Patent Publication Number: US-8971405-B2

Title: Block transform and quantization for image and video coding

Description:
CROSS REFERENCE TO RELATED APPLICATION 
     This application is a continuation of U.S. patent application Ser. No. 11/067,101,filed Feb. 25, 2005, which is a continuation of U.S. patent application Ser. No. 09/955,577,filed Sep. 18, 2001,both of which are incorporated herein by reference. 
    
    
     FIELD OF THE INVENTION 
     The invention relates generally to the encoding and decoding of image and video signals, and more particularly to an improved block transform and inverse transform, along with quantization and de-quantization methods, for encoding and decoding image video signals. 
     BACKGROUND OF THE INVENTION 
     The Discrete Cosine Transform (DCT) is commonly used in block transform image and video (sequences of images) coding, e.g., JPEG and MPEG. Among other reasons, this is because the DCT separates random signals (such as corresponding to image data) into low frequency parts of greater importance and high frequency parts of lesser importance with respect to the image&#39;s visual quality, thus facilitating compression. 
     By way of example, H.26L is an image and/or video codec technology that constructs frames using four-by-four blocks of pixels. Unlike MPEG and JPEG, which uses eight-by-eight blocks, H.26L obtains relatively high-quality compression with four-by-four blocks of pixels by using prediction information of existing other blocks in the same frame, (i.e., intra-frame coding), in addition to estimation and motion compensation between frames, (i.e., inter-frame coding). In general, to accomplish intra-frame prediction, an H.26L encoder tells the corresponding H.26L decoder what other, previous block of pixels in the frame being built is being used as a basis for the prediction, along with identifying which one of six possible predictors (formulas) to use in determining the pixels for the new block (from those of the previous intra-frame block). This leads to a prediction error, which is also provided to the decoder to correct the new block. The prediction error information is encoded with the block transform (DCT) and sent encoded to the decoder, for re-computing the error information, including via an inverse transform, to correct the predicted block. 
     Image and video encoding or decoding takes a significant amount of processing power. As is known, an encoder typically has far more processing power than do many of the decoders that convert the data to images, as decoders are typically implemented in consumer devices. For example, image and video decoding may take place in television set-top boxes, personal digital assistants (PDAs), pocket-sized personal computers, and more advanced cellular telephones. 
     Thus, when considering image and video encoding and decoding methods, keeping the decoding simple is important, even if it means that encoding has to be made more computationally complex. Anything that can simplify decoding is desirable, provided that in doing so, the encoding and decoding processes do not adversely impact existing levels of compression, final image quality and the amount of other resources needed at the decoder in any substantial way relative to existing technology. 
     SUMMARY OF THE INVENTION 
     Briefly, the present invention provides an improved method, system and block transform that significantly simplifies computational complexity for images and video at both the encoder and decoder. At the same time, compression, image/video quality, and other resources are only negligibly affected. 
     More particularly, an orthogonal block transform and corresponding inverse transform with new integer approximations to the Discrete Cosine Transform (DCT) are provided, along with other computing changes that significantly reduce computational complexity at both the encoder and decoder. Indeed, in one implementation, at the encoder and decoder, the number of transform operations, per coefficient, has been reduced to four additions and one shift in 16-bit arithmetic, (from four additions and three multiplications in 32-bit arithmetic required with the H.26L-specified transform). The present invention correctly transforms the pixel information (e.g., error correction data) because quantization (scaling and rounding to an integer) during encoding, and de-quantization during decoding, via the use of one of three tables selected based on each coefficient&#39;s position, have parameter values that already compensate for factors of other transformation multiplications, except for the one performed by the shift operation during the transformation and inverse transformation processes. 
     Further, during decoding, on each quantized coefficient, the decoder can perform a sixteen-bit multiplication to de-quantize that quantized coefficient into a transform coefficient, instead of a thirty-two bit multiplication (as required in H.26L). On many devices, with typical images and video, the processing benefits of the present invention are thus significant relative to H.26L, especially at the decoder, with only negligible if any loss in quality and/or compression. 
     Other benefits and advantages will become apparent from the following detailed description when taken in conjunction with the drawings, in which: 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a block diagram representing an exemplary computer system into which the present invention may be incorporated; 
         FIG. 2  is a block diagram representing components used in encoding pixels of video or image data into encoded video, and decoding the encoded video into pixel information, including transformation, quantization, de-quantization and inverse transformation in accordance with one aspect of the present invention; 
         FIG. 3  is a representation of a butterfly structure that represents the improved transformation matrix in accordance with one aspect of the present invention; and 
         FIG. 4  is a representation of a butterfly structure that represents the improved inverse transformation matrix in accordance with one aspect of the present invention. 
     
