Abstract:
A method and apparatus for detecting a pattern within an image. Image data ( 22 ) is received which is representative of the image. Filter values ( 70 ) are determined which substantially optimizes a first predetermined criterion ( 68 ). The first predetermined criterion ( 68 ) is based upon image data ( 22 ). A correlation output ( 40 ) is determined which is indicative of the presence of the pattern within the image data ( 22 ). The correlation output ( 40 ) is based upon the determined filter values ( 70 ) and the image data ( 22 ) via a non-linear polynomial relationship ( 78 ).

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     The application is based upon and claims priority to U.S. Provisional Patent Application Serial No. 60/043,408 filed Apr. 4, 1997, and entitled Polynomial Filters for Higher Order Correlation and Multi-Input Information Fusion, the specification and drawings of which are herein expressly incorporated by reference. 
    
    
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates generally to pattern recognition, and more particularly, to correlation filters used in pattern recognition. 
     2. Description of Related Art 
     Two-dimensional correlation techniques have used spatial filters (known as correlation filters) to detect, locate and classify targets in observed scenes. A correlation filter should attempt to yield: sharp correlation peaks for targets of interest, high discrimination against unwanted objects, excellent robustness to noise in the input scene and high tolerance to distortions in the input. A variety of filters to address these aspects and other aspects have been proposed (for example, see: B. V. K. Vijaya Kumar, “Tutorial Survey of Composite Filter Designs for Optical Correlators,”  Applied Optics , Vol. 31, pp. 4773-4801, 1992). 
     Linear filters known as Synthetic Discriminant Function (SDF) filters have been introduced by Hester and Casasent as well as by Caulfield and Maloney (see: C. F. Hester and D. Casasent, “Multivariant Techniques for Multiclass Pattern Recognition,”  Applied Optics , Vol. 19, pp. 1758-1761, 1980; H. J. Caulfield and W. T. Maloney, “Improved Discrimination in Optical Character Recognition,”  Applied Optics , Vol. 8, pp. 2354-2356, 1969). 
     Other correlation filters include the minimum squared error Synthetic Discriminant Function (MSE SDF) where the correlation filter is selected that yields the smallest average squared error between the resulting correlation outputs and a specified shape (see: B. V. K. Vijaya Kumar, A. Mahalanobis, S. Song, S. R. F. Sims and J. Epperson, “Minimum Squared Error Synthetic Discriminant Functions,”  Optical Engineering , Vol. 31, pp. 915-922, 1992). 
     Another filter is the maximum average correlation height (MACH) filter that determines and uses the correlation shape yielding the smallest squared error (see: A. Mahalanobis, B. V. K. Vijaya Kumar, S. R. F. Sims, J. Epperson, “Unconstrained Correlation Filters,”  Applied Optics , Vol. 33, pp. 3751-3759, 1994). However, the MACH filter and other current filters generally perform only linear operations on input image data and consequently are limited in their performance to detect patterns within the input image data. Moreover, the current approaches suffer the disadvantage of an inadequate ability to process information from multiple sensors as well as at different resolution levels. 
     SUMMARY OF THE INVENTION 
     The present invention is a method and apparatus for detecting a pattern within an image. Image data is received which is representative of the image. Filter values are determined which substantially optimize a first predetermined criterion. The first predetermined criterion is based upon the image data. A correlation output is generated using a non-linear polynomial relationship based upon the determined filter values and the image data. The correlation output is indicative of the presence of the pattern within the image data. 
     The present invention contains the following features (but is not limited to): improved probability of correct target recognition, clutter tolerance and reduced false alarm rates. The present invention also contains such features as (but is not limited to): detection and recognition of targets with fusion of data from multiple sensors, and the ability to combine optimum correlation filters with multi-resolution information (such as Wavelets and morphological image transforms) for enhanced performance. 
     Additional advantages and features of the present invention will become apparent from the subsequent description and the appended claims, taken in conjunction with the accompanying drawings in which: 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a flow diagram depicting the N-th order polynomial correlation filter; 
     FIGS. 2 a - 2   b  are flow charts depicting the operations involved for the correlation filter; 
     FIG. 3 is a flow diagram depicting the N-th order polynomial correlation filter for multi-sensor fusion; 
     FIG. 4 are perspective views of sample tanks at different angles of perspective; and 
     FIG. 5 is a graph depicting peak-to-sidelobe ratio versus frame number. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     Notation Format 
     The notation employed in the present invention is as follows: images in the space domain are denoted in lower case italics while upper case italics are used to represent the same in the frequency domain. Thus, a two dimensional (2D) image x(m, n) has Fourier transform X(k,l). Vectors are represented by lower case bold characters while matrices are denoted by upper case bold characters. Either x(m,n) or X(k,l) can be expressed as a column vector x by lexicographical scanning. The superscript  T  denotes the transpose operation, and + denotes the complex conjugate transpose of vectors and matrices. 
     Referring to FIG. 1, the correlation filter  20  of the present invention receives input image data  22  from input device  23  in order to detect a pattern within the image data  22 . A first order term  24  of image data  22  is associated with a first filter term  26 . Successive order terms ( 28  and  30 ) of image data  22  are associated with successive filter terms ( 32  and  34 ). Ultimately, the Nth order term  36  is associated with filter term h N    38 . 
     Values for the filter terms are determined which substantially optimize a performance criterion which is based upon the image data, and a spectral quantity. The spectral quantity represents a spectral feature of the image data  22 . For a description of spectral quantities and features, please see: A. Mahalanobis, B. V. K. Vijaya Kumar, and D. Casasent, “Minimum Average Correlation Energy Filters,”  Applied Optics , vol. 26, pp. 3633-3640, 1987. 
     A correlation output g x    40  is produced based upon the determined filter values and the image data  22  using a non-linear polynomial relationship. The non-linear polynomial relationship is a feature of the present invention over other approaches—that is, the present invention treats the output as a non-linear function of the input. In the present invention, the non-linear polynomial relationship of the output is expressed as: 
     
