Abstract:
A system and method for smoothing digital images using a non-orthogonal convolution kernel involves steps for rearranging and remapping input image data and the convolution kernel image from hexagonal coordinate mapping to orthogonal mapping of a memory space thus enabling implementation of a general box filter convolution strategy to smooth the input image.

Description:
TECHNICAL FIELD 
       [0001]    The present disclosure is generally related to nuclear medicine imaging systems and, more particularly, is related to systems and methods for processing image pixels in a nuclear medicine imaging system. 
       BACKGROUND 
       [0002]    Smoothing of a digital image can be effectively implemented by averaging all pixels in a rectangular (box) neighborhood of a region (mask). This is known in the art as box filter. The box filter method runs independent of the box (filter kernel) size. However, the performance optimization of the box filter is based on the box shaped filter kernel. If a different shape filter kernel required smoothing, a 2-dimensional convolution has to be employed. This, however, takes longer time because of the increased calculations and the performance is dependent on the filter kernel size. The larger the filter kernel the longer it will take to smooth the digital image. 
         [0003]    Typically, in a single photon emission computed tomography (SPECT), an iterative image reconstruction method uses models for a point-spread function (PSF) to improve spatial resolution in the reconstructed images. In order to accurately replicate the physical response of a collimator of the SPECT, the PSF models often depend on a bore shape of the collimator channels and distance between the collimator and an object. The bore shape of the collimator channels can be circular, rectangular, or hexagonal. The PSF models are depth dependent with the extent of the modeling function widening two (2) dimensions (e.g., width and height of the detected area) as the distance of the object from the collimator increases. The most time-consuming part of the iterative reconstruction is the computations to model the PSF or convolutions with the distance dependent bore shaped uniform convolution kernel. 
       SUMMARY 
       [0004]    A method for smoothing a digital image with various shape and size of filter kernels comprises the following steps: (a) calculating Skip-Lengths and Run-Lengths for each row of pixels in the non-orthogonal convolution kernel; (b) performing a convolution for the first anchor pixel in the input image to obtain a first smoothed output pixel data and storing the smoothed output pixel data in a data storage unit; (c) remapping the convolution kernel image into an orthogonal coordinate map of a memory space and generate a remapping template; (d) remapping the input image into the orthogonal coordinate of the memory space using the remapping template; (e) dividing the convolution kernel into two sections; (f) further dividing one of the two sections into two subsections and extending the two subsections in the remapped input image to cover all neighborhood pixels corresponding to all of the anchor pixels, the first extended subsection representing pixels that are inside the kernel and the second extended subsection representing pixels that are outside the kernel; (g) realigning the pair of extended subsections within the memory space in decrement order or increment order; (h) for the pair of extended subsections, summing the pixel values in each column and then substracting the summed values from the second extended subsection from the summed values from the first extended subsection and generate an array H first section  representing the difference in the kernel values from one kernel to the next in the first section; (i) repeating the steps (f)-(h) for the second of the two sections and generating an array H second section  representing the differences in the kernel values from one kernel to the next in the second section; (j) adding the arrays H first section  and H second section  to obtain a summed array H representing the differences in the kernel values from one kernel to the next associated with the anchor pixels other than the first anchor pixel; (k) adding the value of the first smoothed output pixel data to the first element of the summed array H and storing the result as the second smoothed output pixel data in the data storage unit; (l) adding the value of the most recently calculated smoothed output pixel data to the next element of the summed array H and storing the result as the next smoothed output pixel data in the sequence; and (m) repeating the step (l) until the last element of the summed array H is processed by the step (l) and storing the result as the smoothed output pixel data for the last anchor pixel. 
         [0005]    The method of the present disclosure focuses on several techniques for organizing data elements into arrays, creating loop blocking by transforming the memory domain of misalignment into smaller chunks and realigned in memory space for maximizing data reuse. These techniques improve the performance for any shape of uniform convolution kernel for image smoothing. 
         [0006]    The method of the present disclosure is useful for processing digital image pixels in a nuclear medicine imaging system. Other systems, devices, methods, features of the invention will be or will become apparent to one skilled in the art upon examination of the following figures and detailed description. It is intended that all such systems, devices, methods, features be included within the scope of the invention, and be protected by the accompanying claims. 