    
    
     DETAILED DESCRIPTION 
     Exemplary Operating Environment 
       FIG. 1  illustrates an example of a suitable operating environment  120  in which the invention may be implemented, particularly for decoding image and/or video data. The operating environment  120  is only one example of a suitable operating environment and is not intended to suggest any limitation as to the scope of use or functionality of the invention. Other well known computing systems, environments, and/or configurations that may be suitable for use with the invention include, but are not limited to, personal computers, server computers, hand-held or laptop devices, multiprocessor systems, microprocessor-based systems, programmable consumer electronics, network PCs, minicomputers, mainframe computers, distributed computing environments that include any of the above systems or devices, and the like. For example, it is likely that encoding image and/or video image data often will be performed on a computer with more processing power than contemporary hand-held personal computers, but there is no reason encoding cannot be performed on the exemplary device, or decoding on a more powerful machine. 
     The invention may be described in the general context of computer-executable instructions, such as program modules, executed by one or more computers or other devices. Generally, program modules include routines, programs, objects, components, data structures and so forth that perform particular tasks or implement particular abstract data types. Typically the functionality of the program modules may be combined or distributed as desired in various embodiments. Computing device  120  typically includes at least some form of computer readable media. Computer-readable media can be any available media that can be accessed by the computing device  120 . By way of example, and not limitation, computer readable media may comprise computer storage media and communication media. Computer storage media includes volatile and nonvolatile, removable and non-removable media implemented in any method or technology for storage of information such as computer readable instructions, data structures, program modules or other data. Computer storage media includes, but is not limited to, RAM, ROM, EEPROM, flash memory or other memory technology, CD-ROM, digital versatile disks (DVD) or other optical storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other medium which can be used to store the desired information and which can accessed by the computing device  120 . Communication media typically embodies computer readable instructions, data structures, program modules or other data in a modulated data signal such as a carrier wave or other transport mechanism and includes any information delivery media. The term “modulated data signal” means a signal that has one or more of its characteristics set or changed in such a manner as to encode information in the signal. By way of example, and not limitation, communication media includes wired media such as a wired network or direct-wired connection, and wireless media such as acoustic, RF, infrared and other wireless media. Combinations of the any of the above should also be included within the scope of computer readable media. 
       FIG. 1  shows functional components of one such handheld computing device  120 , including a processor  122 , a memory  124 , a display  126 , and a keyboard  128  (which may be a physical or virtual keyboard). The memory  124  generally includes both volatile memory (e.g., RAM) and non-volatile memory (e.g., ROM, PCMCIA cards, and so forth). An operating system  130  is resident in the memory  124  and executes on the processor  122 , such as the Windows® CE operating system from Microsoft® Corporation, or another operating system. 
     One or more application programs  132  are loaded into memory  124  and run on the operating system  130 . Examples of applications include email programs, scheduling programs, PIM (personal information management) programs, word processing programs, spreadsheet programs, Internet browser programs, and so forth. The handheld personal computer  120  may also include a notification manager  134  loaded in the memory  124 , which executes on the processor  122 . The notification manager  134  handles notification requests, e.g., from the application programs  132 . 
     The handheld personal computer  120  has a power supply  136 , which is implemented as one or more batteries. The power supply  136  may further include an external power source that overrides or recharges the built-in batteries, such as an AC adapter or a powered docking cradle. 
     The exemplary handheld personal computer  120  represented in  FIG. 1  is shown with three types of external notification mechanisms: one or more light emitting diodes (LEDs)  140  and an audio generator  144 . These devices may be directly coupled to the power supply  136  so that when activated, they remain on for a duration dictated by a notification mechanism even though the handheld personal computer processor  122  and other components might shut down to conserve battery power. The LED  140  preferably remains on indefinitely until the user takes action. Note that contemporary versions of the audio generator  144  use too much power for today&#39;s handheld personal computer batteries, and so it is configured to turn off when the rest of the system does or at some finite duration after activation. 
     Encoding and Decoding 
     By way of background, the Discrete Cosine Transform (DCT) maps a length-N vector x into a new vector X of transform coefficients by a linear transformation X=H x, where the element in the kth row and nth column of H is defined by 
               H   kn     =       H   ⁡     (     k   ,   n     )       =       c   k     ⁢       2   N       ⁢     cos   ⁡     (       (     n   +     1   /   2       )     ⁢       k   ⁢           ⁢   π     N       )                 
for k=0, 1, . . . ,N−1,and n=0, 1, . . . ,N−1, with c 0 =√{square root over (2)} and c k =1 for k&gt;1. The DCT matrix is orthogonal, so its inverse equals its transpose, that is x=H −1  X=H T  X.
 
