Patent Publication Number: US-7912307-B2

Title: Deconvolution method using neighboring-pixel-optical-transfer-function in fourier domain

Description:
This application claims the benefit of U.S. Provisional Application No. 60/900,132, filed Feb. 8, 2007. 
    
    
     FIELD OF THE INVENTION 
     The present invention relates to optical devices, and more specifically to optical imaging sensors and image processing techniques in order to increase spatial resolution of images. 
     BACKGROUND OF THE INVENTION 
     For an optical system, if it is free from optical aberrations, then its spatial resolution is limited by optical diffraction. Optical diffraction is well known phenomenon. When light passes through an aperture, the light spreads out so that a point light sources becomes a pattern of light across the director known as diffraction pattern. Diffraction pattern depends on size and shape of the optical aperture. For example, a circular aperture generates a diffraction pattern known as Airy disc, which has a bright central spot, surrounded by fainter bright and dark rings. These rings gradually fade as the diameter increases. When the center bright spot of the Airy disc becomes large relative to the pixel spacing of detector, the image looks burr. If two Airy disks become closer than half their width, the two spots are no longer resolvable (Rayleigh criterion). An optical system with the ability to produce images with angular separations as small as the instrument&#39;s theoretical limit is called diffraction limited. 
     A basic limitation of any optical systems is the aperture size. High resolution optical systems require very larger apertures. In many situations the large aperture optical systems are very costly, especially for space based optical systems. 
     Deconvolution is specifically used to refer to the process of reversing the optical distortion that takes place in an optical imaging instrument, and creating sharper images. Deconvolution is usually done by software algorithm. Generally speaking, if a point spread function (PSF) can be found, we can compute its inverse function, and restore the original, undistorted image. But in practice, the inverse of the PSFs may become infinite in some areas, and the noise level may become unacceptable. The Optical Transfer Function (OTF) describes the spatial (angular) variation as a function of spatial (angular) frequency. The OTF is the Fourier transform of the PSF. 
     It is desirable for obtain high resolution images with small aperture optical systems. There are a few imaging techniques can claim to increase image resolution beyond diffraction limit. Super-resolution (SR) is one technique that in some way can enhance the resolution of an imaging system. There are some SR techniques that process images in the frequency domain, by assuming that the object on the image is an analytic function, and that we can exactly know the function values in some interval. SR method needs multi-frame images and is limited by the noise that is present in digital imaging systems. 
     SUMMARY OF THE INVENTION 
     The invention provides a deconvolution method to increase resolution of single frame images taken by diffraction limited optical systems. If point spread function (PSF) of an optical system is considered uniform over focal plane of a regularly spaced pixel detector, and if the spacing of the pixels is much smaller than the PSF pattern, then the image resolution can be enhanced beyond diffraction limit after being processed by the present invention. 
     The present invention, deconvolution method using neighboring-pixel-phase-modulation-function in Fourier domain, includes the following steps: (a) calculating initial neighboring-pixel correlation coefficients (NPCCs) using point spread function (PSF) for an input image, (b) applying discrete Fourier transform (FT) to the input image, (c) build a neighboring-pixel-optical-transfer-function (NPOTF) using NPCCs and periodical boundary conditions, (d) applying the NPOTF to the Fourier components, and generating modified new Fourier components, and (e) applying inversed discrete FT to these new Fourier components, and a new, higher resolution image is generated. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG.  1 -( a ) is plot of a typical angular light distribution of diffraction pattern from a circular aperture. 
       FIG.  1 -( b ) illustrates a typical two dimension angular light distribution of diffraction pattern from a circular aperture. 
         FIG. 2  is a diagram of geometry of the neighboring pixel labeling for a squared pixel array. 
         FIG. 3  is a diagram of geometry of the neighboring pixel labeling for a hexagon pixel array. 
         FIG. 4  is a flowchart of the deconvolution using NPOTF for a monochrome image. 
         FIG. 5  is a flowchart of the present invention for color images. 
       FIG.  6 -( a ) is a modeling result of a diffraction pattern of a 1-D optical aperture. 
       FIG.  6 -( b ) is an illustration of two 1D point objects separated by a small angular span. 
       FIG.  6 -( c ) is a modeling result of the image of the two objects though the 1-D optical aperture, the two objects become undistinguishable. 
       FIG.  6 -( d ) is a modeling result of the image  6 -( c ) after processed using the present invention, the two objects are clearly separated from each other. 
       FIG.  7 -( a ) is a modeling result of a diffraction pattern of a 2-D optical aperture. 
       FIG.  7 -( b ) is an illustration of two 2D point objects separated by a small angular span. 
       FIG.  7 -( c ) is a modeling result of the image of the two objects after the 2-D optical aperture, the two objects become undistinguishable. 
       FIG.  7 -( d ) is a modeling result of the image  7 -( c ) after processed by the present invention, the two objects are clearly separated from each other. 
       FIG.  8 -( a ) is a 3-D diagram illustrating the light intensity distribution of FIG.  7 -( c ), the two objects are undistinguishable. 
       FIG.  8 -( a ) is a 3-D diagram illustrating the light intensity distribution of FIG.  7 -( d ), the two objects are clearly separated from each other after processed by the present invention. 
       FIG.  9 -( a ) represents a partial image of space image of Kuwait in 1991 (courtesy of NASA/Goddard Space Flight Center). 
       FIG.  9 -( b ) is the processed image of FIG.  9 -( a ) using the present invention, the image becomes much sharper. 
       FIG.  10 -( a ) represents a partial image of ‘Double Bubble’ of Gas and Dust in the Large Magellanic Cloud taken by Hubble Space Telescope (courtesy of NASA and The Hubble Heritage). 
       FIG.  10 -( b ) is the processed image of FIG.  10 -( a ) using the present invention, some new stars and new features are unveiled. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
     The present invention is an image processing method that is able to achieve image resolutions beyond the diffraction limit. The method employs deconvolution method using neighboring-pixel-optical-transfer-function (NPOTF) in Fourier domain. 
     Resolution of any recorded image is limited by point spread function (PSF) associated with the instrument taking the image. For example, the image of a point light source through a circular aperture has an angular distribution of Airy disc on the recording detector as show in  FIG. 1 . FIG.  1 -( a ) is plot of angular light distribution of diffraction pattern of a typical circular aperture, and FIG.  1 -( b ) illustrates two dimension angular light distribution of diffraction pattern of a typical circular aperture. 
     An image on a detector can be expressed as a convolution of the original image with the PSF as:
 
 I   d ( x )= t ( x ) I   0 ( x )+ n ( x )  (1)
 
where I 0 (x) and I d (x) are the original and observed light distributions, t(x) is the total PSF, and n(x) are the measurement errors. For a perfect optical instrument, n(x)=0, and t(x) is the diffraction function.
 
     Resolution of an imaging instrument is limited by diffraction of its aperture. For photographic film imaging, because the photosensitive grains have irregular sizes and shapes, the diffraction limit barrier is impossible to overcome. Now CCD and CMOS image sensors have replaced films in most of the imaging instruments. All of the pixels in a CCD or a CMOS have the same size and spacing, this property provides us an opportunity to enhance an image resolution higher then the diffraction limit, simply by computing. 
     When a solid state light detector, for example a CCD, is used, the light is sampled at regularly spaced pixels. For a perfect instrument, the equation (1) becomes 
     
       
         
           
             
               
                 
                   
                     I 
                     k 
                     d 
                   
                   = 
                   
                     
                       ∑ 
                       
                         j 
                         = 
                         1 
                       
                       
                         N 
                         t 
                       
                     
                     ⁢ 
                     
                       
                         t 
                         jk 
                       
                       ⁢ 
                       
                         I 
                         j 
                         0 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     Where N t  is the total number of pixels, I d   k , I 0   j  are the vector components of I d (x), I 0 (x), and t jk  is the value of point j of the PSF centered on point k. From eq. (2), we can see that if we can calculate the inverse matrix of {t jk }, then the original image can recovered. But the calculation is almost impossible, because the inverse matrix is difficult to calculate or not even exists. 
     The present invention, deconvolution method using NPOTF in Fourier domain, employs neighboring-pixel model to generate a NPOTF. The NPOTF can generate a one-to-one relation between the Fourier components of the original image and the Fourier components of the detected image. Therefore the present invention is able to increase image resolution, even beyond the diffraction limit. 
     For many imaging systems, the apertures are circular and PSF is symmetric. If the field of view is small, PSF can be considered almost the same all over the detector focal plane. Because of diffraction, each pixel of the detector not only collects light from the corresponding pixel of the original image, but also collects light from the neighboring pixels of the corresponding pixel of the original image. For a 2D detector, light intensity received by any detector pixel I d   jk  can be written as: 
     