     
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         [0007]    Many aspects of the disclosure can be better understood with reference to the following drawings in which: 
           [0008]      FIG. 1  illustrates an example of an input image to be smoothed using a hexagonal convolution kernel according to the present disclosure; 
           [0009]      FIG. 2A  shows a partial coordinate system for a hexagonal convolution kernel showing the Skip-Length array and a Run-Length array associated with the convolution kernel; 
           [0010]      FIG. 2B  illustrates a hexagonal coordinate system for representing a hexagonal kernel; 
           [0011]      FIG. 3A  illustrates the hexagonal convolution kernel of  FIG. 2B  remapped from the hexagonal coordinate system shown in  FIG. 2B  into an orthogonal coordinate map of a memory space; 
           [0012]      FIG. 3B  illustrates the remapped convolution kernel in the memory map divided into two sections, a top section and a bottom section; 
           [0013]      FIG. 4A  illustrates the remapped convolution kernel of  FIG. 3B  in which each of the two sections are further divided into two subsections, one subsection representing the pixels in the anchor pixel&#39;s neighborhood and a second subsection representing the pixels that are outside the neighborhood; 
           [0014]      FIG. 4B  illustrates the remapping of convolution kernel into memory space as shown in  FIG. 3A  extended to cover the whole input image of  FIG. 1  and also showing the two subsections of the top section from  FIG. 4A  extended to cover the neighborhood pixels of all anchor pixels in the input image; 
           [0015]      FIGS. 5 and 6  illustrate the image smoothing process of the present disclosure being applied once the input image pixel data and the hexagonal convolution kernel image are remapped and realigned to enable application of a box filter convolution strategy; 
           [0016]      FIG. 7  is a block diagram that illustrates an example of a SPECT system architecture in which the method of the present disclosure can be implemented; 
           [0017]      FIG. 8  is a high-level flow chart that illustrates an embodiment of the method of the present disclosure. 
           [0018]    While several embodiments are described in connection with these drawings, the invention is not intended to be limited to the embodiment or embodiments disclosed herein and the scope of the invention covers all alternatives, modifications, and equivalents. 
       
    
    
     DETAILED DESCRIPTION 
       [0019]    The method of the present disclosure rearranges the image pixels based on a non-rectangular image convolution kernel shape to organize the pixels to alignment and contiguity of data access patterns so that general box filter convolution strategy can be applied for smoothing the input image without severe performance penalties during memory access. The method of the present disclosure uses two arrays to represent non-rectangular shaped convolution kernel in an orthogonal coordinate system so that a general box filter convolution strategy can be applied. The length of the two arrays is equal to the size of the convolution kernel&#39;s vertical dimension (the height) and the contents of the two arrays define the convolution kernel&#39;s horizontal dimension (the width). One of the two arrays carries the skip length of the kernel (representing the number of pixels not in the convolution kernel) and the other array carries the run length (representing the number of pixels that are in the convolution kernel). Once the non-rectangular convolution kernel is presented in an orthogonal coordinate system, a general box filter convolution strategy can be applied to smooth the input image. This concept will be illustrated using a hexagonal shaped convolution kernel as an example. 
         [0020]      FIG. 1  shows an input image  300  that is 11 pixels high by 22 pixels wide. In actual SPECT operation, an input image could be much larger than the example input image  300  shown here but for the purpose of this description, the example input image  300  is selected to have this limited size. The 242 pixels in the input image  300  are labeled using a row-major convention notation inside a pair of brackets, i.e. [row, column]. 
         [0021]    According to an aspect of the present disclosure, a non-rectangular shaped convolution kernel is applied to smooth the input image  300 , specifically a hexagonal shaped convolution kernel. For example, a hexagonal convolution kernel image hi is shown overlaid onto the input image  300  centered over the pixel [ 6 , 6 ]. The pixel [ 6 , 6 ] is the pixel in the input image  300  that will be smoothed by the hexagonal convolution kernel image h 1  and aligns with the center or the anchor pixel location of the hexagonal convolution kernel image h 1 . 
