Abstract:
Low complexity (16 bit arithmetic) video compression has 8×8 block with transforms using 8×8 integer matrices and quantization with look up table scalar plus constant right shift for all quantization steps. Inverse quantization also a look up table scalar plus right shift dependent upon the quantization step and inverse transform using the 8×8 integer matrices.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS  
       [0001]     This application claims priority from provisional application No. 60/524,831, filed Nov. 25, 2003. The following co-assigned pending patent application discloses related subject matter: application Ser. No. ______, filed Sep. 24, 2004. 
     
    
     BACKGROUND OF THE INVENTION  
       [0002]     The present invention relates to digital image and video signal processing, and more particularly to block transformation and/or quantization plus inverse quantization and/or inverse transformation.  
         [0003]     Various applications for digital video communication and storage exist, and corresponding international standards have been and are continuing to be developed. Low bit rate communications, such as video telephony and conferencing, plus large video file compression, such as motion pictures, led to various video compression standards: H.261, H.263, MPEG-1, MPEG-2, AVS, and so forth. These compression methods rely upon the discrete cosine transform (DCT) or an analogous transform plus quantization of transform coefficients to reduce the number of bits required to encode.  
         [0004]     DCT-based compression methods decompose a picture into macroblocks where each macroblock contains four 8×8 luminance blocks plus two 8×8 chrominance blocks, although other block sizes and transform variants could be used.  FIG. 2  depicts the functional blocks of DCT-based video encoding. In order to reduce the bit-rate, 8×8 DCT is used to convert the 8×8 blocks (luminance and chrominance) into the frequency domain. Then, the 8×8 blocks of DCT-coefficients are quantized, scanned into a 1-D sequence, and coded by using variable length coding (VLC). For predictive coding in which motion compensation (MC) is involved, inverse-quantization and IDCT are needed for the feedback loop. Except for MC, all the function blocks in  FIG. 2  operate on an 8×8 block basis. The rate-control unit in  FIG. 2  is responsible for generating the quantization step (qp) in an allowed range and according to the target bit-rate and buffer-fullness to control the DCT-coefficients quantization unit. Indeed, a larger quantization step implies more vanishing and/or smaller quantized coefficients which means fewer and/or shorter codewords and consequent smaller bit rates and files.  
         [0005]     There are two kinds of coded macroblocks. An INTRA-coded macroblock is coded independently of previous reference frames. In an INTER-coded macroblock, the motion compensated prediction block from the previous reference frame is first generated for each block (of the current macroblock), then the prediction error block (i.e. the difference block between current block and the prediction block) are encoded.  
         [0006]     For INTRA-coded macroblocks, the first (0,0) coefficient in an INTRA-coded 8×8 DCT block is called the DC coefficient, the rest of 63 DCT-coefficients in the block are AC coefficients; while for INTER-coded macroblocks, all 64 DCT-coefficients of an INTER-coded 8×8 DCT block are treated as AC coefficients. The DC coefficients may be quantized with a fixed value of the quantization step, whereas the AC coefficients have quantization steps adjusted according to the bit rate control which compares bit used so far in the encoding of a picture to the allocated number of bits to be used. Further, a quantization matrix (e.g., as in MPEG-4) allows for varying quantization steps among the DCT coefficients.  
         [0007]     In particular, the 8×8 two-dimensional DCT is defined as:  
         F   ⁡     (     u   ,   v     )       =       1   4     ⁢     C   ⁡     (   u   )       ⁢     C   ⁡     (   v   )       ⁢       ∑     x   =   0     7     ⁢           ⁢       ∑     y   =   0     7     ⁢           ⁢       f   ⁡     (     x   ,   y     )       ⁢   cos   ⁢         (       2   ⁢   x     +   1     )     ⁢   u   ⁢           ⁢   π     16     ⁢   cos   ⁢         (       2   ⁢   y     +   1     )     ⁢   v   ⁢           ⁢   π     16                 
 
 where j(x,y) is the input 8×8 sample block and F(u,v) the output 8×8 transformed block where u,v,x,y=0, 1, . . . , 7; and  
         C   ⁡     (   u   )       ,       C   ⁡     (   v   )       =     {           1     2               for   ⁢           ⁢   u     ,     v   =   0               1       otherwise                 
 
 Note that this transforming has the form of 8×8 matrix multiplications, F=D t ×f×D, where D is the 8×8 matrix with u,x element  
         C   ⁡     (   u   )       ⁢   cos   ⁢           (       2   ⁢   x     +   1     )     ⁢   u   ⁢           ⁢   π     16     .         
 