       
         
           
             
               
                 
                   
                     I 
                     jk 
                     d 
                   
                   = 
                   
                     
                       
                         a 
                         0 
                       
                       ⁢ 
                       
                         I 
                         jk 
                         0 
                       
                     
                     + 
                     
                       
                         a 
                         1 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           1 
                           
                             N 
                             1 
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             1 
                             ⁢ 
                             st_closest 
                             ⁢ 
                             _neighbor 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         a 
                         2 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           1 
                           
                             N 
                             2 
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             2 
                             ⁢ 
                             nd_closest 
                             ⁢ 
                             _neighbor 
                           
                           ) 
                         
                       
                     
                     + 
                     … 
                     + 
                     
                       
                         a 
                         p 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           1 
                           
                             N 
                             p 
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             pth_closest 
                             ⁢ 
                             _neighbor 
                           
                           ) 
                         
                       
                     
                     + 
                     … 
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     where I 0   jk  is light intensity of pixel (jk) of the original image; N 1  is total number of the first closest neighboring pixels of pixel (jk), N 2  is the total number of the second closest neighboring pixels of pixel (jk), . . . N p  is the total number of the p th  closest neighboring pixels of pixel (jk), . . . ; and a 0 , a 1 , a 2 , . . . are called as neighboring-pixel-correlation-coefficients (NPCCs). The NPCCs a 0 , a 1 , a 2 , a 3  . . . can be calculated from PSF, the integration of PSF over the center pixel gives a 0 ; the integration of PSF over one of the first closest neighboring pixel gives a 1 ; the integration of PSF over one of the second closest neighboring pixel gives a 2 ; . . . the integration of PSF over one of the pth closest neighboring pixel gives a p ; . . . etc. NPCCs depend on size and shape of the aperture, the incident photon spectrum, the field of view, the shape of the pixels, the spacing between of the pixels, quantum efficiency and some other factors. The NPCCs are constants for a given image system and can be normalized as:
 
 a   0   +a   1   N   1   +a   2   N   2   + . . . a   p   N   p  . . . =1  (4)
 
     The best way to get the NPCCs is to do a real time measurement for the optical instrument with a point light source, the angular span of the light source should be much smaller than the diffraction limit of the aperture. At the focal plane, the pixel spacing should be much smaller than the diffraction pattern on the focal plane. 
     NPCCs can also be calculated if we know the PSF. Light intensity of pixels generated by PSF is proportional to the NPCCs. If the PSF is not available, we can use a blind estimate, starting with reasonable guessed values of NPCCs, then adjust the NPCCs until we reach the maximum sharpness for processed images. 
     For example, for a square array detector with square pixels, as shown in  FIG. 2 , eq. (3) can be written as:
 
 I   d   jk   =a   0   I   0   jk   +a   1 ( I   0   j,k+1   +I   0   j,k−1   +I   0   j+1,k   +I   0   j−1,k )+ a   2 ( I   0   j+1,k+1   +I   0   j+1,k−1   +I   0   j−1,k−1 )+  (5)
 
     When we apply discrete Fourier Transform (FT) to both of the detected image and the original image, the Fourier components of I 0   jk , and I 0   jk  can be written as: 
     
       
         
           
             
               
                 
                   
                     
                       F 
                       lm 
                       d 
                     
                     = 
                     
                       
                         1 
                         N 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               k 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             
                               I 
                               jk 
                               d 
                             
                             ⁢ 
                             
                               ⅇ 
                               
                                 
                                   - 
                                   2 
                                 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 ⅈ 
                                 ⁢ 
                                 
                                   
                                     ( 
                                     
                                       jl 
                                       + 
                                       km 
                                     
                                     ) 
                                   
                                   N 
                                 
                               
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   and 
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
             
               
                 
                   
                     F 
                     lm 
                     0 
                   
                   = 
                   
                     
                       1 
                       N 
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             I 
                             jk 
                             0 
                           
                           ⁢ 
                           
                             ⅇ 
                             
                               
                                 - 
                                 2 
                               
                               ⁢ 
                               πⅈ 
                               ⁢ 
                               
                                 
                                   ( 
                                   
                                     jl 
                                     + 
                                     km 
                                   
                                   ) 
                                 
                                 N 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     From eq. (6) and eq. (5), we have 
     
       
         
           
             
               
                 
                   
                     F 
                     lm 
                     d 
                   
                   = 
                   
                     
                       
                         1 
                         N 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               k 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             
                               [ 
                               
                                 
                                   
                                     a 
                                     0 
                                   
                                   ⁢ 
                                   
                                     I 
                                     jk 
                                     0 
                                   
                                 
                                 + 
                                 
                                   
                                     a 
                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         I 
                                         
                                           j 
                                           , 
                                           
                                             k 
                                             + 
                                             1 
                                           
                                         
                                         0 
                                       
                                       + 
                                       
                                         I 
                                         
                                           j 
                                           , 
                                           
                                             k 
                                             - 
                                             1 
                                           
                                         
                                         0 
                                       
                                       + 
                                       
                                         I 
                                         
                                           
                                             j 
                                             + 
                                             1 
                                           
                                           , 
                                           k 
                                         
                                         0 
                                       
                                       + 
                                       
                                         I 
                                         
                                           
                                             j 
                                             - 
                                             1 
                                           
                                           , 
                                           k 
                                         
                                         0 
                                       
                                     
                                     ) 
                                   
                                 
                                 + 
                                 
                                   
                                     a 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         I 
                                         
                                           
                                             j 
                                             + 
                                             1 
                                           
                                           , 
                                           
                                             k 
                                             + 
                                             1 
                                           
                                         
                                         0 
                                       
                                       + 
                                       
                                         I 
                                         
                                           
                                             j 
                                             + 
                                             1 
                                           
                                           , 
                                           
                                             k 
                                             - 
                                             1 
                                           
                                         
                                         0 
                                       
                                       + 
                                       
                                         I 
                                         
                                           
                                             j 
                                             - 
                                             1 
                                           
                                           , 
                                           
                                             k 
                                             + 
                                             1 
                                           
                                         
                                         0 
                                       
                                       + 
                                       
                                         I 
                                         
                                           
                                             j 
                                             - 
                                             1 
                                           
                                           , 
                                           
                                             k 
                                             - 
                                             1 
                                           
                                         
                                         0 
                                       
                                     
                                     ) 
                                   
                                 
                                 + 
                                 … 
                               
                               ] 
                             
                             ⁢ 
                             
                               ⅇ 
                               
                                 
                                   - 
                                   2 
                                 
                                 ⁢ 
                                 πⅈ 
                                 ⁢ 
                                 
                                   
                                     ( 
                                     
                                       jl 
                                       + 
                                       km 
                                     
                                     ) 
                                   
                                   N 
                                 
                               
                             
                           
                         
                       
                     
                     + 
                     … 
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     Here comes the key point of this present invention: all of the indices in space domain, j, k, (j+1), (j−1), (k+1), (k−1), . . . , become dummy indices when we apply Fourier transform to the eq. (8). Because the indices j, k, (j+1), (j−1), (k+1), (k−1), . . . , are all dummy indices, we can greatly simplify eq. (8) with the following manipulations. 
     Look at the second term in the sum of equation (8), 
                 a   1     N     ⁢       ∑     j   =   0       N   -   1       ⁢       ∑     k   =   0       N   -   1       ⁢       I       j   +   1     ,   k     0     ⁢     ⅇ       -   2     ⁢   πⅈ   ⁢       (     jl   +   km     )     N                     
which only relates to I 0   jk,k+1 , this term can be written as:
 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               a 
                               1 
                             
                             N 
                           
                           ⁢ 
                           
                             
                               ∑ 
                               
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                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               
                                 ∑ 
                                 
                                   k 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 
                                   I 
                                   
