Abstract:
Two implementations of the inverse wavelet transform for use in an image decompression system do not waste computation power on the zero-valued values inserted into the data stream during an upsampling process. The implementation optimized for low-bandwidth applications toggles between even and odd modes each clock cycle. In even/odd mode, the transformed values are multiplied by the even/odd filter coefficients. The implementation optimized for high-bandwidth applications multiplies the transformed values by the even and odd filter coefficients seperately in two sets of multipliers and outputs two different results each clock cycle.

Description:
CROSS-REFERENCES TO RELATED APPLICATIONS 
     The present application is related to U.S. patent application Ser. No. 08/645,575 (now issued as U.S. Pat. No. 5,706,220) and U.S. patent application Ser. No. 08/645,572 (allowed). 
    
    
     FIELD OF THE INVENTION 
     The present invention relates generally to multimedia computing and communication. More particularly, the invention relates to a system and method for implementing the inverse wavelet transform for the digital compression of still images and video signals. 
     BACKGROUND OF THE INVENTION 
     Recently, there has been growing demand for multimedia computing and communication. This growing demand has motivated searches for better bandwidth management techniques, including newer and more efficient compression methods. 
     Today&#39;s mainstream compression methods, such as JPEG for still images and MPEG for video, use the Discrete Cosine Transform (DCT). The sinusoidal basis functions of the DCT have infinite support and so each sinusoidal basis function provides perfect frequency resolution but no spatial resolution. At the other extreme, impulse basis functions have infinitesimal support, so each impulse basis function provides perfect spatial resolution but no frequency resolution. However, neither sinusoidal nor impulse basis functions are very well suited for the purposes of image and video compression. Better suited are basis functions which can trade-off between frequency and spatial resolution. 
     The wavelet basis functions of a wavelet transform are such basis functions in that each wavelet basis function has finite support of a different width. The wider wavelets examine larger regions of the signal and resolve low frequency details accurately, while the narrower wavelets examine a small region of the signal and resolve spatial details accurately. Wavelet-based compression has the potential for better compression ratios and less complexity than sinusoidal-based compression. The potential for wavelet-based compression is illustrated by FIGS. 19,  20 , and  21 . FIG. 19 is an original 8 bits per pixel 512×512 image of “Lena.” FIG. 20 is a reconstructed image of Lena after JPEG compression at a compression ratio of approximately 40. FIG. 21 is a reconstructed image of Lena after wavelet compression at a compression ratio of approximately 40 using a preferred embodiment of the present invention. FIG. 21 appears less “blocky” than FIG. 20 because of the varying widths of the wavelet basis functions. 
     For a practical discussion of wavelet-based compression, see, for example, “Compressing Still and Moving Images with Wavelets,” by Michael, L. Hilton, Bjorn D. Jawerth, and Ayan Sengupta, in  Multimedia Systems,  volume 2, number 3 (1994). Another useful article is “Vector Quantization,” by Robert M. Gray, in  IEEE ASSP Magazine,  April 1984. Both the above articles are herein incorporated by reference in their entirety. 
     Prior systems and methods for inverse wavelet filtering use conventional filters. A conventional filter does not efficiently compute the inverse wavelet transform of an image because it does not take advantage of the fact that its input is an upsampled stream of data. 
     SUMMARY OF THE INVENTION 
     The system and method of the present invention provides two implementations of the inverse wavelet transform for use in an image decompression system. Both implementations do not waste computation power on the zero-valued values inserted into the data stream during an upsampling process. The implementation optimized for low-throughput applications toggles between even and odd modes each clock cycle. In even and odd modes, the transformed values are multiplied by the associated even or odd filter coefficients. The implementation optimized for high-throughput applications multiplies the transformed values by the even and odd filter coefficients separately in two sets of multipliers and outputs two different results each clock cycle. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a block diagram illustrating a wavelet-based image compression and decompression system including a preferred embodiment of the present invention. 
     FIG. 2 is a block diagram illustrating a wavelet filtering device within a compression system designed for a compression ratio of approximately 40 in a preferred embodiment of the present invention. 
     FIG. 3 is a block diagram illustrating a single stage of a wavelet filtering device within a compression system in a preferred embodiment of the present invention. 