         [0008]     The transform is performed in double precision, and the final transform coefficients are rounded to integer values.  
         [0009]     Next, define the quantization of the transform coefficients as  
         QF   ⁡     (     u   ,   v     )       =       F   ⁡     (     u   ,   v     )       QP         
 
 where QP is the quantization factor computed in double precision from the quantization step, qp, as an exponential such as: QP=2 qp/8 . The quantized coefficients are rounded to integer values and are encoded. 
 
         [0010]     The corresponding inverse quantization becomes: 
 
 F ′( u, v )= QF ( u, v )* QP  
 
 with double precision values rounded to integer values. 
 
         [0011]     Lastly, the inverse transformation (reconstructed sample block) is:  
           f   ′     ⁡     (     x   ,   y     )       =       1   4     ⁢       ∑     u   =   0     7     ⁢           ⁢       ∑   v   7     ⁢           ⁢       C   ⁡     (   u   )       ⁢     C   ⁡     (   v   )       ⁢       F   ′     ⁡     (     u   ,   v     )       ⁢   cos   ⁢         (       2   ⁢   x     +   1     )     ⁢   u   ⁢           ⁢   π     16     ⁢   cos   ⁢         (       2   ⁢   y     +   1     )     ⁢   v   ⁢           ⁢   π     16                 
 
 again with double precision values rounded to integer values. 
 
         [0012]     Various alternative approaches, such as the H.264 and AVS standards, simplify the double precision method by using integer transforms and/or different size blocks. In particular, define an 8×8 integer transform matrix, T 8×8 , with elements analogous to the 8×8 DCT transform coefficients matrix D. Then, with f 8×8  and F 8×8  denoting the input 8×8 sample data matrix (block of pixels or residuals) and the output 8×8 transform-coefficients block, respectively, define the forward 8×8 integer transform as: 
 
 F   8×8   =T   8×8   t   ×f   8×8   ×T   8×8  
 
 where “×” denotes 8×8 matrix multiplication, and the 8×8 matrix T 8×8   t  is the transpose of the 8×8 matrix T 8×8 . 
 
         [0013]     The quantization of the transformed coefficients may be exponentials of the quantization step as above or may use lookup tables with integer entries. The inverse quantization mirrors the quantization. And the inverse transform also uses T 8×8 , and its transpose analogous to the DCT using D and its transpose for both the forward and inverse transforms.  
         [0014]     However, these alternative methods still have computational complexity which should be reduced.  
       SUMMARY OF THE INVENTION  
       [0015]     The present invention provides low-complexity 8×8 transformation for image/video processing by partitioning bit shifting and round-off.  
         [0016]     The preferred embodiment methods provide for 16-bit operations useful in video coding with motion compensation. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0017]      FIGS. 1   a - 1   b  are flow diagrams.  
         [0018]      FIG. 2  illustrates a motion compensation video compression with DCT-transformation and quantization.  
         [0019]      FIG. 3  shows method comparisons.  
     
