 
                                       
                                       
                                         N 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         
                                           B 
                                           ~ 
                                         
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         
                                           R 
                                           ~ 
                                         
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                   
               
               
                   
                 
                   
                     
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           
                                             B 
                                             ~ 
                                           
                                           2 
                                         
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           
                                             B 
                                             ~ 
                                           
                                            
                                           
                                             ( 
                                             
                                               m 
                                               , 
                                               n 
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                           
                                             R 
                                             ~ 
                                           
                                            
                                           
                                             ( 
                                             
                                               m 
                                               , 
                                               n 
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           
                                             B 
                                             ~ 
                                           
                                            
                                           
                                             ( 
                                             
                                               m 
                                               , 
                                               n 
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                           
                                             R 
                                             ~ 
                                           
                                            
                                           
                                             ( 
                                             
                                               m 
                                               , 
                                               n 
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           
                                             R 
                                             ~ 
                                           
                                           2 
                                         
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             ] 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     α 
                                     G 
                                   
                                 
                               
                               
                                 
                                   
                                     β 
                                     G 
                                   
                                 
                               
                             
                             ] 
                           
                         
                         = 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     ∑ 
                                     
                                       m 
                                       = 
                                       0 
                                     
                                     
                                       M 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       ∑ 
                                       
                                         n 
                                         = 
                                         0 
                                       
                                       
                                         N 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         
                                           G 
                                           ~ 
                                         
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         
                                           B 
                                           ~ 
                                         
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     ∑ 
                                     
                                       m 
                                       = 
                                       0 
                                     
                                     
                                       M 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       ∑ 
                                       
                                         n 
                                         = 
                                         0 
                                       
                                       
                                         N 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         
                                           R 
                                           ~ 
                                         
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         
                                           G 
                                           ~ 
                                         
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                   
               
               
                   
                 
                   
                     
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           
                                             R 
                                             ~ 
                                           
                                           2 
                                         
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           
                                             R 
                                             ~ 
                                           
                                            
                                           
                                             ( 
                                             
                                               m 
                                               , 
                                               n 
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                           
                                             G 
                                             ~ 
                                           
                                            
                                           
                                             ( 
                                             
                                               m 
                                               , 
                                               n 
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           
                                             R 
                                             ~ 
                                           
                                            
                                           
                                             ( 
                                             
                                               m 
                                               , 
                                               n 
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                           
                                             G 
                                             ~ 
                                           
                                            
                                           
                                             ( 
                                             
                                               m 
                                               , 
                                               n 
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           
                                             G 
                                             ~ 
                                           
                                           2 
                                         
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             ] 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     α 
                                     R 
                                   
                                 
                               
                               
                                 
                                   
                                     β 
                                     R 
                                   
                                 
                               
                             
                             ] 
                           
                         
                         = 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     ∑ 
                                     
                                       m 
                                       = 
                                       0 
                                     
                                     
                                       M 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       ∑ 
                                       
                                         n 
                                         = 
                                         0 
                                       
                                       
                                         N 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         
                                           B 
                                           ~ 
                                         
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         
                                           R 
                                           ~ 
                                         
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     ∑ 
                                     
                                       m 
                                       = 
                                       0 
                                     
                                     
                                       M 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       ∑ 
                                       
                                         n 
                                         = 
                                         0 
                                       
                                       
                                         N 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         
                                           G 
                                           ~ 
                                         
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         
                                           B 
                                           ~ 
                                         
                                          
                                         
                                           ( 
                                           
                                             m 
                                             , 
                                             n 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                   
               
               
                   
                 {tilde over (R)}(m, n) = R(m, n) − μ R , {tilde over (G)}(m, n) = G(m, n) − μ G , {tilde over (B)}(m, n) = B(m, n) − μ B   
               
               
                 Minimum Correlation 
                 
                   
                     
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       RG 
                                     
                                   
                                 
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       BR 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         [ 
                                         
                                           
                                             G 
                                             2 
                                           
                                           - 
                                           GB 
                                         
                                         ] 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         [ 
                                         
                                           GB 
                                           - 
                                           
                                             B 
                                             2 
                                           
                                         
                                         ] 
                                       
                                     
                                   
                                 
                               
                             
                             ] 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     α 
                                     B 
                                   
                                 
                               
                               
                                 
                                   
                                     β 
                                     B 
                                   
                                 
                               
                             
                             ] 
                           
                         
                         = 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         R 
                                         2 
                                       
                                     
                                   
                                    
                                   
