Abstract:
A method of calculating the discrete cosine transform (DCT) of blocks of pixels of a picture includes the steps of defining first subdivision blocks called range blocks, having a fractional and scaleable size N/2 i *N/2 i , where i is an integer number, with respect to a maximum pre-defined size of N*N pixels of blocks of division of the picture, referred to as domain blocks, shiftable by intervals of N/2 i  pixels. The method also includes the step of calculating the DCT on 2 i  range blocks of a subdivision of a domain block of N*N pixels of the picture, in parallel.

Description:
FIELD OF THE INVENTION 
     The invention relates in general to digital processing systems for recording and/or transmitting pictures, and more in particular to systems for compressing and coding pictures by calculating the discrete cosine transform (DCT) of blocks of pixels of a picture. The invention is particularly useful in video coders according to the MPEG2 standard though it is applicable also to other systems. 
     BACKGROUND OF THE INVENTION 
     The calculation of the discrete cosine transform (DCT) of a pixel matrix of a picture is a fundamental step in processing picture data. A division by a quantization matrix is performed on the results of the discrete cosine transform for reducing the amplitude of the DCT coefficients, as a precondition to data compression which occurs during a coding phase, according to a certain transfer protocol of video data to be transmitted or stored. Typically, the calculation of the discrete cosine transform is carried out on blocks or matrices of pixels, in which a whole picture is subdivided for processing purposes. 
     Increasing speed requisites of picture processing systems for storage and/or transmission, imposes the use of hardware architectures to speed up various processing steps among which, primarily, the step of discrete cosine transform calculation by blocks of pixels. Use of hardware processing imposes a pre-definition of few fundamental parameters, namely the dimensions of the blocks of pixels into which a picture is subdivided to meet processing requisites. 
     Such a pre-definition may represent a heavy constraint that limits the possibility of optimizing the processing system, for example a MPEG2 coder, or its adaptability to different conditions of use in terms of different performance requisites. It is also evident the enormous economic advantage in terms of reduction of costs of an integrated data processing system that may be programmed to calculate in parallel the DCT on several blocks of pixels of size selectable among a certain number of available sizes. 
     SUMMARY OF THE INVENTION 
     It is evident that a need exists for a method and of a hardware architecture for calculating the discrete cosine transform (DCT) on a plurality of blocks of pixels, in parallel, which provides for the scalability of the size of the blocks of pixels. For example, the calculation of the discrete cosine transform (DCT) either for one block of 8×8 pixels, or four blocks of 4×4 pixels in parallel, or for sixteen blocks of 2×2 pixels in parallel, operating a selection of the block&#39;s size. 
     Scalability of the size of the block of pixels and the possibility of performing the calculation of the discrete cosine transform in parallel on blocks of congruently reduced size compared to a certain maximum block&#39;s size, by a hardware structure is also instrumental of the implementation of highly efficient “hybrid” picture compression schemes and algorithms. For example, by virtue of the scalability of the block size and of the ability to calculate in parallel the DCT on more blocks, it is possible, according to the present invention, to implement a fractal coding applied in the DCT domain rather than in the space domain of picture data, as customary. 
     Therefore, another important aspect of the invention is a new picture data compressing and coding method that practically is made possible by a hardware structure calculating the DCT on blocks of scaleable size and which includes
         subdividing a picture by defining two distinct types of subdivision blocks: a first type, of N/i*N/i dimension called range blocks that are not overimposable one on another, and a second type, of N*N dimension, called domain blocks, that are transferable by intervals of N/i pixels and overimposable one on another (by transferring on the original picture a window that identifies a domain block by an interval equivalent to the horizontal and/or vertical dimension of a range block);   calculating the discrete cosine transform (DCT) of the 2 i  range blocks and of a relative domain block in parallel;   classifying the transform range blocks according to their relative complexity calculated by summing the three AC coefficients;   applying the fractal transform in the DCT domain to the data of range blocks whose complexity exceeds a pre-defined threshold and storing only the DC coefficient of the range blocks with a complexity lower than said threshold, identifying a relative domain block belonging to the range block being transformed that produces the best fractal approximation of the same range block and calculating its discrete cosine transform;   calculating a difference picture between each range block and its fractal approximation;   quantizing said difference picture in the DCT domain by using a quantization table predisposed in function of human sight characteristics;   coding said quantized difference picture by a method based on the probability of the quantization coefficients;   storing or transmitting the coding code for each range block compressed in the DCT domain and the DC coefficient of each uncompressed range block.       

     By indicating with F R (u,v), the DCT of a generic range block, in the domain of the DCT, the AC coefficients indicate the block&#39;s complexity. For the case of N=8, the four coefficients at the top left of the block are:
         the DC coefficient that occupies the position 00; and   the three AC coefficients that occupy the positions 01, 10 and 11, respectively.       

     The AC coefficients are used to decide to which a certain range block belongs: if the sum of their absolute values is less that a determined threshold T, the block range in question is classified as a “low activity” block, on the contrary, if the sum is equal or greater than T, the range block is classified as a “high activity” block. 
     For a low activity range block, the AC coefficients are small and therefore they may be omitted without significantly affecting fidelity: in this case the block may be approximated by storing only its DC coefficient. 
     For a high activity range block, the progression of fractal coding of the invention includes searching two appropriate linear transform, for example, rotations, φ, overturns, τ, or the like and a domain block DCT, of which being defined by F D (u,v), which at least approximately satisfy the following equation:
 
 F   R ( u,v )=φ( F   D ( u,v ))
 
 F   R ( u,v )=τ( F   D ( u,v ))  (1)
 
     Having so identified the domain block most similar or homologous to the range block that is being processed, its parameters F D  (u,v),τ,φ are stored. The difference picture between the range block and its fractal position is then calculated:
 
 D ( u,v )= F   R ( u,v )−φ( F   D ( u,v ))
 
 D ( u,v )= F   R ( u,v )−τ( F   D ( u,v ))  (2)
 
     By quantizing the difference picture D(u,v),
 
 D   Q ( u,v )= INTEG[D ( u,v )/ Q ( u,v )]  (3)
 
is obtained, where:
         DQ(u,v) is the quantized difference picture in the domain of DCT;   Q(u,v) is a quantization table designed by considering human sight characteristics;   INTEG is a function that approximates its argument to the nearest integer;       

     After quantization, the majority of the D Q (u,v) coefficients are null. Therefore, it is easy to design a coding, for example Huffman coding, based on the probabilities of the coefficients. Finally, the code to be recorded or transmitted is stored. The compression procedure terminates when each range block has been coded. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The different aspects and implementations of the scaleable architecture for calculating the discrete cosine transform of the invention as well as of the method of compression and fractal coding, will be more easily understood through the following detailed description of an embodiment of the architecture of the invention and of the different functioning modes according to a selection of the size selection of the blocks of pixels into which the picture is divided by referring to the annexed drawings, wherein: 
         FIG. 1  is a block diagram of a coder effecting hybrid compression based on fractal coding and DCT, according to the present invention; 
         FIG. 2  is a flow chart of the parallel computation of the DCT of sixteen blocks of 2×2-pixel size; 
         FIG. 3  illustrates the architecture for parallel computation of sixteen 2×2 DCTs; 
         FIG. 4  shows the arrangement of the input data for calculating the sixteen 2×2 DCTs; 
         FIG. 5  shows the PROCESS phase of the calculating procedure of sixteen 2×2 DCTs; 
         FIG. 6  illustrates the architecture for parallel computation of four 4×4 DCTs; 
         FIG. 7  shows the arrangement of the input data for calculating the four 4×4 DCTs; 
         FIG. 8  shows the PROCESS phase of the calculating procedure of four 4×4 DCTs; 
         FIG. 9  illustrates the architecture for parallel computation of an 8×8 DCT; 
         FIG. 10  shows the arrangement of the input data for calculating an 8×8 DCT; 
         FIG. 11  shows the PROCESS phase for the calculating procedure of an 8×8 DCT; 
         FIG. 12  shows the scaleable hardware architecture of the invention for calculating an 8×8 DCT or four 4×4 DCTs in parallel or sixteen 2×2 DCTs in parallel; 
         FIG. 13  shows the INPUT structure of the scaleable architecture of the invention; 
         FIG. 14  shows the PROCESS structure for the scaleable architecture of the invention; 
         FIG. 15  shows the functional schemes of the blocks that implement the PROCESS phase in the scaleable architecture of the invention; 
         FIG. 16  is a detailed scheme of the QA block; 
         FIG. 17  is a detailed scheme of the QB block; 
         FIG. 18  is a detailed scheme of the QC block; 
         FIG. 19  is a detailed scheme of the QD block; 
         FIG. 20  is a detailed scheme of the QE block; 
         FIG. 21  is a detailed scheme of the QF block; 
         FIG. 22  is a detailed scheme of the QG block; 
         FIG. 23  illustrates an implementation of the ORDER phase in the scaleable architecture of the invention; and 
         FIG. 24  illustrates an implementation of the OUTPUT phase in the scaleable architecture of the invention. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     Though referring in some of the scheme illustrated in the figures to a particularly significant and effective implementation of the architecture of parallel computation of the discrete cosine transform (DCT) on blocks of pixels of scaleable size, which comprises a compression phase for the fractal coding of the picture data, it is understood that the method and architecture of parallel calculation of the discrete cosine transformed (DCT) of a bidimensional matrix of input data by blocks of a scaleable size, provide for an exceptional freedom in implementing particularly effective compression algorithms by exploiting the scalability and the possibility of a parallel calculation of DCT. 
     The partitioning steps and the calculation of the discrete cosine transform of a bidimensional matrix of input data will be described separately for each size of range block, according to an embodiment of the invention, starting from the smallest block dimension of 2×2 for which the DCT calculation is performed in parallel, up to the maximum block dimension of 8×8. This description of an architecture scaleable according to needs by changing the value of the global variable size, will follow. 
     The procedure for a parallel DCT computation of the invention may be divided in distinct phases:
     INPUT phase   PROCESS phase   ORDER phase   OUTPUT phase.   

     Each phase is hereinbelow described for each case considered. 
     The DCT operation may be defined as follows. 
     For an input data matrix x N*N =[x i,j ],0≦i,j≦N−1, the output matrix y N*N =[y m,n ],0≦m,n≦N−1, is defined by: 
                     y     m   ,   n       =       2   N     ⁢     ɛ   ⁡     (   m   )       ⁢     ɛ   ⁡     (   n   )       ⁢       ∑     i   =   0       N   -   1       ⁢           ⁢       ∑     j   =   0       N   -   1       ⁢           ⁢       x     i   ,   j       ⁢     cos   ⁡     (           (       2   ⁢   i     +   1     )     ⁢   m       2   ⁢   N       ⁢   π     )       ⁢     cos   ⁡     (           (       2   ⁢   j     +   1     )     ⁢   n       2   ⁢   N       ⁢   π     )                       (   4   )               
where
 
     
       
         
           
             
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     Parallel Computation of Sixteen 2×2 DCTs 
     For N=2 equation (5) becomes: 
     
       
         
           
             
               
                 
                   
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     The flow graph for a 2×2 DCT is shown in  FIG. 2 , in which A=B=C=1 and the input and output data are the pixels in the positions (0,0), (0,1), (1,0), (1,1). 
     Let us now consider how to calculate in parallel sixteen 2×2 DCTs in which an 8×8 block is subdivided. The procedure is divided in many steps, a global view of which is depicted in  FIG. 3 . This figure highlights the transformations performed on the 2×2 block constituted by the pixels (0,6), (0,7), (1,6), (1,7). 
     The pixels that constitute the input block are ordered in the INPUT phase and are processed in the PROCESS phase to obtain the coefficients of the sixteen bidimensional DCTs, or briefly 2-D DCTs, on four samples, for example, the 2-D DCT of the block (0,1) constituted by:
     {l[0],m[0],n[0],o[0]} is {a[0],b[0],c[0],d[0]}   

     The coefficients of the 2D DCT are re-arranged in the ORDER phase into eight vectors of eight components. For example the coefficients {a[0],b[0],c[0],d[0]} constitute the vector l′. The vectors thus obtained proceed to the OUTPUT phase to give the coefficients of the 2×2 DCT, constituting the output block. 
     INPUT Phase 
     The pixels of each block (i,j), with 0≦i≦1 and 0≦j≦3, are ordered to the eight-component vectors l, m, n, o in the following manner:
         the pixels that occupy the position (0,0) in the block constitute the vector l;   the pixels that occupy the position (0,1) in the block constitute the vector m;   the pixels that occupy the position (1,0) in the block constitute the vector n;   the pixels that occupy the position (1,1) in the block constitute the vector o.       

     Similarly, the pixels of each block (i,j), with 2≦i≦3 and 0≦i≦3, are ordered to constitute the eight-component vectors p, q, r, s in the following manner:
         the pixels that occupy the position (0,0) in the block constitute the vector p;   the pixels that occupy the position (0,1) in the block constitute the vector q;   the pixels that occupy the position (1,0) in the block constitute the vector r;   the pixels that occupy the position (1,0) in the block constitute the vector s.       

     This arrangement is detailed in  FIG. 4 . It should be noted, for example, that the pixels of the block (0,3) will constitute the third component of the l, m, n, o vectors. 
     PROCESS Phase 
     The PROCESS phase includes calculating in parallel the sixteen 2-D DCTs by processing the eight-component vectors l, m, . . . , s as shown in  FIG. 5 . It is noted for example, that the coefficients of the 2-D DCT of the block (0,3) will constitute the third component of the vectors a, b, c, d of  FIG. 3 . 
     ORDER Phase 
     The ORDER phase includes arranging the output sequences of the eight 2-D DCTs in eight vectors l′, m′, . . . , s′ thus defined: 
     
       
         
           
             
               
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                   ] 
                 
                 . 
               