     FIG. 4 is a block diagram illustrating a prior transform module, including two shift registers, a QMF pair, and two downsamplers, located within a stage of a wavelet filtering device. 
     FIG. 5 is a flow diagram illustrating a prior transform method using a prior transform module located within a stage of a wavelet filtering device. 
     FIG. 6 is a block diagram illustrating a transform module, including one shift register and a multimode QMF, located within a stage of a wavelet filtering device in a preferred embodiment of the present invention. 
     FIG. 7 is a flow diagram illustrating a transform method using a transform module located within a stage of a wavelet filtering device in a preferred embodiment of the present invention. 
     FIG. 8 is a block diagram illustrating an inverse wavelet filtering device within a decompression system designed to match the wavelet filtering device shown in FIG. 2 in a preferred embodiment of the present invention. 
     FIG. 9 is a block diagram illustrating a stage of an inverse wavelet filtering device within a decompression system in a preferred embodiment of the present invention. 
     FIG. 10 is a block diagram illustrating a prior conventional filter for inverse wavelet filtering. 
     FIG. 11 is a block diagram illustrating a high-throughput filter for inverse wavelet filtering in a preferred embodiment of the present invention. 
     FIG. 12 is a block diagram illustrating a low-throughput filter for inverse wavelet filtering in a preferred embodiment of the present invention. 
     FIG. 13 is a flow diagram illustrating a conventional filtering method. 
     FIG. 14 is a flow diagram illustrating the high-throughput filtering method of a preferred embodiment of the present invention. 
     FIG. 15 is a flow diagram illustrating the low-throughput filtering method of a preferred embodiment of the present invention. 
     FIG. 16 is a diagram illustrating cyclical boundary conditions for a general row i and for a general column j. 
     FIG. 17 is a diagram illustrating a “raster scan” implementation of a technique for overcoming the necessity for side computations in a preferred embodiment of the present invention. 
     FIG. 18 is a diagram illustrating a “zig-zag” implementation of a technique for overcoming the necessity for side computations in a preferred embodiment of the present invention. 
     FIG. 19 is an original 8 bits per pixel 512×512 image of “Lena.” 
     FIG. 20 is a reconstructed image of Lena after JPEG compression at a compression ratio of approximately 40. 
     FIG. 21 is a reconstructed image of Lena after wavelet compression at a compression ratio of approximately 40 using a preferred embodiment of the present invention. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENT 
     The preferred embodiment of the present invention is now described with reference to the figures where like reference numbers indicate identical or functionally similar elements. Also in the figures, the most significant digit(s) of each reference number corresponds to the figure in which the reference number is first used. 
     FIG. 1 is a block diagram illustrating a wavelet-based image compression and decompression system including a preferred embodiment of the present invention. As shown in FIG. 1, a wavelet-based image compression system  110  generally has three main components: a wavelet filtering device  112  to derive select subimages from the image, a quantization device  114  to quantize the resulting transform coefficients, and a coding device  116  to encode the quantized values into a coded bit stream  118  which is transmitted over a network. Quantization is discussed in the article “Vector Quantization” referenced above. A common encoding scheme is conventional Huffman encoding. 
     Image reconstruction by a decompression system  120  is accomplished by inverting the compression operations. The three main components are: a decoding device  122  to determine the quantized values from the coded bit stream, an inverse quantizing device  124  to form the subimages from the quantized values, and an inverse wavelet filtering device  126  to reconstruct the image from the subimages. 
     FIG. 2 is a block diagram illustrating a wavelet filtering device  112  within the compression system  110  in a preferred embodiment of the present invention. The wavelet filtering device  112  utilizes multiple stages, specifically three stages. The number of stages used depends on the compression ratio required, among other factors. The compression ratio required varies depending on the specific application. In this case, a compression ratio of approximately 40 was specified for prototyping purposes. 
     The first stage  202  receives the original image I(i,j) and outputs four subimages: one high-pass filtered in both i- and j-directions (HH), one high-pass filtered in the i-direction and low-pass filtered in the j-direction (HL), one low-pass filtered in the i-direction and high-pass filtered in the j-direction (LH), and one low-pass filtered in both i- and j-directions (LL). Each subimage has ½ the height and ½ the width of the original image and therefore has ¼ the number of pixels as the original image. The first three subimages (HH, HL, and LH) containing higher-frequency “detail” information are ignored, but the fourth subimage (LL) containing lower-frequency “coarse” information is passed onto the second stage  204 . Information is purposefully lost in ignoring the three detail subimages and that the coarse subimage is meant to be a high-quality lossy approximation to the original image I(i,j). In practice, since the three detail subimages are ignored, the first stage  202  need not even generate them. 
