Abstract:
A special purpose processor and method of computation for performing an N-length real-number discrete cosine transform (DCT). The algorithm is called the split-radix discrete cosine transform because an Nth order DCT consists of an (N/2)th order DCT and two (N/4)th order inverse DCT (IDCT), where N is an integer power of 2 and larger than 4. In the embodiments of the invention, signal flow-graphs for both the forward and inverse scaled DCT have been implemented based on fused multiply and add operations in pipeline architectures. In the two-dimensional cases, over 20% arithmetic operations are saved compared to other known fast two-dimensional DCT algorithms. In addition, the split-radix DCT method offers flexibility of processing different input sizes under software control.

Description:
STATEMENT OF GOVERNMENT INTEREST 
     The invention described herein may be manufactured and used by or for the Government of the United States for governmental purposes without the payment of royalty therefor. 
    
    
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The discrete cosine transform (DCT), a special kind of orthonormal transform, has been widely accepted as the preferred method for compressing and decompressing gray-scaled images. A DCT compressor comprises mainly two parts: The first part transforms highly correlated image data into weakly correlated coefficients using a DCT transform and the second part performs adaptive quantization on coefficients to reduce the bit rate for transmission or storage. However, the computational burden in performing a DCT is demanding. For example, to process a one-dimensional DCT of length 8 pixels requires 11 multiplications and 29 additions in currently known fast algorithms. In practice, the image is divided into square blocks of size 8 by 8 pixels, 16 by 16 pixels or 32 by 32 pixels. Each block is often processed by the one-dimensional DCT in row-by-row fashion followed by column-by-column. On the other hand, different image block sizes are selected for compression due to different types of input images and different quality requirements on the decompressed image. A radix-2 DCT algorithm has been described in the article, &#34;A Fast Recursive Algorithm for Computing the Discrete Cosine Transform,&#34; IEEE Trans. on Acoustics, Speech, and Signal Processing, Vol. ASSP-35, No. 10, pp. 1455-1461, by H. S. Hou in October 1987. The purpose is to reduce the number of multiplications as well as to offer design flexibility of processing different sizes of DCT blocks. The references in the above article list the prior arts of the DCT algorithms. 
     In the article, &#34;A Fast DCT-SQ Scheme for Images,&#34; Trans. IEICE, Vol. E-71, No. 11, pp. 1095-1097, Nov. 1988, Y. Arai, T. Agui, and M. Nakajima have proposed that many of the DCT multiplications can be formulated as scaling multipliers to the DCT coefficients. The DCT after the multipliers are factored out is called the scaled DCT. Evidently, the scaled DCT is still orthogonal but no longer normalized, whereas the scaling factors may be restored in the following quantization process. Arai, et al. have demonstrated in their article that only 5 multiplications and 29 additions are required in processing an 8-point scaled DCT. Then E. Feig has mathematically described the scaled DCT, in particular the 8 by 8 scaled DCT, in U.S. Pat. No. 5,293,434 issued on Mar. 8, 1994 and the article, &#34;A Fast Scaled-DCT Algorithm,&#34; presented at the 1990 SPIE/SPSE Symposium of Electronic Imaging Science and Technology, Feb. 12, 1990, Santa Clara, Calif. However, the recursive properties of the scaled DCT have not been mentioned in the previous publications. Subsequently, H. S. Hou described the recursive properties of the scaled DCT in radix-2 formulations in the article, &#34;Recursive Scaled-DCT,&#34; presented at the 1991 SPIE International Symposium, conference 1567, Jul. 22, 1991, San Diego, Calif. 
     The goal of previous DCT algorithms with the scaled DCT included is to reduce the number of multiplications in the processor. But the fastest processors today are based on the fused multiply and add operations in pipeline architectures. In the fused multiply and add operations, a multiplication and an addition in the form of a+bc can be performed in one instruction cycle. For example, according to the current specification of the microprocessor i860 from Intel, a 32-bit fused multiply and add operation takes 20 nsec, whereas a single 32-bit multiply or a single 32-bit add also takes 20 nsec. Hence, there is a net gain in processing speed if we can take advantage of these architectures for implementation of the scaled DCT. E. Feig and E. Linzer have described the result of using the fused multiply and add architecture in performing an 8-point scaled DCT in their article, &#34;Scaled DCT Algorithms for JPEG and MPEG Implementations on Fused Multiply/Add Architectures,&#34; presented in SPIE conferences, 1991. Again the recursive nature of the scaled DCT has not been considered for the selection of different sizes of image blocks under program control. 
     All the recursive DCT algorithms published today are in radix-2 forms, i.e., splitting an Nth order DCT into two (N/2)th order DCT. Yet, it is known in fast Fourier transforms and fast Hartley transforms that split-radix algorithms give the fastest operations. But no corresponding split-radix DCT has been known to exist in the state-of-the-art. This invention discloses the split-radix algorithm and the implementation schemes for processing the regular DCT and the scaled DCT. 
     Due to the fact that the number of arithmetic operations in performing a DCT grows faster than linearly proportional to the number of input, from both the speed performance and the design flexibility viewpoints, it is desirable to use the recursive algorithms for both the DCT and the scaled DCT. In so doing, one can process a combination of lower order DCT instead of a higher order DCT by itself. In the radix-2 recursive DCT algorithm, an Nth order DCT contains two (N/2)th order DCT; whereas in the radix-2 recursive scaled DCT algorithm, an Nth order scaled DCT contains one (N/2)th order scaled DCT and one (N/2)th order scaled IDCT. The disclosed split-radix DCT and the split-radix scaled DCT algorithms are further improvements of the radix-2 algorithms, because in the split-radix DCT algorithm an Nth order DCT contains an (N/2)th order DCT and two (N/4)th order IDCT, whereas in the split-radix scaled DCT algorithms an Nth order scaled DCT contains an (N/2)th order scaled DCT and two (N/4)th order modified scaled IDCT. 
     SUMMARY OF THE INVENTION 
     The present invention is a special purpose processor and method of computation for performing an N-length discrete cosine transform or the scaled discrete cosine transform in split-radix algorithm. In accordance with the invented split-radix algorithm, the N-length DCT can be realized with an (N/2)th order DCT and two (N/4)th order IDCT. Similarly, an N-length IDCT can be realized with an (N/2)th order IDCT and two (N/4)th order DCT. In particular, the signal flow-graphs for both the forward and inverse scaled DCT have been implemented with fused multiply and add operations in pipeline architectures. These embodiments not only have the least number of fused arithmetic operations in processing two-dimensional input blocks but also have the flexibility of processing input blocks of different size. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is an electrical circuit block diagram of the one-dimensional split-radix DCT of order N. 
     FIG. 2 is an electrical circuit block diagram of the one-dimensional split-radix IDCT of order N. 
     FIG. 3 is the signal flow-graph for the 4th order split-radix forward scaled DCT implemented by fused multiply and add operations; each is being represented by a circle. The solid line represents a direct connection. The dashed line represents a direct connection with sign reversal of data. 
     FIG. 4 is the signal flow-graph for the 8th order split-radix forward scaled DCT implemented by fused multiply and add operations. 
     FIG. 5 is the signal flow-graph for the 16th order split-radix forward scaled DCT implemented by fused multiply and add operations. 
     FIG. 6 is the signal flow-graph for the 32nd order split-radix forward scaled DCT implemented by fused multiply and add operations. 
     FIG. 7 is the signal flow-graph for the 4th order split-radix inverse scaled DCT implemented by fused multiply and add operations. 
     FIG. 8 is the signal flow-graph for the 8th order split-radix inverse scaled DCT implemented by fused multiply and add operations. 
     FIG. 9 is the signal flow-graph for the 16th order split-radix inverse scaled DCT implemented by fused multiply and add operations. 
     FIG. 10 is the signal flow-graph for the 32nd order split-radix inverse scaled DCT implemented by fused multiply and add operations. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     In the following we provide a rigorous derivation of the algorithm to be implemented in the reduced instruction set computer (RISC) architecture, in which the multiplication and addition pair can be performed in one instruction cycle. Therefore, the goal of the following derivations is to obtain as few multiplications and additions as possible, to be operated in pairs. 
     The derivations of the algorithm start with the regular DCT in radix-2 form. We then obtain the split-radix DCT algorithm which constitutes the theoretical foundation of this invention. Finally, both the forward and inverse split-radix scaled DCT algorithms are derived for physical implementations in RISC architecture. The signal flow-graphs accompanied by the formulations further illustrate the electrical connections between the fused multiply and add processing units. 
     1. REGULAR RECURSIVE DCT 
     According to the definition of DCT, for a given data sequence {x n  ; n=0, 1, 2, . . . , N-1}, the DCT data sequence {z k  ; k=0, 1, 2, . . . , N-1} is given by the following relation: ##EQU1## 
     The DCT defined in Eq. (1) can be written in matrix form as 
     