             
           
         
       
     
     It is noted, for example, that the coefficients of the 2-D DCT of the block (0,3) will constitute the components  4 ,  5 ,  6 ,  7  of the vector m′. 
     OUTPUT Phase 
     This phase includes rearranging the output data: starting from the eight-component vectors a, b, . . . , h, a 64 component vector defined as follows, is constructed: 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             y 
                             ⁡ 
                             
                               [ 
                               0 
                               ] 
                             
                           
                         
                         
                           
                             y 
                             ⁡ 
                             
                               [ 
                               1 
                               ] 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             y 
                             ⁡ 
                             
                               [ 
                               7 
                               ] 
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             y 
                             ⁡ 
                             
                               [ 
                               54 
                               ] 
                             
                           
                         
                         
                           
                             y 
                             ⁡ 
                             
                               [ 
                               55 
                               ] 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             y 
                             ⁡ 
                             
                               [ 
                               63 
                               ] 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             l 
                             ⁡ 
                             
                               [ 
                               0 
                               ] 
                             
                           
                         
                         
                           
                             l 
                             ⁡ 
                             
                               [ 
                               1 
                               ] 
                             
                           
                         
                         
                           
                             l 
                             ⁡ 
                             
                               [ 
                               4 
                               ] 
                             
                           
                         
                         
                           
                             l 
                             ⁡ 
                             
                               [ 
                               5 
                               ] 
                             
                           
                         
                         
                           
                             m 
                             ⁡ 
                             
                               [ 
                               0 
                               ] 
                             
                           
                         
                         
                           
                             m 
                             ⁡ 
                             
                               [ 
                               1 
                               ] 
                             
                           
                         
                         
                           
                             m 
                             ⁡ 
                             
                               [ 
                               4 
                               ] 
                             
                           
                         
                         
                           
                             m 
                             ⁡ 
                             
                               [ 
                               5 
                               ] 
                             
                           
                         
                       
                       
                         
                           
                             l 
                             ⁡ 
                             
                               [ 
                               2 
                               ] 
                             
                           
                         
                         
                           
                             l 
                             ⁡ 
                             
                               [ 
                               3 
                               ] 
                             
                           
                         
                         
                           
                             l 
                             ⁡ 
                             
                               [ 
                               6 
                               ] 
                             
                           
                         
                         
                           
                             l 
                             ⁡ 
                             
                               [ 
                               7 
                               ] 
                             
                           
                         
                         
                           
                             m 
                             ⁡ 
                             
                               [ 
                               2 
                               ] 
                             
                           
                         
                         
                           
                             m 
                             ⁡ 
                             
                               [ 
                               3 
                               ] 
                             
                           
                         
                         
                           
                             m 
                             ⁡ 
                             
                               [ 
                               6 
                               ] 
                             
                           
                         
                         
                           
                             m 
                             ⁡ 
                             
                               [ 
                               7 
                               ] 
                             
                           
                         
                       
                       
                         
                           
                             n 
                             ⁡ 
                             
                               [ 
                               0 
                               ] 
                             
                           
                         
                         
                           
                             n 
                             ⁡ 
                             
                               [ 
                               1 
                               ] 
                             
                           
                         
                         
                           
                             n 
                             ⁡ 
                             
                               [ 
                               4 
                               ] 
                             
                           
                         
                         
                           
                             n 
                             ⁡ 
                             
                               [ 
                               5 
                               ] 
                             
                           
                         
                         
                           
                             o 
                             ⁡ 
                             
                               [ 
                               0 
                               ] 
                             
                           
                         
                         
                           
                             o 
                             ⁡ 
                             
                               [ 
                               1 
                               ] 
                             
                           
                         
                         
                           
                             o 
                             ⁡ 
                             
                               [ 
                               4 
                               ] 
                             
                           
                         
                         
                           
                             o 
                             ⁡ 
                             
                               [ 
                               5 
                               ] 
                             
                           
                         
                       
                       
                         
                           
                             n 
                             ⁡ 
                             
                               [ 
                               2 
                               ] 
                             
                           
                         
                         
                           
                             n 
                             ⁡ 
                             
                               [ 
                               3 
                               ] 
                             
                           
                         
                         
                           
                             n 
                             ⁡ 
                             
                               [ 
                               6 
                               ] 
                             
                           
                         
                         
                           
                             n 
                             ⁡ 
                             
                               [ 
                               7 
                               ] 
                             
                           
                         
                         
                           
                             o 
                             ⁡ 
                             
                               [ 
                               2 
                               ] 
                             
                           
                         
                         
                           
                             o 
                             ⁡ 
                             
                               [ 
                               3 
                               ] 
                             
                           
                         
                         
                           
                             o 
                             ⁡ 
                             
                               [ 
                               6 
                               ] 
                             
                           
                         
                         
                           
                             o 
                             ⁡ 
                             
                               [ 
                               7 
                               ] 
                             
                           
                         
                       
                       
                         
                           
                             p 
                             ⁡ 
                             
                               [ 
                               0 
                               ] 
                             
                           
                         
                         
                           
                             p 
                             ⁡ 
                             
                               [ 
                               1 
                               ] 
                             
                           
                         
                         
                           
                             p 
                             ⁡ 
                             
                               [ 
                               4 
                               ] 
                             
                           
                         
                         
                           
                             p 
                             ⁡ 
                             
                               [ 
                               5 
                               ] 
                             
                           
                         
                         
                           
                             q 
                             ⁡ 
                             
                               [ 
                               0 
                               ] 
                             
                           
                         
                         
                           
                             q 
                             ⁡ 
                             
                               [ 
                               1 
                               ] 
                             
                           
                         
                         
                           
                             q 
                             ⁡ 
                             
                               [ 
                               4 
                               ] 
                             
                           
                         
                         
                           
                             q 
                             ⁡ 
                             
                               [ 
                               5 
                               ] 
                             
                           
                         
                       
                       
                         
                           
                             p 
                             ⁡ 
                             
                               [ 
                               2 
                               ] 
                             
                           
                         
                         
                           
                             p 
                             ⁡ 
                             
                               [ 
                               2 
                               ] 
                             
                           
                         
                         
                           
                             p 
                             ⁡ 
                             
                               [ 
                               6 
                               ] 
                             
                           
                         
                         
                           
                             p 
                             ⁡ 
                             
                               [ 
                               7 
                               ] 
                             
                           
                         
                         
                           
                             q 
                             ⁡ 
                             
                               [ 
                               2 
                               ] 
                             
                           
                         
                         
                           
                             q 
                             ⁡ 
                             
                               [ 
                               3 
                               ] 
                             
                           
                         
                         
                           
                             q 
                             ⁡ 
                             
                               [ 
                               6 
                               ] 
                             
                           
                         
                         
                           
                             q 
                             ⁡ 
                             
                               [ 
                               7 
                               ] 
                             
                           
                         
                       
                       
                         
                           
                             r 
                             ⁡ 
                             
                               [ 
                               0 
                               ] 
                             
                           
                         
                         
                           
                             r 
                             ⁡ 
                             
                               [ 
                               1 
                               ] 
                             
                           
                         
                         
                           
                             r 
                             ⁡ 
                             
                               [ 
                               4 
                               ] 
                             
                           
                         
                         
                           
                             r 
                             ⁡ 
                             
                               [ 
                               5 
                               ] 
                             
                           
                         
                         
                           
                             s 
                             ⁡ 
                             
                               [ 
                               0 
                               ] 
                             
                           
                         
                         
                           
                             s 
                             ⁡ 
                             
                               [ 
                               1 
                               ] 
                             
                           
                         
                         
                           
                             s 
                             ⁡ 
                             
                               [ 
                               4 
                               ] 
                             
                           
                         
                         
                           
                             s 
                             ⁡ 
                             
                               [ 
                               5 
                               ] 
                             
                           
                         
                       
                       
                         
                           
                             r 
                             ⁡ 
                             
                               [ 
                               2 
                               ] 
                             
                           
                         
                         
                           
                             r 
                             ⁡ 
                             
                               [ 
                               2 
                               ] 
                             
                           
                         
                         
                           
                             r 
                             ⁡ 
                             
                               [ 
                               6 
                               ] 
                             
                           
                         
                         
                           
                             r 
                             ⁡ 
                             
                               [ 
                               7 
                               ] 
                             
                           
                         
                         
                           
                             s 
                             ⁡ 
                             
                               [ 
                               2 
                               ] 
                             
                           
                         
                         
                           
                             s 
                             ⁡ 
                             
                               [ 
                               3 
                               ] 
                             
                           
                         
                         
                           
                             s 
                             ⁡ 
                             
                               [ 
                               6 
                               ] 
                             
                           
                         
                         
                           
                             s 
                             ⁡ 
                             
                               [ 
                               7 
                               ] 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     Parallel Computation of Four 4×4 DCTs 
     For N=4, equation (5) becomes 
     
       
         
           
             
               
                 
                   
                     y 
                     
                       m 
                       , 
                       n 
                     
                   
                   ⁢ 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         0 
                       
                       3 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           0 
                         
                         3 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           x 
                           
                             i 
                             , 
                             j 
                           
                         
                         ⁢ 
                         
                           cos 
                           ⁡ 
                           
                             [ 
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         2 
                                         ⁢ 
                                         i 
                                       
                                       + 
                                       1 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   m 
                                 
                                 8 
                               
                               ⁢ 
                               π 
                             
                             ] 
                           
                         
                         ⁢ 
                         
                           cos 
                           ⁡ 
                           
                             [ 
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         2 
                                         ⁢ 
                                         j 
                                       
                                       + 
                                       1 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   n 
                                 
                                 8 
                               
                               ⁢ 
                               π 
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     If: 
                 Y       16   *     ⁢   1       =     [           y   0               y   1               y   2               y   3           ]       ,       F       64   *     ⁢   1       =     [           f   0               f   1               f   2               f   3           ]             
where:
     y i =[y i,0 ,y i,1 ,y i,2   ,y   i,3 ]′   {f 0,r } r=0   3 =DCT({A 1,i } i=0   3 ), {f 2,r } r=0   3 =DCT({B 3,i } i=0   3     {f 1,r } r=0   3 =DCT({A 3,i } i=0   3 ), {f 3,r } r=0   3 =DCT({B 1,i } i=0   3 )   {A 1,i } i=0   3 ={x 0,0 ,x 1,1 ,x 2,2 ,x 3,3 },   {A 3,i } i=0   3 ={x 1,0 ,x 3,1 ,x 0,2 ,x 2,3 },   {B 3,i } i=0   3 ={x 2,0 ,x 0,1 ,x 3,2 ,x 1,3 ,
 
it may be demonstrated that:
 
 Y   16*1 =( E   4 ) 16   F   16*1   (9)
 
where:
   
     
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         E 
                         4 
                       
                       ) 
                     
                     16 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             
                               ( 
                               
                                 H 
                                 0 
                               
                               ) 
                             
                             4 
                           
                         
                         
                           
                             
                               ( 
                               
                                 H 
                                 0 
                               
                               ) 
                             
                             4 
                           
                         
                         
                           
                             
                               ( 
                               
                                 H 
                                 0 
                               
                               ) 
                             
                             4 
                           
                         
                         
                           
                             
                               ( 
                               
                                 H 
                                 0 
                               
                               ) 
                             
                             4 
                           
                         
                       
                       
                         
                           
                             
                               ( 
                               
                                 H 
                                 1 
                               
                               ) 
                             
                             4 
                           
                         
                         
                           
                             
                               ( 
                               
                                 H 
                                 3 
                               
                               ) 
                             
                             4 
                           
                         
                         
                           
                             - 
                             
                               
                                 ( 
                                 
                                   H 
                                   3 
                                 
                                 ) 
                               
                               4 
                             
                           
                         
                         
                           
                             - 
                             
                               
                                 ( 
                                 
                                   H 
                                   1 
                                 
                                 ) 
                               
                               4 
                             
                           
                         
                       
                       
                         
                           
                             
                               ( 
                               
                                 H 
                                 2 
                               
                               ) 
                             
                             4 
                           
                         
                         
                           
                             - 
                             
                               
                                 ( 
                                 
                                   H 
                                   2 
                                 
                                 ) 
                               
                               4 
                             
                           
                         
                         
                           
                             - 
                             
                               
                                 ( 
                                 
                                   H 
                                   2 
                                 
                                 ) 
                               
                               4 
                             
                           
                         
                         
                           
                             
                               ( 
                               
                                 H 
                                 2 
                               
                               ) 
                             
                             4 
                           
                         
                       
                       
                         
                           
                             
                               ( 
                               
                                 H 
                                 3 
                               
                               ) 
                             
                             4 
                           
                         
                         
                           
                             - 
                             
                               
                                 ( 
                                 
                                   H 
                                   1 
                                 
                                 ) 
                               
                               4 
                             
                           
                         
                         
                           
                             
                               ( 
                               
                                 H 
                                 1 
                               
                               ) 
                             
                             4 
                           
                         
                         
                           
                             - 
                             
                               
                                 ( 
                                 
                                   H 
                                   3 
                                 
                                 ) 
                               
                               4 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     The matrices (H i ) 4 , i=0, 1, 2, 3 are as follows: 
     
       
         
           
             
               
                 
                   ( 
                   
                     H 
                     0 
                   
                   ) 
                 
                 4 
               
               = 
               
                 [ 
                 
                   
                     
                       1 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       1 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                   
                 
                 ] 
               
             
             , 
             
               
                 
                   ( 
                   
                     H 
                     1 
                   
                   ) 
                 
                 4 
               
               = 
               
                 [ 
                 
                   
                     
                       0 
                     
                     
                       1 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                   
                 
                 ] 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 
                   ( 
                   
                     H 
                     2 
                   
                   ) 
                 
                 4 
               
               = 
               
                 [ 
                 
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                   
                   
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       
                         - 
                         
                           1 
                           2 
                         
                       
                     
                   
                 
                 ] 
               
             
             , 
             
               
                 
                   ( 
                   
                     H 
                     3 
                   
                   ) 
                 
                 4 
               
               = 
               
                 
                   [ 
                   
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           1 
                           2 
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         
                           1 
                           2 
                         
                       
                       
                         0 
                       
                       
                         
                           - 
                           
                             1 
                             2 
                           
                         
                       
                     
                     
                       
                         
                           1 
                           2 
                         
                       
                       
                         0 
                       
                       
                         
                           - 
                           
                             1 
                             2 
                           
                         
                       
                       
                         0 
                       
                     
                   
                   ] 
                 
                 . 
               
             
           
         
       
     
     The monodimensional DCT, or briefly the 1-D DCT, is expressed by the matrix (1-D DCT) 4  given by: 
     
       
         
           
             
               C 
               4 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         1 
                       
                       
                         1 
                       
                       
                         1 
                       
                     
                     
                       
                         
                           C 
                           8 
                           1 
                         
                       
                       
                         
                           C 
                           8 
                           3 
                         
                       
                       
                         
                           - 
                           
                             C 
                             8 
                             3 
                           
                         
                       
                       
                         
                           - 
                           
                             C 
                             8 
                             1 
                           
                         
                       
                     
                     
                       
                         
                           C 
                           4 
                           1 
                         
                       
                       
                         
                           - 
                           
                             C 
                             4 
                             1 
                           
                         
                       
                       
                         
                           - 
                           
                             C 
                             4 
                             1 
                           
                         
                       
                       
                         
                           C 
                           4 
                           1 
                         
                       
                     
                     
                       
                         
                           C 
                           8 
                           3 
                         
                       
                       
                         
                           - 
                           
                             C 
                             8 
                             1 
                           
                         
                       
                       
                         
                           C 
                           8 
                           1 
                         
                       
                       
                         
                           - 
                           
                             C 
                             8 
                             3 
                           
                         
                       
                     
                   
                   ] 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 where 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   C 
                   n 
                   m 
                 
               
               = 
               
                 cos 
                 ⁡ 
                 
                   ( 
                   
                     
                       m 
                       n 
                     
                     ⁢ 
                     π 
                   
                   ) 
                 
               
             
           
         
       
     
     From these equations it may be said that the computation of one 4×4 DCT may be divided into two steps:
         computation of four 1-D DCT, each performed on an appropriate sequence of four pixels.   computation of the 2-D DCT starting from the four 1-D DCT.       