     The second stage  204  receives the coarse subimage from the first stage  202  and, like the first stage, outputs four subimages: one high-pass filtered in both i- and j-directions (HH), one high-pass filtered in the i-direction and low-pass filtered in the j-direction (HL), one low-pass filtered in the i-direction and high-pass filtered in the j-direction (LH), and one low-pass filtered in both i- and j-directions (LL). Each subimage has ¼ the height and ¼ the width of the original image and therefore has {fraction (1/16)} the number of pixels as the original image. Like the first stage  202 , the first three subimages (HH, HL, and LH) containing higher-frequency detail information are ignored, while the fourth subimage (LL) containing lower-frequency coarse information is passed onto the third stage  206 . Information is purposefully lost in ignoring the three detail subimages from the second stage and the coarse subimage from the second stage is meant to be a high-quality lossy approximation to the original image I(i,j). In practice, since the three detail subimages are ignored, the second stage  204  need not even generate them. The third stage  206  receives the coarse subimage from the second stage  204  and, like the first and second stages, outputs four subimages including one high-pass filtered in both i- and j-directions (HH), one high-pass filtered in the i-direction and low-pass filtered in the j-direction (HL), one low-pass filtered in the i-direction and high-pass filtered in the j-direction (LH), and one low-pass filtered in both i- and j-directions (LL). Each subimage has ⅛ the height and ⅛ the width of the original image and therefore has {fraction (1/64)} the number of pixels as the original image I(i,j). Unlike the first and second stages, however, none of the four subimages are ignored since all four subimages are output to the quantization device  114 . 
     The bit budget for the subimages was determined by the compression ratio desired and was part of the specification to which the wavelet filtering device  112  was designed. In this case, the system was designed to have a compression ratio of approximately 40. The bit budget chosen influenced the determination of the number of stages (three) in the wavelet filtering device  112  and the ignoring of the three detail subimages of the first  202  and second  204  stages. 
     The particular bit budgets used were chosen for prototyping purposes only and corresponds to a compression ratio of approximately 40. Depending on the specific application, different bit budgets are desirable. Changing the bit budgets would affect the number of stages needed and the detail subimages to be ignored. 
     The fractions of the “bit budget” allocated to each subimage in the first stage are 0 for HH, ⅙ for HL, ⅙ for LH, and ⅔ for LL. For the second and third stages the fractions are {fraction (1/9)} for HH, {fraction (2/9)} for HL, {fraction (2/9)} for LH, and {fraction (4/9)} for LL. These particular fractions were chosen for prototyping purposes only. Other fractions are possible. 
     FIG. 3 is a block diagram illustrating a single stage  202 ,  204 , or  206  of the wavelet filtering device  112  within the compression system  110  in a preferred embodiment of the present invention. The stage  202 ,  204 , or  206  is implemented using three modules  303 ,  305 , and  311 , each module comprising a Quadrature Mirror Filters (QMF) pair and a pair of downsamplers. Each QMF pair consists of two parts: a high-pass (HP) filter and a low-pass (LP) filter. Each downsampler downsamples by two (effectively removing every other pixel). One downsampler acts on the output of the HP filter and the other downsampler acts on the output of the LP filter. The input to the stage is an image in the form of a matrix of pixels represented by P(i,j)  302 . If the stage is the first stage  202 , then P(i,j)=I(i,j). 
     The image P(i,j)  302  is first input into a horizontal module  303 . The horizontal module  303  high-pass filters the image P(i,j)  302  along the i-direction, then downsamples by two in the i-direction, resulting in a high-pass subimage H(i,j)  304 . Due to the downsampling by two, the high-pass subimage H(i,j)  304  contains only one-half the number of pixels as P(i,j)  302 . 