       
           g   x   =A   1   x   1   +A   2   x   2   + . . . +A   N   x   N   (1)  
       
     
     where x i  represents the vector x with each of its element raised to the power i, and A i  is a matrix of coefficients associated with the ith term of the polynomial. It should be noted that the output g x  is also a vector. 
     Equation (1) is termed the polynomial correlation filter (or PCF). Thus if x represents the input image in vector notation, then g x  is a vector which represents the output correlation plane as a polynomial function of x. To ensure that the output is shift invariant, all the coefficient matrices are in a Toeplitz format. For a description of the Toeplitz format, see the following reference: Matrix Computations, Gene H. Golub, Charles F. Van Loan, Johns Hopkins Press, 1989. Each term in the polynomial is computed as a linear shift-invariant filtering operation: 
     
       
           A   i   x   i   =h   i ( m, n )⊕ x   i ( m, n )  (2)  
       
     
     or that filtering x i (m,n) by h i (m,n) is equivalent to multiplying x i  by A i . The symbol “⊕” is used to indicate spatial filtering. The output of the polynomial correlation filter is mathematically expressed as:                  g   x          (     m   ,   n     )       =       ∑     i   =   1     N              h   i          (     m   ,   n     )       ⊕       x   i          (     m   ,   n     )                   (   3   )                                
     The filters h i (m, n) are determined such that the structure shown in FIG. 1 optimizes a performance criterion of choice. For the preferred embodiment, the Optimal Trade-off (OT) performance criterion is selected (for a discussion of the OT performance criterion, see Ph. Refregier, “Filter Design for Optical Pattern Recognition: Multicriteria Optimization Approach,”  Optics Letters , Vol. 15, pp. 854-856, 1990). The OT performance criterion is as expressed as:                J        (   h   )       =                m   +        h          2         h   +        Bh               (   4   )                                
     where h is the filter vector in the frequency domain, B is a diagonal matrix related to a spectral quantity, and m is the mean image vector, also in the frequency domain. The following spectral quantities can be used in the OT performance criterion: average correlation energy (ACE); average similarity measure (ASM); output noise variance (ONV); or combinations of these performance criterion can be used which are all of the same quadratic form as the denominator of Eq. (4). However, it is to be understood that the present invention is not limited to only these spectral quantities, but includes those which will function for the application at hand. An alternate embodiment of the present invention includes optimizing the same class of performance criteria. 
     Sample Second Order Correlation Filter 
     By way of example, the operations involved in a second order correlation filter of the present invention is discussed herein. However, it is to be understood that the present invention is not limited to only second order correlation filters but includes any higher order correlation filter. 
     Accordingly in this example, the polynomial has only two terms and the output is expressed as: 
     
       
           g ( m, n )= x ( m, n )⊕ h   i ( m, n )+ x   2 ( m, n )⊕ h   2 ( m,n )  (5)  
       