     These two steps are carried out in a similar manner, and are implemented with the same hardware that is used twice. Let us consider now how to calculate in parallel four 4×4 DCTs. The total 64 samples are obtained from the 4×4 blocks into which an 8×8 block is subdivided. The procedure is subdivided in distinct phases to each of which corresponds an architectural block. A whole view is shown in  FIG. 6 . This figure highlights the transformations carried out on each 4×4 block. 
     INPUT Phase 
     The pixel of each quadrant (i,j), 0≦i,j≦1 are ordered to constitute the vectors: 
     
       
         
           
             
               
                 A 
                 1 
                 
                   i 
                   , 
                   j 
                 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         x 
                         
                           0 
                           , 
                           0 
                         
                         
                           i 
                           , 
                           j 
                         
                       
                     
                   
                   
                     
                       
                         x 
                         
                           1 
                           , 
                           1 
                         
                         
                           i 
                           , 
                           j 
                         
                       
                     
                   
                   
                     
                       
                         x 
                         
                           2 
                           , 
                           2 
                         
                         
                           i 
                           , 
                           j 
                         
                       
                     
                   
                   
                     
                       
                         x 
                         
                           3 
                           , 
                           3 
                         
                         
                           i 
                           , 
                           j 
                         
                       
                     
                   
                 
                 ] 
               
             
             , 
             
               
                 A 
                 3 
                 
                   i 
                   , 
                   j 
                 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         x 
                         
                           1 
                           , 
                           0 
                         
                         
                           i 
                           , 
                           j 
                         
                       
                     
                   
                   
                     
                       
                         x 
                         
                           3 
                           , 
                           1 
                         
                         
                           i 
                           , 
                           j 
                         
                       
                     
                   
                   
                     
                       
                         x 
                         
                           0 
                           , 
                           2 
                         
                         
                           i 
                           , 
                           j 
                         
                       
                     
                   
                   
                     
                       
                         x 
                         
                           2 
                           , 
                           3 
                         
                         
                           i 
                           , 
                           j 
                         
                       
                     
                   
                 
                 ] 
               
             
             , 
             
               
                 B 
                 3 
                 
                   i 
                   , 
                   j 
                 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         x 
                         
                           2 
                           , 
                           0 
                         
                         
                           i 
                           , 
                           j 
                         
                       
                     
                   
                   
                     
                       
                         x 
                         
                           0 
                           , 
                           1 
                         
                         
                           i 
                           , 
                           j 
                         
                       
                     
                   
                   
                     
                       
                         x 
                         
                           3 
                           , 
                           2 
                         
                         
                           i 
                           , 
                           j 
                         
                       
                     
                   
                   
                     
                       
                         x 
                         
                           1 
                           , 
                           3 
                         
                         
                           i 
                           , 
                           j 
                         
                       
                     
                   
                 
                 ] 
               
             
             , 
             
               
                 B 
                 1 
                 
                   i 
                   , 
                   j 
                 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         x 
                         
                           3 
                           , 
                           0 
                         
                         
                           i 
                           , 
                           j 
                         
                       
                     
                   
                   
                     
                       
                         x 
                         
                           2 
                           , 
                           1 
                         
                         
                           i 
                           , 
                           j 
                         
                       
                     
                   
                   
                     
                       
                         x 
                         
                           1 
                           , 
                           2 
                         
                         
                           i 
                           , 
                           j 
                         
                       
                     
                   
                   
                     
                       
                         x 
                         
                           0 
                           , 
                           3 
                         
                         
                           i 
                           , 
                           j 
                         
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
     After arranging the data in 16 four-component vectors, we define the eight-component vectors l, m, n, o, constituted by the first, second, third and fourth components, respectively, of the initial vectors constituted by the pixels of the 00 and 01 quadrants, and the p, q, r, s, vectors constituted by the first, second, third and fourth components, respectively, of the initial vectors constituted by the pixels of the 10 and 11 quadrants. Precisely: 
     
       
         
           
             
               l 
               = 
               
                 [ 
                 
                   
                     
                       
                         
                           
                             A 
                             1 
                           
                           ⁡ 
                           
                             [ 
                             0 
                             ] 
                           
                         
                         
                           0 
                           , 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             A 
                             3 
                           
                           ⁡ 
                           
                             [ 
                             0 
                             ] 
                           
                         
                         
                           0 
                           , 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             B 
                             3 
                           
                           ⁡ 
                           
                             [ 
                             0 
                             ] 
                           
                         
                         
                           0 
                           , 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             B 
                             1 
                           
                           ⁡ 
                           
                             [ 
                             0 
                             ] 
                           
                         
                         
                           0 
                           , 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             A 
                             1 
                           
                           ⁡ 
                           
                             [ 
                             0 
                             ] 
                           
                         
                         
                           0 
                           , 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             A 
                             3 
                           
                           ⁡ 
                           
                             [ 
                             0 
                             ] 
                           
                         
                         
                           0 
                           , 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             B 
                             3 
                           
                           ⁡ 
                           
                             [ 
                             0 
                             ] 
                           
                         
                         
                           0 
                           , 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             B 
                             1 
                           
                           ⁡ 
                           
                             [ 
                             0 
                             ] 
                           
                         
                         
                           0 
                           , 
                           1 
                         
                       
                     
                   
                 
                 ] 
               
             
             , 
             
               m 
               = 
               
                 [ 
                 
                   
                     
                       
                         
                           
                             A 
                             1 
                           
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         
                           0 
                           , 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             A 
                             3 
                           
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         
                           0 
                           , 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             B 
                             3 
                           
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         
                           0 
                           , 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             B 
                             1 
                           
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         
                           0 
                           , 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             A 
                             1 
                           
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         
                           0 
                           , 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             A 
                             3 
                           
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         
                           0 
                           , 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             B 
                             3 
                           
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         
                           0 
                           , 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             B 
                             1 
                           
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         
                           0 
                           , 
                           1 
                         
                       
                     
                   
                 
                 ] 
               
             
             , 
             
               n 
               = 
               
                 [ 
                 
                   
                     
                       
                         
                           
                             A 
                             1 
                           
                           ⁡ 
                           
                             [ 
                             2 
                             ] 
                           
                         
                         
                           0 
                           , 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             A 
                             3 
                           
                           ⁡ 
                           
                             [ 
                             2 
                             ] 
                           
                         
                         
                           0 
                           , 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             B 
                             3 
                           
                           ⁡ 
                           
                             [ 
                             2 
                             ] 
                           
                         
                         
                           0 
                           , 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             B 
                             1 
                           
                           ⁡ 
                           
                             [ 
                             2 
                             ] 
                           
                         
                         
                           0 
                           , 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             A 
                             1 
                           
                           ⁡ 
                           
                             [ 
                             2 
                             ] 
                           
                         
                         
                           0 
                           , 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             A 
                             3 
                           
                           ⁡ 
                           
                             [ 
                             2 
                             ] 
                           
                         
                         
                           0 
                           , 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             B 
                             3 
                           
                           ⁡ 
                           
                             [ 
                             2 
                             ] 
                           
                         
                         
                           0 
                           , 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             B 
                             1 
                           
                           ⁡ 
                           
                             [ 
                             2 
                             ] 
                           
                         
                         
                           0 
                           , 
                           1 
                         
                       
                     
                   
                 
                 ] 
               
             
             , 
             
               o 
               = 
               
                 [ 
                 
                   
                     
                       
                         
                           
                             A 
                             1 
                           
                           ⁡ 
                           
                             [ 
                             3 
                             ] 
                           
                         
                         
                           0 
                           , 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             A 
                             3 
                           
                           ⁡ 
                           
                             [ 
                             3 
                             ] 
                           
                         
                         
                           0 
                           , 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             B 
                             3 
                           
                           ⁡ 
                           
                             [ 
                             3 
                             ] 
                           
                         
                         
                           0 
                           , 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             B 
                             1 
                           
                           ⁡ 
                           
                             [ 
                             3 
                             ] 
                           
                         
                         
                           0 
                           , 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             A 
                             1 
                           
                           ⁡ 
                           
                             [ 
                             3 
                             ] 
                           
                         
                         
                           0 
                           , 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             A 
                             3 
                           
                           ⁡ 
                           
                             [ 
                             3 
                             ] 
                           
                         
                         
                           0 
                           , 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             B 
                             3 
                           
                           ⁡ 
                           
                             [ 
                             3 
                             ] 
                           
                         
                         
                           0 
                           , 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             B 
                             1 
                           
                           ⁡ 
                           
                             [ 
                             3 
                             ] 
                           
                         
                         
                           0 
                           , 
                           1 
                         
                       
                     
                   
                 
                 ] 
               
             
           
         
       
       
         
           
             
               p 
               = 
               
                 [ 
                 
                   
                     
                       
                         
                           
                             A 
                             1 
                           
                           ⁡ 
                           
                             [ 
                             0 
                             ] 
                           
                         
                         
                           1 
                           , 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             A 
                             3 
                           
                           ⁡ 
                           
                             [ 
                             0 
                             ] 
                           
                         
                         
                           1 
                           , 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             B 
                             3 
                           
                           ⁡ 
                           
                             [ 
                             0 
                             ] 
                           
                         
                         
                           1 
                           , 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             B 
                             1 
                           
                           ⁡ 
                           
                             [ 
                             0 
                             ] 
                           
                         
                         
                           1 
                           , 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             A 
                             1 
                           
                           ⁡ 
                           
                             [ 
                             0 
                             ] 
                           
                         
                         
                           1 
                           , 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             A 
                             3 
                           
                           ⁡ 
                           
                             [ 
                             0 
                             ] 
                           
                         
                         
                           1 
                           , 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             B 
                             3 
                           
                           ⁡ 
                           
                             [ 
                             0 
                             ] 
                           
                         
                         
                           1 
                           , 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             B 
                             1 
                           
                           ⁡ 
                           
                             [ 
                             0 
                             ] 
                           
                         
                         
                           1 
                           , 
                           1 
                         
                       
                     
                   
                 
                 ] 
               
             
             , 
             
               p 
               = 
               
                 [ 
                 
                   
                     
                       
                         
                           
                             A 
                             1 
                           
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         
                           1 
                           , 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             A 
                             3 
                           
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         
                           1 
                           , 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             B 
                             3 
                           
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         
                           1 
                           , 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             B 
                             1 
                           
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         
                           1 
                           , 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             A 
                             1 
                           
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         
                           1 
                           , 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             A 
                             3 
                           
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         
                           1 
                           , 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             B 
                             3 
                           
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         
                           1 
                           , 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             B 
                             1 
                           
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                         
                           1 
                           , 
                           1 
                         
                       
                     
                   
                 
                 ] 
               
             
             , 
             
               q 
               = 
               
                 [ 
                 
                   
                     
                       
                         
                           
                             A 
                             1 
                           
                           ⁡ 
                           
                             [ 
                             2 
                             ] 
                           
                         
                         
                           1 
                           , 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             A 
                             3 
                           
                           ⁡ 
                           
                             [ 
                             2 
                             ] 
                           
                         
                         
                           1 
                           , 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             B 
                             3 
                           
                           ⁡ 
                           
                             [ 
                             2 
                             ] 
                           
                         
                         
                           1 
                           , 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             B 
                             1 
                           
                           ⁡ 
                           
                             [ 
                             2 
                             ] 
                           
                         
                         
                           1 
                           , 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             A 
                             1 
                           
                           ⁡ 
                           
                             [ 
                             2 
                             ] 
                           
                         
                         
                           1 
                           , 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             A 
                             3 
                           
                           ⁡ 
                           
                             [ 
                             2 
                             ] 
                           
                         
                         
                           1 
                           , 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             B 
                             3 
                           
                           ⁡ 
                           
                             [ 
                             2 
                             ] 
                           
                         
                         
                           1 
                           , 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             B 
                             1 
                           
                           ⁡ 
                           
                             [ 
                             2 
                             ] 
                           
                         
                         
                           1 
                           , 
                           1 
                         
                       
                     
                   
                 
                 ] 
               
             
             , 
             
               s 
               = 
               
                 [ 
                 
                   
                     
                       
                         
                           
                             A 
                             1 
                           
                           ⁡ 
                           
                             [ 
                             3 
                             ] 
                           
                         
                         
                           1 
                           , 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             A 
                             3 
                           
                           ⁡ 
                           
                             [ 
                             3 
                             ] 
                           
                         
                         
                           1 
                           , 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             B 
                             3 
                           
                           ⁡ 
                           
                             [ 
                             3 
                             ] 
                           
                         
                         
                           1 
                           , 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             B 
                             1 
                           
                           ⁡ 
                           
                             [ 
                             3 
                             ] 
                           
                         
                         
                           1 
                           , 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             A 
                             1 
                           
                           ⁡ 
                           
                             [ 
                             3 
                             ] 
                           
                         
                         
                           1 
                           , 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             A 
                             3 
                           
                           ⁡ 
                           
                             [ 
                             3 
                             ] 
                           
                         
                         
                           1 
                           , 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             B 
                             3 
                           
                           ⁡ 
                           
                             [ 
                             3 
                             ] 
                           
                         
                         
                           1 
                           , 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             B 
                             1 
                           
                           ⁡ 
                           
                             [ 
                             3 
                             ] 
                           
                         
                         
                           1 
                           , 
                           1 
                         
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
     By taking into account the way in which the vectors A 1   i,j , A 3   i,j , B 3   i,j , B 1   i,j  are defined, the arrangement detailed in  FIG. 7  is obtained. It should be noted that in this figure the original 8×8 block is subdivided in the four 4×4 quadrants, within each quadrant (i,j), the pixels belonging to the respective vectors A 1   i,j , A 3   i,j , B 3   i,j , B 1   i,j  have different shadings in the figure. 
     According to what has been described above, the computation of a 4×4 DCT may be subdivided in two stages: consequently, the PROCESS phase that is the only phase in which arithmetical operations are performed, is done twice:
         a first time, to compute in parallel the sixteen 1-D DCTs;   a second time, to compute in parallel four 4×4 DCT starting from the coefficients of the 1-D DCTs.       

     The variable stage indicates whether the first or second calculation stage is being performed. 
     During the INPUT phase the variable stage is updated to the value 0. 
     At the input in the PROCESS phase, there are 64 input MUXes that are controlled by the variable stage. Each MUX receives two inputs:
         a pixel of the original picture, coming from the INPUT phase (this input is selected when stage=0);   a coefficient of a 1-D DCT, coming from the ORDER phase (this input is selected when stage=1).       

     PROCESS Phase 
     This phase includes processing the l, m, . . . , s vectors as shown in  FIG. 8 . In this figure the following symbols are used: 
     
       
         
           
             
               
                 
                   A 
                   = 
                   
                     { 
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                               C 
                               8 
                               1 
                             
                             × 
                             
                               I 
                               
                                 8 
                                 × 
                                 8 
                               
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             0 
                           
                         
                       
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         H 
                                         1 
                                       
                                       ) 
                                     
                                     4 
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   
                                     
                                       ( 
                                       
                                         H 
                                         1 
                                       
                                       ) 
                                     
                                     4 
                                   
                                 
                               
                             
                             ] 
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             1 
                           
                         
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
             
               
                 
                   B 
                   = 
                   
                     { 
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                               C 
                               8 
                               3 
                             
                             × 
                             
                               I 
                               
                                 8 
                                 × 
                                 8 
                               
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             0 
                           
                         
                       
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     - 
                                     
                                       
                                         ( 
                                         
                                           H 
                                           3 
                                         
                                         ) 
                                       
                                       4 
                                     
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   
                                     - 
                                     
                                       
                                         ( 
                                         
                                           H 
                                           3 
                                         
                                         ) 
                                       
                                       4 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             1 
                           
                         
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
             
               
                 
                   C 
                   = 
                   
                     { 
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                               C 
                               4 
                               1 
                             
                             × 
                             
                               I 
                               
                                 8 
                                 × 
                                 8 
                               
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             0 
                           
                         
                       
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         H 
                                         2 
                                       
                                       ) 
                                     
                                     4 
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   
                                     
                                       ( 
                                       
                                         H 
                                         2 
                                       
                                       ) 
                                     
                                     4 
                                   
                                 
                               
                             
                             ] 
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             1 
                           
                         
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
             
               
                 
                   t 
                   = 
                   
                     { 
                     
                       
                         
                           1 
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             0 
                           
                         
                       
                       
                         
                           2 
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     At the output of the PROCESS structure there are 64 DEMUXes controlled by the variable stage. The DEMUX address the data according to two conditions:
         if stage=0, the input datum to each DEMUX is a coefficient of a 1-D DCT; therefore the datum must be further processed and, for this purpose, is conveyed to the ORDER phase;   if stage=1, the input datum to each DEMUX is a coefficient of a 2-D DCT; therefore the datum must not be processed further and therefore is conveyed to the OUTPUT phase.       