                                       
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     ∑ 
                                     
                                       m 
                                       = 
                                       0 
                                     
                                     
                                       M 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       ∑ 
                                       
                                         n 
                                         = 
                                         0 
                                       
                                       
                                         N 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       [ 
                                       
                                         RG 
                                         - 
                                         BR 
                                       
                                       ] 
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                   
               
               
                   
                 
                   
                     
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       GB 
                                     
                                   
                                 
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       RG 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         [ 
                                         
                                           
                                             B 
                                             2 
                                           
                                           - 
                                           BR 
                                         
                                         ] 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         [ 
                                         
                                           BR 
                                           - 
                                           
                                             R 
                                             2 
                                           
                                         
                                         ] 
                                       
                                     
                                   
                                 
                               
                             
                             ] 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     α 
                                     G 
                                   
                                 
                               
                               
                                 
                                   
                                     β 
                                     G 
                                   
                                 
                               
                             
                             ] 
                           
                         
                         = 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         G 
                                         2 
                                       
                                     
                                   
                                    
                                   
                                       
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     ∑ 
                                     
                                       m 
                                       = 
                                       0 
                                     
                                     
                                       M 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       ∑ 
                                       
                                         n 
                                         = 
                                         0 
                                       
                                       
                                         N 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       [ 
                                       
                                         GB 
                                         - 
                                         RG 
                                       
                                       ] 
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                   
               
               
                   
                 
                   
                     
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       BR 
                                     
                                   
                                 
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       GB 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         [ 
                                         
                                           
                                             R 
                                             2 
                                           
                                           - 
                                           RG 
                                         
                                         ] 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         [ 
                                         
                                           RG 
                                           - 
                                           
                                             G 
                                             2 
                                           
                                         
                                         ] 
                                       
                                     
                                   
                                 
                               
                             
                             ] 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     α 
                                     B 
                                   
                                 
                               
                               
                                 
                                   
                                     β 
                                     B 
                                   
                                 
                               
                             
                             ] 
                           
                         
                         = 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         B 
                                         2 
                                       
                                     
                                   
                                    
                                   
                                       
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     ∑ 
                                     
                                       m 
                                       = 
                                       0 
                                     
                                     
                                       M 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       ∑ 
                                       
                                         n 
                                         = 
                                         0 
                                       
                                       
                                         N 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       [ 
                                       
                                         BR 
                                         - 
                                         GB 
                                       
                                       ] 
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                   
               
               
                 Minimum Covariance 
                 
                   
                     
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           R 
                                           ~ 
                                         
                                          
                                         
                                           G 
                                           ~ 
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           B 
                                           ~ 
                                         
                                          
                                         
                                           R 
                                           ~ 
                                         
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         [ 
                                         
                                           
                                             
                                               G 
                                               ~ 
                                             
                                             2 
                                           
                                           - 
                                           
                                             
                                               G 
                                               ~ 
                                             
                                              
                                             
                                               B 
                                               ~ 
                                             
                                           
                                         
                                         ] 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         [ 
                                         
                                           
                                             
                                               G 
                                               ~ 
                                             
                                              
                                             
                                               B 
                                               ~ 
                                             
                                           
                                           - 
                                           
                                             
                                               B 
                                               ~ 
                                             
                                             2 
                                           
                                         
                                         ] 
                                       
                                     
                                   
                                 
                               
                             
                             ] 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     α 
                                     R 
                                   
                                 
                               
                               
                                 
                                   
                                     β 
                                     R 
                                   
                                 
                               
                             
                             ] 
                           
                         
                         = 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           R 
                                           ~ 
                                         
                                         2 
                                       
                                     
                                   
                                    
                                   
                                       
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     ∑ 
                                     
                                       m 
                                       = 
                                       0 
                                     
                                     
                                       M 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       ∑ 
                                       
                                         n 
                                         = 
                                         0 
                                       
                                       
                                         N 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       [ 
                                       
                                         
                                           
                                             R 
                                             ~ 
                                           
                                            
                                           
                                             G 
                                             ~ 
                                           
                                         
                                         - 
                                         
                                           
                                             B 
                                             ~ 
                                           
                                            
                                           
                                             R 
                                             ~ 
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                   
               
               
                   
                 
                   
                     
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           G 
                                           ~ 
                                         
                                          
                                         
                                           B 
                                           ~ 
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           R 
                                           ~ 
                                         
                                          
                                         
                                           G 
                                           ~ 
                                         
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         [ 
                                         
                                           
                                             