                                     
                                       j 
                                       + 
                                       1 
                                     
                                     , 
                                     k 
                                   
                                   0 
                                 
                                 ⁢ 
                                 
                                   ⅇ 
                                   
                                     
                                       - 
                                       2 
                                     
                                     ⁢ 
                                     πⅈ 
                                     ⁢ 
                                     
                                       
                                         ( 
                                         
                                           jl 
                                           + 
                                           km 
                                         
                                         ) 
                                       
                                       N 
                                     
                                   
                                 
                               
                             
                           
                         
                         = 
                         
                           
                             
                               a 
                               1 
                             
                             N 
                           
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 
                                   j 
                                   ′ 
                                 
                                 = 
                                 1 
                               
                               N 
                             
                             ⁢ 
                             
                               
                                 ∑ 
                                 
                                   k 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 
                                   I 
                                   
                                     
                                       j 
                                       ′ 
                                     
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                                     k 
                                   
                                   0 
                                 
                                 ⁢ 
                                 
                                   ⅇ 
                                   
                                     
                                       - 
                                       2 
                                     
                                     ⁢ 
                                     πⅈ 
                                     ⁢ 
                                     
                                       
                                         [ 
                                         
                                           
                                             
                                               ( 
                                               
                                                 
                                                   j 
                                                   ′ 
                                                 
                                                 + 
                                                 1 
                                               
                                               ) 
                                             
                                             ⁢ 
                                             l 
                                           
                                           + 
                                           km 
                                         
                                         ] 
                                       
                                       N 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               a 
                               1 
                             
                             N 
                           
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 
                                   j 
                                   ′ 
                                 
                                 = 
                                 1 
                               
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                             ⁢ 
                             
                               
                                 ∑ 
                                 
                                   k 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 
                                   I 
                                   
                                     
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                                       ′ 
                                     
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                                     k 
                                   
                                   0 
                                 
                                 ⁢ 
                                 
                                   ⅇ 
                                   
                                     
                                       - 
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                                     ⁢ 
                                     πⅈ 
                                     ⁢ 
                                     
                                       
                                         ( 
                                         
                                           
                                             
                                               j 
                                               ′ 
                                             
                                             ⁢ 
                                             l 
                                           
                                           + 
                                           km 
                                         
                                         ) 
                                       
                                       N 
                                     
                                   
                                 
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                                       - 
                                       
                                         
                                           ( 
                                           
                                             
                                               
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                                                 ′ 
                                               
                                               ⁢ 
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                                             + 
                                             km 
                                           
                                           ) 
                                         
                                         N 
                                       
                                     
                                   
                                   . 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     Here j′=j+1. Then 
     
       
         
           
             
               
                 
                   
                     
                       
                         a 
                         1 
                       
                       N 
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           
                             j 
                             ′ 
                           
                           = 
                           1 
                         
                         N 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             I 
                             
                               
                                 j 
                                 ′ 
                               
                               , 
                               k 
                             
                             0 
                           
                           ⁢ 
                           
                             ⅇ 
                             
                               
                                 - 
                                 2 
                               
                               ⁢ 
                               πⅈ 
                               ⁢ 
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         j 
                                         ′ 
                                       
                                       ⁢ 
                                       l 
                                     
                                     + 
                                     km 
                                   
                                   ) 
                                 
                                 N 
                               
                             
                           
                           ⁢ 
                           
                             ⅇ 
                             
                               - 
                               
                                 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   ⅈ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   l 
                                 
                                 N 
                               
                             
                           
                         
                       
                     
                   
                   = 
                   
                     
                       
                         
                           a 
                           1 
                         
                         N 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             
                               j 
                               ′ 
                             
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               k 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             
                               I 
                               
                                 
                                   j 
                                   ′ 
                                 
                                 , 
                                 k 
                               
                               0 
                             
                             ⁢ 
                             
                               ⅇ 
                               
                                 
                                   - 
                                   2 
                                 
                                 ⁢ 
                                 πⅈ 
                                 ⁢ 
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           j 
                                           ′ 
                                         
                                         ⁢ 
                                         l 
                                       
                                       + 
                                       km 
                                     
                                     ) 
                                   
                                   N 
                                 
                               
                             
                             ⁢ 
                             
                               ⅇ 
                               
                                 
                                   - 
                                   2 
                                 
                                 ⁢ 
                                 πⅈ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 l 
                               
                             
                           
                         
                       
                     
                     + 
                     
                       
                         
                           a 
                           1 
                         
                         N 
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               
                                 I 
                                 
                                   N 
                                   , 
                                   k 
                                 
                                 0 
                               
                               ⁢ 
                               
                                 ⅇ 
                                 
                                   
                                     - 
                                     2 
                                   
                                   ⁢ 
                                   πⅈ 
                                 
                               
                             
                             - 
                             
                               I 
                               
                                 0 
                                 , 
                                 k 
                               
                               0 
                             
                           
                           ) 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     I 0   Nk  does not exist. But if we use periodical boundary conditions, which means that the image repeats itself when the pixels are out of the detector, then
 
I 0   Nk =I 0   0k , and
 
I 0   jN =I 0   j0 .
 
This periodical boundary conditions also give us:
 
 I   0   −1,k   =I   0   N−1,k   , I   0   j,−1   =I   0   j,N−1   , I   0   −2,k   =I   0   N−2,k   , I   0   j,−2   =I   0   j,N−2   , I   0   −3,k   =I   0   N−3,k   , I   0   j,−3   =I   0   j,N−3, . . . ;  
 
Then from (7) and (10), we have:
 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               a 
                               1 
                             
                             N 
                           
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 j 
                                 = 
                                 1 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               
                                 ∑ 
                                 
                                   k 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 
                                   I 
                                   
                                     
                                       j 
                                       + 
                                       1 
                                     
                                     , 
                                     k 
                                   
                                   0 
                                 
                                 ⁢ 
                                 
                                   ⅇ 
                                   
                                     
                                       - 
                                       2 
                                     
                                     ⁢ 
                                     πⅈ 
                                     ⁢ 
                                     
                                       
                                         ( 
                                         
                                           jl 
                                           + 
                                           km 
                                         
                                         ) 
                                       
                                       N 
                                     
                                   
                                 
                               
                             
                           
                         
                         = 
                         
                           
                             
                               a 
                               1 
                             
                             N 
                           
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 j 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               
                                 ∑ 
                                 
                                   k 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 
                                   I 
                                   
                                     
                                       j 
                                       ′ 
                                     
                                     , 
                                     k 
                                   
                                   0 
                                 
                                 ⁢ 
                                 
                                   ⅇ 
                                   
                                     
                                       - 
                                       2 
                                     
                                     ⁢ 
                                     πⅈ 
                                     ⁢ 
                                     
                                       
                                         ( 
                                         
                                           jl 
                                           + 
                                           km 
                                         
                                         ) 
                                       
                                       N 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   ⅇ 
                                   
                                     - 
                                     
                                       
                                         2 
                                         ⁢ 
                                         π 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         ⅈ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         l 
                                       
                                       N 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               a 
                               1 
                             
                             N 
                           
                           ⁢ 
                           
                             F 
                             lm 
                             0 
                           
                           ⁢ 
                           
                             
                               ⅇ 
                               
                                 - 
                                 
                                   
                                     2 
                                     ⁢ 
                                     πⅈ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     l 
                                   
                                   N 
                                 
                               
                             
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     From (11), we can see that the Fourier components of the first closest neighboring pixel I 0   j+1,k  can be expressed by the Fourier component of the center pixel I 0   jk  with an addition phase factor. 
     Further more, we can combine the 2 nd  and 3 rd  terms of Eq. (8) as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             a 
                             1 
                           
                           N 
                         
                         = 
                         
                           
                             ∑ 
                             
                               j 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 k 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               
                                 ( 
                                 
                                   
                                     I 
                                     
                                       
                                         j 
                                         + 
                                         1 
                                       
                                       , 
                                       k 
                                     
                                     0 
                                   
                                   + 
                                   
                                     I 
                                     
                                       
                                         j 
                                         - 
                                         1 
                                       
                                       , 
                                       k 
                                     
                                     0 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 ⅇ 
                                 
                                   
                                     - 
                                     2 
                                   
                                   ⁢ 
                                   πⅈ 
                                   ⁢ 
                                   
                                     
                                       ( 
                                       
                                         jl 
                                         + 
                                         km 
                                       