     ORDER Phase 
     The ORDER phase includes arranging the output sequence of the eight 1-D DCTs in eight l′, m′, . . . , s′ vectors, thus defined: 
     
       
         
           
             
               
                 l 
                 ′ 
               
               = 
               
                 
                   [ 
                   
                     
                       
                         
                           f 
                           0 
                           
                             0 
                             , 
                             0 
                           
                         
                       
                     
                     
                       
                         
                           f 
                           0 
                           
                             0 
                             , 
                             1 
                           
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   [ 
                   
                     
                       
                         
                           a 
                           ⁡ 
                           
                             [ 
                             0 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           b 
                           ⁡ 
                           
                             [ 
                             0 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           c 
                           ⁡ 
                           
                             [ 
                             0 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           d 
                           ⁡ 
                           
                             [ 
                             0 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           a 
                           ⁡ 
                           
                             [ 
                             4 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           b 
                           ⁡ 
                           
                             [ 
                             4 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           c 
                           ⁡ 
                           
                             [ 
                             4 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           d 
                           ⁡ 
                           
                             [ 
                             4 
                             ] 
                           
                         
                       
                     
                   
                   ] 
                 
               
             
             , 
             
               
                 m 
                 ′ 
               
               = 
               
                 
                   [ 
                   
                     
                       
                         
                           f 
                           1 
                           
                             0 
                             , 
                             0 
                           
                         
                       
                     
                     
                       
                         
                           f 
                           1 
                           
                             0 
                             , 
                             1 
                           
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   [ 
                   
                     
                       
                         
                           a 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           b 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           c 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           d 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           a 
                           ⁡ 
                           
                             [ 
                             5 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           b 
                           ⁡ 
                           
                             [ 
                             5 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           c 
                           ⁡ 
                           
                             [ 
                             5 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           d 
                           ⁡ 
                           
                             [ 
                             5 
                             ] 
                           
                         
                       
                     
                   
                   ] 
                 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 n 
                 ′ 
               
               = 
               
                 
                   [ 
                   
                     
                       
                         
                           f 
                           2 
                           
                             0 
                             , 
                             0 
                           
                         
                       
                     
                     
                       
                         
                           f 
                           2 
                           
                             0 
                             , 
                             1 
                           
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   [ 
                   
                     
                       
                         
                           a 
                           ⁡ 
                           
                             [ 
                             2 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           b 
                           ⁡ 
                           
                             [ 
                             2 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           c 
                           ⁡ 
                           
                             [ 
                             2 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           d 
                           ⁡ 
                           
                             [ 
                             2 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           a 
                           ⁡ 
                           
                             [ 
                             6 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           b 
                           ⁡ 
                           
                             [ 
                             6 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           c 
                           ⁡ 
                           
                             [ 
                             6 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           d 
                           ⁡ 
                           
                             [ 
                             6 
                             ] 
                           
                         
                       
                     
                   
                   ] 
                 
               
             
             , 
             
               
                 o 
                 ′ 
               
               = 
               
                 
                   [ 
                   
                     
                       
                         
                           f 
                           3 
                           
                             0 
                             , 
                             0 
                           
                         
                       
                     
                     
                       
                         
                           f 
                           3 
                           
                             0 
                             , 
                             1 
                           
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   [ 
                   
                     
                       
                         
                           a 
                           ⁡ 
                           
                             [ 
                             3 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           b 
                           ⁡ 
                           
                             [ 
                             3 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           c 
                           ⁡ 
                           
                             [ 
                             3 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           d 
                           ⁡ 
                           
                             [ 
                             3 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           a 
                           ⁡ 
                           
                             [ 
                             7 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           b 
                           ⁡ 
                           
                             [ 
                             7 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           c 
                           ⁡ 
                           
                             [ 
                             7 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           d 
                           ⁡ 
                           
                             [ 
                             7 
                             ] 
                           
                         
                       
                     
                   
                   ] 
                 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 p 
                 ′ 
               
               = 
               
                 
                   [ 
                   
                     
                       
                         
                           f 
                           0 
                           
                             1 
                             , 
                             0 
                           
                         
                       
                     
                     
                       
                         
                           f 
                           0 
                           
                             1 
                             , 
                             1 
                           
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   [ 
                   
                     
                       
                         
                           e 
                           ⁡ 
                           
                             [ 
                             0 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           f 
                           ⁡ 
                           
                             [ 
                             0 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           g 
                           ⁡ 
                           
                             [ 
                             0 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           h 
                           ⁡ 
                           
                             [ 
                             0 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           e 
                           ⁡ 
                           
                             [ 
                             4 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           f 
                           ⁡ 
                           
                             [ 
                             4 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           g 
                           ⁡ 
                           
                             [ 
                             4 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           h 
                           ⁡ 
                           
                             [ 
                             4 
                             ] 
                           
                         
                       
                     
                   
                   ] 
                 
               
             
             , 
             
               
                 q 
                 ′ 
               
               = 
               
                 
                   [ 
                   
                     
                       
                         
                           f 
                           1 
                           
                             1 
                             , 
                             0 
                           
                         
                       
                     
                     
                       
                         
                           f 
                           1 
                           
                             1 
                             , 
                             1 
                           
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   [ 
                   
                     
                       
                         
                           e 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           f 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           g 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           h 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           e 
                           ⁡ 
                           
                             [ 
                             5 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           f 
                           ⁡ 
                           
                             [ 
                             5 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           g 
                           ⁡ 
                           
                             [ 
                             5 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           h 
                           ⁡ 
                           
                             [ 
                             5 
                             ] 
                           
                         
                       
                     
                   
                   ] 
                 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 r 
                 ′ 
               
               = 
               
                 
                   [ 
                   
                     
                       
                         
                           f 
                           2 
                           
                             1 
                             , 
                             0 
                           
                         
                       
                     
                     
                       
                         
                           f 
                           2 
                           
                             1 
                             , 
                             1 
                           
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   [ 
                   
                     
                       
                         
                           e 
                           ⁡ 
                           
                             [ 
                             2 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           f 
                           ⁡ 
                           
                             [ 
                             2 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           g 
                           ⁡ 
                           
                             [ 
                             2 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           h 
                           ⁡ 
                           
                             [ 
                             2 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           e 
                           ⁡ 
                           
                             [ 
                             6 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           f 
                           ⁡ 
                           
                             [ 
                             6 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           g 
                           ⁡ 
                           
                             [ 
                             6 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           h 
                           ⁡ 
                           
                             [ 
                             6 
                             ] 
                           
                         
                       
                     
                   
                   ] 
                 
               
             
             , 
             
               
                 s 
                 ′ 
               
               = 
               
                 
                   [ 
                   
                     
                       
                         
                           f 
                           3 
                           
                             1 
                             , 
                             0 
                           
                         
                       
                     
                     
                       
                         
                           f 
                           3 
                           
                             1 
                             , 
                             1 
                           
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   [ 
                   
                     
                       
                         
                           e 
                           ⁡ 
                           
                             [ 
                             3 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           f 
                           ⁡ 
                           
                             [ 
                             3 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           g 
                           ⁡ 
                           
                             [ 
                             3 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           h 
                           ⁡ 
                           
                             [ 
                             3 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           e 
                           ⁡ 
                           
                             [ 
                             7 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           f 
                           ⁡ 
                           
                             [ 
                             7 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           g 
                           ⁡ 
                           
                             [ 
                             7 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           h 
                           ⁡ 
                           
                             [ 
                             7 
                             ] 
                           
                         
                       
                     
                   
                   ] 
                 
               
             
             , 
           
         
       
     
     After the ORDER phase the variable stage is updated to the value 1. The output data from the ORDER phase are sent to the PROCESS phase. 
     OUTPUT Phase 
     This phase includes rearranging the data originating from the second (stage=1) execution of the PROCESS step: starting from these data, which constitute the eight-component vectors: a, b, . . . , h, the output block Y N*N  is thus defined: 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             y 
                             ⁡ 
                             
                               [ 
                               0 
                               ] 
                             
                           
                         
                         
                           
                             y 
                             ⁡ 
                             
                               [ 
                               1 
                               ] 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             y 
                             ⁡ 
                             
                               [ 
                               7 
                               ] 
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             y 
                             ⁡ 
                             
                               [ 
                               54 
                               ] 
                             
                           
                         
                         
                           
                             y 
                             ⁡ 
                             
                               [ 
                               55 
                               ] 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             y 
                             ⁡ 
                             
                               [ 
                               63 
                               ] 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                       
                     
                       [ 
                       
                         
                           
                             
                               a 
                               ⁡ 
                               
                                 [ 
                                 0 
                                 ] 
                               
                             
                           
                           
                             
                               b 
                               ⁡ 
                               
                                 [ 
                                 0 
                                 ] 
                               
                             
                           
                           
                             
                               c 
                               ⁡ 
                               
                                 [ 
                                 0 
                                 ] 
                               
                             
                           
                           
                             
                               d 
                               ⁡ 
                               
                                 [ 
                                 0 
                                 ] 
                               
                             
                           
                           
                             
                               e 
                               ⁡ 
                               
                                 [ 
                                 0 
                                 ] 
                               
                             
                           
                           
                             
                               f 
                               ⁡ 
                               
                                 [ 
                                 0 
                                 ] 
                               
                             
                           
                           
                             
                               g 
                               ⁡ 
                               
                                 [ 
                                 0 
                                 ] 
                               
                             
                           
                           
                             
                               h 
                               ⁡ 
                               
                                 [ 
                                 0 
                                 ] 
                               
                             
                           
                         
                         
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               a 
                               ⁡ 
                               
                                 [ 
                                 3 
                                 ] 
                               
                             
                           
                           
                             
                               b 
                               ⁡ 
                               
                                 [ 
                                 3 
                                 ] 
                               
                             
                           
                           
                             
                               c 
                               ⁡ 
                               
                                 [ 
                                 3 
                                 ] 
                               
                             
                           
                           
                             
                               d 
                               ⁡ 
                               
                                 [ 
                                 3 
                                 ] 
                               
                             
                           
                           
                             
                               e 
                               ⁡ 
                               
                                 [ 
                                 3 
                                 ] 
                               
                             
                           
                           
                             
                               f 
                               ⁡ 
                               
                                 [ 
                                 3 
                                 ] 
                               
                             
                           
                           
                             
                               g 
                               ⁡ 
                               
                                 [ 
                                 3 
                                 ] 
                               
                             
                           
                           
                             
                               h 
                               ⁡ 
                               
                                 [ 
                                 3 
                                 ] 
                               
                             
                           
                         
                         
                           
                             
                               e 
                               ⁡ 
                               
                                 [ 
                                 4 
                                 ] 
                               
                             
                           
                           
                             
                               f 
                               ⁡ 
                               
                                 [ 
                                 4 
                                 ] 
                               
                             
                           
                           
                             
                               g 
                               ⁡ 
                               
                                 [ 
                                 4 
                                 ] 
                               
                             
                           
                           
                             
                               h 
                               ⁡ 
                               
                                 [ 
                                 4 
                                 ] 
                               
                             
                           
                           
                             
                               a 
                               ⁡ 
                               
                                 [ 
                                 4 
                                 ] 
                               
                             
                           
                           
                             
                               b 
                               ⁡ 
                               
                                 [ 
                                 4 
                                 ] 
                               
                             
                           
                           
                             
                               c 
                               ⁡ 
                               
                                 [ 
                                 4 
                                 ] 
                               
                             
                           
                           
                             
                               d 
                               ⁡ 
                               
                                 [ 
                                 4 
                                 ] 
                               
                             
                           
                         
                         
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               e 
                               ⁡ 
                               
                                 [ 
                                 7 
                                 ] 
                               
                             
                           
                           
                             
                               f 
                               ⁡ 
                               
                                 [ 
                                 7 
                                 ] 
                               
                             
                           
                           
                             
                               g 
                               ⁡ 
                               
                                 [ 
                                 7 
                                 ] 
                               
                             
                           
                           
                             
                               h 
                               ⁡ 
                               
                                 [ 
                                 7 
                                 ] 
                               
                             
                           
                           
                             
                               a 
                               ⁡ 
                               
                                 [ 
                                 7 
                                 ] 
                               
                             
                           
                           
                             
                               b 
                               ⁡ 
                               
                                 [ 
                                 7 
                                 ] 
                               
                             
                           
                           
                             
                               c 
                               ⁡ 
                               
                                 [ 
                                 7 
                                 ] 
                               
                             
                           
                           
                             
                               d 
                               ⁡ 
                               
                                 [ 
                                 7 
                                 ] 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     The main differences between the hardware for calculating the four 4×4 DCTs and the hardware needed for the sixteen 2×2 DCTs are the following:
         the ordering sequences of the pixels of the block of the original picture depend on the chosen DCT size;   to execute the sixteen 2×2 DCTs the PROCESS step must be carried out only once; instead, to execute the four 4×4 DCTs the PROCESS step must be repeated two times;   the operations executed during the PROCESS phase are not always the same for the two cases.       

     Computation of an 8×8 DCT 
     For N=8equation (5) becomes: 
     
       
         
           
             
               
                 
                   
                     y 
                     
                       m 
                       , 
                       n 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         0 
                       
                       7 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           0 
                         
                         7 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           x 
                           
                             i 
                             , 
                             j 
                           
                         
                         ⁢ 
                         
                           cos 
                           ⁡ 
                           
                             [ 
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         2 
                                         ⁢ 
                                         i 
                                       
                                       + 
                                       1 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   m 
                                 
                                 16 
                               
                               ⁢ 
                               π 
                             
                             ] 
                           
                         
                         ⁢ 
                         
                           cos 
                           ⁡ 
                           
                             [ 
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         2 
                                         ⁢ 
                                         j 
                                       
                                       + 
                                       1 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   n 
                                 
                                 16 
                               
                               ⁢ 
                               π 
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     Putting: 
                 Y     64   ×   1       =     [           y   0               y   1             ⋮             y   7           ]       ,       F     64   ×   1       =     [           f   0               f   1             ⋮             f   7           ]             
where:
     y i =[y i,0  y i,1  . . . y i,7 ]   {f 0,r } r=0   7 =DCT({A 1,i } i=0   7 ), {f 4,r } r=0   7 =DCT({B 7,i } i=0   7 ),   {f 1,r } r=0   7 =DCT({A 3,i } i=0   7 ), {f 5,r } r=0   7 =DCT({B 5,i } i=0   7 ),   {f 2,r } r=0   7 =DCT({A 5,i } i=0   7 , {f 6,r } r=0   7 =DCT({B 3,i } i=0   7 ),   {f 3,r } r=0   7 =DCT({A 7,i } i=0   7 ), {f 7,r } r=0   7 =DCT({B 1,i } i=0   7 ,   {A 1,i } i=0   7 ={x 0,0  x 1,1  x 2,2  x 3,3  x 4,4  x 5,5  x 6,6  x 7,7 ,   {A 3,i } i=0   7 ={x 1,0  x 4,1  x 7,2  x 5,3  x 2,4  x 0,5  x 3,6  x 6,7 ,   {A 5,i } i=0   7 ={x 2,0  x 7,1  x 3,2  x 1,3  x 6,4  x 4,5  x 0,6  x 5,7 ,   {A 7,i } i=0   7 ={x 3,0  x 5,1  x 1,2  x 7,3  x 0,4  x 6,5  x 2,6  x 4,7 ,   {B 7,i } i=0   7 ={x 4,0  x 2,1  x 6,2  x 0,3  x 7,4  x 1,5  x 5,6  x 3,7 ,   {B 5,i } i=0   7 ={x 5,0  x 0,1  x 4,2  x 6,3  x 1,4  x 3,5  x 7,6  x 2,7 },   {B 3,i } i=0   7 ={x 6,0  x 3,1  x 0,2  x 2,3  x 5,4  x 7,5  x 4,6  x 1,7 },   {B 1,i } i=0   7 ={x 7,0  x 6,1  x 5,2  x 4,3  x 3,4  x 2,5  x 1,6  x 0,7 }
 
it may be demonstrated that:
 
 Y   64×1 =( E   8 ) 64   F   64×1    (17)
 
where:
   