     One disadvantage of the DCT is that the entries H(k, n) are irrational numbers, and so integer input data x(n) will map to irrational transform coefficients X(k). As a result, with digital computers, when the direct and inverse transforms are computed in cascade, the output data does not exactly equal the input data. In other words, if computed as X=H x, and u=round(H T  X), then it is not true that u(n)=x(n) for all n. However, by introducing appropriate scale factors α, γ, e.g., X=γ H x and u=round(α H T  X), then u(n)=G x(n), where G is an integer, for almost all n, when α and γ are chosen to be large enough, but this does not guarantee an exact result. 
     In a motion-compensated video encoder, for example, the data of past decoded frames are used as reference information for prediction information that will be used to generate the current frame. Therefore, as part of encoding, the encoder generates its own decoded frames, whereby the encoder needs to compute inverse transforms. If the formula u=round(α H T  X) is used, then different floating-point formats and rounding strategies in different processors lead to different results. As a result, there will be a drift between the decoded data at the encoder versus that decoded by decoders (which have different processors), whereby the image worsens more and more per each new frame, since the encoder is basing prediction/motion information on frames of blocks that are increasingly less and less like the frames of blocks that the decoder is producing. 
     One solution to the data drift problem approximates the matrix H by a matrix containing only integers, whereby rounding errors are eliminated. If the rows of H are orthogonal and have the same norm (sum of the squares), then it follows that u can be computed exactly in integer arithmetic for all integer x. In other words, when the direct transform is computed by X=H x and the inverse transform by u=H T  X, then u=G x results, where G is an integer equal to the squared norm of any of the rows in H. 
     One way to generate integer approximations to the DCT is by using the general formula:
 
 Q ( k, n )=round (α  H ( k, n ))
 
where α is a scaling parameter.
 
     In the H.26L video compression standard, the image is made up of blocks of four pixels by four pixels, whereby N=4 in the DCT formula in H.26L. This results in the DCT matrix equaling: 
             H   =       1   2     ⁡     [         1       1       1       1           c       s         -   s           -   c             1         -   1           -   1         1           s         -   c         c         -   s           ]             
where c √{square root over (2)} cos(π/8) and s √{square root over (2)} sin(π/8).
 