         [0022]    The hexagonal convolution kernel employed here has 11×11 pixel size and, thus, twelve kernel images hi through h 12  will be required to smooth the input image  300 . The twelve anchor pixels associated with the twelve kernel images hi through h 12  are shown with shading in  FIG. 1 . The twelve anchor pixels correspond to the pixels [ 6 , 6 ], [ 6 , 7 ], [ 6 , 8 ], [ 6 , 9 ], [ 6 , 10 ], [ 6 , 11 ], [ 6 , 12 ], [ 6 , 13 ], [ 6 , 1 . 4 ], [ 6 , 15 ], [ 6 , 16 ], and [ 6 , 17 ] of the input image  300 . For purposes of illustration, only three of the twelve convolution kernel images h 1 , h 2  and h 12  are shown in  FIG. 1 . Thus, the particular number of convolution kernel images required to smooth an input image would depend on the size of the input image. For illustration purposes, only three of the twelve hexagonal convolution kernel images h 1 , h 2  and h 12  are identified in  FIG. 1 . 
         [0023]      FIG. 2A  shows one hexagonal convolution kernel image  402  of the present example in a boundary box area  400 . The convolution kernel image  402  has a size that is defined by a 11×11 pixels boundary box area  400 . Because the hexagonal convolution kernels is not box-shaped and does not occupy all of the pixels in the 11×11 boundary box area  400 , the shape of the convolution kernel inside the boundary box area  400  is defined with Skip-Length array  405  and Run-Length arrays  410 . The Run-Length array  410  contains the number of neighborhood pixels in each row, i.e. the pixels that are in the convolution kernel image. The Skip-Length array  405  contains values that represent half of the number of pixels in each row that are not the neighborhood pixels. For example, in the first row on top of the image shown in  FIG. 2A , there are five neighborhood pixels in the center that are inside the convolution kernel image and, thus, the Run-Length value for the first row in the Run-Length array  410  is “5”. There are six pixels that are not the neighborhood pixels, three on each side of the five neighborhood pixels. Thus, the Skip-Length value for the first row in the Skip-Length array  405  is “3”. The Skip-Length array and Run-Length array of the convolution kernel image is calculated once by the image processor  240  and stored in the data storage unit  220  during the initialization step before the image smoothing process is executed. 
         [0024]    The method of the present disclosure takes the non-rectangular shape of the convolution kernel image, remaps the convolution kernel image pixels from a hexagonal coordinate map to an orthogonal coordinate map using pre-calculated Skip-Length and Run-Length array information. The orthogonal coordinate map corresponds to the orthogonal coordinate system of the memory space in the data storage unit  220 . Then, the pixels in the input image  300  is also rearranged using the hexagonal coordinate conversion described herein into an orthogonally aligned memory space based on the Skip-Length and Run-Length array information. This rearrangement and alignment of the input image pixel data into orthogonally aligned memory space allows the use of a general box filter convolution strategy for smoothing the input image  300  even when the filter kernel is non-rectangular. In other words, regardless of the particular shape of the non-rectangular filter kernel shape, a general box filter method can be used to smooth the input image. 
         [0025]    In order to explain the remapping of the convolution kernel image pixels from a hexagonal coordinate map to an orthogonal coordinate map of a memory space,  FIG. 2B  shows the hexagonal convolution kernel images defined in a hexagonal coordinate system. The hexagonal coordinate system is a non-orthogonal coordinate system defined by two axes A and B that form an angle of 120° with each other. In the example shown in  FIG. 2B , two hexagonal kernel images are illustrated in the hexagonal coordinate system along with their respective group of 121 image pixels that define each kernel image&#39;s boundary box area associated with their respective anchor pixels. The anchor pixels are marked in dark color and the periphery pixels outlining the hexagonal kernel shape are shown in cross-hatches. The pixels that correspond to the convolution kernel  402  are labeled using (row, column) convention denoting their eventual location in the orthogonal coordinate map of the memory space in the data storage unit  220  after being remapped. 
         [0026]      FIG. 3A  shows the remapped convolution kernel  402 A. In the remapped convolution kernel  402 A, the pixels in the convolution kernel  402  in  FIG. 2B  have been remapped into an 11×11 orthogonal coordinate map of a memory space  500 . The pixels in the memory space  500  are labeled using a row-major convention in where the rows and columns refer to the rows and columns in the memory space  500 . This is not to be confused with the “[row, column]” labeling of the pixels in the input image  300  in FIG. I. In order to distinguish the two, the row, column labeling of the memory space  500  are shown without the brackets “[ ]” in the figure and are referred to using parentheses “(row, column)” in this description. Using the remapped convolution kernel  402 A as a remapping template, the input image  300  pixel data is remapped into the memory space&#39;s orthogonal coordinate map of the memory space also. 