                                               B 
                                               ~ 
                                             
                                             2 
                                           
                                           - 
                                           
                                             
                                               B 
                                               ~ 
                                             
                                              
                                             
                                               R 
                                               ~ 
                                             
                                           
                                         
                                         ] 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         [ 
                                         
                                           
                                             
                                               B 
                                               ~ 
                                             
                                              
                                             
                                               R 
                                               ~ 
                                             
                                           
                                           - 
                                           
                                             
                                               R 
                                               ~ 
                                             
                                             2 
                                           
                                         
                                         ] 
                                       
                                     
                                   
                                 
                               
                             
                             ] 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     α 
                                     G 
                                   
                                 
                               
                               
                                 
                                   
                                     β 
                                     G 
                                   
                                 
                               
                             
                             ] 
                           
                         
                         = 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           G 
                                           ~ 
                                         
                                         2 
                                       
                                     
                                   
                                    
                                   
                                       
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     ∑ 
                                     
                                       m 
                                       = 
                                       0 
                                     
                                     
                                       M 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       ∑ 
                                       
                                         n 
                                         = 
                                         0 
                                       
                                       
                                         N 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       [ 
                                       
                                         
                                           
                                             G 
                                             ~ 
                                           
                                            
                                           
                                             B 
                                             ~ 
                                           
                                         
                                         - 
                                         
                                           
                                             R 
                                             ~ 
                                           
                                            
                                           
                                             G 
                                             ~ 
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                   
               
               
                   
                 
                   
                     
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           B 
                                           ~ 
                                         
                                          
                                         
                                           R 
                                           ~ 
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           G 
                                           ~ 
                                         
                                          
                                         
                                           B 
                                           ~ 
                                         
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         [ 
                                         
                                           
                                             
                                               R 
                                               ~ 
                                             
                                             2 
                                           
                                           - 
                                           
                                             
                                               R 
                                               ~ 
                                             
                                              
                                             
                                               G 
                                               ~ 
                                             
                                           
                                         
                                         ] 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         [ 
                                         
                                           
                                             
                                               R 
                                               ~ 
                                             
                                              
                                             
                                               G 
                                               ~ 
                                             
                                           
                                           - 
                                           
                                             
                                               G 
                                               ~ 
                                             
                                             2 
                                           
                                         
                                         ] 
                                       
                                     
                                   
                                 
                               
                             
                             ] 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     α 
                                     B 
                                   
                                 
                               
                               
                                 
                                   
                                     β 
                                     B 
                                   
                                 
                               
                             
                             ] 
                           
                         
                         = 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         0 
                                       
                                       
                                         M 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           B 
                                           ~ 
                                         
                                         2 
                                       
                                     
                                   
                                    
                                   
                                       
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     ∑ 
                                     
                                       m 
                                       = 
                                       0 
                                     
                                     
                                       M 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       ∑ 
                                       
                                         n 
                                         = 
                                         0 
                                       
                                       
                                         N 
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       [ 
                                       
                                         
                                           
                                             B 
                                             ~ 
                                           
                                            
                                           
                                             R 
                                             ~ 
                                           
                                         
                                         - 
                                         
                                           
                                             G 
                                             ~ 
                                           
                                            
                                           
                                             B 
                                             ~ 
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                   
               
               
                   
                 {tilde over (R)}(m, n) = R(m, n) − μ R , {tilde over (G)}(m, n) = G(m, n) − μ G , {tilde over (B)}(m, n) = B(m, n) − μ B   
               
               
                   
               
             
          
         
       
     
         [0047]    In suboptimal processing there is no cost function involved, the coefficient value is chosen on an ad hoc basis and the resulting mean square error or error variance is not necessarily the minimum. Further, only two components are used in the differencing process. 
         [0048]    The difference images are obtained by selecting the weights as the ratio of the mean values of the respective image components involved, as given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           e 
                           R 
                         
                          
                         
                           ( 
                           
                             m 
                             , 
                             n 
                           
                           ) 
                         
                       
                       = 
                       
                         
                           R 
                            
                           
                             ( 
                             
                               m 
                               , 
                               n 
                             
                             ) 
                           
                         
                         - 
                         
                           
                             α 
                             
                               R 
                                
                               
                                   
                               
                             
                           
                            
                           
                             G 
                              
                             
                               ( 
                               
                                 m 
                                 , 
                                 n 
                               
                               ) 
                             
                           
                         
                       
                     
                     ; 
                   
                    
                   
                     
 
                   
                    
                   
                     