     
     The expression for J(h) is obtained by deriving the numerator and the denominator of Eq. (4). In vector notation, the average intensity of the correlation peak for a second order filter is 
     
       
         |AveragePeak| 2   =|h   1   30   m   1 | 2   +|h   2   +   m   2 | 2   (6)  
       
     
     where h 1  and h 2  are vector representations of the filters associated with the first and second terms of the polynomial, and                m   k     =       1   L                       ∑     i   =   1     L          x   i   k                 (   7   )                                
     is the mean of the training images x i , 1≦i≦L, raised to the kth power. For illustration purposes only, the denominator of the performance criterion in Eq. (4) is chosen to be the ASM metric while noting that the present invention includes any other quadratic form such as ONV or ACE or any combination thereof. The ASM for the second order non-linear filter is expressed as:              ASM   =       1   L                       ∑     i   =   1     L                     h   1   *          X   i   1       +       h   2   *          X   i   2       -       h   1   *          M   1       -       h   2   *          M   2              2                 (   8   )                                
     where X i   k , 1≦i≦L, is the ith training image raised to the kth power expressed as a diagonal matrix, and M k  is their average (also a diagonal matrix). After algebraic manipulations, the expression for ASM is: 
     
       
           ASM=h   1   +   S   11   h   1   +h   2   +   S   22   h   2   +h   1   +   S   12   h   2   +h   2   +   S   21   h   1   (9)  
       
     
     where                  S   k1     =         1   L                       ∑     i   =   1     L              X   i   k          (     X   i   1     )       *         -         M   k          (     M   1     )       *         ,     1   ≤   k     ,     1   ≤   2             (   10   )                                
     are all diagonal matrices. The block vectors and matrices are expressed as:                h   =     [           h   1               h   2           ]       ,     m   =     ⌊           m   1               m   2           ⌋       ,       and                 S     =     [           S   11           S   12               S   21           S   22           ]               (   11   )                                
     The expression for J(h) for the second order filter is expressed as:                      J        (   h   )       =              average                 peak          2     ASM                 =                  h   1   +          m   1            2     +              h   2   +          m   2            2             h   1   +          S   11          h   1       +       h   2   +          S   22          h   2       +       h   1   +          S   12          h   2       +       h   2   +          S   21          h   1                       =                m   +        h          2         h   +        Sh                     (   12   )                                
     The following equation maximizes J(h): 
     
       
         h=S −1 m  (13)  
       