                                       ) 
                                     
                                     N 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               a 
                               1 
                             
                             N 
                           
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 
                                   j 
                                   ′ 
                                 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               
                                 ∑ 
                                 
                                   
                                     k 
                                     ′ 
                                   
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 
                                   I 
                                   
                                     
                                       j 
                                       ′ 
                                     
                                     , 
                                     k 
                                   
                                   0 
                                 
                                 ⁢ 
                                 
                                   
                                     ⅇ 
                                     
                                       
                                         - 
                                         2 
                                       
                                       ⁢ 
                                       πⅈ 
                                       ⁢ 
                                       
                                         
                                           ( 
                                           
                                             
                                               
                                                 j 
                                                 ′ 
                                               
                                               ⁢ 
                                               l 
                                             
                                             + 
                                             km 
                                           
                                           ) 
                                         
                                         N 
                                       
                                     
                                   
                                   ( 
                                   
                                     
                                       ⅇ 
                                       
                                         
                                           
                                             - 
                                             2 
                                           
                                           ⁢ 
                                           πⅈ 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           l 
                                         
                                         N 
                                       
                                     
                                     + 
                                     
                                       ⅇ 
                                       
                                         
                                           2 
                                           ⁢ 
                                           πⅈ 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           l 
                                         
                                         N 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               a 
                               1 
                             
                             N 
                           
                           ⁢ 
                           
                             
                               F 
                               lm 
                               0 
                             
                             · 
                             2 
                           
                           ⁢ 
                           
                             
                               cos 
                               ( 
                               
                                 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   l 
                                 
                                 N 
                               
                               ) 
                             
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     And we can combine the 4 th  and 5 th  terms of Eq. (8) as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         a 
                         1 
                       
                       N 
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 I 
                                 
                                   
                                     j 
                                     + 
                                     1 
                                   
                                   , 
                                   k 
                                 
                                 0 
                               
                               + 
                               
                                 I 
                                 
                                   
                                     j 
                                     - 
                                     1 
                                   
                                   , 
                                   k 
                                 
                                 0 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             ⅇ 
                             
                               
                                 - 
                                 2 
                               
                               ⁢ 
                               πⅈ 
                               ⁢ 
                               
                                 
                                   ( 
                                   
                                     jl 
                                     + 
                                     km 
                                   
                                   ) 
                                 
                                 N 
                               
                             
                           
                         
                       
                     
                   
                   = 
                   
                     
                       
                         
                           a 
                           1 
                         
                         N 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             
                               j 
                               ′ 
                             
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               
                                 k 
                                 ′ 
                               
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             
                               I 
                               
                                 
                                   j 
                                   ′ 
                                 
                                 , 
                                 k 
                               
                               0 
                             
                             ⁢ 
                             
                               
                                 ⅇ 
                                 
                                   
                                     - 
                                     2 
                                   
                                   ⁢ 
                                   πⅈ 
                                   ⁢ 
                                   
                                     
                                       ( 
                                       
                                         
                                           
                                             j 
                                             ′ 
                                           
                                           ⁢ 
                                           l 
                                         
                                         + 
                                         km 
                                       
                                       ) 
                                     
                                     N 
                                   
                                 
                               
                               ( 
                               
                                 
                                   ⅇ 
                                   
                                     
                                       
                                         - 
                                         2 
                                       
                                       ⁢ 
                                       πⅈ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       l 
                                     
                                     N 
                                   
                                 
                                 + 
                                 
                                   ⅇ 
                                   
                                     
                                       2 
                                       ⁢ 
                                       πⅈ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       l 
                                     
                                     N 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     = 
                     
                       
                         
                           a 
                           1 
                         
                         N 
                       
                       ⁢ 
                       
                         
                           F 
                           lm 
                           0 
                         
                         · 
                         2 
                       
                       ⁢ 
                       
                         
                           cos 
                           ( 
                           
                             
                               2 
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               l 
                             
                             N 
                           
                           ) 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     therefore, from all of the 1 st  closest neighboring pixels terms in Eq. (8), we have: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           a 
                           1 
                         
                         N 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             1 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               k 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   I 
                                   
                                     j 
                                     , 
                                     
                                       k 
                                       + 
                                       1 
                                     
                                   
                                   0 
                                 
                                 + 
                                 
                                   I 
                                   
                                     j 
                                     , 
                                     
                                       k 
                                       - 
                                       1 
                                     
                                   
                                   0 
                                 
                                 + 
                                 
                                   I 
                                   
                                     
                                       j 
                                       + 
                                       1 
                                     
                                     , 
                                     k 
                                   
                                   0 
                                 
                                 + 
                                 
                                   I 
                                   
                                     
                                       j 
                                       - 
                                       1 
                                     
                                     , 
                                     k 
                                   
                                   0 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               ⅇ 
                               
                                 
                                   - 
                                   2 
                                 
                                 ⁢ 
                                 πⅈ 
                                 ⁢ 
                                 
                                   
                                     ( 
                                     
                                       jl 
                                       + 
                                       km 
                                     
                                     ) 
                                   
                                   N 
                                 
                               
                             
                           
                         
                       
                     
                     = 
                     
                       
                         
                           a 
                           1 
                         
                         N 
                       
                       ⁢ 
                       
                         
                           F 
                           lm 
                           0 
                         
                         · 
                         
                           
                             2 
                             [ 
                             
                               
                                 cos 
                                 ( 
                                 
                                   
                                     2 
                                     ⁢ 
                                     π 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     l 
                                   
                                   N 
                                 
                                 ) 
                               
                               + 
                               
                                 cos 
                                 ( 
                                 
                                   
                                     2 
                                     ⁢ 
                                     π 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     m 
                                   
                                   N 
                                 
                                 ) 
                               
                             
                             ] 
                           
                           . 
                           
                             
 
                           
                           ⁢ 
                           Similarly 
                         
                       
                     
                   
                   , 
                   
                     we 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     have 
                     ⁢ 
                     
                       : 
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         a 
                         2 
                       
                       N 
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           1 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 I 
                                 
                                   
                                     j 
                                     + 
                                     1 
                                   
                                   , 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                 
                                 0 
                               
                               + 
                               
                                 I 
                                 
                                   
                                     j 
                                     + 
                                     1 
                                   
                                   , 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                 
                                 0 
                               
                               + 
                               
                                 I 
                                 
                                   
                                     j 
                                     - 
                                     1 
                                   
                                   , 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                 
                                 0 
                               
                               + 
                               
                                 I 
                                 
                                   
                                     j 
                                     - 
                                     1 
                                   
                                   , 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                 
                                 0 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             ⅇ 
                             
                               
                                 - 
                                 2 
                               
                               ⁢ 
                               πⅈ 
                               ⁢ 
                               
                                 
                                   ( 
                                   
                                     jl 
                                     + 
                                     km 
                                   
                                   ) 
                                 
                                 N 
                               
                             
                           
                         
                       
                     
                   
                   = 
                   
                     
                       
                         a 
                         2 
                       
                       N 
                     
                     ⁢ 
                     
                       
                         F 
                         lm 
                         0 
                       
                       · 
                       
                         
                           2 
                           [ 
                           
                             
                               cos 
                               ( 
                               
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     l 
                                     + 
                                     m 
                                   
                                   N 
                                 
                               
                               ) 
                             
                             + 
                             
                               cos 
                               ( 
                               
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 
                                   
                                     l 
                                     - 
                                     m 
                                   
                                   N 
                                 
                               
                               ) 
                             
                           
                           ] 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         a 
                         3 
                       
                       N 
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           1 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 I 
                                 
                                   j 
                                   , 
                                   
                                     k 
                                     + 
                                     2 
                                   
                                 
                                 0 
                               
                               + 
                               
                                 I 
                                 
                                   j 
                                   , 
                                   
                                     k 
                                     - 
                                     2 
                                   
                                 
                                 0 
                               
                               + 
                               
                                 I 
                                 
                                   
                                     j 
                                     + 
                                     2 
                                   
                                   , 
                                   k 
                                 
                                 0 
                               
                               + 
                               
                                 I 
                                 
                                   
                                     j 
                                     - 
                                     2 
                                   