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS  
       [0000]     1. Overview  
         [0020]     The preferred embodiment low-complexity methods provide simplified 8×8 forward transform which applies to the 16-bit AVS method.  
         [0021]     The methods have application to video compression which operates on 8×8 blocks of (motion-compensated) pixels with DCT transformation and quantization of the DCT-coefficients where the quantization can vary widely. As illustrated in  FIG. 2 , fullness feedback from the bitstream buffer may determine the quantization factor, which typically varies in the range from 1 to 200-500.  FIGS. 1   a - 1   b  are transform/quantization of encode and decode flows.  
         [0022]     Preferred embodiment systems perform preferred embodiment methods with digital signal processors (DSPs) or general purpose programmable processors or application specific circuitry or systems on a chip (SoC) such as both a DSP and RISC processor on the same chip with the RISC processor controlling. In particular, digital still cameras (DSCs) with video clip capabilities or cell phones with video capabilities could include the preferred embodiment methods. A stored program could be in an onboard ROM or external flash EEPROM for a DSP or programmable processor to perform the signal processing of the preferred embodiment methods. Analog-to-digital converters and digital-to-analog converters provide coupling to the real world, and modulators and demodulators (plus antennas for air interfaces) provide coupling for transmission waveforms.  
         [0000]     2. AVS  
         [0023]     Initially, consider the AVS transform, quantization, and inverses; the preferred embodiment methods will provide simplifications of the forward transform of AVS.  
         [0024]     (a) AVS Forward Transform  
         [0025]     The AVS forward 8×8 transform uses the following 8×8 transform matrix, T 8×8 , for matrix multiplications with 8×8 sample data matrix (blocks of image pixels or motion residuals) plus an 8×8 scaling matrix, SM 8×8 , for scaling the resulting matrix elements. The transform matrix is:  
         T     8   ×   8       =     [         8       10       10       9       8       6       4       2           8       9       4         -   2           -   8           -   10           -   10           -   6             8       6         -   4           -   10           -   8         2       10       9           8       2         -   10           -   6         8       9         -   4           -   10             8         -   2           -   10         6       8         -   9           -   4         10           8         -   6           -   4         10         -   8           -   2         10         -   9             8         -   9         4       2         -   8         10         -   10         6           8         -   10         10         -   9         8         -   6         4         -   2           ]         
 
 And scaling matrix SM 8×8 ={SM i,j : i,j=0, 1, 2, . . . 7} is:  
         SM     8   ×   8       =     [         32768       37958       36158       37958       32768       37958       36158       37958           37958       43969       41884       43969       37958       43969       41884       43969           36158       41884       39898       41884       36158       41884       39898       41884           37958       43969       41884       43969       37958       43969       41884       43969           32768       37958       36158       37958       32768       37958       36158       37958           37958       43969       41884       43969       37958       43969       41884       43969           36158       41884       39898       41884       36158       41884       39898       41884           37958       43969       41884       43969       37958       43969       41884       43969         ]         
 
 The transform proceeds as follows. First, let f 8×8 ={f i,j : i,j=0, 1, 2, . . . , 7} denote the input 8×8 data matrix and let F 8×8 ={F i,j : i, j=0, 1, 2, . . . , 7} denote the 8×8 output DCT coefficients matrix. The AVS forward transform has two steps and uses an intermediate 8×8 matrix X 8×8 : 
 
 X   8×8   ={T   8×8   t   ×f   8×8   ×T   8×8 }         5 
 
 F   i,j   =sign ( X   i,j )*((| X   i,j   |*SM   i,j +2 18 )&gt;&gt;19) i, j=0, 1, 2, . . . , 7 
 
 The following notation is being used here and in the following: 
        T 8×8    t  is the transpose of the transform matrix T 8×8       X 8×8  ={X i,j : i,j=0, 1, 2, . . . , 7} is the intermediate matrix after matrix with the transform matrix and its transpose plus a rounding bit shown above     × is matrix multiplication              is scalar multiplication     |x| is the absolute value of x     Sign(x) is defined as  
         sign   ⁡     (   x   )       =     {         1           if   ⁢           ⁢   x     &gt;   0               -   1         otherwise               
              n is matrix right rounding by n bits; more explicitly, for a matrix M 8×8 ={M i,j : i,j=0, 1, 2, . . . , 7} the operation m 8×8 =M 8×8           n is defined by m 8×8 ={m i,j : i,j=0, 1, 2, . . . , 7} where m i,j =(M i,j +2 n−1 )&gt;&gt;n.     &gt;&gt; denotes right shifting, which applies to the numbers when expressed in binary notation (e.g., two&#39;s complement). 
 
 Thus the transform matrix T 8×8  is analogous to the 8×8 DCT matrix and SM 8×8  is a scaling adjustment. 
       