                       α 
                       R 
                     
                     = 
                     
                       
                         μ 
                         R 
                       
                       
                         μ 
                         G 
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   42 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           e 
                           G 
                         
                          
                         
                           ( 
                           
                             m 
                             , 
                             n 
                           
                           ) 
                         
                       
                       = 
                       
                         
                           G 
                            
                           
                             ( 
                             
                               m 
                               , 
                               n 
                             
                             ) 
                           
                         
                         - 
                         
                           
                             α 
                             G 
                           
                            
                           
                             B 
                              
                             
                               ( 
                               
                                 m 
                                 , 
                                 n 
                               
                               ) 
                             
                           
                         
                       
                     
                     ; 
                   
                    
                   
                     
 
                   
                    
                   
                     
                       α 
                       G 
                     
                     = 
                     
                       
                         μ 
                         G 
                       
                       
                         μ 
                         B 
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   43 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           e 
                           B 
                         
                          
                         
                           ( 
                           
                             m 
                             , 
                             n 
                           
                           ) 
                         
                       
                       = 
                       
                         
                           B 
                            
                           
                             ( 
                             
                               m 
                               , 
                               n 
                             
                             ) 
                           
                         
                         - 
                         
                           
                             α 
                             B 
                           
                            
                           
                             R 
                              
                             
                               ( 
                               
                                 m 
                                 , 
                                 n 
                               
                               ) 
                             
                           
                         
                       
                     
                     ; 
                   
                    
                   
                     
 
                   
                    
                   
                     
                       α 
                       B 
                     
                     = 
                     
                       
                         μ 
                         B 
                       
                       
                         μ 
                         
                           R 
                            
                           
                               
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   44 
                 
               
             
           
         
       
     
         [0000]    In the second case, the coefficients are chosen as the ratio of the standard deviations of the component images and the weights are given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       α 
                       R 
                     
                     = 
                     
                       
                         σ 
                         R 
                       
                       
                         σ 
                         
                           G 
                            
                           
                               
                           
                         
                       
                     
                   
                   ; 
                   
                     
                       α 
                       G 
                     
                     = 
                     
                       
                         σ 
                         G 
                       
                       
                         σ 
                         B 
                       
                     
                   
                   ; 
                   
                     
                       α 
                       B 
                     
                     = 
                     
                       
                         σ 
                         B 
                       
                       
                         σ 
                         R 
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   45 
                 
               
             
           
         
       
     
         [0000]    For a third case we choose the ratio of the maximum component values as the coefficient values. Thus, 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         α 
                         R 
                       
                       = 
                       
                         
                           max 
                            
                           
                             ( 
                             
                               R 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                             ) 
                           
                         
                         
                           max 
                            
                           
                             ( 
                             
                               G 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                             ) 
                           
                         
                       
                     
                     ; 
                     
                       
                         α 
                         G 
                       
                       = 
                       
                         
                           max 
                            
                           
                             ( 
                             
                               G 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                             ) 
                           
                         
                         
                           max 
                            
                           
                             ( 
                             
                               B 
                                
                               
                                 ( 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 ) 
                               
                             
                             ) 
                           
                         
                       
                     
                     ; 
                   
                    
                   
                     
 
                   
                    
                   
                     
                       α 
                       B 
                     
                     = 
                     
                       
                         max 
                          
                         
                           ( 
                           
                             B 
                              
                             
                               ( 
                               
                                 m 
                                 , 
                                 n 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                       
                         max 
                          
                         
                           ( 
                           
                             R 
                              
                             
                               ( 
                               
                                 m 
                                 , 
                                 n 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   46 
                 
               
             
           
         
       
     