         [0027]    Comparing the (row, column) labels in the remapped convolution kernel  402 A in  FIG. 3A  to those in the convolution kernel  402  in  FIG. 2B , one can see that all of the pixel data in the convolution kernel  402  have been remapped into the remapped convolution kernel  402 A&#39;s orthogonal grid. One can see that the 121 pixels in the convolution kernel have been rearranged from the parallelogram shaped arrangement in  FIG. 2B  to the square shaped arrangement of  FIG. 3A . The remapping will be referred to herein as the “hexagonal coordinate remapping.” The shaded pixels represent the neighborhood pixels associated with the anchor pixel ( 6 , 6 ) defined by the Run-Length array  410  and they contain a uniform value. Each value in the Run-Length array  410  represents the number of pixels covered by the convolution kernel in each row. 
         [0028]    In order to ensure a valid operation when image boundary pixels are smoothed, an assumption is made that all referred neighborhood pixels necessary for the operation are available. The term “image boundary pixels” refer to the pixel locations near the edge of the image that are not in the image. For example, in  FIG. 2B , if the hexagonal convolution kernel on the left side were the convolution kernel h 1  in  FIG. 1 , the pixel locations in the region C in  FIG. 2B  would be located outside the boundary of the input image  300  and not part of the actual input image  300 . For image smoothing calculation, these pixel locations are deemed to have null values. 
         [0029]    The image smoothing process of the present disclosure is further described in connection with  FIGS. 3B through 6 . The anchor pixel location ( 6 , 6 ) of the hexagonal convolution kernel h 1  is indexed at the first input image pixel [ 6 , 6 ] to be processed. (See  FIG. 1 ). Smoothing of the first input image pixel [ 6 , 6 ] is calculated first during the initialization procedure applying conventional convolution using the hexagonal kernel  402 . The smoothed output data for the input image&#39;s first anchor pixel [ 6 , 6 ] is then stored in the result array R as the first output pixel SO 1 . 
         [0030]    In order to complete the smoothing of the input image  300 , the remaining input image pixels [ 6 , 7 ] through [ 6 , 17 ] corresponding to the anchor pixels of the convolution kernels h 2  through h 12  need to be processed. The last pixel to be processed in the input image  300  is input image pixel [ 6 , 17 ] corresponding to the anchor pixel of the hexagonal filter kernel h 12 . The image smoothing process will be described from this point in terms of the input image data that has been remapped into the memory space via the hexagonal coordinate remapping and the pixel data locations will reference the coordinate in the memory space&#39;s orthogonal map. 
         [0031]    Referring to  FIG. 3B , the neighborhood pixels of the anchor pixel ( 6 , 6 ) in the convolution kernel  402  can be conceptually separated into two sections, a top section  605  (size 5×11) and a bottom section  610  (size 6×11) in the memory space  500 . As discussed above, the neighborhood pixels are those defined by the Run-Length array  410  and are shown shaded in  FIG. 3B . Separating the hexagonal convolution kernel  402  into two sections reduces demand on data processing memory space (usually a fast cache memory space) especially when the convolution kernel and the input image are large. It should be noted here that the particular sizes of the convolution kernel  402  and the input image  300  being used here as examples are arbitrarily chosen to be not too large for simplification. Thus, the method described herein is applicable to images that are smaller or larger than the examples. Furthermore, one skilled in the art would appreciate that the hexagonal convolution kernel  402  can be divided into more than two sections in order to further reduce the demand on data processing memory space. 
         [0032]    As shown in  FIG. 4A , the top section  605  and the bottom section  610  of each of the hexagonal filter kernel data in the memory space are further divided into two subsections based on the initially calculated Run-Length array data.  FIG. 4A  shows the top section  605  associated with the last anchor pixel ( 6 , 17 ) further divided into subsections  605 A and  605 B. The bottom section  610  associated with the first anchor pixel ( 6 , 6 ) is shown further divided into subsections  610 A and  610 B. The top subsection  605 A represents the neighborhood pixels in the kernel image associated with the last anchor pixel ( 6 , 17 ) in the top section of the remapped input image  300 , as represented in the memory space. In other words, as in  FIG. 3B , the notation in each pixel shown in  FIG. 4A  represents the (row, column) location of the pixel data in the memory space of the data storage unit  220 . The top subsection  605 B represents the pixels that are outside the kernel image associated with the last anchor pixel ( 6 , 17 ) in the top section of the remapped input image  300 , as represented in the memory space. 