                 (     E   8     )     64     =           ⁢     (   18   )                       [             (     H   0     )     8             (     H   0     )     8             (     H   0     )     8             (     H   0     )     8             (     H   0     )     8             (     H   0     )     8             (     H   0     )     8             (     H   0     )     8                 (     H   1     )     8             (     H   3     )     8             (     H   5     )     8             (     H   7     )     8           -       (     H   7     )     8             -       (     H   5     )     8             -       (     H   3     )     8             -       (     H   1     )     8                   (     H   2     )     8             (     H   6     )     8           -       (     H   6     )     8             -       (     H   2     )     8             -       (     H   2     )     8             -       (     H   6     )     8               (     H   6     )     8             (     H   2     )     8                 (     H   3     )     8           -       (     H   7     )     8             -       (     H   1     )     8             -       (     H   5     )     8               (     H   5     )     8             (     H   1     )     8             (     H   7     )     8           -       (     H   3     )     8                   (     H   4     )     8           -       (     H   4     )     8             -       (     H   4     )     8               (     H   4     )     8             (     H   4     )     8           -       (     H   4     )     8             -       (     H   4     )     8               (     H   4     )     8                 (     H   5     )     8           -       (     H   1     )     8               (     H   7     )     8             (     H   3     )     8           -       (     H   3     )     8             -       (     H   7     )     8               (     H   1     )     8           -       (     H   5     )     8                   (     H   6     )     8           -       (     H   2     )     8               (     H   2     )     8           -       (     H   6     )     8             -       (     H   6     )     8               (     H   2     )     8           -       (     H   2     )     8               (     H   6     )     8                 (     H   7     )     8           -       (     H   5     )     8               (     H   3     )     8           -       (     H   1     )     8               (     H   1     )     8           -       (     H   3     )     8               (     H   5     )     8           -       (     H   7     )     8             ]           
the matrices (Hi)s i=0, 1, . . . , 7 have the f following expressions:
 
     
       
         
           
             
               
                 
                   ( 
                   
                     H 
                     0 
                   
                   ) 
                 
                 8 
               
               = 
               
                 [ 
                 
                   
                     
                       1 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       1 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                   
                 
                 ] 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 
                   ( 
                   
                     H 
                     1 
                   
                   ) 
                 
                 8 
               
               = 
               
                 [ 
                 
                   
                     
                       0 
                     
                     
                       1 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                   
                 
                 ] 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 
                   ( 
                   
                     H 
                     2 
                   
                   ) 
                 
                 8 
               
               = 
               
                 [ 
                 
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       
                         - 
                         
                           1 
                           2 
                         
                       
                     
                   
                 
                 ] 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 
                   ( 
                   
                     H 
                     3 
                   
                   ) 
                 
                 8 
               
               = 
               
                 [ 
                 
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         - 
                         
                           1 
                           2 
                         
                       
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       
                         - 
                         
                           1 
                           2 
                         
                       
                     
                     
                       0 
                     
                   
                 
                 ] 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 
                   ( 
                   
                     H 
                     4 
                   
                   ) 
                 
                 8 
               
               = 
               
                 [ 
                 
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                   
                   
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         - 
                         
                           1 
                           2 
                         
                       
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         - 
                         
                           1 
                           2 
                         
                       
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       
                         - 
                         
                           1 
                           2 
                         
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                 
                 ] 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 
                   ( 
                   
                     H 
                     5 
                   
                   ) 
                 
                 8 
               
               = 
               
                 [ 
                 
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         - 
                         
                           1 
                           2 
                         
                       
                     
                   
                   
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         - 
                         
                           1 
                           2 
                         
                       
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         - 
                         
                           1 
                           2 
                         
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       
                         - 
                         
                           1 
                           2 
                         
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                 
                 ] 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 
                   ( 
                   
                     H 
                     6 
                   
                   ) 
                 
                 8 
               
               = 
               
                 [ 
                 
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         - 
                         
                           1 
                           2 
                         
                       
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         - 
                         
                           1 
                           2 
                         
                       
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         - 
                         
                           1 
                           2 
                         
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         - 
                         
                           1 
                           2 
                         
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       
                         - 
                         
                           1 
                           2 
                         
                       
                     
                     
                       0 
                     
                     
                       
                         - 
                         
                           1 
                           2 
                         
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                 
                 ] 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 
                   ( 
                   
                     H 
                     7 
                   
                   ) 
                 
                 8 
               
               = 
               
                 [ 
                 
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       
                         - 
                         
                           1 
                           2 
                         
                       
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       
                         - 
                         
                           1 
                           2 
                         
                       
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       
                         - 
                         
                           1 
                           2 
                         
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       
                         - 
                         
                           1 
                           2 
                         
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       
                         - 
                         
                           1 
                           2 
                         
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       
                         1 
                         2 
                       
                     
                     
                       0 
                     
                     
                       
                         - 
                         
                           1 
                           2 
                         
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                 
                 ] 
               
             
           
         
       
     
     The 1-D DCT is expressed by the matrix: 
     
       
         
           
             
               
                 ( 
                 
                   1 
                   - 
                   DDCT 
                 
                 ) 
               
               8 
             
             = 
             
               
                 C 
                 8 
               
               = 
               
                 [ 
                 
                   
                     
                       1 
                     
                     
                       1 
                     
                     
                       1 
                     
                     
                       1 
                     
                     
                       1 
                     
                     
                       1 
                     
                     
                       1 
                     
                     
                       1 
                     
                   
                   
                     
                       
                         C 
                         16 
                         1 
                       
                     
                     
                       
                         C 
                         16 
                         3 
                       
                     
                     
                       
                         C 
                         16 
                         5 
                       
                     
                     
                       
                         C 
                         16 
                         7 
                       
                     
                     
                       
                         - 
                         
                           C 
                           16 
                           7 
                         
                       
                     
                     
                       
                         - 
                         
                           C 
                           16 
                           5 
                         
                       
                     
                     
                       
                         - 
                         
                           C 
                           16 
                           3 
                         
                       
                     
                     
                       
                         - 
                         
                           C 
                           16 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         C 
                         8 
                         3 
                       
                     
                     
                       
                         C 
                         8 
                         3 
                       
                     
                     
                       
                         C 
                         8 
                         5 
                       
                     
                     
                       
                         C 
                         8 
                         7 
                       
                     
                     
                       
                         C 
                         8 
                         7 
                       
                     
                     
                       
                         C 
                         8 
                         5 
                       
                     
                     
                       
                         C 
                         8 
                         3 
                       
                     
                     
                       
                         C 
                         8 
                         1 
                       
                     
                   
                   
                     
                       
                         C 
                         16 
                         3 
                       
                     
                     
                       
                         - 
                         
                           C 
                           16 
                           7 
                         
                       
                     
                     
                       
                         - 
                         
                           C 
                           16 
                           1 
                         
                       
                     
                     
                       
                         - 
                         
                           C 
                           16 
                           5 
                         
                       
                     
                     
                       
                         C 
                         16 
                         5 
                       
                     
                     
                       
                         C 
                         16 
                         1 
                       
                     
                     
                       
                         C 
                         16 
                         7 
                       
                     
                     
                       
                         - 
                         
                           C 
                           16 
                           3 
                         
                       
                     
                   
                   
                     
                       
                         C 
                         4 
                         1 
                       
                     
                     
                       
                         - 
                         
                           C 
                           4 
                           1 
                         
                       
                     
                     
                       
                         - 
                         
                           C 
                           4 
                           1 
                         
                       
                     
                     
                       
                         C 
                         4 
                         1 
                       
                     
                     
                       
                         C 
                         4 
                         1 
                       
                     
                     
                       
                         - 
                         
                           C 
                           4 
                           1 
                         
                       
                     
                     
                       
                         - 
                         
                           C 
                           4 
                           1 
                         
                       
                     
                     
                       
                         C 
                         4 
                         1 
                       
                     
                   
                   
                     
                       
                         C 
                         16 
                         5 
                       
                     
                     
                       
                         - 
                         
                           C 
                           16 
                           1 
                         
                       
                     
                     
                       
                         C 
                         16 
                         7 
                       
                     
                     
                       
                         C 
                         16 
                         3 
                       
                     
                     
                       
                         - 
                         
                           C 
                           16 
                           3 
                         
                       
                     
                     
                       
                         - 
                         
                           C 
                           16 
                           7 
                         
                       
                     
                     
                       
                         C 
                         16 
                         1 
                       
                     
                     
                       
                         - 
                         
                           C 
                           16 
                           5 
                         
                       
                     
                   
                   
                     
                       
                         C 
                         8 
                         3 
                       
                     
                     
                       
                         - 
                         
                           C 
                           8 
                           1 
                         
                       
                     
                     
                       
                         C 
                         8 
                         1 
                       
                     
                     
                       
                         C 
                         8 
                         5 
                       
                     
                     
                       
                         C 
                         8 
                         5 
                       
                     
                     
                       
                         C 
                         8 
                         1 
                       
                     
                     
                       
                         C 
                         8 
                         1 
                       
                     
                     
                       
                         C 
                         8 
                         3 
                       
                     
                   
                   
                     
                       
                         C 
                         16 
                         7 
                       
                     
                     
                       
                         - 
                         
                           C 
                           16 
                           5 
                         
                       
                     
                     
                       
                         C 
                         16 
                         3 
                       
                     
                     
                       
                         - 
                         
                           C 
                           16 
                           1 
                         
                       
                     
                     
                       
                         C 
                         16 
                         1 
                       
                     
                     
                       
                         - 
                         
                           C 
                           16 
                           3 
                         
                       
                     
                     
                       
                         - 
                         
                           C 
                           16 
                           5 
                         
                       
                     
                     
                       
                         - 
                         
                           C 
                           16 
                           7 
                         
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
     Where we put: 
     
       
         
           
             
               C 
               n 
               m 
             
             = 
             
               cos 
               ⁡ 
               
                 ( 
                 
                   
                     m 
                     n 
                   
                   ⁢ 
                   π 
                 
                 ) 
               
             
           
         
       
     
     From the above equations it is evident that the computation of an 8×8 DCT may be subdivided in two stages:
         calculating eight 1-D DCTs, each for a certain sequence of eight pixels;   calculating the 2-D DCT, starting from the eight 1-D DCTs.       

     These two stages may be executed through the same hardware using it twice. The processing is subdivided in different steps, to each of which corresponds an architectural block. A whole view of the hardware is shown in  FIG. 9 . 
     INPUT Phase 
     The pixels of the 8×8 input block are ordered to constitute the eight-component vectors l, m, n, o, p, q, r, s: 
     
       
         
           
             
               l 
               = 
               
                 [ 
                 
                   
                     
                       
                         
                           A 
                           1 
                         
                         ⁡ 
                         
                           [ 
                           0 
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           A 
                           3 
                         
                         ⁡ 
                         
                           [ 
                           0 
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           A 
                           5 
                         
                         ⁡ 
                         
                           [ 
                           0 
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           A 
                           7 
                         
                         ⁡ 
                         
                           [ 
                           0 
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           B 
                           7 
                         
                         ⁡ 
                         
                           [ 
                           0 
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           B 
                           5 
                         
                         ⁡ 
                         
                           [ 
                           0 
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           B 
                           3 
                         
                         ⁡ 
                         
                           [ 
                           0 
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           B 
                           1 
                         
                         ⁡ 
                         
                           [ 
                           0 
                           ] 
                         
                       
                     
                   
                 
                 ] 
               
             
             , 
             
               m 
               = 
               
                 [ 
                 
                   
                     
                       
                         
                           A 
                           1 
                         
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           A 
                           3 
                         
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           A 
                           5 
                         
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           A 
                           7 
                         
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           B 
                           7 
                         
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           B 
                           5 
                         
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           B 
                           3 
                         
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           B 
                           1 
                         
                         ⁡ 
                         
                           [ 
                           1 
                           ] 
                         
                       
                     
                   
                 
                 ] 
               
             
             , 
             … 
             ⁢ 
             
                 
             
             , 
             
               s 
               = 
               
                 [ 
                 
                   
                     
                       
                         
                           A 
                           1 
                         
                         ⁡ 
                         
                           [ 
                           7 
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           A 
                           3 
                         
                         ⁡ 
                         
                           [ 
                           7 
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           A 
                           5 
                         
                         ⁡ 
                         
                           [ 
                           7 
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           A 
                           7 
                         
                         ⁡ 
                         
                           [ 
                           7 
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           B 
                           7 
                         
                         ⁡ 
                         
                           [ 
                           7 
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           B 
                           5 
                         
                         ⁡ 
                         
                           [ 
                           7 
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           B 
                           3 
                         
                         ⁡ 
                         
                           [ 
                           7 
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           B 
                           1 
                         
                         ⁡ 
                         
                           [ 
                           7 
                           ] 
                         
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
     By taking into account the way in which the vectors A 1 , A 3 , A 5 , A 7 , B 7 , B 5 , B 3 , B 1  are defined, we obtain the detailed arrangement of  FIG. 10 . It should be noticed that in this figure the pixels belonging to the vectors A 1 , A 3 , A 5 , A 7 , B 7 , B 5 , B 3 , B 1  are countersigned by different shadings. 
     As shown above, the computation of an 8×8 DCT may be subdivided into two stage. The PROCESS step, which is the only phase in which mathematical operations are performed, is performed twice:
         the first time, to compute in parallel sixteen 1-D DCTs;   the second time, to compute the 8×8 DCT starting from the coefficients of the sixteen 1-D DCTs.       

     The variable stage indicates whether the first or second calculation step is being performed. During the INPUT phase, the variable stage is updated to the value 0. 
     At the input of the PROCESS structure, there are 64 MUXes controlled by the variable stage. Each MUX receives two inputs:
         a pixel of the original picture, originating from the INPUT phase (this input is selected when stage=0);   a coefficient of a 1-D DCT, originating from the ORDER phase (this input is selected when stage=1).       

     PROCESS Phase 
     This phase includes processing the l, m, . . . , s vectors as shown in  FIG. 11 . In this figure, the following symbols are used: 
     
       
         
           
             
               
                 
                   A 
                   = 
                   
                     { 
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                               C 
                               8 
                               1 
                             
                             × 
                             
                               I 
                               
                                 8 
                                 × 
                                 8 
                               
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             0 
                           
                         
                       
                       
                         
                           
                             
                               ( 
                               
                                 H 
                                 2 
                               
                               ) 
                             
                             8 
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
             
               
                 
                   B 
                   = 
                   
                     { 
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                               C 
                               8 
                               3 
                             
                             × 
                             
                               I 
                               
                                 8 
                                 × 
                                 8 
                               
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             0 
                           
                         
                       
                       
                         
                           
                             - 
                             
                               
                                 ( 
                                 
                                   H 
                                   6 
                                 
                                 ) 
                               
                               8 
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
             
               
                 
                   C 
                   = 
                   
                     { 
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                               C 
                               4 
                               1 
                             
                             × 
                             
                               I 
                               
                                 8 
                                 × 
                                 8 
                               
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             0 
                           
                         
                       
                       
                         
                           
                             
                               ( 
                               
                                 H 
                                 4 
                               
                               ) 
                             
                             8 
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
             
               
                 
                   t 
                   = 
                   
                     { 
                     
                       
                         
                           1 
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             0 
                           
                         
                       
                       
                         
                           2 
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
     At the output of the PROCESS structure there are 64 DEMUXes controlled by the variable stage. The DEMUXes address the data according to two possibilities:
         if stage=0, the input datum to each DEMUX is a coefficient of a 1-D DCT; therefore, the datum must be further processed and, for this purpose, is sent to the ORDER phase;   if stage=1, the input datum to each DEMUX is a coefficient of a 2-D DCT; therefore, the datum does not need any further processing and therefore is sent to the OUTPUT phase.       