     Subsequently, the high-pass subimage H(i,j)  304  is input into a first vertical module  305 . The first vertical module  305  high-pass filters the high-pass subimage H(i,j)  304  along the j-direction, then downsampled by two in the j-direction, resulting in a high-pass/high-pass subimage HH(i,j)  306 . The high-pass subimage H(i,j)  304  is also low-pass filtered along the j-direction, then downsampled by two in the j-direction, resulting in a high-pass/low-pass subimage HL(i,j)  308 . Due to the downsampling, the high-pass/high-pass subimage HH(i,j)  306  and the high-pass/low-pass subimage HL(i,j)  308  each contain only one-quarter the number of pixels as the image P(i,j)  302 . 
     Similarly, the horizontal module  303  low-pass filters the image I(i,j)  302  along the i-direction, then downsamples by two in the i-direction, resulting in a low-pass subimage L(i,j)  310 . Due to the downsampling, the low-pass subimage L(ij)  310  contains only one-half the number of pixels as P(i,j)  302 . 
     Subsequently, the low-pass subimage L(i,j)  310  is input into a second vertical module  311 . The second vertical module  311  high-pass filters the low-pass subimage L( i,j)  310  along the j-direction, then downsampled by two in the j-direction, resulting in a low-pass/high-pass subimage LH(ij)  312 . The low-pass subimage L(i,j)  310  is also low-pass filtered along the j-direction, then downsampled by two in the j-direction, resulting in a low-pass/low-pass subimage LL(ij)  314 . Due to the downsampling, the low-pass/high-pass subimage LH(i,j)  312  and the low-pass/low-pass subimage LL(i,j)  308  each contain only one-quarter the number of pixels as the image P(i,j)  302 . 
     Note that the horizontal module  303  acts substantially simultaneously to high-pass filter and low-pass filter the image P(i,j)  302  into the high-pass image H(i,j)  304  and the low-pass image L(i,j)  310 . 
     FIG. 4 is a block diagram illustrating a prior transform module  303 ,  305 , or  311 , including two shift registers, a QMF pair, and two downsamplers by two, located within a single stage  202 ,  204 , or  206  of the wavelet filtering device  112 . Let P(i,j) represent the pixels of the image input into the module. If the module is the horizontal module  303  of the first stage  202 , then P(i,j)=I(i,j). The order in which the pixels is input is along the i-direction if the module is the horizontal module  303  and is along the j-direction if the module is the first or the second vertical modules  305  or  311 . The number of filter taps of the QMFs is defined to be the integer M. The filter coefficients of the QMFs are defined to be C 0 , C 1 , C 2 , . . . , C M−2 , and C M−1 . When the QMF is a high-pass QMF, then C 0 =H 0 , C 1 =H 1 , C 2 =H 2 , . . . , C M−2 =H M−2 , and C M−1 =H M−1 . When the QMF is a low-pass QMF, then C 0 =L 0 , C 1 =L 1 , C 2 =L 2 , . . . , C M−2 =L M−2 , and C M−1 =L M−1 . 
     Consider that M pixels from row y [P(i,j), P(i,j+1), P(i,j+2), . . . , P(i,j+M−2), and P(i,j+M−1)] are shifted into a first and a second shift register  404  and  414 , respectively. The pixels are then transferred from the first shift register  404  to a high-pass QMF  406  and from the second shift register  414  to a low-pass QMF  416 . The high-pass QMF  406  outputs                H        (     i   ,   j     )       =       ∑     k   =   0       M   -   1              H   k            P        (     i   ,     j   +   k       )       .                 (     Eq   .              1     )                                
     to the downsampler  408 . The downsampler  408  downsamples by two the output of the high-pass QMF  406 , meaning that it removes or ignores every other output from the high-pass QMF  406 . The output of the downsampler  408  is the high-pass output of the horizontal module  303 . Similarly, the low-pass QMF  416  outputs:                L        (     i   ,   j     )       =       ∑     k   =   0       M   -   1              L   k            P        (     i   ,     j   +   k       )       .                 (     Eq   .              2     )                                
     to the downsampler  418 . The downsampler  418  downsamples by two the output of the low-pass QMF  416 , meaning that it removes or ignores every other output from the low-pass QMF  416 . The output of the downsampler  418  is the low-pass output of the horizontal module  303 . 