     The transform matrix in the current version of H.26L is obtained by setting α=26,whereby the values compute to: 
     
       
         
           
             
               Q 
               0 
             
             = 
             
               
                 round 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   
                     26 
                     ⁢ 
                     H 
                   
                   ) 
                 
               
               = 
               
                 [ 
                 
                   
                     
                       13 
                     
                     
                       13 
                     
                     
                       13 
                     
                     
                       13 
                     
                   
                   
                     
                       17 
                     
                     
                       7 
                     
                     
                       
                         - 
                         7 
                       
                     
                     
                       
                         - 
                         17 
                       
                     
                   
                   
                     
                       13 
                     
                     
                       
                         - 
                         13 
                       
                     
                     
                       
                         - 
                         13 
                       
                     
                     
                       13 
                     
                   
                   
                     
                       7 
                     
                     
                       
                         - 
                         17 
                       
                     
                     
                       17 
                     
                     
                       
                         - 
                         7 
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
     With a scaling parameter value of 26, the rows and columns of Q 0  are orthogonal to each other (i.e., the inner product of any two columns is zero), and all rows and columns have a norm equal to 676. In fact, for values wherein α&lt;50, only α=2 or α=26 provide orthogonal matrices with equal norm rows. However, the solution for α=2 does not lead to good compression, and larger values for α are not attractive because of the increase in the computational complexity (e.g., word length) required to compute the results of the direct transform X=Q 0  x. Hence, α=26 has been heretofore chosen in H.26L. 
     The inverse transform is defined by x′=Q 0  X, so that it can also be computed with integer arithmetic. From the definition above, x′=676 x, i.e. the reconstructed data x′ is equal to the original data x amplified by an integer gain of 676 (which is the norm of any of the rows in Q 0 ). 
     In accordance with one aspect of the present invention, a block transform matrix is provided that uses integer approximations that are orthogonal, yet have different values that significantly reduce computational complexity when encoding and decoding. More particularly, using α=2.5, the following matrix is generated: 
     
       
         
           
             
               Q 
               D 
             
             ⁢ 
             
               = 
               Δ 
             
             ⁢ 
             
               
                 round 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   
                     2.5 
                     ⁢ 
                     H 
                   
                   ) 
                 
               
               = 
               
                 [ 
                 
                   
                     
                       1 
                     
                     
                       1 
                     
                     
                       1 
                     
                     
                       1 
                     
                   
                   
                     
                       2 
                     
                     
                       1 
                     
                     
                       
                         - 
                         1 
                       
                     
                     
                       
                         - 
                         2 
                       
                     
                   
                   
                     
                       1 
                     
                     
                       
                         - 
                         1 
                       
                     
                     
                       
                         - 
                         1 
                       
                     
                     
                       1 
                     
                   
                   
                     
                       1 
                     
                     
                       
                         - 
                         2 
                       
                     
                     
                       2 
                     
                     
                       
                         - 
                         1 
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
     Note that the rows of Q D  are orthogonal to each other, and, while their norms are different (rows zero and two have norms equal to four, while rows one and three have norms equal to ten), as described below this is handled in quantization and de-quantization in a manner that does not significantly increase computational complexity. 
     Turning to  FIG. 2  of the drawings, there is shown a block transform encoder  200  and block transform decoder  222 , connectable via by some type of medium and/or intermediate devices  220 . As is understood, an encoder  200  need not make its output directly available to the decoder  222 , and thus the line labeled  220  in  FIG. 2  may represent virtually any type of medium or devices, such as a network cable, wireless transmission medium, telephone line, storage device, router, and/or virtually any combination thereof. 
     In general, the encoder  200  operates by transforming each block of N×N input pixel data  202  (e.g., representing error correction information) via a separable two-dimensional transforms  204 . More particularly, first the rows of pixel data within the block are transformed by a row transformation process  206 , and then the results of the row transformation (not specifically represented in  FIG. 2 ) are transformed by a column transformation process  208  into the resulting transform coefficients  210 . In other words, the two-dimensional DCT essentially performs a one-dimensional DCT on each row of a block of pixels, followed by a one-dimensional DCT on each column of the block of pixels that was produced by the one-dimensional DCTs on the rows. The row and column transformations can be performed in reversed order, with the same result obtained. 
     In H.26L, the specified transform matrix results in the following formulas for computing the transform coefficients:
 