                                   , 
                                   k 
                                 
                                 0 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             ⅇ 
                             
                               
                                 - 
                                 2 
                               
                               ⁢ 
                               πⅈ 
                               ⁢ 
                               
                                 
                                   ( 
                                   
                                     jl 
                                     - 
                                     km 
                                   
                                   ) 
                                 
                                 N 
                               
                             
                           
                         
                       
                     
                   
                   = 
                   
                     
                       
                         a 
                         3 
                       
                       N 
                     
                     ⁢ 
                     
                       
                         F 
                         lm 
                         0 
                       
                       · 
                       
                         
                           2 
                           [ 
                           
                             
                               cos 
                               ( 
                               
                                 
                                   4 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   l 
                                 
                                 N 
                               
                               ) 
                             
                             + 
                             
                               cos 
                               ( 
                               
                                 
                                   4 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   m 
                                 
                                 N 
                               
                               ) 
                             
                           
                           ] 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         a 
                         4 
                       
                       N 
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           1 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 I 
                                 
                                   
                                     j 
                                     + 
                                     1 
                                   
                                   , 
                                   
                                     k 
                                     + 
                                     2 
                                   
                                 
                                 0 
                               
                               + 
                               
                                 I 
                                 
                                   
                                     j 
                                     + 
                                     1 
                                   
                                   , 
                                   
                                     k 
                                     - 
                                     2 
                                   
                                 
                                 0 
                               
                               + 
                               
                                 I 
                                 
                                   
                                     j 
                                     + 
                                     2 
                                   
                                   , 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                 
                                 0 
                               
                               + 
                               
                                 I 
                                 
                                   
                                     j 
                                     + 
                                     2 
                                   
                                   , 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                 
                                 0 
                               
                               + 
                               
                                 I 
                                 
                                   
                                     j 
                                     - 
                                     1 
                                   
                                   , 
                                   
                                     k 
                                     + 
                                     2 
                                   
                                 
                                 0 
                               
                               + 
                               
                                 I 
                                 
                                   
                                     j 
                                     - 
                                     1 
                                   
                                   , 
                                   
                                     k 
                                     - 
                                     2 
                                   
                                 
                                 0 
                               
                               + 
                               
                                 I 
                                 
                                   
                                     j 
                                     - 
                                     2 
                                   
                                   , 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                 
                                 0 
                               
                               + 
                               
                                 I 
                                 
                                   
                                     j 
                                     - 
                                     2 
                                   
                                   , 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                 
                                 0 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             ⅇ 
                             
                               
                                 - 
                                 2 
                               
                               ⁢ 
                               πⅈ 
                               ⁢ 
                               
                                 
                                   ( 
                                   
                                     jl 
                                     + 
                                     km 
                                   
                                   ) 
                                 
                                 N 
                               
                             
                           
                         
                       
                     
                   
                   = 
                   
                     
                       
                         a 
                         4 
                       
                       N 
                     
                     ⁢ 
                     
                       
                         F 
                         lm 
                         0 
                       
                       · 
                       
                         
                           2 
                           [ 
                           
                             
                               cos 
                               ⁢ 
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                     π 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           2 
                                           ⁢ 
                                           l 
                                         
                                         + 
                                         m 
                                       
                                       ) 
                                     
                                   
                                 
                                 N 
                               
                             
                             + 
                             
                               cos 
                               ⁢ 
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                     π 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           2 
                                           ⁢ 
                                           l 
                                         
                                         - 
                                         m 
                                       
                                       ) 
                                     
                                   
                                 
                                 N 
                               
                             
                             + 
                             
                               cos 
                               ⁢ 
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                     π 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         l 
                                         + 
                                         
                                           2 
                                           ⁢ 
                                           m 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 N 
                               
                             
                             + 
                             
                               cos 
                               ⁢ 
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                     π 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         l 
                                         - 
                                         
                                           2 
                                           ⁢ 
                                           m 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 N 
                               
                             
                           
                           ] 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     Let&#39;s introducing a neighboring-pixel-optical-transfer-function (NPOTF) Δ lm  as: 
     
       
         
           
             
               
                 
                   
                     
                       Δ 
                       lm 
                     
                     = 
                     
                       
                         a 
                         0 
                       
                       + 
                       
                         Δ 
                         lm 
                         1 
                       
                       + 
                       
                         Δ 
                         lm 
                         2 
                       
                       + 
                       
                         Δ 
                         lm 
                         3 
                       
                       + 
                       
                         Δ 
                         lm 
                         4 
                       
                       + 
                       
                         Δ 
                         lm 
                         5 
                       
                       + 
                       
                         Δ 
                         lm 
                         6 
                       
                       + 
                       
                         Δ 
                         lm 
                         7 
                       
                       + 
                       
                         Δ 
                         lm 
                         8 
                       
                       + 
                       … 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   where 
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
             
               
                 
                   
                     Δ 
                     lm 
                     1 
                   
                   = 
                   
                     2 
                     ⁢ 
                     
                       
                         a 
                         1 
                       
                       [ 
                       
                         
                           cos 
                           ( 
                           
                             
                               2 
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               d 
                             
                             N 
                           
                           ) 
                         
                         + 
                         
                           cos 
                           ( 
                           
                             
                               2 
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               m 
                             
                             N 
                           
                           ) 
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
             
               
                 
                   
                     Δ 
                     lm 
                     2 
                   
                   = 
                   
                     2 
                     ⁢ 
                     
                       
                         a 
                         2 
                       
                       [ 
                       
                         
                           cos 
                           ( 
                           
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             
                               
                                 l 
                                 + 
                                 m 
                               
                               N 
                             
                           
                           ) 
                         
                         + 
                         
                           cos 
                           ( 
                           
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             
                               
                                 l 
                                 - 
                                 m 
                               
                               N 
                             
                           
                           ) 
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
             
               
                 
                   
                     Δ 
                     lm 
                     3 
                   
                   = 
                   
                     2 
                     ⁢ 
                     
                       
                         a 
                         3 
                       
                       [ 
                       
                         
                           cos 
                           ( 
                           
                             
                               2 
                               ⁢ 
                               
                                 π 
                                 · 
                                 2 
                               
                               ⁢ 
                               l 
                             
                             N 
                           
                           ) 
                         
                         + 
                         
                           cos 
                           ( 
                           
                             
                               2 
                               ⁢ 
                               
                                 π 
                                 · 
                                 2 
                               
                               ⁢ 
                               m 
                             
                             N 
                           
                           ) 
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
             
               
                 
                   
                     Δ 
                     lm 
                     4 
                   
                   = 
                   
                     2 
                     ⁢ 
                     
                       
                         a 
                         4 
                       
                       [ 
                       
                         
                           cos 
                           ⁢ 
                           
                             
                               2 
                               ⁢ 
                               
                                 π 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       2 
                                       ⁢ 
                                       l 
                                     
                                     + 
                                     m 
                                   
                                   ) 
                                 
                               
                             
                             N 
                           
                         
                         + 
                         
                           cos 
                           ⁢ 
                           
                             
                               2 
                               ⁢ 
                               
                                 π 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       2 
                                       ⁢ 
                                       l 
                                     
                                     - 
                                     m 
                                   
                                   ) 
                                 
                               
                             
                             N 
                           
                         
                         + 
                         
                           cos 
                           ⁢ 
                           
                             
                               2 
                               ⁢ 
                               
                                 π 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     l 
                                     + 
                                     
                                       2 
                                       ⁢ 
                                       m 
                                     
                                   
                                   ) 
                                 
                               
                             
                             N 
                           
                         
                         + 
                         
                           cos 
                           ⁢ 
                           
                             
                               2 
                               ⁢ 
                               
                                 π 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     l 
                                     - 
                                     m 
                                   
                                   ) 
                                 
                               
                             
                             N 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
             
               
                 
                   
                     Δ 
                     lm 
                     5 
                   
                   = 
                   
                     2 
                     ⁢ 
                     
                       
                         a 
                         5 
                       
                       [ 
                       
                         
                           cos 
                           ( 
                           
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             
                               
                                 
                                   2 
                                   ⁢ 
                                   l 
                                 
                                 + 
                                 
                                   2 
                                   ⁢ 
                                   m 
                                 
                               
                               N 
                             
                           
                           ) 
                         