         [0033]    Similarly, the bottom subsection  610 A represents the neighborhood pixels in the kernel image associated with the first anchor pixel ( 6 , 6 ) in the bottom section of the remapped input image  300 , as represented in the memory space. The subsection  610 B represents the pixels that are outside the kernel image associated with the first anchor pixel ( 6 , 6 ) in the bottom section of the remapped input image  300 ; as represented in the memory space. The notation A 1  refers to the first anchor pixel at memory location ( 6 , 6 ) and A 12  refers to the twelfth anchor pixel at memory location ( 6 , 17 ). 
         [0034]    In order to carry out the image smoothing process of the present disclosure, each of the four subsections  605 A,  605 B,  610 A and  610 B need to be extended to cover all neighborhood pixels for all of the eleven remaining anchor pixels ( 6 , 7 ) to ( 6 , 17 ) in the remapped input image  300 A (i.e., the input image  300  after it has been remapped into the orthogonal coordinate map of the memory space as discussed above). This is illustrated in  FIG. 4B . There are eleven anchor pixels to be considered because the first image pixel [ 6 , 6 ], which is now located at memory space location ( 6 , 6 ), has already been smoothed during the initialization step as discussed above. When considering the input image  300 &#39;s pixel data remapped into the memory space using the hexagonal coordinate remapping discussed above, the position of the twelve anchor pixels in the memory space&#39;s orthogonal map are ( 6 , 6 ) to ( 6 , 17 ) in the memory space&#39;s (row, column) convention. 
         [0035]    FIG.  4 B( a ) shows a schematic illustration of the remapped input image  300 A in which the extended top subsection  605 A is shown in dotted line and labeled as  605 A-extended. FIG.  4 B( b ) shows a schematic illustration of the remapped input image  300 A in which the extended top subsection  605 B is shown in dotted line and labeled as  605 A-extended. The  605 A-extended is aligned to the right side of the memory space or memory index decrement (from right to left) and the top subsection  605 B-extended is aligned to the left side of the memory space or memory index increment (from left to right). However, in order to apply the general box filter convolution, the subsection  605 B-extended need to be aligned to the right side of the memory space or memory index decrement, same as the subsection  605 A-extended. This can be realized by loading the non-aligned pixel data for the subsection  605 B-extended from a main memory space (such as the data storage unit  220  shown in  FIG. 7 ) location to a temporary cache memory (which is generally smaller memory but has much faster access time) and aligning the pixel data from the cache memory in order to same computation time. The cache memory can be provided as part of the data storage unit  220  or as a separate memory device. 
         [0036]    The two subsections  605 A-extended and  605 B-extended can have overlap of the memory space as shown. For reference, FIGS.  4 B( a ) and ( b ) also show the outlines of the neighborhood pixels N (6,6)  and N (6,17)  under the convolution kernel (per the calculated Run-Length array) associated with the first anchor pixel ( 6 , 6 ) to be smoothed and the last anchor pixel ( 6 , 17 ) to be smoothed, respectively. 
         [0037]      FIG. 5  shows both the top subsections  605 A-extended and  605 B-extended aligned to the right side of the memory space in the decremented order and ready for further processing. With both of the top subsections aligned in decremented order, we can calculate the convolution kernel values for the top portions of the neighborhood pixels associated with each of the eleven remaining anchor pixels efficiently. First, summation operations  705  and  710  are performed to sum the values of the pixels in each column in the top subsections  605 A-extended and in the top subsection  605 B-extended, respectively. The number of columns in each of the two top subsections  605 A-extended and  605 B-extended is same as the total number of anchor pixels in the input image  300  minus 1, which in this case is 11. Then, the summed column values of the top subsection  605 B-extended are subtracted from the summed column values of the top subsection  605 A-extended and the subtraction results are placed in a temporary buffer array H top . The temporary buffer array H top  holds eleven values, each value representing the difference in the kernel values for the top portion of the kernel between one kernel and the next kernel in the sequence. These differences represent the difference in the kernel values from one kernel to the next associated with the anchor pixels ( 6 , 7 ) to ( 6 , 17 ) if one were to perform convolution for each of the anchor pixels individually in sequence. For example, referring to the illustration in FIG.  4 B( a ), after a convolution is performed on the neighborhood pixels N (6,6)  for the first kernel, in order to perform a convolution on the neighborhood pixels for the next kernel in sequence, i.e. N (6,7) , the kernel image is shifted one pixel to the right so that the kernel image is now centered over the next anchor pixel ( 6 , 7 ). Shifting the kernel image one pixel to the right is same thing as adding a column of neighborhood pixels on the right side of the kernel image While subtracting a column of pixels on the left side of the kernel image. Therefore, the effective change between the value of the first kernel to the next is effectively the difference between the column of neighborhood pixels added on the right side and the column of neighborhood pixels subtracted on the left side. Thus, the values in the array H top  represent the differences in the kernel values from one kernel to the next associated with the anchor pixels ( 6 , 7 ) to ( 6 , 17 ) if one were to perform convolution for each of the anchor pixels individually in sequence. 