 
         [0034]     (b) AVS Quantization  
         [0035]     The AVS quantization supports  64  quantization steps, qp=0, 1, . . . , 63, and uses the following quantization table Q_TAB[64]:  
                                                                   qp   0   1   2   3   4   5   6   7       Q_TAB[qp]   32768   29775   27554   25268   23170   21247   19369   17770       qp   8   9   10   11   12   13   14   15       Q_TAB[qp]   16302   15024   13777   12634   11626   10624    9742    8958       qp   16   17   18   19   20   21   22   23       Q_TAB[qp]    8192    7512    6889    6305    5793    5303    4878    4467       qp   24   25   26   27   28   29   30   31       Q_TAB[qp]    4091    3756    3444    3161    2894    2654    2435    2235       qp   32   33   34   35   36   37   38   39       Q_TAB[qp]    2048    1878    1722    1579    1449    1329    1218    1117       qp   40   41   42   43   44   45   46   47       Q_TAB[qp]    1024    939    861    790    724    664    609    558       qp   48   49   50   51   52   53   54   55       Q_TAB[qp]    512    470    430    395    362    332    304    279       qp   56   57   58   59   60   61   63   63       Q_TAB[qp]    256    235    215    197    181    166    152    140                  
 
 Thus the quantization factor Q_TAB[qp]: is essentially 2 15−qp/8  and the quantization of the transformed matrix F 8×8  is: 
 
 QF   i,j   =sign ( F   i,j )*(| F   i,j   |*Q   —   TAB[qp]+α* 2 15 )&gt;&gt;15 i,j=0, 1, 2, . . . , 7 
 
 where α is quantization control parameter, such as ⅓ for INTRA-coded macroblocks and ⅙ for INTER-coded macroblocks. These quantized coefficients are encoded. 
 
         [0036]     (c) AVS Inverse Quantization  
         [0037]     The AVS inverse quantization for an 8×8 quantized DCT coefficient block QF 8×8 ={QF ij : i,j=1, 2, . . . , 7} is defined as: 
 
 F   ij =( QF   ij   *IQ   —   TAB[qp]+ 2 IQ     —     SHIFT[qp] )&lt;&lt; IQ   —   SHIFT[qp ] i,j=0, 1, 2, . . . , 7 
 
         [0038]     where F′ 8×8 ={F′ i,j : i,j=1, 2, . . . , 7}is the inverse-quantized DCT coefficients block and the IQ_TAB and IQ_SHIFT tables are defined as:  
                                                                   qp   0   1   2   3   4   5   6   7       IQ_TAB [qp]   32768   36061   38968   42495   46341   50535   55437   60424       IQ_SHIFT [qp]   14   14   14   14   14   14   14   14       qp   8   9   10   11   12   13   14   15       IQ_TAB [qp]   32932   35734   38968   42495   46177   50535   55109   59933       IQ_SHIFT [qp]   13   13   13   13   13   13   13   13       qp   16   17   18   19   20   21   22   23       IQ_TAB [qp]   65535   35734   38968   42577   46341   50617   55027   60097       IQ_SHIFT [qp]   13   12   12   12   12   12   12   12       qp   24   25   26   27   28   29   30   31       IQ_TAB [qp]   32809   35734   38968   42454   46382   50576   55109   60056       IQ_SHIFT [qp]   11   11   11   11   11   11   11   11       qp   32   33   34   35   36   37   38   39       IQ_TAB [qp]   65535   35734   38968   42495   46320   50515   55109   60076       IQ_SHIFT [qp]   11   10   10   10   10   10   10   10       qp   40   41   42   43   44   45   46   47       IQ_TAB [qp]   65535   35744   38968   42495   46341   50535   55099   60087       IQ_SHIFT [qp]   10   9   9   9   9   9   9   9       qp   48   49   50   51   52   53   54   55       IQ_TAB [qp]   65535   35734   38973   42500   46341   50535   55109   60097       IQ_SHIFT [qp]   9   8   8   8   8   8   8   8       qp   56   57   58   59   60   61   62   63       IQ_TAB [qp]   32771   35734   38965   42497   46341   50535   55109   60099       IQ_SHIFT [qp]   7   7   7   7   7   7   7   7                  
 
 Note that IQ_TAB[qp] is a 16-bit positive integer (no sign bit) with a most significant bit (MSB) equal to 1 for all qp, and IQ_SHIFT[qp] is in the range 7-14. 
 