     ORDER Phase 
     This phase includes arranging the output sequence of the eight 1-D DCTs in eight l′, m′, . . . , s′ vectors, thus defined: 
     
       
         
           
             
               
                 l 
                 ′ 
               
               = 
               
                 
                   f 
                   0 
                 
                 = 
                 
                   [ 
                   
                     
                       
                         
                           a 
                           ⁡ 
                           
                             [ 
                             0 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           b 
                           ⁡ 
                           
                             [ 
                             0 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           c 
                           ⁡ 
                           
                             [ 
                             0 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           d 
                           ⁡ 
                           
                             [ 
                             0 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           e 
                           ⁡ 
                           
                             [ 
                             0 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           f 
                           ⁡ 
                           
                             [ 
                             0 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           g 
                           ⁡ 
                           
                             [ 
                             0 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           h 
                           ⁡ 
                           
                             [ 
                             0 
                             ] 
                           
                         
                       
                     
                   
                   ] 
                 
               
             
             , 
             
               
                 m 
                 ′ 
               
               = 
               
                 
                   f 
                   1 
                 
                 = 
                 
                   [ 
                   
                     
                       
                         
                           a 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           b 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           c 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           d 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           e 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           f 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           g 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           h 
                           ⁡ 
                           
                             [ 
                             1 
                             ] 
                           
                         
                       
                     
                   
                   ] 
                 
               
             
             , 
             … 
             ⁢ 
             
                 
             
             , 
             
               
                 q 
                 ′ 
               
               = 
               
                 
                   f 
                   7 
                 
                 = 
                 
                   [ 
                   
                     
                       
                         
                           a 
                           ⁡ 
                           
                             [ 
                             7 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           b 
                           ⁡ 
                           
                             [ 
                             7 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           c 
                           ⁡ 
                           
                             [ 
                             7 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           d 
                           ⁡ 
                           
                             [ 
                             7 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           e 
                           ⁡ 
                           
                             [ 
                             7 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           f 
                           ⁡ 
                           
                             [ 
                             7 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           g 
                           ⁡ 
                           
                             [ 
                             7 
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           h 
                           ⁡ 
                           
                             [ 
                             7 
                             ] 
                           
                         
                       
                     
                   
                   ] 
                 
               
             
           
         
       
     
     Following the ORDER phase the variable stage is updated to the value 1. The output data from the ORDER phase are sent to the PROCESS phase. 
     OUTPUT Phase 
     This phase includes rearranging the data originating from the second execution of the PROCESS step (that is, with stage=1): starting from these data, which constitute the eight-component vectors a, b, . . . , h, the output block Y N*N  defined as follows is constituted: 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             y 
                             ⁡ 
                             
                               [ 
                               0 
                               ] 
                             
                           
                         
                         
                           
                             y 
                             ⁡ 
                             
                               [ 
                               1 
                               ] 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             y 
                             ⁡ 
                             
                               [ 
                               7 
                               ] 
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             y 
                             ⁡ 
                             
                               [ 
                               54 
                               ] 
                             
                           
                         
                         
                           
                             y 
                             ⁡ 
                             
                               [ 
                               55 
                               ] 
                             
                           
                         
                         
                           ⋮ 
                         
                         
                           
                             y 
                             ⁡ 
                             
                               [ 
                               63 
                               ] 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             a 
                             ⁡ 
                             
                               [ 
                               0 
                               ] 
                             
                           
                         
                         
                           
                             b 
                             ⁡ 
                             
                               [ 
                               0 
                               ] 
                             
                           
                         
                         
                           
                             c 
                             ⁡ 
                             
                               [ 
                               0 
                               ] 
                             
                           
                         
                         
                           
                             d 
                             ⁡ 
                             
                               [ 
                               0 
                               ] 
                             
                           
                         
                         
                           
                             e 
                             ⁡ 
                             
                               [ 
                               0 
                               ] 
                             
                           
                         
                         
                           
                             f 
                             ⁡ 
                             
                               [ 
                               0 
                               ] 
                             
                           
                         
                         
                           
                             g 
                             ⁡ 
                             
                               [ 
                               0 
                               ] 
                             
                           
                         
                         
                           
                             h 
                             ⁡ 
                             
                               [ 
                               0 
                               ] 
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             a 
                             ⁡ 
                             
                               [ 
                               7 
                               ] 
                             
                           
                         
                         
                           
                             b 
                             ⁡ 
                             
                               [ 
                               7 
                               ] 
                             
                           
                         
                         
                           
                             c 
                             ⁡ 
                             
                               [ 
                               7 
                               ] 
                             
                           
                         
                         
                           
                             d 
                             ⁡ 
                             
                               [ 
                               7 
                               ] 
                             
                           
                         
                         
                           
                             e 
                             ⁡ 
                             
                               [ 
                               7 
                               ] 
                             
                           
                         
                         
                           
                             f 
                             ⁡ 
                             
                               [ 
                               7 
                               ] 
                             
                           
                         
                         
                           
                             g 
                             ⁡ 
                             
                               [ 
                               7 
                               ] 
                             
                           
                         
                         
                           
                             h 
                             ⁡ 
                             
                               [ 
                               7 
                               ] 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
     The main differences between the hardware that calculates an 8×8 DCT and the hardware that calculates the four 4×4 DCTs are:
         the sequences into which must be arranged the pixels of a block of the original picture depend on the chosen size of the DCT;   the operations executed during the PROCESS step are not always the same for the two cases.       

     Procedure for Calculating the DCT for Blocks of Scaleable Size (8×8 DCTs, 4×4 DCTs and 2×2 DCTs) 
     From the above described procedures, an algorithm for calculating a chosen one of 8×8 DCT of four 4×4 DCTs (in parallel) or sixteen 2×2 DCTs (in parallel) may be derived. The selection is made by the user by assigning a certain value to the global variable size: 
     
       
         
           
             size 
             = 
             
               { 
               
                 
                   
                     
                       0 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       for 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       an 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       8 
                       × 
                       8 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       DCT 
                     
                   
                 
                 
                   
                     
                       1 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       for 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       four 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       DCTs 
                     
                   
                 
                 
                   
                     
                       2 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       for 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       sixteen 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       2 
                       × 
                       2 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       DCTs 
                     
                   
                 
               
             
           
         
       
     
     The procedure is subdivided in various phases (regardless of the value of the variable size), to each of which corresponds an architectural block. A whole view is shown in  FIG. 12 . Each phase has been organized in order to provide for partial results corresponding to the chosen value, minimizing redundancies. Sometimes the operations performed are different depending on the value of size. In these cases, the architecture considers a MUX whose control input is size. Let us examine now the various phases and highlight the differences in respect to the architectures that have already been described above: 
     INPUT Phase 
     The object of this phase, depicted in  FIG. 13 , is to arrange the data to allow the computation starting from the arranged data of the 1-D DCTs. This is done by inputting the luminance values of the pixels (8×8 matrix) and arranging them in eight-component vectors l, m, . . . , s. 
     For example: 
     
       
         
           
             
               1 
               ⁡ 
               
                 [ 
                 0 
                 ] 
               
             
             = 
             
               x 
               
                 0 
                 , 
                 0 
               
             
           
         
       
       
         
           
             
               1 
               ⁡ 
               
                 [ 
                 1 
                 ] 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         
                           x 
                           
                             1 
                             , 
                             2 
                           
                         
                       
                       
                         
                           
                             per 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             size 
                           
                           = 
                           
                             0 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             or 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                         
                       
                     
                     
                       
                         
                           x 
                           
                             2 
                             , 
                             0 
                           
                         
                       
                       
                         
                           
                             per 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             size 
                           
                           = 
                           2 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     s 
                     ⁡ 
                     
                       [ 
                       7 
                       ] 
                     
                   
                 
                 = 
                 
                   { 
                   
                     
                       
                         
                           x 
                           
                             0 
                             , 
                             7 
                           
                         
                       
                       
                         
                           
                             per 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             size 
                           
                           = 
                           0 
                         
                       
                     
                     
                       
                         
                           x 
                           
                             4 
                             , 
                             7 
                           
                         
                       
                       
                         
                           
                             per 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             size 
                           
                           = 
                           1 
                         
                       
                     
                     
                       
                         
                           x 
                           
                             7 
                             , 
                             7 
                           
                         
                       
                       
                         
                           
                             per 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             size 
                           
                           = 
                           2 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     PROCESS Phase with Stage=0 
     This phase includes calculating in parallel the eight 1-D DCTs by processing the vectors l, m, . . . , s as shown in  FIG. 14 . In this figure may be observed the use of 16 MUXes controlled by the variable size. The eight MUXes on the left serve to bypass the operations required for the computation of the 8×8 DCT. Thus, the bypass occurs when size=1 or 2, while it does not occur for size=0. The eight MUXes on the right serve to output only the result that corresponds to the pre-selected value of size. 
     
       
         
           
             
               
                 
                   t 
                   = 
                   
                     { 
                     
                       
                         
                           1 
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             0 
                           
                         
                       
                       
                         
                           2 
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
             
               
                 
                   A 
                   = 
                   
                     { 
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                               C 
                               8 
                               1 
                             
                             × 
                             
                               I 
                               
                                 8 
                                 × 
                                 8 
                               
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             , 
                             
                               size 
                               = 
                               
                                 
                                   ( 
                                   
                                     0 
                                     , 
                                     0 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 or 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     0 
                                     , 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             I 
                             
                               8 
                               × 
                               8 
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             , 
                             
                               size 
                               = 
                               
                                 
                                   ( 
                                   
                                     0 
                                     , 
                                     2 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 or 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     1 
                                     , 
                                     2 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               ( 
                               
                                 H 
                                 2 
                               
                               ) 
                             
                             8 
                           
                         
                         
                           
                             
                               
                                 per 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 stage 
                               
                               , 
                               
                                 size 
                                 = 
                                 
                                   ( 
                                   
                                     1 
                                     , 
                                     0 
                                   
                                   ) 
                                 
                               
                             
                             ⁢ 
                             
                                 
                             
                           
                         
                       
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         H 
                                         1 
                                       
                                       ) 
                                     
                                     4 
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   
                                     
                                       ( 
                                       
                                         H 
                                         1 
                                       
                                       ) 
                                     
                                     4 
                                   
                                 
                               
                             
                             ] 
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             , 
                             
                               size 
                               = 
                               
                                 ( 
                                 
                                   1 
                                   , 
                                   1 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
             
               
                 
                   B 
                   = 
                   
                     { 
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                               C 
                               8 
                               3 
                             
                             × 
                             
                               I 
                               
                                 8 
                                 × 
                                 8 
                               
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             , 
                             
                               size 
                               = 
                               
                                 
                                   ( 
                                   
                                     0 
                                     , 
                                     0 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 or 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     0 
                                     , 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             I 
                             
                               8 
                               × 
                               8 
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             , 
                             
                               size 
                               = 
                               
                                 
                                   ( 
                                   
                                     0 
                                     , 
                                     2 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 or 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     1 
                                     , 
                                     2 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             - 
                             
                               
                                 ( 
                                 
                                   H 
                                   6 
                                 
                                 ) 
                               
                               8 
                             
                           
                         
                         
                           
                             
                               
                                 per 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 stage 
                               
                               , 
                               
                                 size 
                                 = 
                                 
                                   ( 
                                   
                                     1 
                                     , 
                                     0 
                                   
                                   ) 
                                 
                               
                             
                             ⁢ 
                             
                                 
                             
                           
                         
                       
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     - 
                                     
                                       
                                         ( 
                                         
                                           H 
                                           3 
                                         
                                         ) 
                                       
                                       4 
                                     
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   
                                     - 
                                     
                                       
                                         ( 
                                         
                                           H 
                                           3 
                                         
                                         ) 
                                       
                                       4 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             , 
                             
                               size 
                               = 
                               
                                 ( 
                                 
                                   1 
                                   , 
                                   1 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
             
               
                 
                   C 
                   = 
                   
                     { 
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                               C 
                               4 
                               1 
                             
                             × 
                             
                               I 
                               
                                 8 
                                 × 
                                 8 
                               
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             , 
                             
                               size 
                               = 
                               
                                 
                                   ( 
                                   
                                     0 
                                     , 
                                     0 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 or 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     0 
                                     , 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             I 
                             
                               8 
                               × 
                               8 
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             , 
                             
                               size 
                               = 
                               
                                 
                                   ( 
                                   
                                     0 
                                     , 
                                     2 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 or 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     1 
                                     , 
                                     2 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               ( 
                               
                                 H 
                                 4 
                               
                               ) 
                             
                             8 
                           
                         
                         
                           
                             
                               
                                 per 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 stage 
                               
                               , 
                               
                                 size 
                                 = 
                                 
                                   ( 
                                   
                                     1 
                                     , 
                                     0 
                                   
                                   ) 
                                 
                               
                             
                             ⁢ 
                             
                                 
                             
                           
                         
                       
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         H 
                                         2 
                                       
                                       ) 
                                     
                                     4 
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   
                                     
                                       ( 
                                       
                                         H 
                                         2 
                                       
                                       ) 
                                     
                                     4 
                                   
                                 
                               
                             
                             ] 
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             , 
                             
                               size 
                               = 
                               
                                 ( 
                                 
                                   1 
                                   , 
                                   1 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
             
               
                 
                   D 
                   = 
                   
                     { 
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                               C 
                               16 
                               1 
                             
                             × 
                             
                               I 
                               
                                 8 
                                 × 
                                 8 
                               
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             0 
                           
                         
                       
                       
                         
                           
                             
                               ( 
                               
                                 H 
                                 1 
                               
                               ) 
                             
                             8 
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
             
               
                 
                   E 
                   = 
                   
                     { 
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                               C 
                               8 
                               3 
                             
                             × 
                             
                               I 
                               
                                 8 
                                 × 
                                 8 
                               
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             0 
                           
                         
                       
                       
                         
                           
                             
                               ( 
                               
                                 H 
                                 5 
                               
                               ) 
                             
                             8 
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
             
               
                 
                   F 
                   = 
                   
                     { 
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                               C 
                               16 
                               7 
                             
                             × 
                             
                               I 
                               
                                 8 
                                 × 
                                 8 
                               
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             0 
                           
                         
                       
                       
                         
                           
                             - 
                             
                               
                                 ( 
                                 
                                   H 
                                   7 
                                 
                                 ) 
                               
                               8 
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
             
               
                 
                   G 
                   = 
                   
                     { 
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                               C 
                               16 
                               5 
                             
                             × 
                             
                               I 
                               
                                 8 
                                 × 
                                 8 
                               
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             0 
                           
                         
                       
                       
                         
                           
                             - 
                             
                               
                                 ( 
                                 
                                   H 
                                   3 
                                 
                                 ) 
                               
                               8 
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
           
         
       
     
     The scheme in  FIG. 14  may be subdivided into the architectural blocks shown in  FIG. 15 . For example, two vectors each of eight components (each component being a pixel, that may have been processed already) are input to the QA block, which outputs two vectors of eight components: the first vector is the sum of the two input vectors, while the second vector is the difference between the two input vectors that is successively processed with the linear operator A. It should be noted that the operators A, B, C, D, E, F, G are 8×8 matrices. 
     By considering a lower level of generalization, the QA, QB, QC blocks are shown in detail in  FIGS. 16 ,  17  and  18 , respectively. In these figures the MUXes are controlled by three bits, which correspond to the variable stage (which may take the value 0 or 1, and thus is represented by a bit) and the variable size (which may take the value 0, 1 or 2, and thus is represented by two bits). The blocks QD, QE, QF, QG are shown in detail in  FIGS. 19 ,  20 ,  21  and  22 , respectively. In these figures the MUXes are controlled by a bit that corresponds to the variable stage. 
     ORDER Phase 
     The ORDER phase, depicted in  FIG. 23 , includes arranging the output sequences of the eight 1-D DCTs in eight vectors l′, m′, . . . , s′. For example: 
     
       
         
           
             
               
                 
                   
                     
                       
                         l 
                         ′ 
                       
                       ⁡ 
                       
                         [ 
                         0 
                         ] 
                       
                     
                     = 
                     
                       a 
                       ⁡ 
                       
                         [ 
                         0 
                         ] 
                       
                     
                   
                   ; 
                 
               
             
             
               
                 
                   
                     
                       
                         l 
                         ′ 
                       
                       ⁡ 
                       
                         [ 
                         1 
                         ] 
                       
                     
                     = 
                     
                       b 
                       ⁡ 
                       
                         [ 
                         0 
                         ] 
                       
                     
                   
                   ; 
                 
               
             
             
               
                 ⋮ 
               
             
             
               
                 
                   
                     
                       l 
                       ′ 
                     
                     ⁡ 
                     
                       [ 
                       4 
                       ] 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             e 
                             ⁡ 
                             
                               [ 
                               0 
                               ] 
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               size 
                             
                             = 
                             0 
                           
                         
                       
                       
                         
                           
                             a 
                             ⁡ 
                             
                               [ 
                               4 
                               ] 
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               size 
                             
                             = 
                             1 
                           
                         
                       
                       
                         