 A= 13 a+ 13 b+ 13 c+ 13 d  
 
 B= 17 a+ 7 b− 7 c− 17 d  
 
 C= 13 a− 13 b− 13 c+ 13 d  
 
 D= 7 a− 17 b+ 17 c− 7 d  
 
where [abcd] first represents a row of pixel values, and then, after row transformation, represents a column of those row-transformed data values, in a four-by-four block. However, in practice, these formulas can be simplified, and the DCT matrix has a recursive structure that reduces the number of required multiplication and addition operations. Nevertheless, the above matrix requires at least four additions and three multiplications to compute the transform coefficients for each pixel. Moreover, because each pixel may be a signed nine-bit value, when multiplied by the row and column factors (the gain equals the norm of 676) the computation requires 32-bit arithmetic, in which a single computation takes as much time as many 16-bit computations. While these considerations are not normally as significant during encoding, with these H.26L specified matrix values, the extra operations and 32-b Type equation here, it arithmetic also occur during decoding, wherein the expense is significant.
 
     In contrast to the H.26L-specified matrix, using the same general formula representation, as described above the present invention&#39;s block transform matrix is: 
               Q   D     ⁢     =   Δ     ⁢       round   ⁢           ⁢     (     2.5   ⁢   H     )       =     [         1       1       1       1           2       1         -   1           -   2             1         -   1           -   1         1           1         -   2         2         -   1           ]             
which provides the following formulas for computing the transform coefficients:
 
 A=a+b+c+d  
 
 B= 2 a+b−c− 2 d  
 
 C=a−b−c+d  
 
 D=a− 2 b+ 2 c−d  
 
Part of the reason that this formula/matrix works more optimally is that instead of performing individual multiplications in the transformation stage, the multiplication factors are essentially handled in the quantization phase, by simply changing the values used in scaling. As a result, the only multiplication with these formulas is by a power of two, which in processors is accomplished via a simple shift left operation, not an actual multiplication. With u, v, y and z used as auxiliary variables, and wherein “&lt;&lt;1” means shift left one bit (equivalent to multiplying by two but computationally faster), the above formulas simplify to the following formulas:
 
 u=a+d;  
 
 v=b+c;  
 
 y=b−c;  
 
 z=a−d;  
 
 A=u+v;  
 
 C=u−v;  
 
 B=y+ ( z&lt;&lt; 1);
 
 D=z− ( y&lt;&lt; 1);
 
     Via the butterfly structure represented in  FIG. 3 , (wherein a line between two points indicates a sum, and the numbers on the lines (the −1s, 2 and −2 indicate a multiplication factor) finding these transform coefficients actually requires only four additions and one shift for each given pixel, providing a very substantial performance improvement relative to the H.26L-specified factors. Moreover, because of the lower coefficient values, all transformation operations may be performed in 16-bit arithmetic (with 9-bit pixel data). More particularly, with the transform matrix Q D  defined above, after a two-dimensional 2-D transform, the maximum signal amplification is 36, whereby the output coefficients after the two-dimensional transform will span 6 more bits than the input. Therefore, for 9-bit input, the output coefficients have a dynamic range of 15 bits, meaning that the two-dimensional direct transform can be computed with 16-bit arithmetic. Moreover, as described below, the performance increase resulting from far fewer operations, each of which requires only 16-bit arithmetic, is essentially repeated during decoding operations with the inverse transform, providing a significant increase in decoding performance. 
     Once the transform coefficients  210  are computed, the block of coefficients  210  is quantized by scaling the values and rounding them to their nearest integers. This is represented in  FIG. 2  by the quantization process  212 , which selects from among quantization values from quantization parameters  214  to scale the transform coefficients  210  into quantized coefficients  216 . As described above, because there is not one norm, but three in the improved matrix, three quantization tables Q0, Q1 and Q2 contain these values, depending on where the transformed coefficient to be quantized is positioned in the block. 
     The quantization formula for scaling and rounding is as follows, which should be computed with 32-bit precision:
 
 L=[K×A ( QP, r )+ fX]&gt;&gt; 20
 
where L is the quantized coefficient, K is the transformed coefficient, A(QP, r) is the scale factor indexed by the quantization parameter QP and r, wherein r identifies which table (Q0, Q1 or Q2 to use), fX is in the range [0-0.5]×2 20  (fX has the same sign as K), and &gt;&gt;20 means shift right twenty places (divide by 1,048,576), to scale the number down. The quantization process thus introduces errors. Note that although 32-bit arithmetic is used for quantization, this is only required when encoding, which is not as significant as when decoding (wherein de-quantization needs only 16-bit precision, as described below).
 