     FIG. 5 is a flow diagram illustrating a “prior” transform method using the prior transform module  303 ,  305 , or  311  located within a stage  202 ,  204 , or  206  of a wavelet filtering device  112 . First, the first memory cell of both shift registers  404  and  414  each receive  502  a single pixel. (Of course, at substantially the same time, a single pixel is also shifted out of the last cell of each shift register.) Next, the M pixels currently in each of the two shift registers  404  and  414  are transferred and filtered  504  through both the high-pass and low-pass QMFs  406  and  416 . Finally, the outputs of the two QMFs are downsampled by two  506 . The process then repeats itself with the first cell of both shift registers  404  and  414  each receiving the next single pixel. 
     FIG. 6 is a block diagram illustrating a transform module  303 ,  305 , or  311 , including one shift register and a multimode QMF, located within  206  of a wavelet filtering device  112  in a preferred embodiment of the present invention. Let P(i,j) represent the pixels of the image input into the module. The order in which the pixels is input is along the i-direction if the module is the horizontal module  303  and is along the j-direction if the module is the first or the second vertical modules  305  or  311 . The number of filter taps of the QMFs is defined to be the integer M. For example, six filter taps may be used, in which case M=6. The filter coefficients of the QMFs are defined to be C 0 , C 1 , C 2 , . . . , C M−2 , and C M−1 . When the QMF is a high-pass QMF, then C 0 =H 0 , C 1 =H 1 , C 2 =H 2 , . . . , C M−2 =H M−2 , C M−1 =H M−1 . When the QMF is a low-pass QMF, then C 0 =L 0 , C 1 =L 1 , C 2 =L 2 , . . . , C M−2 =L M−2 , C M−1 =L M−1 . 
     Consider that M pixels from row y [P(i,j), P(i,j+1), P(i,j+2), . . . , P(i,j+M−2), and P(i,j+M−1)] are shifted two-at-a-time into a single shift register  602 . A multimode QMF  604  is toggled from low-pass to high-pass mode or vice versa. The M pixels currently in the shift register  602  are transferred to the multimode QMF  604 . If the multimode QMF  604  is in high-pass mode, then the multimode QMF  604  outputs                H        (     i   ,   j     )       =       ∑     k   =   0       M   -   1              H   k            P        (     i   ,     j   +   k       )       .                 (     Eq   .              3     )                                
     If the multimode QMF  604  is in low-pass mode, then the multimode QMF  604  outputs                L        (     i   ,   j     )       =       ∑     k   =   0       M   -   1              L   k            P        (     i   ,     j   +   k       )       .                 (     Eq   .              4     )                                
     Switching between high-pass and low-pass modes requires switching certain filter coefficients (or equivalently switching the corresponding pixels) and multiplying other certain filter coefficients by negative one (or equivalently multiplying the corresponding pixels by negative one). The implementation of such a multimode QMF  604  is well known to those of ordinary skill in the art. 
     The output of the multimode QMF  604  is the output of the transform module  303 ,  305 , or  311  and no downsampling by two is necessary because the pixels are shifted two-at-a-time into the shift register. 
     The low-pass output of the multimode QMF  604  is stored in a low-pass storage buffer at location (t+M−1)/2. In contrast, the high-pass output of the multimode QMF  604  is stored in a high-pass storage buffer at location (t+1)/2. This is because the low-pass filter uses “back” pixel values while the high-pass filter uses “future” pixel values. 
     FIG. 7 is a flow diagram illustrating a transform method using the transform module located within a stage of a wavelet filtering device in a preferred embodiment of the present invention. First, pixels are shifted two-at-a-time  702  into the first two cells of the single shift register  602 . Second, the multimode QMF  604  is toggled from low-pass mode to high-pass mode (or vice versa). Third, the M pixels in the single shift register  602  are filtered  706  through the multimode QMF  604 . Fourth, the multimode QMF  604  is toggled from high-pass mode to low-pass mode (or vice versa). Fifth, the M pixels in the single shift register  602  are filtered through the multimode QMF  604 . The process is then repeated starting with the next two pixels being shifted  702  into the first two cells of the single shift register  602 , and the multimode QMF  604  being toggled  704  from high-pass mode to low-pass mode or vice versa. 
     FIG. 8 is a block diagram illustrating an inverse wavelet filtering device  126  within a decompression system  120  in a preferred embodiment of the present invention. The inverse wavelet filtering device  126  utilizes the same number of stages (three) as the wavelet filtering device  112 . 