                           
                             a 
                             ⁡ 
                             
                               [ 
                               1 
                               ] 
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               size 
                             
                             = 
                             2 
                           
                         
                       
                     
                   
                 
               
             
             
               
                 ⋮ 
               
             
             
               
                 
                   
                     
                       
                         s 
                         ′ 
                       
                       ⁡ 
                       
                         [ 
                         7 
                         ] 
                       
                     
                     = 
                     
                       h 
                       ⁡ 
                       
                         [ 
                         7 
                         ] 
                       
                     
                   
                   ; 
                 
               
             
           
         
       
     
     OUTPUT Phase 
     This phase, depicted in  FIG. 24 , includes rearranging the data coming from the second (that is with stage=1) execution of the PROCESS step. Starting from these data, constituting the eight-component vectors a, b, . . . , h, the output block yN*N is constituted. 
     For example: 
     
       
         
           
             
               
                 
                   
                     y 
                     ⁡ 
                     
                       [ 
                       0 
                       ] 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             a 
                             ⁡ 
                             
                               [ 
                               0 
                               ] 
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               size 
                             
                             = 
                             
                               0 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               or 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                           
                         
                       
                       
                         
                           
                             l 
                             ⁡ 
                             
                               [ 
                               2 
                               ] 
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               size 
                             
                             = 
                             2 
                           
                         
                       
                     
                     } 
                   
                 
               
             
             
               
                 
                   
                     y 
                     ⁡ 
                     
                       [ 
                       1 
                       ] 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             b 
                             ⁡ 
                             
                               [ 
                               0 
                               ] 
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               size 
                             
                             = 
                             
                               0 
                               ⁢ 
                               o1 
                             
                           
                         
                       
                       
                         
                           
                             l 
                             ⁡ 
                             
                               [ 
                               1 
                               ] 
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               size 
                             
                             = 
                             2 
                           
                         
                       
                     
                     } 
                   
                 
               
             
             
               
                 ⋮ 
               
             
             
               
                 
                   
                     y 
                     ⁡ 
                     
                       [ 
                       63 
                       ] 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             h 
                             ⁡ 
                             
                               [ 
                               7 
                               ] 
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               size 
                             
                             = 
                             0 
                           
                         
                       
                       
                         
                           
                             d 
                             ⁡ 
                             
                               [ 
                               7 
                               ] 
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               size 
                             
                             = 
                             1 
                           
                         
                       
                       
                         
                           
                             s 
                             ⁡ 
                             
                               [ 
                               7 
                               ] 
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               size 
                             
                             = 
                             2 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
     Description of the Drawings 
     A functional block diagram of a picture compressor-coder according to the present invention may be represented as shown in  FIG. 1 . 
     Essentially, the compressor-coder performs a hybrid compression based on a fractal coding in the DCT domain. This is made possible by the peculiar architecture of parallel calculation of the DCT on blocks of scaleable size of pixels, as described above. 
     Hereinbelow, the remaining figures are described one by one: 
       FIG. 2  is a flow graph of the 2×2 DCT generating block. 
     This block is the “base” block that is repeatedly used in the PROCESS phase of all the N×N DCTs, where N is a power of 2. 
     In particular:
         the flow graph for a 2×2 DCT is shown in  FIG. 2 , wherein A=B=C=1 and the input and output data are pixels in the positions (0,0), (0,1), (1,0), (1,1);   for sixteen 2×2 DCTs, the inputs and the outputs are eight-component vectors and the following symbols are used, considering A=B=C=I 8×8 ;   for four 4×4 DCTs the inputs and outputs are eight-component vectors and the following symbols are used:       

     
       
         
           
             
               
                 
                   A 
                   = 
                   
                     { 
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                               C 
                               8 
                               1 
                             
                             × 
                             
                               I 
                               
                                 8 
                                 × 
                                 8 
                               
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             0 
                           
                         
                       
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         H 
                                         1 
                                       
                                       ) 
                                     
                                     4 
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   
                                     
                                       ( 
                                       
                                         H 
                                         1 
                                       
                                       ) 
                                     
                                     4 
                                   
                                 
                               
                             
                             ] 
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
             
               
                 
                   B 
                   = 
                   
                     { 
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                               C 
                               8 
                               3 
                             
                             × 
                             
                               I 
                               
                                 8 
                                 × 
                                 8 
                               
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             0 
                           
                         
                       
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     - 
                                     
                                       
                                         ( 
                                         
                                           H 
                                           3 
                                         
                                         ) 
                                       
                                       4 
                                     
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   
                                     - 
                                     
                                       
                                         ( 
                                         
                                           H 
                                           3 
                                         
                                         ) 
                                       
                                       4 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
             
               
                 
                   C 
                   = 
                   
                     { 
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                               C 
                               4 
                               1 
                             
                             × 
                             
                               I 
                               
                                 8 
                                 × 
                                 8 
                               
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             0 
                           
                         
                       
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         H 
                                         2 
                                       
                                       ) 
                                     
                                     4 
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   
                                     
                                       ( 
                                       
                                         H 
                                         2 
                                       
                                       ) 
                                     
                                     4 
                                   
                                 
                               
                             
                             ] 
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
           
         
       
         
         
           
             for an 8×8 DCT, the inputs and outputs are eight-component vectors and the flowing symbols are used: 
           
         
       
    
     
       
         
           
             
               
                 
                   A 
                   = 
                   
                     { 
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                               C 
                               8 
                               1 
                             
                             × 
                             
                               I 
                               
                                 8 
                                 × 
                                 8 
                               
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             0 
                           
                         
                       
                       
                         
                           
                             
                               ( 
                               
                                 H 
                                 2 
                               
                               ) 
                             
                             8 
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
             
               
                 
                   B 
                   = 
                   
                     { 
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                               C 
                               8 
                               3 
                             
                             × 
                             
                               I 
                               
                                 8 
                                 × 
                                 8 
                               
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             0 
                           
                         
                       
                       
                         
                           
                             - 
                             
                               
                                 ( 
                                 
                                   H 
                                   6 
                                 
                                 ) 
                               
                               8 
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   36 
                   ) 
                 
               
             
             
               
                 
                   C 
                   = 
                   
                     { 
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                               C 
                               4 
                               1 
                             
                             × 
                             
                               I 
                               
                                 8 
                                 × 
                                 8 
                               
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             0 
                           
                         
                       
                       
                         
                           
                             
                               ( 
                               
                                 H 
                                 4 
                               
                               ) 
                             
                             8 
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               stage 
                             
                             = 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
           
         
       
     
     In the scaleable architecture for calculating an 8×8 DCT or four 4×4 DCTs (in parallel) or sixteen 2×2 DCTs (in parallel), the inputs and the outputs are vectors of eight components and the following symbols are used: 
     
       
         
           
             
               
                 
                   A 
                   = 
                   
                     { 
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                               C 
                               8 
                               1 
                             
                             × 
                             
                               I 
                               
                                 8 
                                 × 
                                 8 
                               
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   stage 
                                   , 
                                   size 
                                 
                                 ) 
                               
                             
                             = 
                             
                               
                                 ( 
                                 
                                   0 
                                   , 
                                   0 
                                 
                                 ) 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               or 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   0 
                                   , 
                                   1 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           
                             I 
                             
                               8 
                               × 
                               8 
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   stage 
                                   , 
                                   size 
                                 
                                 ) 
                               
                             
                             = 
                             
                               
                                 ( 
                                 
                                   0 
                                   , 
                                   2 
                                 
                                 ) 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               or 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   1 
                                   , 
                                   2 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               ( 
                               
                                 H 
                                 2 
                               
                               ) 
                             
                             8 
                           
                         
                         
                           
                             per 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               
                                 stage 
                                 , 
                                 size 
                               
                               ) 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               
                                 1 
                                 , 
                                 0 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         H 
                                         1 
                                       
                                       ) 
                                     
                                     4 
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   
                                     
                                       ( 
                                       
                                         H 
                                         1 
                                       
                                       ) 
                                     
                                     4 
                                   
                                 
                               
                             
                             ] 
                           
                         
                         
                           
                             per 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               
                                 stage 
                                 , 
                                 size 
                               
                               ) 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               
                                 1 
                                 , 
                                 1 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
             
               
                 
                   B 
                   = 
                   
                     { 
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                               C 
                               8 
                               3 
                             
                             × 
                             
                               I 
                               
                                 8 
                                 × 
                                 8 
                               
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   stage 
                                   , 
                                   size 
                                 
                                 ) 
                               
                             
                             = 
                             
                               
                                 ( 
                                 
                                   0 
                                   , 
                                   0 
                                 
                                 ) 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               or 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   0 
                                   , 
                                   1 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           
                             I 
                             
                               8 
                               × 
                               8 
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   stage 
                                   , 
                                   size 
                                 
                                 ) 
                               
                             
                             = 
                             
                               
                                 ( 
                                 
                                   0 
                                   , 
                                   2 
                                 
                                 ) 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               or 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   1 
                                   , 
                                   2 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           
                             - 
                             
                               
                                 ( 
                                 
                                   H 
                                   6 
                                 
                                 ) 
                               
                               8 
                             
                           
                         
                         
                           
                             per 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               
                                 stage 
                                 , 
                                 size 
                               
                               ) 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               
                                 1 
                                 , 
                                 0 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     - 
                                     
                                       
                                         ( 
                                         
                                           H 
                                           3 
                                         
                                         ) 
                                       
                                       4 
                                     
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   
                                     - 
                                     
                                       
                                         ( 
                                         
                                           H 
                                           3 
                                         
                                         ) 
                                       
                                       4 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                         
                           
                             per 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               
                                 stage 
                                 , 
                                 size 
                               
                               ) 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               
                                 1 
                                 , 
                                 1 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
             
               
                 
                   C 
                   = 
                   
                     { 
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                               C 
                               4 
                               1 
                             
                             × 
                             
                               I 
                               
                                 8 
                                 × 
                                 8 
                               
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   stage 
                                   , 
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                                 ) 
                               
                             
                             = 
                             
                               
                                 ( 
                                 
                                   0 
                                   , 
                                   0 
                                 
                                 ) 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               or 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   0 
                                   , 
                                   1 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           
                             I 
                             
                               8 
                               × 
                               8 
                             
                           
                         
                         
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   stage 
                                   , 
                                   size 
                                 
                                 ) 
                               
                             
                             = 
                             
                               
                                 ( 
                                 
                                   0 
                                   , 
                                   2 
                                 
                                 ) 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               or 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   1 
                                   , 
                                   2 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               ( 
                               
                                 H 
                                 4 
                               
                               ) 
                             
                             8 
                           
                         
                         
                           
                             per 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               
                                 stage 
                                 , 
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                               ) 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               
                                 1 
                                 , 
                                 0 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         H 
                                         2 
                                       
                                       ) 
                                     
                                     4 
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   
                                     
                                       ( 
                                       
                                         H 
                                         2 
                                       
                                       ) 
                                     
                                     4 
                                   
                                 
                               
                             
                             ] 
                           
                         
                         
                           
                             per 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               
                                 stage 
                                 , 
                                 size 
                               
                               ) 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               
                                 1 
                                 , 
                                 1 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   40 
                   ) 
                 
               
             
           
         
       
     
       FIG. 3  illustrates the architecture for calculating sixteen 2×2 DCTs in parallel. 
     The pixels that constitute the input block are ordered during the INPUT phase and processed during the PROCESS phase to obtain the coefficients of the sixteen 2-D DCTs on four samples. For example, the 2-D DCT of the block (0,1) constituted by
     {l[0],m[0],n[0],o[0]} is }a[0],b[0],c[0],d[0]}.   

     The coefficients of the 2-D DCTs are rearranged during the ORDER phase in eight vectors of eight components. For example the coefficients }a[0],b[0],c[0],d[0]} will constitute the vector l′. 
     The sixteen two-component vectors so obtained are sent to the PROCESS phase to obtain the coefficients of the 2×2 DCT. These coefficients, reordered during the OUTPUT phase, constitute the output block. 
       FIG. 4  shows the ordering of the input data for calculating sixteen 2×2 DCTs. 
     This figure shows the way the pixels of the 8×8 input block are ordered to constitute the vectors of 8 components l, m, . . . , s. In each quadrant (i,j), with 0≦i,j≦3, the pixels belonging to the vectors are symbolized by different shadings. For example:
     {A i,k   0,0 } k=0   1 ={x 0,0 ,x 1,1 }   

     From each of these vectors, the components with the same index (that is the pixels with the same column index) will form a vector of four components. For example the vector l is constituted by the elements {A 1 [0], B 1 [0]}. 
     Therefore, each pixel of the 8×8 input block will constitute a component of one of the vectors l, m, n, o, p, q, r, s. 
       FIG. 5  shows the process phase for calculating sixteen 2×2 DCTs. 
     This phase includes processing the eight-component vectors l, m, . . . , s. The PROCESS phase, which is the only phase in which arithmetical operations are performed, is executed only once to calculate in parallel the sixteen 2-D DCTs. 
       FIG. 6  illustrates the architecture for calculating four 4×4 DCTs. 
     The pixels that constitute the input block are ordered in the INPUT phase and processed in the PROCESS phases to obtain the coefficients of the sixteen 1-D DCTs on 4 samples. For example, the 1-D DCT of the sequence {l[0],m[0],n[0], or [0]} is {a[0],b[0],c[0],d[0]}. 
     The coefficients of the 1-D DCTs are reordered in the ORDER phase in 8 vectors of eight components. For example the coefficients {a[0],b[0],c[0],d[0]} will constitute the vector l′. 
     The 4 four-component vectors so obtained are sent to the PROCESS phase to obtain the coefficients of the 4×4 DCT. These coefficients, reordered in the OUTPUT phase, constitute the output block. 
       FIG. 7  shows the arrangement of the input data for calculating four 4×4 DCTs. 
     This figure shows how the pixels of the 8×8 input block are ordered to constitute the eight-component vectors l, m, . . . , s. 
     In each quadrant (i,j), with 0≦i,j≦3, the pixels belonging to the different vectors have different shadings. For example:
     {A i,k   0,0 } k=0   3 ={x 0,0 ,x 1,1 ,x 2,2 ,x 3,3 }   

     From each of these vectors, the components with the same index (that is, the pixels with the same column index) will form a vector of four components. For example the vector l is constituted by the elements {A 1 [0], A 3 [0], B 3 [0], B 1 [0]}. 
     The outcome is that each pixel of the input 8×8 block will constitute one component of one of the vectors l, m, n, o, p, q, r, s. 
       FIG. 8  depicts the PROCESS phase for calculating the four 4×4 DCTs. 
     This phase includes processing the eight-component vectors: l, m, . . . , s. 
     The PROCESS phase, which is the only phase wherein arithmetical operations are performed, is carried out twice:
         the first time (stage=0), to calculate in parallel the sixteen 1-D DCTs;   the second time (stage=1), to calculate the 8×8 DCT starting from the coefficients of the 1-D DCTs.       