     The index r thus selects which of the three quantization tables Q0, Q1 or Q2 to use to adjust for the three different norms, based on the position of the coefficient K in the block:
         r=0 (use Q0) if the coefficient came from one of the positions {(0, 0), (0, 1), (1, 0), (1, 1)};   r=1 (use Q1) if the coefficient came from one of the positions {(0, 2), (0, 3), (1, 2), (1, 3), (2, 0), (2, 1), (3, 0), (3, 1)};   r=2 (use Q2) if the coefficient came from one of the positions {(2, 2), (2, 3), (3, 2), (3, 3)}.
 
After the right shift by 20 bits, each quantized result L fits into a 16-bit integer.
       

     In an exemplary implementation, the scaling factor A(QP, r) depends on the quantization parameter QP and coefficient position group r according to the following tables:
         A(QP=0 . . . 31, r=0)={104858, 93418, 83226, 74146, 66056, 58849, 52429, 46709, 41613, 37073, 33028, 29425, 26214, 23354, 20806, 18536, 16514, 14712, 13107, 11677, 10403, 9268, 8257, 7356, 6554, 5839, 5202, 4634, 4129, 3678, 3277, 2919};   A(QP=0 . . . 31, r=1)={66318, 59082, 52636, 46894, 41778, 37220, 33159, 29541, 26318, 23447, 20889, 18610, 16579, 14771, 13159, 11723, 10444, 9305, 8290, 7385, 6580, 5862, 5222, 4652, 4145, 3693, 3290, 2931, 2611, 2326, 2072, 1846};   A(QP=0 . . . 31, r=2)={41943, 37367, 33290, 29658, 26422, 23540, 20972, 18684, 16645, 14829, 13211, 11770, 10486, 9342, 8323, 7415, 6606, 5885, 5243, 4671, 4161, 3707, 3303, 2942, 2621, 2335, 2081, 1854, 1651, 1471, 1311, 1168}.
 
The particular values above were designed to satisfy the design specification in H.26L that the quantization step sizes should double for every increment of six in the table index. For other video or still image coding applications, other quantization tables can be designed, in view of the desired fidelity increments. Note that to store the quantization tables at the encoder, some extra table space is needed to store three tables instead of one, but this amount of extra space is only 64 bytes, which is negligible.
       

     Returning to  FIG. 2 , following the quantization process, the quantized coefficients  216  are fed to an entropy encoder  218  which, in general, further reduces the number of bits needed to encode the block. Entropy encoders (and decoders) are known, and are thus not described herein, as the present invention operates before and essentially independent of entropy encoding, and after and essentially independent of entropy decoding. 
     Turning to a consideration of decoding in the block transform decoder  222 , at some time, regardless of how delivered, the entropy encoded output bits are fed as input bits to an entropy decoder  224 . In general, such entropy decoders are known, and thus among other possible operations, it suffices to state that the entropy encoder  224  reproduces the quantized coefficients  226  for a given block. Note that typically the encoding and decoding are not lossy, i.e., the quantized coefficients  216  fed into the entropy encoder  218  will be identical to the quantized coefficients  226  produced by entropy decoder  224 . 
     In general, the block transform decoder  222  mirrors the operation of the block transform encoder  200 , although as described below and in keeping with the present invention, a modified inverse transform matrix is provided that enables 16-bit arithmetic to be used throughout the decoding process, thus greatly simplifying the decoder&#39;s computational complexity. 
     Once the quantized coefficients  226  are retrieved from the entropy decoder  224 , each quantized coefficient L is converted to a reconstructed (de-quantized) value K′ by a de-quantization process  228  implementing the formula:
 
 K′=L×B ( QP, r )
 
where the scaling factor B depends on the QP index used when encoding, and r determines the de-quantization parameters  230  by selecting a parameter from one of the tables D0, D1, or D2 (to adjust for the three norms). Note that r can be deduced from the position of the coefficient being de-quantized in the block, as described above with respect to quantization, although it is feasible to send the r value from the encoder, which would possibly reduce computational complexity on the decoder but would require at least two more bits be sent per coefficient to identify r.
 