         [0038]    The computation of the temporary buffer array H top  is shown in Equation 1 as follows: 
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         [0000]    where hex refers to hexagonal kernel size and i refers to a number of anchor pixels minus 1, range from 17 to 7 (i.e., in decremental order from right to left starting from the position of the last of the twelve anchor pixels to the second anchor pixel in the input image  300 ). “TopSection 1 ” refers to the top subsection  605 A-extended and “TopSection 2 ” refers to the top sub-section  605 B-extended. 
         [0039]    Similarly, the two bottom subsections  610 A-extended,  610 B-extended are aligned in the incremental order (from left to right) to facilitate implementing the box filter convolution. Similar to the summing operations performed on the two top subsections  605 A-extended and  605 B-extended, the columns in each of the bottom subsections  610 A-extended,  610 B-extended are also summed. Then, the summed column values of the bottom subsection  610 B-extended are subtracted from the summed column values of the bottom subsection  610 A-extended and the subtraction results are placed in a temporary buffer array H bot . The length of the array H bot  is the total number of anchor pixels in the pixel row  6  in the input image  300  minus 1. The computation of the temporary buffer H bot  is shown in Equation 2 as follows: 
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         [0000]    where hex refers to hexagonal kernel size and i refers to the number of anchor pixels minus 1, range from 7 to 17 (i.e., in incremental order from left to right starting from the second anchor pixel position to the last of the twelve anchor pixels in the input image  300 ). “BottomSection 1 ” refers to the bottom subsection  610 A-extended and “BottomSection 2 ” refers to the bottom subsection MOB-extended. 
         [0040]    Next, referring to  FIG. 6 , since the arrays H top  and H bot  contain the differences in the kernel values from one kernel to the next associated with the anchor pixels ( 6 , 7 ) to ( 6 , 17 ) for the top portion and the bottom portion of the input image data, the arrays H top  and H bot  are added together to obtain a temporary summed array H. Thus, the array H contains the differences in the kernel values from one kernel to the next associated with the anchor pixels ( 6 , 7 ) to ( 6 , 17 ). This process can be shown in Equation 3 as follows: 
         [0000]        H[i]=H   bot   [i]+H   top   [N−i], N:  ImageSize−hex   (Eq. 3)
 
         [0000]    where hex refers to hexagonal kernel size and i refers to the number of anchor pixels minus 1, range from 7 to 17. Because the total number of anchor pixels in this example is 12, the N−i indexing means that the H top  is aligned from left to right when H bot  is aligned from right to left, so array H top  is flipped before being added to array H bot . The summed array H consists of elements H[ 7 ] through H[ 17 ]. 
         [0041]    Once the array H is obtained, a box filter convolution strategy can be applied to complete the image smoothing. As mentioned above, smoothing of the input image&#39;s first anchor pixel [ 6 , 6 ] was pre-calculated during the initialization step and already stored in the result array R as the first output pixel SO 1 . Such output pixel SO 1  is the output image for the first kernel image h 1 . The value of the first smoothed output pixel SO 1  is added to the first element H[ 7 ] of the summed array H to obtain the second smoothed output pixel SO 2 , which is the output image for the second kernel image h 2 . The value of the second smoothed output pixel SO 2  is then added to the second element H[ 8 ] to obtain the third smoothed output pixel SO 3 , which is the output image for the third kernel image h 3 . This operation is repeated until all anchor pixels are smoothed, generating smoothed output pixels SO 1  to SO 12  for kernel images h 1  through h 12  corresponding to the anchor pixels [ 6 , 6 ] through [ 6 , 17 ] in the input image  300 . 