 (d) AVS Inverse Transform 
 
         [0039]     The AVS inverse 8×8 transform uses matrix multiplications with the same 8×8 transform matrix, T 8×8 , and its transpose: 
 
 f′   8×8   ={T   8×8 ×(( F′   8×8   ×T   8×8   t )         3)}         7 
 
 where f′ 8×8 ={f′ i,j : i,j=1, 2, . . . , 7} is the reconstructed 8×8 sample data matrix. 
 
 3. First Preferred Embodiment 
 
         [0040]     In order to reduce the transform and quantization complexity of the AVS of section 2, the preferred embodiment methods provide a modified forward transform to use together the quantization, inverse quantization, and inverse transformation of section 2. The preferred embodiment methods simplify the computations by eliminating a sign( ) operation and limiting the bit shifting so a 16-bit-based processor operates more efficiently. That is, only the forward transform is modified, and comparisons of the AVS of section with the preferred embodiment methods appear in section 4.  
         [0041]     (a) Preferred Embodiment Forward Transform  
         [0042]     Recall the AVS forward transform as described in section 2 is: 
 
 X   8×8   ={T   8×8   t   ×f   8×8   ×T   8×8 }         5 
 
 F   i,j   =sign ( X   i,j )*(| X   i,j   |*SM   i,j +2 18 )&gt;&gt;19) i,j=0, 1, 2, . . . , 7 
 
 The second step is computationally expensive, especially for 16-bit devices. In order to reduce the complexity, the preferred embodiment methods modify the forward transform second step to essentially split the shift of 19 bits into a shift of N bits plus a shift of 19−N bits in the scaling matrix:  
             F     i   ,   j       =     (         X     i   ,   j       *     SM     i   ,   j       (   N   )         +     2     N   -   1         )       &gt;&gt;     N   ⁢           ⁢   i       ,     j   =   1     ,   2   ,   …   ⁢           ,           ⁢     7   ⁢           ⁢   where   ⁢           ⁢     SM     i   ,   j       (   N   )             
 
 is defined as  
             SM     i   ,   j       (   N   )       =       (       SM     i   ,   j       +     2     18   -   N         )     ⪢       (     19   -   N     )     ⁢           ⁢   i         ,     j   =   1     ,   2   ,   …   ⁢           ,           ⁢   7     }       
 
 where SM 8×8 ={SM i,j : i,j=1, 2, . . . , 7} is the scaling matrix defined in section 2 and  
         SM     8   ×   8       (   N   )       =     {         SM     i   ,   j       (   N   )       ⁢     :     ⁢           ⁢   i     ,     j   =   1     ,   2   ,   …   ⁢           ,   7     }         
 
 is the new scaling matrix. 
 
         [0043]     In this transform N is the number of shift bits and the performance improves as N increases (see the next section); but for 16-bit processor complexity reduction, N is taken to be less than or equal to 16.  
         [0044]     For example, with N=16:  
         SM     8   ×   8     16     =     [         4096       4745       4520       4745       4096       4745       4520       4745           4745       5496       5236       5496       4745       5496       5236       5496           4520       5236       4987       5236       4520       5236       4987       5236           4745       5496       5236       5496       4745       5496       5236       5496           4096       4745       4520       4745       4096       4745       4520       4745           4745       5496       5236       5496       4745       5496       5236       5496           4520       5236       4987       5236       4520       5236       4987       5236           4745       5496       5236       5496       4745       5496       5236       5496         ]         
 
 Note that  
       SM     8   ×   8       (   N   )         
 
 is essentially equal to SM 8×8  of section 2 when N=19, and for each decrement of N by 1 the matrix elements are all divided by 2 with a final round off. 
 