     In this manner, the scaling factor B(QP, r) depends on the quantization parameter index for QP used when encoding, and the coefficient position group r. In an exemplary implementation, that dependency is specified according to the following tables:
     B(QP=0 . . . 31, r=0)={80, 90, 101, 113, 127, 143, 160, 180, 202, 226, 254, 285, 320, 359, 403, 453, 508, 570, 640, 718, 806, 905, 1016, 1140, 1280, 1437, 1613, 1810, 2032, 2281, 2560, 2874};   B(QP=0 . . . 31, r=1)={101, 114, 127, 143, 161, 180, 202, 227, 255, 286, 321, 361, 405, 454, 510, 572, 643, 721, 810, 909, 1020, 1145, 1285, 1443, 1619, 1817, 2040, 2290, 2570, 2885, 3239, 3635};   B(QP=0 . . . 31, r=2)={128, 144, 161, 181, 203, 228, 256, 287, 323, 362, 406, 456, 512, 575, 645, 724, 813, 912, 1024, 1149, 1290, 1448, 1625, 1825, 2048, 2299, 2580, 2896, 3252, 3650, 4095, 4596}.
 
As with the encoding tables, the particular values above were designed to satisfy the design specification in H.26L that the quantization step sizes should double for every increment of six in the table index. For other video or still image coding applications, other quantization tables can be designed, in view of the desired fidelity increments.
   

     As can be appreciated, the de-quantization formula scales the coefficients back up to larger values, although each of these scaling values is low enough to ensure that only 16-bit arithmetic is needed at the decoder. It should be noted that since each entry in a table is mathematically related (based on the norm) to corresponding QP-indexed table entries in the other two tables, it is alternatively possible have only one table of QP values, with an appropriate mathematical adjustment based on the r value, rather than have three tables. However, lookups in small tables are relatively efficient, and the number of bytes required to store three tables of thirty-two entries at two bytes per entry instead of one table of thirty-two entries at four bits per entry (required with the specified H.26L transform) is sixty-four extra bytes, which is negligible in contemporary computing devices, especially considering that the transform coefficients generated by de-quantization fit into sixteen bit words instead of 32-bit DWords, thereby reducing the total amount of memory needed. 
     As also represented in  FIG. 2 , once the quantized coefficients  226  have been de-quantized via the above formula and tables, a four-by-four matrix of reconstructed transform coefficients  232  is present. From those transform coefficients  232 , the reconstructed pixels are generated by feeding them to a two-dimensional inverse transform process  234  comprising column and row inverse transform processes  238  and  238 , respectively. 
     In keeping with the present invention, instead of using the inverse transform Q I =Q D   T , as would normally be done, to allow for 16-bit de-quantization and inverse transform computation, the present invention defines the following inverse transform matrix for use: 
               Q   I     ⁢     =   Δ     ⁢     [         1       1       1         1   /   2             1         1   /   2           -   1           -   1             1           -   1     /   2           -   1         1           1         -   1         1           -   1     /   2           ]           
Note that the columns of Q I  are orthogonal to each other, but their norms are different. However, this was handled via the “r” value used in selecting the de-quantization parameter, as described above. Further, note that in generating Q I  from Q D , Q D  was transposed, with columns one and three multiplied by one-half. Again, however, the QP values already present in the tables D0, D1 and D2 have been adjusted for this in advance, eliminating the need for any additional computational complexity to compensate. Moreover, the only “multiplication” that will need to be done when computing the inverse transforms with such a matrix is by one-half, which is actually performed by a highly-efficient shift right operation. Note that the small amount of noise introduced by such imprecise division has essentially no impact in the rate-distortion performance.
 