         [0042]    There are many ways to implement a box filter efficiently. For example, the image pixels of the hexagonal convolution kernel images h 1  . . . h 12  can be moved in the memory space from left to right by subtracting pixels at the left-most side in the column of the box kernel and adding pixels at the next column of the image at the right side of the kernel box. In general, the operations by subtraction and addition can be added up by a number of elements in the column (kernel size). If the box kernel size is K pixels, the total operations per output pixel is 2*K without counting overhead. In other words, the output pixel is the weighted sum of neighboring input pixels. In contrast, the convolution takes K 2  operations (multiply-additions). By organizing pixel data elements into arrays and creating loop blocking by transforming the memory domain of misalignment into smaller chunks of aligned memory space to maximize data reuse, the performance for image smoothing can be improved for any shape of convolution kernel. Given the box kernel size K, the total operation per output pixel is 2*(K+1) without counting overhead of rearranged pixels. The box filter is able to accelerate the performance of a geometric shape point-spread-function (G-PSF) in the 3D iterative SPECT reconstruction. The rearranged image pixels can be applied to a general box filter convolution without substantial performance penalties during memory access. The box filter attenuates the high-spatial frequency components of an image and has little effect on the low-frequency components. The effect of passing an image through a low-pass filter is a slight blurring. 
         [0043]    Referring to a schematic diagram of  FIG. 7 , the method of the present disclosure is envisioned as being carried out in a SPECT system by an image processor  240  of a SPECT system controller  200 . The SPECT system controller  200  is a computer that can comprise a central processor  210  for managing and executing various programs necessary for the operation of the SPECT system  100 , the image processor  240  for executing the image processing described here, and a data storage unit  220 . The data storage unit  220  can be a single component unit or, if appropriate, can comprise multiple components that collectively provide the data storage unit  220  the ability to store information permanently and/or temporarily as necessary. For example, the data storage unit  220  can include one or more suitable storage components for holding firmware and other programs required for the operation and management of the SPECT system  100 . The data storage unit  220  can also include fast access data storage hardware such as cache memory devices for temporarily holding data for purposes of performing convolution calculations on the data fast. In the present case, the image pixel data would be temporarily stored in the data storage unit  220  in order to perform the image convolution described in this disclosure. The data storage unit  220  can also include more permanent data storage devices for storing information. Regardless, the data storage unit  220  can include any appropriate computer readable data storage medium in which a set of instructions (e.g, a software program) is tangibly embodied thereon. The set of instructions when executed by a computer processor such as the image processor  240 , the image processor performs the image smoothing method of the present disclosure. By implementing the method of the present disclosure, the image processor  240  is able to smooth the input image  300  using the non-rectangular convolution kernel  402  using a general box filter convolution strategy without substantial performance penalties during memory access. The SPECT system controller  200  receives a SPECT image data  80  from the SPECT system  100 , processes the image data  80  and the image processor  240  converts the image data into an input image  300  and can display the image on the display  150 . The input image  300  is stored in the data storage unit  220 . The image data may include one or more kernel images discussed above. 
         [0044]    The image smoothing method disclosed herein can be implemented in software, hardware, or a combination thereof as a set of instructions tangibly embodied in any computer readable storage medium such as the data storage unit  220  or other portable medium, e.g. compact disks, flash memory devices, etc., provided external to the SPECT system controller  200 . When the image processor  240  executes the instructions, the image processor  240  performs the image processing method described herein. 