     The first inverse stage  802  receives from the inverse quantizer  802  four subimages including one high-pass filtered in both i- and j-directions (HH), one high-pass filtered in the i-direction and low-pass filtered in the j-direction (HL), one low-pass filtered in the i-direction and high-pass filtered in the j-direction (LH), and one low-pass filtered in both i- and j-directions (LL). Each subimage has ⅛ the height and ⅛ the width of the original image and therefore has {fraction (1/64)} the number of pixels as the original image. These four subimages correspond to the four subimages output by the third stage  206  of the wavelet filtering device  112 . The first inverse stage  802  combines the four subimages and outputs the combined image to the LL input of the second inverse stage  804 . The combined image has ¼ the height and ¼ the width of the original image and therefore has {fraction (1/16)} the number of pixels as the original image. 
     Since the three detail subimages output by the second stage  204  of the wavelet filtering device  112  were ignored, the only necessary input to the second inverse stage  804  is the output of the first inverse stage  802 . The second inverse stage  804  takes the output of the first inverse stage  802  and outputs an image which has ½ the height and ½ the width of the original image and therefore has ¼ the number of pixels as the original image. 
     Since the three detail subimages output by the first stage  202  of the wavelet filtering device  112  were ignored, the only necessary input to the third inverse stage  806  is the output of the second inverse stage  804 . The third inverse stage  806  takes the output of the second inverse stage  804  and outputs a reconstructed image which has the same height and the same width of the original image; it has the same number of pixels as the original image. 
     FIG. 9 is a block diagram illustrating a single inverse stage  802 ,  804 , or  806  of an inverse wavelet filtering  126  device within a decompression system  120  in a preferred embodiment of the present invention. The inverse stage  802 ,  804 , or  806  is implemented using three modules  902 ,  904 , and  906 , each module comprising a pair of upsamplers and a filter pair. Each upsampler upsamples by two (effectively inserting a zero valued pixel between every two pixels). Each filter pair includes a high-pass (HP) filter and a low-pass (LP) filter. The HP filter acts on the output of one upsampler, and the LP filter acts on the output of the other upsampler. Four subimages (HH, HL, LH, and LL) are input into the inverse stage  802 ,  804 , or  806 . 
     Two of the subimages (HH and HL) are input into the first inverse vertical module  902 . Each of the two subimages are upsampled. The HH subimage is then filtered through the HP filter, while the HL subimage is filtered through the LP filter. After filtering, the two subimages are added together to produce an H subimage. 
     Similarly, two of the subimages (LH and LL) are input into the second inverse vertical module  904 . Each of the two subimages are upsampled. The LH subimage is then filtered through the HP filter, while the LL subimage is filtered through the LP filter. After filtering, the two subimages are added together to produce an L subimage. 
     The H and L subimages are input into the inverse horizontal module  906 . Each of the two subimages are upsampled. The H subimage is then filtered through the HP filter, while the L subimage is filtered through the LP filter. After filtering, the two subimages are added together to produce an output image. 
     FIG. 10 is a block diagram illustrating a prior conventional filter  1000  for inverse wavelet filtering. In conjunction with FIG. 10, FIG. 13 is a flow diagram illustrating the conventional filtering method. 
     The conventional filter  1000  is a standard implementation intended for general purpose FIR filtering applications. The conventional filter  1000  is an M-point filter, where M is an even integer. The values of the M cells  1002  of the register at an instant in time are shown in FIG.  10 . 
     First, the next filtered value from the upsampled stream is shifted  1302  into the M cells  1002  every clock cycle. The filtered values from M cells  1002  are multiplied  1304  by the filter coefficients  1004  in the multipliers  1006 . The resulting products are summed together in adders  1008  and then outputted  1306 . 
     Although the conventional filter  1000  does implement inverse wavelet filtering, it does so inefficiently because it does not take into account a special characteristic of the upsampled stream of filtered values. The special characteristic of the upsampled stream is that every alternate value is zero. Because of this special characteristic, the following inefficiencies exist in the conventional filter  1000 : 
     1. At any instant in time, half of the registers contain a value of zero. 
     2. At every alternate clock cycle, a zero value is shifted into the filter. 
     3. Given two consecutive clock cycles, the same set of nonzero values are multiplied by two different sets of coefficients. 