                         + 
                         
                           cos 
                           ( 
                           
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             
                               
                                 
                                   2 
                                   ⁢ 
                                   l 
                                 
                                 - 
                                 
                                   2 
                                   ⁢ 
                                   m 
                                 
                               
                               N 
                             
                           
                           ) 
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
             
               
                 
                   
                     Δ 
                     lm 
                     6 
                   
                   = 
                   
                     2 
                     ⁢ 
                     
                       
                         a 
                         6 
                       
                       [ 
                       
                         
                           cos 
                           ( 
                           
                             
                               2 
                               ⁢ 
                               
                                 π 
                                 · 
                                 3 
                               
                               ⁢ 
                               l 
                             
                             N 
                           
                           ) 
                         
                         + 
                         
                           cos 
                           ( 
                           
                             
                               2 
                               ⁢ 
                               
                                 π 
                                 · 
                                 3 
                               
                               ⁢ 
                               m 
                             
                             N 
                           
                           ) 
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
             
               
                 
                   
                     Δ 
                     lm 
                     7 
                   
                   = 
                   
                     2 
                     ⁢ 
                     
                       
                         a 
                         7 
                       
                       [ 
                       
                         
                           cos 
                           ⁢ 
                           
                             
                               2 
                               ⁢ 
                               
                                 π 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       3 
                                       ⁢ 
                                       l 
                                     
                                     + 
                                     m 
                                   
                                   ) 
                                 
                               
                             
                             N 
                           
                         
                         + 
                         
                           cos 
                           ⁢ 
                           
                             
                               2 
                               ⁢ 
                               
                                 π 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       3 
                                       ⁢ 
                                       l 
                                     
                                     - 
                                     m 
                                   
                                   ) 
                                 
                               
                             
                             N 
                           
                         
                         + 
                         
                           cos 
                           ⁢ 
                           
                             
                               2 
                               ⁢ 
                               
                                 π 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     l 
                                     + 
                                     
                                       3 
                                       ⁢ 
                                       m 
                                     
                                   
                                   ) 
                                 
                               
                             
                             N 
                           
                         
                         + 
                         
                           cos 
                           ⁢ 
                           
                             
                               2 
                               ⁢ 
                               
                                 π 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     l 
                                     - 
                                     
                                       3 
                                       ⁢ 
                                       m 
                                     
                                   
                                   ) 
                                 
                               
                             
                             N 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
             
               
                 
                   
                     Δ 
                     lm 
                     8 
                   
                   = 
                   
                     2 
                     ⁢ 
                     
                       
                         a 
                         7 
                       
                       [ 
                       
                         
                           cos 
                           ⁢ 
                           
                             
                               2 
                               ⁢ 
                               
                                 π 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       3 
                                       ⁢ 
                                       l 
                                     
                                     + 
                                     
                                       2 
                                       ⁢ 
                                       m 
                                     
                                   
                                   ) 
                                 
                               
                             
                             N 
                           
                         
                         + 
                         
                           cos 
                           ⁢ 
                           
                             
                               2 
                               ⁢ 
                               
                                 π 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       3 
                                       ⁢ 
                                       l 
                                     
                                     - 
                                     
                                       2 
                                       ⁢ 
                                       m 
                                     
                                   
                                   ) 
                                 
                               
                             
                             N 
                           
                         
                         + 
                         
                           cos 
                           ⁢ 
                           
                             
                               2 
                               ⁢ 
                               
                                 π 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       2 
                                       ⁢ 
                                       l 
                                     
                                     + 
                                     
                                       3 
                                       ⁢ 
                                       m 
                                     
                                   
                                   ) 
                                 
                               
                             
                             N 
                           
                         
                         + 
                         
                           cos 
                           ⁢ 
                           
                             
                               2 
                               ⁢ 
                               
                                 π 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       2 
                                       ⁢ 
                                       l 
                                     
                                     - 
                                     
                                       3 
                                       ⁢ 
                                       m 
                                     
                                   
                                   ) 
                                 
                               
                             
                             N 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
     Therefore, From eq. (8), the relation between the Fourier component of the detected image and the original image can be written as:
 
 F   lm   d   =F   lm   0 ·Δ lm ·.  (27)
 
and the Fourier components of the original image can be written as
 
     
       
         
           
             
               
                 
                   
                     F 
                     lm 
                     0 
                   
                   = 
                   
                     
                       
                         F 
                         lm 
                         d 
                       
                       
                         Δ 
                         lm 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
     Eq. (28) shows that we now have a one-to-one relations between Fourier components of the original image and the Fourier components of the detected image by using NPOTF. 
     And finally, we can regenerate the original image with inversed discrete Fourier transform. 
                     I   jk   0     =         1   N     ⁢       ∑     l   =   0       N   -   1       ⁢       ∑     m   =   0       N   -   1       ⁢       F   lm   0     ⁢     ⅇ     2   ⁢   πⅈ   ⁢       (     jl   +   km     )     N                 =       1   N     ⁢       ∑     l   =   0       N   -   1       ⁢       ∑     m   =   0       N   -   1       ⁢         F   lm   d       Δ   lm       ⁢     ⅇ     2   ⁢   πⅈ   ⁢       (     jl   +   km     )     N                           (   29   )               
And this is the image with enhanced image with resolution beyond the diffraction limit.
 
     Look at formula (29), we can see that I 0   jk  may have singular points. This means that if the diffraction pattern is not relatively small, the NPOTF Δ lm , may become zero for some Fourier components and noise goes to infinity. To avoid this problem, when the diffraction pattern is not relatively small, then we need to set up a threshold for the NPOTF Δ lm , which means that when Δ lm  is smaller than the threshold, we use the threshold value or some other value to replace Δ lm . The threshold can be chosen as either a constant or a function of the FT order (l,m). General speaking, the threshold is related to the noise level of the processed image, the larger the threshold, the lower the noise level. But when the threshold is too large, the image becomes blurred again. 
     Procedure for computing the present invention is illustrated in flow charts as shown in FIG. ( 4 ) and FIG. ( 5 ). FIG. ( 4 ) shows the flow chat of applying the deconvolution method of the present invention to a monochrome image, which has the procedure as the following:
         (a) selecting a reasonable PSF, and calculating initial NPCCs for the monochrome image;   (b) applying FT for the input image;   (c) building a NPOTF using the NPCCs and initial threshold in frequency domain;   (d) applying the NPOTF to the Fourier components, and generating modified new Fourier components;   (e) applying inversed FT to these new Fourier components to generate a new, higher resolution image; and   (f) adjusting NPCCs and thresholds, and repeating steps (b)-(e) until we achieve best possible resolution with acceptable noise level.       

     Because PSF depends on the light wavelength, images of different colors have different NPCCs and NPOTFs. For a full color image, for example a RGB image, we need to separate the color image into monochrome grayscale images, and process each monochrome image using the steps shown in FIG. ( 4 ). The flow chat of processing color images is shown in FIG. ( 5 ), which includes the following steps:
         (a) separating the color image into monochrome images;   (b) applying the deconvolution method, as shown in FIG. ( 4 ), to each of the monochrome images;   (c) adjusting brightness of the processed monochrome images, the brightness of these monochrome images should be adjusted this way, so that the color of any bulk areas of the unprocessed image should match the same color for the same bulk area of the processed image; and   (d) recombining the monochrome images for an output color image.       