     
     Using the definitions in Eq.(11), the solution for the two filters of the second order polynomial is:                [           h   1               h   2           ]     =         [           S   11           S   12               S   21           S   11           ]       -   1            ⌊           m   1               m   2           ⌋               (   14   )                                
     The inverse of the block matrix is expressed as:                [           h   1               h   2           ]     =     ⌊                 S   12          m   2       -       S   22          m   1                    S   12          2     -       S   11          S   22                           S   21          m   1       -       S   11          m   2                    S   12          2     -       S   11          S   22                 ⌋             (   15   )                                
     The solution in Eq. (14) is extended to the general Nth order case. Following the same analysis as for the second order case, the N-th order solution is expressed as:                [           h   1               h   2             ⋮             h   N           ]     =         [           S   11           S   12         ⋯         S     1      N                 S   21           S   22         ⋯         S     2      N               ⋮       ⋮       ⋰       ⋮             S   N1           S   N2         ⋯         S   NN           ]       -   1            [           m   1               m   2             ⋮             m   N           ]               (   16   )                                
     The block matrix to be inverted in Eq. (16) can be quite large depending on the size of the images. However, because all S k1  are diagonal and S k1 =(S 1k )*, the inverse can be efficiently computed using a recursive formula for inverting block matrices. 
     The present invention is not limited to only a power series representation of the polynomial correlation filter as used for deriving the solution in Eq. (16). The analysis and the form of the solution remain substantially the same irrespective of the non-linearities used to obtain the terms of the polynomial. Thus, the correlation output is generally expressed as:                g   N     =       ∑     i   =   1     N                       A   i            f   i          (   x   )                   (   17   )                                
     where f(.) is any non-linear function of x. For example, possible choices for the non-linearities include absolute magnitude and sigmoid functions. The selection of the proper non-linear terms depends on the specific application of the correlation filter of the present invention. For example, it may be detrimental to use logarithms when bipolar noise is present since the logarithm of a negative number is not defined. 
     FIG. 2 a  depicts the sequence of operations for the correlation filter of the present invention to determine the filter values. The preferred embodiment performs these operations “off-line.” 
     Start indication block  60  indicates that block  62  is to be executed first. Block  62  receives the exemplar image data from single or multiple sensor sources. Block  64  processes the exemplar image data nonlinearly and/or at different resolution levels. Processing the data nonlinearly refers to the calculation of the “f(.)” terms of equation 17 above. 
     Block  64  may use Wavelets and morphological image transforms in order to process information at different resolution levels. For a description of Wavelets and morphological image transforms, see the following reference: “Morphological Methods in Image and Signal Processing,” Giardine and Dougherty, Prentice Hall, Englewood Cliffs, 1988; and C. K. Chui, “An Introduction to Wavelets” Academic Press, New York, 1992. 
     Block  66  determines the filter values through execution of the subfunction optimizer block  68 . The subfunction optimizer block  68  determines the filter values which substantially optimize a predetermined criterion (such as the Optimal trade-off performance criterion). The function of the predetermined criterion interrelates filter values  70 , exemplar image data  71  and a spectral quantity  72  (such as average correlation energy (ACE), average similarity measure (ASM), output noise variance (ONV), and combinations thereof). Processing for determining the filter values terminates at termination block  73 . 
     FIG. 2 b  depicts the operational steps for determining correlation outputs based upon the filter values. The preferred embodiment performs these operations “on-line.” 
     Start indication block  80  indicates that block  82  is to be executed first. Block  82  receives image data from single or multiple sensor sources. Block  84  processes the image data non-linearly and/or at different resolution levels. Processing the data nonlinearly refers to the calculation of the “f(.)” terms of equation 17 above. 
     Block  86  determines the correlation output  40 . The correlation output  40  is indicative of the presence of the pattern within the image data  22 . A non-linear polynomial relationship  78  interrelates the correlation output  40 , the determined filter values  70 , and the image data  22 . Processing terminates at termination block  88 . 
     As discussed in connection to FIG. 2 b , the present invention can be used to simultaneously correlate data from different image sensors. In this case, the sensor imaging process and its transfer function itself are viewed as the non-linear mapping function. The different terms of the polynomial do not have to be from the same sensor or versions of the same data. 
     FIG. 3 depicts input image data from different sensors which is directly injected into the correlation filter  20  of the present invention resulting in a fused correlation output  40 . For example, image sensor  100  is an Infrared (IR) sensor; image sensor  102  is a Laser Radar (LADAR) sensor; image sensor  104  is a Synthetic Aperture Radar (SAR) sensor; and image sensor  106  is millimeter wave (MMW) sensor. 
     The analysis and the form of the solution remain the same as that in Eq. (16). Accordingly, each image sensor ( 100 ,  102 ,  104 , and  106 ) has their individual input image data fed into their respective non-linear polynomial relationship ( 108 ,  110 ,  112 , and  114 ). Each non-linear polynomial relationship ( 108 ,  110 ,  112 , and  114 ) depicts a pixel by pixel nonlinear operation on the data. 
     Each image sensor ( 100 ,  102 ,  104 , and  106 ) has their respective filter terms ( 116 ,  118 ,  120 , and  122 ) determined in accordance to the optimization principles described above. The determined filter values are then used along with the input image data to produce a fused correlation output  40 . 
     Moreover, FIG. 3 depicts the present invention&#39;s extension to multi-sensor and multi-resolution inputs. In other words, the terms of the polynomial are the multi-spectral data represented at different resolutions levels, as for example to achieve correlation in Wavelet type transform domains. Wavelet type transform domains are described in the following reference: C. K. Chui, “An Introduction to Wavelets” Academic Press, New York, 1992. 
     EXAMPLE 
     Sample images of a tank from a database are shown in FIG.  4 . The images were available at intervals of three degrees in azimuth. The end views of the tank are generally depicted at  140 . The broadside views of the tank are generally depicted at  142 . 
     The sample images were used for training and testing a conventional linear MACH filter versus a fourth order (N=4) PCF. The peak-to-sidelobe ratio (PSR) of the correlation peaks defined as              PSR   =         p   -   mean       standard                 deviation       =       p   -   μ     σ               (   18   )                                
     was computed and used for evaluating the performance of the filters. In each case, Gaussian white noise was added to the test images to simulate a per pixel signal to noise ratio (SNR) of 10dB. 
     The PSR outputs of the conventional linear MACH filter  150  and the 4th order MACH PCF  152  are shown in FIG. 5 for comparison. FIG. 5 shows the behavior of PSR over the range of aspect angles. While the PSR is fundamentally low at end views (where there are fewer pixels on the target), the PSR output of the MACH PCF is always higher than its linear counterpart. 
     A detection threshold  154  is used to determine if the tank pattern has been detected within the image frame number. As seen from FIG. 5, the 4th order MACH PCF missed fewer detections of the pattern than the conventional linear MACH filter. 
     The embodiments which have been set forth above for the purpose of illustration were not intended to limit the invention. It will be appreciated by those skilled in the art that various changes and modifications may be made to the embodiments discussed in the specification without departing from the spirit and scope of the invention as defined by the appended claims.