     Based on the above inefficiencies, the filter may be further optimized depending on the type of application (high or low throughput applications). 
     FIG. 11 is a block diagram illustrating a high-throughput filter  1100  for inverse wavelet filtering in a preferred embodiment of the present invention. In conjunction with FIG. 11, FIG. 14 is a flow diagram illustrating the high-throughput filtering method of a preferred embodiment of the present invention. The high-throughput filter  1100  uses the fact that the input stream is an upsampled stream and not a random stream of values to increase the throughput of the filter. 
     The high-throughput filter  1100  is an M/2-point filter, where M is an even integer. Compared to the conventional filter  1000 , half the registers are eliminated in the high-throughput filter. The values of the M/2 cells  1102  of the register at an instant in time are shown in FIG.  11 . 
     First, a new nonzero value is shifted  1402  into the filter every clock cycle while the zero values in between the nonzero values are not shifted into the filter. The values of the M/2 cells  1102  are multiplied  1404  and  1406  by two sets of constants  1104  and  1110  in two sets of multipliers  1106  and  1112 . The first set of constants  1104  comprise the even filter coefficients (C 0 , C 2 , C 4 , . . . , C M−4 , and C M−2 ); the second set of constants  1110  comprise the odd filter coefficients (C 1 , C 3 , C 5 , . . . , C M−3 , and C M−1 ). The resulting products are summed together  1408  and  1410  in two sets of adders  1108  and  1114 . The two sets of adders  1108  and  1114  output two different results each clock cycle. Thus, the high-throughput filter  1100  is able to achieve double the throughput as the conventional filter  1000  while keeping the area needed on an integrated circuit chip to implement the filter approximately the same. 
     If the high-throughput filter  1100  is an inverse high-pass filter, then the outputs of the sets of adders  1108  and  1114  are stored in a high-pass storage buffer at locations 2t+M−2 and 2t+M−1, respectively. In contrast, if the high-throughput filter  1100  is an inverse low-pass filter, then the output of the sets of adders  1108  and  1114  are stored in a low-pass storage buffer at locations 2t and 2t+1, respectively. This is because the inverse high-pass filter uses “back” values while the inverse low-pass filter uses “future” values. 
     FIG. 12 is a block diagram illustrating a low-throughput filter  1200  in a preferred embodiment of the present invention. In conjunction with FIG. 12, FIG. 15 is a flow diagram illustrating the low-throughput filtering method of a preferred embodiment of the present invention. The low-throughput filter  1200  uses the fact that the input stream is an upsampled stream and not a random stream of pixels to reduce the area required on an integrated circuit chip to implement the filter. 
     Like the high-throughput filter  1100 , the low-throughput filter  1200  is an M/2-point filter, where M is an even integer. Compared to the conventional filter  1000 , only a register of half the size is needed to store the filtered values in the low-throughput filter  1200 . The values of the M/2 memory cells  1202  at an instant in time are shown in FIG.  12 . 
     First, a nonzero value is shifted  1502  into the filter every alternate clock cycle while the zero values in between the nonzero values are not shifted into the filter. At every clock cycle the mode of the multiplexers  1206  is toggled between even and odd modes  1504  and  1510 . The values of the memory cells  1202  are multiplied in the set of multipliers  1208  by two sets of constants  1204  depending on the mode of the multiplexers  1206 . When the multiplexers  1206  are in the even mode, the values are multiplied  1506  by the even filter coefficients (C 0 , C 2 , C 4 , . . . , C M−4 , and C M− 2). When the multiplexers  1206  are in the odd mode, the values are multiplied  1512  by the odd filter coefficients (C 1 , C 3 , C 5 , . . . , C M−3 , and C M−1 ). For either the even mode or the odd mode, the resulting products of the multipliers  1208  are summed together in adders  1210  and the sum is outputted  1508  and  1514 . 
     If the low-throughput filter  1200  is an inverse high-pass filter, then the output of the set of adders  1210  in even and odd modes are stored in a high-pass storage buffer at locations 2t+M−2 and 2t+M−1, respectively. In contrast, if the low-throughput filter  1200  is an inverse low-pass filter, then the output of the set of adders  1210  in even and odd modes are stored in a low-pass storage buffer at locations 2t and 2t+1, respectively. This is because the inverse high-pass filter uses “back” values while the inverse low-pass filter uses “future” values. 