     For geometries of detector arrays other than square array, Eq. (18), Eq. (28) and Eq. (29) still valid, only the NPOTF components are different. For example, for an N x ×N y  rectangular array pixel detector, we have: 
     
       
         
           
             
               
                 
                   
                     Δ 
                     lm 
                     1 
                   
                   = 
                   
                     2 
                     ⁢ 
                     
                       
                         a 
                         1 
                       
                       [ 
                       
                         
                           cos 
                           ( 
                           
                             
                               2 
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               d 
                             
                             
                               N 
                               x 
                             
                           
                           ) 
                         
                         + 
                         
                           cos 
                           ( 
                           
                             
                               2 
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               m 
                             
                             
                               N 
                               y 
                             
                           
                           ) 
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
             
               
                 
                   
                     Δ 
                     lm 
                     2 
                   
                   = 
                   
                     2 
                     ⁢ 
                     
                       a 
                       2 
                     
                     ⁢ 
                     
                       { 
                       
                         
                           cos 
                           [ 
                           
                             2 
                             ⁢ 
                             
                               π 
                               ( 
                               
                                 
                                   l 
                                   
                                     N 
                                     x 
                                   
                                 
                                 + 
                                 
                                   m 
                                   
                                     N 
                                     y 
                                   
                                 
                               
                               ) 
                             
                           
                           ] 
                         
                         + 
                         
                           cos 
                           [ 
                           
                             2 
                             ⁢ 
                             
                               π 
                               ( 
                               
                                 
                                   l 
                                   
                                     N 
                                     x 
                                   
                                 
                                 - 
                                 
                                   m 
                                   
                                     N 
                                     y 
                                   
                                 
                               
                               ) 
                             
                           
                           ] 
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
             
               
                 
                   
                     Δ 
                     lm 
                     3 
                   
                   = 
                   
                     2 
                     ⁢ 
                     
                       
                         a 
                         3 
                       
                       [ 
                       
                         
                           cos 
                           ( 
                           
                             
                               2 
                               ⁢ 
                               
                                 π 
                                 · 
                                 2 
                               
                               ⁢ 
                               l 
                             
                             
                               N 
                               x 
                             
                           
                           ) 
                         
                         + 
                         
                           cos 
                           ( 
                           
                             
                               2 
                               ⁢ 
                               
                                 π 
                                 · 
                                 2 
                               
                               ⁢ 
                               m 
                             
                             
                               N 
                               y 
                             
                           
                           ) 
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
             
               
                 
                   
                     Δ 
                     lm 
                     4 
                   
                   = 
                   
                     2 
                     ⁢ 
                     
                       a 
                       4 
                     
                     ⁢ 
                     
                       { 
                       
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           [ 
                           
                             2 
                             ⁢ 
                             
                               π 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     
                                       2 
                                       ⁢ 
                                       l 
                                     
                                     
                                       N 
                                       x 
                                     
                                   
                                   + 
                                   
                                     m 
                                     
                                       N 
                                       y 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                         + 
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           [ 
                           
                             2 
                             ⁢ 
                             
                               π 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     
                                       2 
                                       ⁢ 
                                       l 
                                     
                                     
                                       N 
                                       x 
                                     
                                   
                                   - 
                                   
                                     m 
                                     
                                       N 
                                       y 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                         + 
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           [ 
                           
                             2 
                             ⁢ 
                             
                               π 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     l 
                                     
                                       N 
                                       x 
                                     
                                   
                                   + 
                                   
                                     
                                       2 
                                       ⁢ 
                                       m 
                                     
                                     
                                       N 
                                       y 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                         + 
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           [ 
                           
                             2 
                             ⁢ 
                             
                               π 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     l 
                                     
                                       N 
                                       x 
                                     
                                   
                                   - 
                                   
                                     
                                       2 
                                       ⁢ 
                                       m 
                                     
                                     
                                       N 
                                       y 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
             
               
                 
                   
                     Δ 
                     lm 
                     5 
                   
                   = 
                   
                     2 
                     ⁢ 
                     
                       
                         a 
                         5 
                       
                       [ 
                       
                         
                           cos 
                           ( 
                           
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             
                               
                                 
                                   2 
                                   ⁢ 
                                   l 
                                 
                                 + 
                                 
                                   2 
                                   ⁢ 
                                   m 
                                 
                               
                               
                                 N 
                                 x 
                               
                             
                           
                           ) 
                         
                         + 
                         
                           cos 
                           ( 
                           
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             
                               
                                 
                                   2 
                                   ⁢ 
                                   l 
                                 
                                 - 
                                 
                                   2 
                                   ⁢ 
                                   m 
                                 
                               
                               
                                 N 
                                 y 
                               
                             
                           
                           ) 
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
             
               
                 
                   
                     Δ 
                     lm 
                     6 
                   
                   = 
                   
                     2 
                     ⁢ 
                     
                       
                         
                           a 
                           6 
                         
                         [ 
                         
                           
                             cos 
                             ( 
                             
                               
                                 2 
                                 ⁢ 
                                 
                                   π 
                                   · 
                                   3 
                                 
                                 ⁢ 
                                 l 
                               
                               
                                 N 
                                 x 
                               
                             
                             ) 
                           
                           + 
                           
                             cos 
                             ( 
                             
                               
                                 2 
                                 ⁢ 
                                 
                                   π 
                                   · 
                                   3 
                                 
                                 ⁢ 
                                 m 
                               
                               
                                 N 
                                 y 
                               
                             
                             ) 
                           
                         
                         ] 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
             
               
                 
                   
                     Δ 
                     lm 
                     7 
                   
                   = 
                   
                     2 
                     ⁢ 
                     
                       a 
                       7 
                     
                     ⁢ 
                     
                       
                         { 
                         
                           
                             cos 
                             [ 
                             
                               2 
                               ⁢ 
                               
                                 π 
                                 ( 
                                 
                                   
                                     
                                       3 
                                       ⁢ 
                                       l 
                                     
                                     
                                       N 
                                       x 
                                     
                                   
                                   + 
                                   
                                     m 
                                     
                                       N 
                                       y 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ] 
                           
                           + 
                           
                             cos 
                             [ 
                             
                               2 
                               ⁢ 
                               
                                 π 
                                 ( 
                                 
                                   
                                     
                                       3 
                                       ⁢ 
                                       l 
                                     
                                     
                                       N 
                                       x 
                                     
                                   
                                   - 
                                   
                                     m 
                                     
                                       N 
                                       y 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ] 
                           
                           + 
                           
                             cos 
                             [ 
                             
                               2 
                               ⁢ 
                               
                                 π 
                                 ( 
                                 
                                   
                                     l 
                                     
                                       N 
                                       x 
                                     
                                   
                                   + 
                                   
                                     
                                       3 
                                       ⁢ 
                                       m 
                                     
                                     
                                       N 
                                       y 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ] 
                           
                           + 
                           
                             cos 
                             [ 
                             
                               2 
                               ⁢ 
                               
                                 π 
                                 ( 
                                 
                                   
                                     l 
                                     
                                       N 
                                       x 
                                     
                                   
                                   - 
                                   
                                     
                                       3 
                                       ⁢ 
                                       m 
                                     
                                     
                                       N 
                                       y 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ] 
                           
                         
                         } 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   36 
                   ) 
                 
               
             
             
               
                 
                   
                     Δ 
                     lm 
                     8 
                   
                   = 
                   
                     2 
                     ⁢ 
                     
                       a 
                       8 
                     
                     ⁢ 
                     
                       
                         { 
                         
                           
                             cos 
                             [ 
                             
                               2 
                               ⁢ 
                               
                                 π 
                                 ( 
                                 
                                   
                                     
                                       3 
                                       ⁢ 
                                       l 
                                     
                                     
                                       N 
                                       x 
                                     
                                   
                                   + 
                                   
                                     
                                       2 
                                       ⁢ 
                                       m 
                                     
                                     
                                       N 
                                       y 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ] 
                           
                           + 
                           
                             cos 
                             [ 
                             
                               2 
                               ⁢ 
                               
                                 π 
                                 ( 
                                 
                                   
                                     
                                       3 
                                       ⁢ 
                                       l 
                                     
                                     
                                       N 
                                       x 
                                     
                                   
                                   - 
                                   
                                     
                                       2 
                                       ⁢ 
                                       m 
                                     
                                     
                                       N 
                                       y 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ] 
                           
                           + 
                           
                             cos 
                             [ 
                             
                               2 
                               ⁢ 
                               
                                 π 
                                 ( 
                                 
                                   
                                     
                                       2 
                                       ⁢ 
                                       l 
                                     
                                     
                                       N 
                                       x 
                                     
                                   
                                   + 
                                   
                                     
                                       3 
                                       ⁢ 
                                       m 
                                     
                                     
                                       N 
                                       y 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ] 
                           
                           + 
                           
                             cos 
                             [ 
                             
                               2 
                               ⁢ 
                               
                                 π 
                                 ( 
                                 
                                   
                                     
                                       2 
                                       ⁢ 
                                       l 
                                     