     With Q I  defined as above, its maximum gain equals four. Therefore, the two-dimensional inverse transformation process only expands the dynamic range by four bits, allowing for computation in 16-bit arithmetic. 
     The column inverse transform process  236  and row inverse transform process  238  (that operates on the result of the column inverse transform process) represented in  FIG. 2  generate the output pixel data  240 . To this end, using the above inverse transform matrix and calling [A B C D] a particular row or column to be inverse transformed, the corresponding inverse transformed values [a′ b′ c′ d′] are computed by:
 
 u=A+C;  
 
 v=A−C;  
 
 y= ( B&gt;&gt; 1)− D;  
 
 z= ( D&gt;&gt; 1)+ B;  
 
 a′=u+z;  
 
 b′=v+y;  
 
 c′=v−y;  
 
 d′=u−z;  
 
where u, v, y and z are auxiliary variables and &gt;&gt;1 means shift one bit to the right (equivalent to multiplying by one-half). Similar to encoding, via the butterfly structure represented in  FIG. 4 , these equations actually reduce to four addition operations and one shift operation per coefficient, all in 16-bit arithmetic. Lastly, the inverse transform process  234  ( FIG. 2 ) scales the reconstructed pixel values by a factor of 2 −7  (performed via a 7-bit shift-right operation).
 
     The reconstructed pixel values after the 7-bit shift have a 9-bit range, so the inverse transform can be computed in 16-bit arithmetic. The de-quantization tables D0, D1 and D2 set forth above are such that the de-quantization equations will not overflow when computed in 16-bit arithmetic. The following tables summarizes some of the computational advantages when using the transform and inverse transform of the present invention relative to those specified in H.26L: 
                                                 Operations per           Operations per           pixel, direct   Quanti-   De-quanti-   pixel, inverse       Transform   transform   zation   zation   transform                  H.26L   4 additions, 3   1 mult. in   1 mult. in   4 additions, 3       Specified   multiplications   32-bit   32-bit   multiplications           in 32-bit   arithmetic   arithmetic   in 32-bit           arithmetic           arithmetic       Present   4 additions, 1   1 mult. in   1 mult. in   4 additions, 1       Invention   shift, in   32-bit   16-bit   shift, in           16-bit   arithmetic   arithmetic   16-bit           arithmetic           arithmetic                    
As can be readily appreciated, such savings are significant, particularly on 16-bit capable processors. For example, in practice, a typical 16-bit PDA processor may see an improvement in decoding of roughly double the speed with the present invention.
 
     Moreover, the transform/inverse transform have been tested with respect to its impact on quality and compression. Results for coding gain (usually defined as an increase in signal-to-noise ratio) show that the use of the transforms of the present invention with test data results in a loss of only about 0.01 dB relative to the H26L transforms, which is negligible, and in fact is likely far smaller than that with actual video signal information, such as pixel prediction errors. Further, overall performance was tested relative to the H.26L specified transforms across each of the thirty-two quantization parameters, resulting in only a plus or minus half percent range of differences, with an average very close to zero, thereby indicating that the transform of the present invention performed as well as the transform specified in H.26L. 
     As can be seen from the foregoing detailed description, there is provided an improved transform and inverse transform matrices for image or video encoding and decoding, respectively, that significantly reduce computational complexity with respect to other known transforms without adversely impacting compression or quality. Transformation multiplications are eliminated yet correct results are obtained because quantization and de-quantization parameter values compensate for those multiplication factors, except for one performed by a shift operation during the transformation and inverse transformation processes. Because of the values used, transformation operations during encoding, and de-quantization and transformation operations during decoding can be performed in 16-bit arithmetic (for pixel data represented by nine bits or less). 
     While the invention is susceptible to various modifications and alternative constructions, certain illustrated embodiments thereof are shown in the drawings and have been described above in detail. It should be understood, however, that there is no intention to limit the invention to the specific forms disclosed, but on the contrary, the intention is to cover all modifications, alternative constructions, and equivalents falling within the spirit and scope of the invention.