     Thus, during a clock cycle in the even mode the filtering is performed with the even filter coefficients, and during a clock cycle in the odd mode the filtering is performed with the odd filter coefficients. Compared with the conventional filter  1000 , the low-throughput filter  1200  uses only half the number of memory cells, multipliers, and adders. However, an extra set of multiplexers is required. In conclusion, the low-throughput filter  1200  is able to achieve the same throughput as the conventional filter  1000  while reducing the area required for implementation by approximately a factor of two. 
     Now consider a two-dimensional image or filtered image represented by an I by J matrix of values. The top left value having coordinate (0,0) and the bottom right value having coordinate (I- 1 ,J- 1 ). 
     With wavelet filtering or inverse wavelet filtering, it is necessary to use values from outside the matrix boundaries. When applying the high-pass filter on row i, “future” pixel values after the end of the row are needed. When applying the low-pass filter, “back” pixel values before the start of the row are required. Conversely, when applying the inverse high-pass filter, “back” filtered values before the start of the row are needed, while when applying the inverse low-pass filter, “future” filtered values after the end of the row are required. 
     There is a certain derivable relationship between the extended pixel values and the extended filtered values. Assuming values for either, values for the other can be computed. This computation, however, has to be done on the side and is a nuisance during implementation. 
     A prior technique to eliminate the need for the side computation is to assume that a row (or column) of values (whether pixel values or filtered values) is cyclical. Following this technique, the “future” values for a row (or column) start at the beginning of the same row (or column), and the “back” values for a row (or column) start at the end of the same row (or column). This assumption is illustrated in FIG.  16 . The cyclical boundary conditions are illustrated for a general row i  1602  and for a general column j  1604 . 
     FIGS. 17 and 18 are diagrams illustrating implementations of a technique for overcoming the necessity of the side computations. The technique treats the two-dimensional matrix of values as a one-dimensional array of values. 
     Treating the two-dimensional matrix as a one-dimensional array has an advantage over transforming individual rows in that it saves the overhead required to flush the registers of prior values from the row just processed and to load the registers with the initial values of the row to be processed. The only flushing and loading needed is between frames, instead of between rows (or columns). 
     FIG. 17 is a diagram illustrating one implementation of the technique of treating a two-dimensional matrix of values as a one-dimensional array according to a preferred embodiment of the present invention. As shown in FIG. 17, each row is transversed from left-to-right, in “raster scan” order. 
     The “future” extension of each row continues to the beginning of the next row. The “back” extension of each row continues from the end of the preceding row. For the first row  1702 , the “back” extension continues from the end of the last row  1710 . For the last row  1710 , the “future” extension continues to the beginning of the first row  1702 . 
     While FIG. 17 illustrates the implementation applied to rows, the same implementation is of course applicable to columns as well. When the two-dimensional matrix of values is transversed horizontally, the implementation as applied to rows is used. When the matrix is traversed vertically, the implementation as applied to columns is used. 
     An advantage of this raster scan implementation is that it makes it easy to maintain the correlation of values along each column for when the matrix is traversed vertically. A disadvantage is that any correlation between values at the edges of adjacent rows is not utilized. 
     FIG. 18 is a diagram illustrating another implementation of the technique of treating the two-dimensional matrix of values as a one-dimensional array. In the implementation of FIG. 18, the rows are traversed in “zig-zag” order. Each row with an even i-value (an even row) is traversed from left-to-right, while each row with an odd i-value (an odd row) is traversed from right-to-left. 
     The “future” extension of each even row continues to the end of the next row, while the “future” extension of each odd row continues to the beginning of the next row. The “back” extension of each even row continues from the beginning of the previous row, while the “back” extension of each odd row continues from the end of the previous row. For the first row  1802 , the “back” extension continues from the beginning of the last row  1810 . For the last row  1810 , the “future” extension continues to the beginning of the first row  1802 . 
     An advantage of this zig-zag implementation is that it takes advantage of any correlation between values at the edges of adjacent rows. A disadvantage is that it makes it difficult to maintain the correlation of values along each column for when the matrix is traversed vertically. 
     FIGS. 19-21 are described above in the Background of the Invention.