         [0045]      FIG. 8  is a high-level flow diagram  1000  that illustrates the image smoothing process of the present disclosure referring to the exemplary input image  300  and the hexagonal filter kernel image  402 . The image processor  240  first determines a valid output image dimension based the input image  300  and the filter kernel image  402 . (See Box  1005 ). For example, a valid output image dimension for the input image  300  is 1×12 based on a 11×22 input image and an 11×11 kernel image. The image processor  240  calculates the Skip-Lengths and Run-Length values for each row of pixels in the convolution kernel image  402  for the Skip-Length array  405  and the Run-Length array  410 . (See box  1010 ). The calculated Skip-Length array  405  and Run-Length array  410  data is stored as a data table in a suitable memory such as the data storage unit  220  accessible to the image processor  240 . As part of the initialization process, the image processor  240  performs convolution for the first anchor pixel in the input image using the convolution kernel  402  and stores the smoothed output pixel data in the result array R as smoothed output pixel SO 1 . (See box  1020 ). The convolution kernel  402  is remapped into the orthogonal coordinate map of a memory space using the hexagonal coordinate remapping described above to generate a remapping template. (See box  1030 ). Using the remapping template, the input image  300  pixel data is remapped into the orthogonal coordinate map of the memory space. (See box  1040 ). Next, the remapped convolution kernel in the memory space is defined into at least two sections, a top section  605  and a bottom section  610  so that operation on the convolution kernel can be conducted on one section at a time to reduce the demand on the memory space during computation. (See block  1050 ). Next, one of the two sections, the top section  605  or the bottom section  610 , is further defined into two subsections, grouping the pixels in each of the two sections into a first subsection that are inside the kernel image  402  and a second subsection that are not in the kernel image  402 . These subsections are then extended to subsections (e.g.  605 A-extended,  605 B-extended,  610 A-extended,  610 B-extended) to cover all neighborhood pixels corresponding to all of the anchor pixels in the remapped input image  300 A. (See block  1060 ). Next, the extended subsections are realigned within the memory space so that each pair of subsections (i.e.  605 A-extended and  605 B-extended;  610 A-extended and  610 B-extended) in each of the top and bottom sections in the remapped input image  300 A are aligned in decremental order or incremental order. (See block  1070 ). For each pair of extended subsections, the pixel values in each column are summed first and then the summed values from the extended subsection ( 605 B-extended,  610 B-extended) representing the pixels that are outside the kernel image are subtracted from the summed values from the extended subsection ( 605 A-extended,  610 A-extended) representing the pixels that are inside the kernel image, thus, generating an array (one of H top  or H bottom  depending on which pair of subsections were processed) representing the differences in the kernel values from one kernel to the next associated with the remaining anchor pixels (i.e. all anchor pixels minus the first one for which the smoothed value has already been calculated) for the corresponding top or bottom portion of the input image data. (See block  1080 ). Next, the steps in blocks  1060 ,  1070  and  1080  are repeated for the second section ( 605  or  610  depending on which section was processed first) generating a second array (the other of H top  or H bottom ). (See block  1090 ). Next, the arrays H top  and H bottom  are added together to obtain an array H which represents the differences in the kernel values from one kernel to the next associated with the anchor pixels ( 6 , 7 ) to ( 6 , 17 ), i.e. the remaining anchor pixels other than the first anchor pixel ( 6 , 6 ). (See block  1100 ). 
         [0046]    Now the box filter convolution strategy is applied to the array H to obtain the smoothed data for the remaining anchor pixels. The value of the first smoothed output pixel SO 1  is added to the first element of the summed array H, representing the difference in the kernel value between the first kernel h 1  and the second kernel h 2 , and obtain the second smoothed output pixel SO 2 . (See block  1110 ). The value of the second smoothed output pixel SO 2  is then added to the second element of the summed array H, representing the difference in the kernel value between the second kernel h 2  and the third kernel h 3 . (See block  1120 ). This operation is repeated until the last smoothed output pixel data SO 12  is processed by the step  1120  and the result is stored in the data storage unit  220  as the last smoothed output pixel data for the last anchor pixel, i.e. pixel ( 6 , 17 ). This process produces the results array R containing the smoothed output pixels SO 1  to SO 12  corresponding to the anchor pixels ( 6 , 6 ) to ( 6 , 17 ) for the hexagonal convolution kernel images h 1  to h 12 . (See block  1130 ). 
         [0047]    It should be noted that any process descriptions or blocks in flowcharts should be understood as representing modules, segments, or portions of code which include one or more executable instructions for implementing specific logical functions or steps in the process. As would be understood by those of ordinary skill in the art of the software development, alternate embodiments are also included within the scope of the disclosure. In these alternate embodiments, functions may be executed out of order from that shown or discussed, including substantially concurrently or in reverse order, depending on the functionality involved. 
         [0048]    This description has been presented for purposes of illustration and description. It is not intended to be exhaustive or to limit the disclosure to the precise forms disclosed. Obvious modifications or variations are possible in light of the above teachings. The embodiments discussed, however, were chosen to illustrate the principles of the disclosure, and its practical application. The disclosure is thus intended to enable one of ordinary skill in the art to use the disclosure, in various embodiments and with various modifications, as are suited to the particular use contemplated. All such modifications and variation are within the scope of this disclosure, as determined by the appended claims when interpreted in accordance with the breadth to which they are fairly and legally entitled.