         [0045]     Compared to the AVS forward transform described in section 2, the preferred embodiment has much lower complexity because of the elimination of the sign(x) operation and the limitation of memory accesses and right shifts all within  16  bits. Thus, the preferred embodiment method makes the section 2 described AVS forward transform more computationally-cost-effective.  
         [0046]     (b) Preferred Embodiment Quantization  
         [0047]     The preferred embodiment methods use the same quantization as described in section 2.  
         [0048]     (c) Preferred Embodiment Inverse Quantization  
         [0049]     The preferred embodiment methods use the same inverse quantization as decribed in section 2.  
         [0050]     (d) Preferred Embodiment Inverse Transform  
         [0051]     The preferred embodiment methods use the same inverse transform as decribed in section 2.  
         [0000]     Experimental Results  
         [0052]     Simulations ere carried out to test the efficiency of the preferred embodiment simplified forward transform. In the following Table, the column “Anchor T&amp;Q” shows the signal-to-noise ratio (SNR0) for applications of the AVS transform plus quantization followed by inverse quantization plus inverse transform described in section 2. The “Simplified T&amp;Q” columns show the signal-to-noise ratios (SNR1) and differences from the SNR0 of the same blocks for applications of the preferred embodiment forward transform for various values of N together with AVS quantization followed by AVS inverse quantization and AVS inverse transform; that is, only the forward transform is changed in these cases, everything else remains the same. All quantization steps (qp=0, 1, 2, . . . 63) are tested. Each qp is tested with 6000 random 8×8 blocks, the pixel values lie in the range of [−255:255]. The SNR values between the input sample blocks and their reconstructed blocks are computed (see  FIG. 3 ) over all the test sample blocks for each qp. The results for N=16, 15, 14, 13, 12, 11 are listed in the Table.  
                                                                                             Simplified T&amp;Q   Simplified T&amp;Q   Simplified T&amp;Q   Simplified T&amp;Q   Simplified T&amp;Q           Anchor   N = 16   N = 15, 14   N = 13   N = 12   N = 11       qp   SNR0[dB]   SNR1[dB](Δ dB)   SNR1[dB](Δ dB)   SNR1[dB](Δ dB)   SNR1[dB](Δ dB)   SNR1[dB](Δ dB)                                0   60.088   60.125 (0.037)   60.072 (−0.016)   60.033 (−0.055)   59.075 (−1.013)   58.307 (−1.780)       1   54.157   54.129 (−0.028)   54.149 (−0.008)   54.160 (0.004)   53.972 (−0.184)   53.789 (−0.368)       2   54.077   54.070 (−0.007)   54.061 (−0.016)   54.042 (−0.035)   53.861 (−0.216)   53.704 (−0.373)       3   53.898   53.907 (0.009)   53.892 (−0.007)   53.868 (−0.031)   53.783 (−0.115)   53.648 (−0.250)       4   53.723   53.700 (−0.023)   53.709 (−0.014)   53.705 (−0.018)   53.588 (−0.135)   53.436 (−0.286)       5   53.376   53.380 (0.004)   53.369 (−0.008)   53.367 (−0.009)   53.260 (−0.116)   53.105 (−0.271)       6   52.461   52.456 (−0.006)   52.466 (0.004)   52.454 (−0.008)   52.380 (−0.081)   52.253 (−0.209)       7   51.873   51.882 (0.010)   51.883 (0.010)   51.897 (0.024)   51.805 (−0.068)   51.747 (−0.125)       8   51.472   51.460 (−0.012)   