                                     
                                       N 
                                       x 
                                     
                                   
                                   - 
                                   
                                     
                                       3 
                                       ⁢ 
                                       m 
                                     
                                     
                                       N 
                                       y 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ] 
                           
                         
                         } 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
           
         
       
     
     For other detector array geometry, we can setup a 2-D x-y coordinates using the center pixel (a 0 ) as origin (0, 0). For a 2D detector with the N x  pixels in x-direction and N y  pixels in y-direction, any neighboring pixel can be labeled by its coordinate (x, y). Because the pixels are regularly spaced, N x /x and N y /y are integers. The NPOTF can be written as: 
                     Δ   lm     =       a   0     +       a   1     ⁢       ∑   1     N   1       ⁢     exp   [     2   ⁢     πⅈ   (         x   ·   l         N   x     ·     n   x         +       y   ·   m         N   y     ·     n   y           )       ]         +       a   2     ⁢       ∑   1     N   2       ⁢     exp   [     2   ⁢     πⅈ   (         x   ·   l         N   x     ·     n   x         +       y   ·   m         N   y     ·     n   y           )       ]         +       a   p     ⁢       ∑   1     N   p       ⁢     exp   [     2   ⁢     πⅈ   (         x   ·   l         N   x     ·     n   x         +       y   ·   m         N   y     ·     n   y           )       ]                   (   41   )               
Here x, y are coordinates of pixels in x and y directions respectively. N 1  is the total number of the first closest neighboring pixels; N 2  is the total number of pixels of the second closed neighboring pixels; . . . N is the total number of the nth closest neighboring pixels; . . . etc. The first summation is over all of the first closest neighboring pixels; the second summation is over all of the second closest neighboring pixels; . . . the pth summation is over all of the pth closest neighboring pixels . . . etc. Note x′s and y′s have different values for different pixels. For example, for the first closest neighboring pixels of a square spacing detector, we have four pixels with coordinates of (1, 1), (1,−1), (−1, 1) and (−1, −1). If a director array has a symmetry that for each pixel of coordinate (x, y), there is a pixel with coordinate (−x, −y), then these exponential terms in eq. (41) can be written in terms of cosine functions.
 
     Another example is for a hexagon detector array, the NPCCs arrangement is shown in  FIG. 3 . We can use formula (41) to calculate the NPOTF Δ lm . For example, the coordinates of the center of the six first closest neighboring pixels are: 
               (         3     2     ,     1   2       )     ,           ⁢     (     0   ,   1     )     ,           ⁢     (       -       3     2       ,     1   2       )     ,           ⁢     
     ⁢     (       -       3     2       ,     -     1   2         )     ,           ⁢       (     0   ,     -   1       )     ⁢           ⁢   and   ⁢           ⁢       (         3     2     ,     -     1   2         )     .             
Therefore, for the first order component of Δ lm , we have:
 
                     Δ   lm   1     =       a   1     [       ⅇ       -   2     ⁢     πⅈ   (           3     ⁢   l       N   x       +     m     2   ⁢     N   y           )         +     ⅇ       -   2     ⁢     πⅈ   (     m     N   y       )         +     ⅇ       -   2     ⁢     πⅈ   (           -     3       ⁢   l       2   ⁢     N   x         +     m     2   ⁢     N   y           )         +     ⅇ     2   ⁢     πⅈ   (           3     ⁢   l       2   ⁢     N   x         +     m     2   ⁢     N   y           )         +     ⅇ     2   ⁢     πⅈ   (     m     N   y       )         +     ⅇ     2   ⁢     πⅈ   (           3     ⁢   l       2   ⁢     N   x         +     m     2   ⁢     N   y           )           ]             (   42   )               
or it can be written as cosine form as:
 
                     Δ   lm   1     =     2   ⁢         a   1     [       cos   ⁢           ⁢   2   ⁢     π   (           3     ⁢   l       2   ⁢     N   x         +     m     2   ⁢     N   y           )       +     cos   ⁢           ⁢   2   ⁢     π   (           3     ⁢   l       2   ⁢     N   x         -     m     2   ⁢     N   y           )       +     cos   ⁢           ⁢   2   ⁢     π   (     m     N   y       )         ]     .               (   43   )               
Similarly, we can calculate other order of components of NPOTF.
 
     Feasibility of the present invention has been approved by modeling and processing real images. 
       FIG. 6  shows modeling result of the present invention with linear (1D) array pixel detector. For a diffraction limited optical system with PSF shown as FIG.  6 -( a ), light from a point source is spread over many pixels, and the light intensity in these pixels gives us the NPCCs. The light intensity of the center pixel, which has the maximum light intensity, gives the coefficient a 0 ; the light intensity of the two pixels next to the center pixel give the coefficient a 1 ; the light intensity of the two second-closet pixels to the center pixel give the coefficient a 2 ; . . . etc. For two point objects separated by a small angular distance shown as FIG.  6 -( b ), due to the diffraction effect, the two point objects become undistinguishable as shown in FIG.  6 -( c ), the detected image only shows a single peak. But when we process the detected image ( 6 -( c )) with the deconvolution method of the present invention, the two point objects are clearly separated in the processed image, as shown in FIG.  6 -( d ). The two peaks have the same positions with the original image, with some processing induced noise. 
       FIG. 7  shows modeling result of the present invention for with square array and square pixel detector. For a diffraction limited circular aperture optical system with PSF shown as FIG.  7 -( a ), light from a point source is spread over as Any disc pattern. Light from a point source is spread over many pixels, and the light intensity in the pixels gives us the NPCCs. The light intensity in the center pixel, which has the maximum light intensity, gives the coefficient a 0 ; the light intensity of the four pixels that are the closest to the center pixel give the coefficient a 1 ; the light intensity of the four second-closet pixels to the center pixel give the coefficient a 2 ; . . . etc. For two point objects separated by a small angular distance shown as FIG.  7 -( b ), due to the diffraction effect, the two point objects become undistinguishable as shown in FIG.  7 -( c ), the detected image only shows a single peak. But when we process the detected image ( 7 -( c )) with the present deconvolution method, the two point objects become distinguishable in the processed image as shown in FIG.  7 -( d ). 3-D plots of FIG.  7 -( c ) and  7 -( d ) are shown in FIG.  8 -( a ) and  8 -( b ). FIG.  8 -( a ) is a 3-D diagram illustrating the light intensity distribution of FIG.  7 -( c ), the two objects are undistinguishable; FIG.  8 -( a ) is a 3-D diagram illustrating the light intensity distribution of FIG.  7 -( d ), the two objects are clearly separated from each other. From FIG.  7 -( d ) and FIG.  8 -( b ), we can see that the two objects have the same positions with the original image, with some processing induced noise. 
     Two real images were processed to test the present invention. Because PSFs of the two images were not available, blind estimation was used to set the NPCCs. The two images were RGB images, only the green color images were processed. 
     One example of using this invention to increase resolution of a given image is shown in  FIG. 9 . FIG.  9 -( a ) is an enlarged portion of the image of Kuwait oil field (courtesy of NASA/Goddard Space Flight Center) in 1991 (http://sys.gsfc.nasa.gov/vis/a000000/a002700/a002715/image1.tif). We can see that the image is somehow blurred due to diffraction effect. We processed the image of FIG.  9 -( a ) using the present invention with the NPCC&#39;s of a 0 =0.26, a 1 =0.075, a 2 =0.054, a 3 =0.027, and a 4 =0.014. The threshold was set at 1/256. The processed image is shown in FIG.  9 -( b ), the image becomes much sharper, and some new features showed up. 
     Another example of using this invention to increase resolution of a given image is shown in  FIG. 10 . FIG.  10 -( a ) is an enlarged portion from the image of ‘Double Bubble’ of Gas and Dust in the Large Magellanic Cloud taken by Hubble Space Telescope (courtesy of NASA and The Hubble Heritage) (http://imgsrc.hubblesite.org/hu/db/2002/29/images/a/formats/full_tif.tif). We can see that the image is smoky due to diffraction effect. We processed the image of FIG.  10 -( a ) using the present invention with the NPCC&#39;s of a 0 =0.24, a 1 =0.096, a 2 =0.059, a 3 =0.022, and a 4 =0.0073. The threshold was set at 1/256. The processed image is shown in FIG.  10 -( b ), many new stars and new features are unveiled.