         [0049]    The image may be processed in whole or in sub-blocks. The pre-detection processing methods just described can be applied to the whole image or they can be applied to sub-blocks of an image. Images in general and video sequence images in particular are non-stationary. This implies that the statistics change over regions within an image. To exploit non-stationarity of images, it is advantageous to apply the pre-detection processing methods to subimages. As shown in  FIG. 1 , one can process sub-blocks of fixed size or variable size or one row at a time. In processing sub-blocks of varying size, one has to adopt a suitable criterion for choosing sub-block size. This is described in the next subsection. 
         [0050]    Quad-tree decomposition is an efficient procedure to decompose an image into sub-blocks. The only requirement is that the image height and width be integer powers of 2. In this procedure one starts with a given image as the root node of a tree and a suitable metric for the whole image is calculated. Next the metrics for the four sub-blocks, each of size ¼ th  the size of the whole image are computed. A decision is made to split each sub-block sequentially depending on whether that sub-block metric is greater than the metric of its parent node. This process is continued until the specified sub-block size is reached. The minimum number of sub-blocks is 4.  FIG. 2  shows a typical quad-tree structure. 
         [0051]    The contrast of the processed component images can be improved by contrast stretching. The contrast regions are automatically selected using the image statistics such as the image histogram. 
         [0052]    Once the images are processed we need to draw the boundaries of the targets present in the scene for further data collection such as target classification. This involves first smoothing of the processed images. Smoothing generally tends to blur the image. Hence we use special, non-blurring filtering such as the diffusion filtering to smooth the processed images. Next an edge detection processing method, typically Canny&#39;s method, is used followed by morphological processing to calculate properties of the object boundaries. 
         [0053]    The acquired raw images usually contain a number of pixels that are saturated. These may occur as isolated single pixels or as clusters of pixels scattered over the image. These clusters of saturated pixels are known as glints. The presence of glints reduces the contrast of areas of interest, especially if the glints occupy a sizeable fraction of the image area. Moreover, glints dominate the statistical quantities of interest due to their very large values. Therefore, images must be preprocessed to remove the presence of glints. Preprocessing is also necessary to reduce image noise. 
         [0054]    Preprocessing images in the wavelet domain is effective in removing glints and noise from the images. It also has the advantage of computational savings. Specifically, applications of two-dimensional Discrete Wavelet Transform (DWT) once to the image decomposes it into one approximation and three detail coefficients, each ¼ th  the size of the original image. It has been found that biorthogonal wavelet performs better than orthogonal wavelet. The approximation coefficients contain the information in the original image at a lower resolution. Hence glint removal operation is performed on the approximation coefficients. As pointed out, since the approximation component of the DWT coefficients is only ¼ th  the size of the original image, computational saving is achieved. As the detail coefficients carry edge information in the original image at lower resolutions, image denoising (a term used to imply the removal of noise) is effected in these components. Once glint removal and denoising operations are performed, the image is reconstructed by applying inverse 2D DWT (IDWT) to the DWT coefficients. 
         [0055]    The detail components of the 2D DWT of each channel image contain edge information as well as noise. Noise coefficients have amplitudes according to a Laplacian distribution and are usually large while edges have smaller amplitudes. Thus noise is reduced by thresholding the detail coefficients in the wavelet domain. These thresholds are determined based on the statistics of the detail coefficients. 
         [0056]    Glints are specular reflections of the IR source by the surface being imaged. Therefore, glints produce highly saturated pixels in all the channels whose size depend on various properties of the reflecting surface and it is random. Consequently, the approximation coefficients of the 2D DWT must be inspected. In one approach, glints may be removed or reduced using morphological processing to determine the boundaries of the glint regions, which in turn are determined by hard or soft thresholding. The thresholds are determined from the statistics of the approximation coefficients of the 2D DWT. Thresholding can be performed on each channel or on the dominant channel. Thresholding yields a binary image from which the boundaries of the glints are calculated. Glint regions are eliminated by replacing them with corresponding mean values in all the channels. 
         [0057]    Moreover, aspects of the claimed subject matter may be implemented as a method, apparatus, or article of manufacture using standard programming and/or engineering techniques to produce software, firmware, hardware, or any combination thereof to control a computer or computing components to implement various aspects of the claimed subject matter. The term “article of manufacture” as used herein is intended to encompass a computer program accessible from any computer-readable device, carrier, or media. For example, computer readable media can include but are not limited to magnetic storage devices (e.g., hard disk, floppy disk, magnetic strips . . . ), optical disks (e.g., compact disk (CD), digital versatile disk (DVD) . . . ), smart cards, and flash memory devices (e.g., card, stick, key drive Of course, those skilled in the art will recognize many modifications may be made to this configuration without departing from the scope or spirit of what is described herein. 
         [0058]    What has been described above includes examples of one or more embodiments. It is, of course, not possible to describe every conceivable combination of components or methodologies for purposes of describing the aforementioned embodiments, but one of ordinary skill in the art may recognize that many further combinations and permutations of various embodiments are possible. Accordingly, the described embodiments are intended to embrace all such alterations, modifications and variations that fall within the spirit and scope of the appended claims. Furthermore, to the extent that the term “includes” is used in either the detailed description or the claims, such term is intended to be inclusive in a manner similar to the term “comprising” as “comprising” is interpreted when employed as a transitional word in a claim.