     
       
         
           
             
               
                 
                   A 
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               2 
                               ⁢ 
                               
                                 C 
                                 8 
                                 1 
                               
                               × 
                               
                                 I 
                                 
                                   8 
                                   × 
                                   8 
                                 
                               
                             
                           
                           
                             
                               
                                 per 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 stage 
                               
                               = 
                               0 
                             
                           
                         
                         
                           
                             
                               [ 
                               
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           H 
                                           1 
                                         
                                         ) 
                                       
                                       4 
                                     
                                   
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                   
                                     
                                       
                                         ( 
                                         
                                           H 
                                           1 
                                         
                                         ) 
                                       
                                       4 
                                     
                                   
                                 
                               
                               ] 
                             
                           
                           
                             
                               
                                 per 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 stage 
                               
                               = 
                               1 
                             
                           
                         
                       
                       , 
                     
                   
                 
               
               
                 
                   ( 
                   41 
                   ) 
                 
               
             
             
               
                 
                   B 
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               2 
                               ⁢ 
                               
                                 C 
                                 8 
                                 3 
                               
                               × 
                               
                                 I 
                                 
                                   8 
                                   × 
                                   8 
                                 
                               
                             
                           
                           
                             
                               
                                 per 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 stage 
                               
                               = 
                               0 
                             
                           
                         
                         
                           
                             
                               [ 
                               
                                 
                                   
                                     
                                       - 
                                       
                                         
                                           ( 
                                           
                                             H 
                                             3 
                                           
                                           ) 
                                         
                                         4 
                                       
                                     
                                   
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                   
                                     
                                       - 
                                       
                                         
                                           ( 
                                           
                                             H 
                                             3 
                                           
                                           ) 
                                         
                                         4 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                           
                           
                             
                               
                                 per 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 stage 
                               
                               = 
                               1 
                             
                           
                         
                       
                       , 
                     
                   
                 
               
               
                 
                   ( 
                   42 
                   ) 
                 
               
             
             
               
                 
                   C 
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               2 
                               ⁢ 
                               
                                 C 
                                 4 
                                 1 
                               
                               × 
                               
                                 I 
                                 
                                   8 
                                   × 
                                   8 
                                 
                               
                             
                           
                           
                             
                               
                                 per 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 stage 
                               
                               = 
                               0 
                             
                           
                         
                         
                           
                             
                               [ 
                               
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           H 
                                           2 
                                         
                                         ) 
                                       
                                       4 
                                     
                                   
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                   
                                     
                                       
                                         ( 
                                         
                                           H 
                                           2 
                                         
                                         ) 
                                       
                                       4 
                                     
                                   
                                 
                               
                               ] 
                             
                           
                           
                             
                               
                                 per 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 stage 
                               
                               = 
                               1 
                             
                           
                         
                       
                       , 
                     
                   
                 
               
               
                 
                   ( 
                   43 
                   ) 
                 
               
             
             
               
                 
                   t 
                   = 
                   
                     { 
                     
                       
                         
                           
                             1 
                           
                           
                             
                               
                                 per 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
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                               = 
                               0 
                             
                           
                         
                         
                           
                             2 
                           
                           
                             
                               
                                 per 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 stage 
                               
                               = 
                               1 
                             
                           
                         
                       
                       , 
                     
                   
                 
               
               
                 
                   ( 
                   44 
                   ) 
                 
               
             
           
         
       
     
       FIG. 9  illustrates the architecture for calculating an 8×8 DCT. 
     The pixels that constitute the input block are ordered during the INPUT phase and are processed in the PROCESS phase to obtain the coefficients of the eight 1-D DCTs on 8 samples. For example, the 1-D DCT of the sequence {l[0],m[0], . . . , s[0]} is {a[0],b[0], . . . , h[0]}. 
     The coefficients of the 1-D DCTs are rearranged during the ORDER phase in 8 vectors of eight components. For example the coefficients {a[0],b[0], . . . , h[7]} will constitute the l′vector. 
     The 8 eight-component vectors so obtained are sent to the PROCESS phase to obtain the 8×8 DCT coefficients. These coefficients, rearranged during the OUTPUT phase, constitute the output block. 
       FIG. 10  shows the arrangement of the input data for calculating an 8×8 DCT. 
     This figure shows how the pixels of the input 8×8 block are arranged to constitute the 8 eight-component vectors l, m, . . . , s. The pixels belonging to the vectors A 1 , A 3 , A 5 , A 7 , B 7 , B 5 , B 3 , B 1  are symbolized with different shadings, for example:
     {A 1,1 } i=0   7 ={x 0,0 ,x 1,1 ,x 2,2 ,x 3,3 ,x 4,4 ,x 5,5 ,x 6,6 ,x 7,7 }   

     From each of these vectors, the components with the same index (that is, the pixels with the same column index) will form a vector of eight components. For example, the vector l is constituted by the elements {A 1 [0], A 3 [0], . . . , B 1 [0]}. 
     The result is that each pixel of the input 8×8 block will constitute a component of one of the vectors l, m, n, o, p, q, r, s. 
       FIG. 11  depicts the PROCESS phase for calculating an 8×8 DCT. 
     This phase includes processing the eight-component vectors l, m, . . . , s. 
     The PROCESS phase in which arithmetical operations are performed is executed twice:
         the first time (stage=0), to calculate in parallel the sixteen 1-D DCTs;   the second time (stage=1), to calculate the 8×8 DCT starting from the coefficients of the 1-D DCTs. In  FIG. 11  the following symbols have been used:       

     
       
         
           
             
               
                 
                   A 
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               2 
                               ⁢ 
                               
                                 C 
                                 8 
                                 1 
                               
                               × 
                               
                                 I 
                                 
                                   8 
                                   × 
                                   8 
                                 
                               
                             
                           
                           
                             
                               
                                 per 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 stage 
                               
                               = 
                               0 
                             
                           
                         
                         
                           
                             
                               
                                 ( 
                                 
                                   H 
                                   2 
                                 
                                 ) 
                               
                               8 
                             
                           
                           
                             
                               
                                 per 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 stage 
                               
                               = 
                               1 
                             
                           
                         
                       
                       , 
                     
                   
                 
               
               
                 
                   ( 
                   45 
                   ) 
                 
               
             
             
               
                 
                   B 
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               2 
                               ⁢ 
                               
                                 C 
                                 8 
                                 3 
                               
                               × 
                               
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                                   8 
                                   × 
                                   8 
                                 
                               
                             
                           
                           
                             
                               
                                 per 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 stage 
                               
                               = 
                               0 
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 
                                   ( 
                                   
                                     H 
                                     6 
                                   
                                   ) 
                                 
                                 8 
                               
                             
                           
                           
                             
                               
                                 per 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 stage 
                               
                               = 
                               1 
                             
                           
                         
                       
                       , 
                     
                   
                 
               
               
                 
                   ( 
                   46 
                   ) 
                 
               
             
             
               
                 
                   C 
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               2 
                               ⁢ 
                               
                                 C 
                                 4 
                                 1 
                               
                               × 
                               
                                 I 
                                 
                                   8 
                                   × 
                                   8 
                                 
                               
                             
                           
                           
                             
                               
                                 per 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 stage 
                               
                               = 
                               0 
                             
                           
                         
                         
                           
                             
                               
                                 ( 
                                 
                                   H 
                                   4 
                                 
                                 ) 
                               
                               8 
                             
                           
                           
                             
                               
                                 per 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 stage 
                               
                               = 
                               1 
                             
                           
                         
                       
                       , 
                     
                   
                 
               
               
                 
                   ( 
                   47 
                   ) 
                 
               
             
             
               
                 
                   t 
                   = 
                   
                     { 
                     
                       
                         
                           
                             1 
                           
                           
                             
                               
                                 per 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 stage 
                               
                               = 
                               0 
                             
                           
                         
                         
                           
                             2 
                           
                           
                             
                               
                                 per 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 stage 
                               
                               = 
                               1 
                             
                           
                         
                       
                       , 
                     
                   
                 
               
               
                 
                   ( 
                   48 
                   ) 
                 
               
             
           
         
       
     
       FIG. 12  illustrates a scaleable architecture for calculating an 8×8 DCT or four 4×4 DCTs or sixteen 2×2 DCTs. 
     The pixels that constitute the input block are ordered during the INPUT phase and processed during the PROCESS phase, which calculates:
         the 1-D DCTs (for stage=0, that is for the 8×8 DCT, and for stage=1, that is for the 4×4 DCTs;   the 2-D DCTs for stage=2 directly, that is for the 2×2 DCTs;       

     When stage=0 and stage=1 the coefficients are then rearranged in the ORDER phase in 8 eight-component vectors, which are sent to the PROCESS phase to obtain the coefficients of the 2-D DCT. These coefficients, rearranged in the OUTPUT phase, constitute the output block. 
     If stage=2 the coefficients are transmitted directly to the OUTPUT phase, where they are rearranged to constitute the output block. 
       FIG. 13  depicts the INPUT phase for a scaleable architecture. 
     The inputs are the 64 pixels that constitute the input block. 
     The arrangement of the inputs is operated through the MUXes controlled by the size variable. 
     The 64 outputs are the 8 vectors of eight components l, m, . . . , s. 
       FIG. 14  depicts the PROCESS phase for a scaleable architecture. 
     This phase includes calculating in parallel the eight 1-D DCTs by processing the vectors l, m, . . . , s as shown in  FIG. 11 . 
     In this figure we may notice that the use of 16 MUXes controlled by size. 
     The eight MUXes on the left serve to bypass the necessary operations only for calculating the 8×8 DCT; therefore, the bypass takes place for stage=1 or 2, while it does not occur when stage=0. 
     The eight MUXes on the right serve to output only the result corresponding to the pre-selected size. 
     In  FIG. 14  the following symbols are used: 
     
       
         
           
             
               
                 
                   t 
                   = 
                   
                     { 
                     
                       
                         
                           
                             1 
                           
                           
                             
                               
                                 per 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 stage 
                               
                               = 
                               0 
                             
                           
                         
                         
                           
                             2 
                           
                           
                             
                               
                                 per 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 stage 
                               
                               = 
                               1 
                             
                           
                         
                       
                       , 
                     
                   
                 
               
               
                 
                   ( 
                   49 
                   ) 
                 
               
             
             
               
                 
                   A 
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               2 
                               ⁢ 
                               
                                 C 
                                 8 
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                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     stage 
                                     , 
                                     size 
                                   
                                   ) 
                                 
                               
                               = 
                               
                                 
                                   ( 
                                   
                                     0 
                                     , 
                                     0 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 or 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     0 
                                     , 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               I 
                               
                                 8 
                                 × 
                                 8 
                               
                             
                           
                           
                             
                               
                                 per 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     stage 
                                     , 
                                     size 
                                   
                                   ) 
                                 
                               
                               = 
                               
                                 
                                   ( 
                                   
                                     0 
                                     , 
                                     2 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                     
                                 
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                                 or 
                                 ⁢ 
                                 
                                     
                                 
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                                   ( 
                                   
                                     1 
                                     , 
                                     2 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 ( 
                                 
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                                   2 
                                 
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                               8 
                             
                           
                           
                             
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                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   stage 
                                   , 
                                   size 
                                 
                                 ) 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   1 
                                   , 
                                   0 
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               [ 
                               
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           H 
                                           1 
                                         
                                         ) 
                                       
                                       4 
                                     
                                   
                                   
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                                         ( 
                                         
                                           H 
                                           1 
                                         
                                         ) 
                                       
                                       4 
                                     
                                   
                                 
                               
                               ] 
                             
                           
                           
                             
                               per 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   stage 
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                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   1 
                                   , 
                                   1 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       , 
                     
                   
                 
               
               
                 
                   ( 
                   50 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     = 
                     
                       { 
                       
                         
                           
                             
                               
                                 2 
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                                 × 
                                 
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                                     ( 
                                     
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                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
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                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       0 
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                                 I 
                                 
                                   8 
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                                     ( 
                                     
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                                 = 
                                 
                                   
                                     ( 
                                     
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                                     ( 
                                     
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                                   ( 
                                   
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                                 [ 
                                 
                                   
                                     
                                       
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                                             ( 
                                             
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                                             ( 
                                             
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                                 ⁢ 
                                 
                                   ( 
                                   
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                                 ⁢ 
                                 
                                   ( 
                                   
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                         , 
                       
                     
                   
                 
               
               
                 
                   ( 
                   51 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     = 
                     
                       { 
                       
                         
                           
                             
                               
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                                     ( 
                                     
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                                   ⁢ 
                                   
                                     ( 
                                     
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                                     ( 
                                     
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                                 ⁢ 
                                 
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                                 ⁢ 
                                 
                                     
                                 
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                                   ( 
                                   
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                         , 
                       
                     
                   
                 
               
               
                 
                   ( 
                   52 
                   ) 
                 
               
             
             
               
                 
                   D 
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               2 
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                               = 
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                               = 
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                   ( 
                   53 
                   ) 
                 
               
             
             
               
                 
                   E 
                   = 
                   
                     { 
                     
                       
                         
                           
                             
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                                 ⁢ 
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                               = 
                               1 
                             
                           
                         
                       
                       , 
                     
                   
                 
               
               
                 
                   ( 
                   54 
                   ) 
                 
               
             
             
               
                 
                   F 
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               2 
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                                 ⁢ 
                                 stage 
                               
                               = 
                               1 
                             
                           
                         
                       
                       , 
                     
                   
                 
               
               
                 
                   ( 
                   55 
                   ) 
                 
               
             
             
               
                 
                   G 
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               2 
                               ⁢ 
                               
                                 C 
                                 16 
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                                   × 
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                                 per 
                                 ⁢ 
                                 
                                     
                                 
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                               0 
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 
                                   ( 
                                   
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                                 8 
                               
                             
                           
                           
                             
                               
                                 per 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
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                               = 
                               1 
                             
                           
                         
                       
                       , 
                     
                   
                 
               
               
                 
                   ( 
                   56 
                   ) 
                 
               
             
           
         
       
     
       FIG. 15  is a block diagram of the structure that implements the PROCESS phase. 
     For example, the QA block receives as an input two vectors of eight components (each component is a pixel, that may have already been processed) and outputs two vectors of eight components. The first vector is the sum of the two input vectors, while the second vector is the difference between the two input vectors, successively processed with the linear operator A. It should be noticed that the A, B, C, D, E, F, G operators are 8×8 matrices. 
       FIG. 16  is a detailed scheme of the QA block. 
     This scheme shows the details of the single components of the two input vectors and the arithmetical operators (adders etc.) which act on each component. The results are sent to the MUXes depicted on the right side of the figure, each of which, depending on the control variables stage and size, select only one result, which constitutes one component of the output vector. 
       FIG. 17  is a detailed scheme of the QB block. 
     This scheme shows the details of the single components of the two input vectors and the arithmetical operators (adders etc.) which act on each component. The results are sent to the MUXes depicted on the right side of the figure, each of which, depending on the control variables stage and size, select only one result, which constitute a component of the output vector. 
       FIG. 18  is a detailed scheme of the QC block. 
     This scheme shows the details of the single components of the two input vectors and the arithmetical operators (adders etc.) acting on each component. The results are sent to the MUXes depicted on the right side of the figure, each of which, depending on the control variable stage and size, select only one result, which constitute one component of the output vector. 
       FIG. 19  is a detailed scheme of the QD block. 
     This scheme shows the details of the single components of the two input vectors and the arithmetical operators (adders etc.) which act on each component. The results are sent to the MUXes depicted on the right side of the figure, each of which, depending on the control variable stage and size, select only one result, which constitute a component of the output vector. 
       FIG. 20  is a detailed scheme of the QE block. 
     This scheme shows the details of the single components of the two input vectors and the arithmetical operators (adders etc.) which act on each component. The results are sent to the MUXes depicted on the right side of the figure, each of which, depending on the control variable stage, select only one result, which constitute one component of the output vector. 
       FIG. 21  is a detailed scheme of the QF block. 
     This scheme shows the details of the single components of the two input vectors and the arithmetical operators (adders etc.) which act on each component. The results are sent to the MUXes depicted on the right side of the figure, each of which, depending on the control variable {\em stage} select only one result, which constitute a component of the output vector. 
       FIG. 22  is a detailed scheme of the QG block. 
     This scheme shows the details of the single components of the two input vectors and the arithmetical operators (adders etc.) which act on each component. The results are sent to the MUXes depicted on the right side of the figure, each of which, depending on the control variable stage, select only one result, which constitute a component of the output vector. 
       FIG. 23  depicts the ORDER phase for the scaleable architecture. 
     The inputs are constituted by the 64 pixels after they have been processed through the PROCESS phase. 
     The inputs arrangement is effected by the MUXes controlled by the variable size. 
     The 64 outputs are the components of the eight-component vectors l, m, . . . , s. 
       FIG. 24  depicts the OUTPUT phase for the scaleable architecture. 
     The inputs are constituted by the 64 2-D DCT coefficients. The input arrangement is effected by the MUXes controlled by the variable size. 
     The 64 outputs are the pixels that constitute the output block.