51.440 (−0.032)   51.450 (−0.021)   51.388 (−0.084)   51.232 (−0.239)       9   51.071   51.079 (0.009)   51.075 (0.005)   51.060 (−0.011)   51.028 (−0.043)   50.970 (−0.100)       10   50.727   50.734 (0.007)   50.714 (−0.012)   50.700 (−0.027)   50.617 (−0.110)   50.563 (−0.163)       11   50.189   50.171 (−0.019)   50.163 (−0.026)   50.157 (−0.033)   50.108 (−0.082)   50.049 (−0.141)       12   49.445   49.451 (0.006)   49.451 (0.006)   49.450 (0.005)   49.396 (−0.049)   49.327 (−0.118)       13   48.857   48.847 (−0.010)   48.860 (0.003)   48.843 (−0.014)   48.798 (−0.059)   48.753 (−0.104)       14   48.386   48.397 (0.011)   48.387 (0.001)   48.391 (0.005)   48.361 (−0.025)   48.321 (−0.065)       15   47.783   47.776 (−0.007)   47.778 (−0.006)   47.778 (−0.006)   47.758 (−0.026)   47.704 (−0.079)       16   46.495   46.495 (0.001)   46.502 (0.007)   46.553 (0.058)   46.430 (−0.065)   46.416 (−0.078)       17   46.554   46.548 (−0.005)   46.544 (−0.010)   46.542 (−0.012)   46.521 (−0.033)   46.497 (−0.057)       18   45.753   45.751 (−0.002)   45.757 (0.004)   45.756 (0.003)   45.735 (−0.018)   45.715 (−0.038)       19   45.074   45.081 (0.007)   45.082 (0.007)   45.079 (0.004)   45.039 (−0.036)   45.027 (−0.048)       20   44.468   44.467 (−0.001)   44.470 (0.003)   44.462 (−0.006)   44.441 (−0.026)   44.429 (−0.039)       21   43.666   43.672 (0.005)   43.674 (0.008)   43.671 (0.004)   43.650 (−0.017)   43.647 (−0.020)       22   42.961   42.961 (−0.000)   42.963 (0.001)   42.961 (−0.000)   42.955 (−0.007)   42.939 (−0.023)       23   42.217   42.216 (−0.000)   42.216 (−0.000)   42.212 (−0.004)   42.214 (−0.002)   42.209 (−0.007)       24   41.471   41.470 (−0.001)   41.470 (−0.001)   41.448 (−0.023)   41.470 (−0.001)   41.466 (−0.005)       25   40.983   40.985 (0.002)   40.983 (0.000)   40.983 (0.000)   40.977 (−0.006)   40.964 (−0.019)       26   40.087   40.088 (0.001)   40.087 (0.000)   40.093 (0.006)   40.093 (0.006)   40.078 (−0.008)       27   39.423   39.424 (0.001)   39.425 (0.002)   39.426 (0.003)   39.418 (−0.005)   39.411 (−0.012)       28   38.602   38.606 (0.004)   38.606 (0.003)   38.607 (0.005)   38.597 (−0.005)   38.597 (−0.005)       29   37.875   37.876 (0.001)   37.877 (0.002)   37.878 (0.003)   37.871 (−0.004)   37.866 (−0.009)       30   37.141   37.140 (−0.001)   37.140 (−0.001)   37.140 (−0.001)   37.137 (−0.004)   37.130 (−0.011)       31   36.396   36.396 (−0.000)   36.398 (0.001)   36.400 (0.003)   36.390 (−0.007)   36.388 (−0.008)       32   35.526   35.526 (0.000)   35.526 (0.000)   35.529 (0.003)   35.521 (−0.005)   35.520 (−0.006)       33   34.853   34.853 (0.000)   34.853 (0.000)   34.854 (0.001)   34.851 (−0.002)   34.852 (−0.001)       34   34.180   34.179 (−0.000)   34.178 (−0.001)   34.179 (−0.000)   34.180 (0.001)   34.176 (−0.004)       35   33.388   33.388 (−0.000)   33.387 (−0.000)   33.386 (−0.002)   33.382 (−0.006)   33.382 (−0.006)       36   32.660   32.660 (0.000)   32.660 (0.000)   32.660 (−0.000)   32.659 (−0.001)   32.659 (−0.002)       37   31.881   31.880 (−0.000)   31.881 (−0.000)   31.881 (0.001)   31.880 (−0.001)   31.879 (−0.001)       38   31.149   31.149 (0.000)   31.149 (0.000)   31.148 (−0.000)   31.148 (−0.001)   31.147 (−0.002)       39   30.382   30.382 (0.001)   30.382 (0.001)   30.382 (0.000)   30.381 (−0.001)   30.379 (−0.003)       40   29.635   29.635 (−0.000)   29.635 (0.000)   29.636 (0.001)   29.634 (−0.001)   29.633 (−0.002)       41   28.872   28.873 (0.000)   28.872 (−0.000)   28.872 (−0.000)   28.874 (0.002)   28.872 (0.000)       42   28.170   28.170 (−0.000)   28.170 (−0.000)   28.170 (−0.001)   28.171 (0.001)   28.169 (−0.001)       43   27.382   27.382 (−0.000)   27.382 (0.000)   27.383 (0.000)   27.383 (0.000)   27.383 (0.000)       44   26.616   26.616 (0.000)   26.616 (0.000)   26.616 (0.000)   26.616 (−0.000)   26.616 (−0.000)       45   25.884   25.884 (0.000)   25.884 (0.000)   25.884 (0.000)   25.883 (−0.001)   25.883 (−0.001)       46   25.131   25.131 (0.000)   25.131 (0.000)   25.131 (0.000)   25.131 (0.000)   25.131 (−0.000)       47   24.371   24.371 (−0.000)   24.371 (−0.000)   24.371 (−0.000)   24.371 (−0.000)   24.370 (−0.001)       48   23.637   23.637 (0.000)   23.637 (0.000)   23.637 (0.000)   23.636 (−0.001)   23.636 (−0.001)       49   22.890   22.890 (−0.000)   22.890 (−0.000)   22.890 (0.000)   22.890 (−0.000)   22.889 (−0.001)       50   22.165   22.164 (−0.000)   22.164 (−0.000)   22.164 (−0.000)   22.164 (−0.000)   22.164 (−0.000)       51   21.418   21.418 (0.000)   21.418 (0.000)   21.418 (0.000)   21.418 (−0.000)   21.418 (−0.000)       52   20.657   20.657 (−0.000)   20.657 (−0.000)   20.657 (0.000)   20.657 (0.000)   20.657 (−0.000)       53   19.892   19.892 (−0.000)   19.892 (−0.000)   19.892 (−0.000)   19.892 (−0.000)   19.892 (−0.000)       54   19.117   19.117 (0.000)   19.117 (−0.000)   19.117 (−0.000)   19.117 (−0.000)   19.117 (−0.000)       55   18.353   18.353 (−0.000)   18.353 (0.000)   18.353 (−0.000)   18.353 (−0.000)   18.353 (−0.000)       56   17.616   17.616 (0.000)   17.616 (0.000)   17.616 (0.000)   17.615 (−0.000)   17.615 (−0.000)       57   16.872   16.872 (0.000)   16.872 (0.000)   16.872 (0.000)   16.872 (−0.000)   16.872 (0.000)       58   16.129   16.129 (−0.000)   16.129 (−0.000)   16.129 (−0.000)   16.129 (0.000)   16.129 (0.000)       59   15.422   15.422 (0.000)   15.422 (0.000)   15.422 (0.000)   15.422 (0.000)   15.422 (0.000)       60   14.709   14.709 (−0.000)   14.709 (−0.000)   14.709 (−0.000)   14.709 (−0.000)   14.709 (−0.000)       61   14.045   14.045 (0.000)   14.045 (0.000)   14.045 (0.000)   14.045 (−0.000)   14.045 (−0.000)       62   13.404   13.404 (0.000)   13.404 (0.000)   13.404 (0.000)   13.404 (−0.000)   13.404 (0.000)       63   12.853   12.853 (0.000)   12.853 (−0.000)   12.853 (−0.000)   12.853 (−0.000)   12.853 (−0.000)                  
 
         [0053]     As shown in the Table, as long as N≧13, the preferred embodiment simplified forward transform method performs almost identically to the AVS forward transform. However, significant loss at high-end blocks (&gt;50 dB area) begins to appear when N≦12.  
         [0054]     Since for 16-bit devices the complexity is almost the same as long as N≦16, the preferred embodiment simplified transform method (16≧N≧13) provides the same compression efficient as the current AVS transform design, but at lower computational complexity.  
         [0000]     5. Modifications  
         [0055]     The preferred embodiment methods can be modified in various ways while retaining the feature of the simplified forward transform.  
         [0056]     For example, the round-off could varied or . . . ???.