         z=T(N)x                                                    (2) 
    
     where x and z are column vectors denoting the input and DCT data sequences arranged in natural order. T(N) is used to designate the DCT matrix of order N where we have assumed that N is a power of 2. Clearly, T(N) is unitary. Based on the derivations in Reference 1, we obtain the orthonormal DCT defined in Eq. (2) as ##EQU2## where x p  and x r  are the previous half and the recent half of the input data sequence for which x is arranged in natural order; and 
     
         z.sub.e =[z.sub.0,z.sub.2,z.sub.4, . . . z.sub.N-2 ].sup.t 
    
     
         z.sub.o =[z.sub.1,z.sub.3,z.sub.5, . . . z.sub.N-1 ].sup.t 
    
     The z e  and z o  are the even half and the odd half of the output data vector z for which z is arranged in natural sequence, and I denotes the identify matrix that has been rotated by 90°, i.e., ##EQU3## Clearly, the first recursive form for the DCT matrix as defined by Eq. (2) is ##EQU4## where the permutation matrix P(N) is to bring the even numbered elements to the upper half and the odd numbered to the lower half, i.e., ##EQU5## For example ##STR1## 
     2. THE BASIC SPLIT-RADIX DCT ALGORITHM 
     Based on the basic definition of DCT given in Eq. (1), the transformation corresponding to D in Eqs. (3) or (4) is given by ##EQU6## for p=0, 1, 2, . . . , M-1 (M=N/2), where in Eq. (7) 
     
         f.sub.m =x.sub.m -x.sub.N-m-1 
    
     The right-hand side of Eq. (7) can be decomposed into two parts as ##EQU7## Now let 
     
         g=P(M) B(M)f                                               (9) 
    
     and 
     
         d=R(M)y                                                    (10) 
    
     where d is a column vector with components d p , and matrices P(M), B(M), and R(M) are defined respectively by ##STR2## where in Eq. (13) 
     
         c.sub.p =cos(pπ/2N) 
    
     and 
     
         s.sub.p =sin(pπ/2N) 
    
     The column vector y in Eq. (10) is related to g in Eq. (9) because of Eq. (8). Thus y may be written in two parts; the first part is ##EQU8## for p=0, 1, 2, . . . , M/2 -1. The second part is ##EQU9## Let us change m into M/2--m inside the summation in Eq. (15) to obtain ##EQU10## Now combining Eqs. (14) and (16) we have the following matrix equation ##EQU11## Thus ##EQU12## where in Eq. (18) ##STR3## Substituting Eq. (9) into Eq. (18) then into Eq. (10), we obtain ##EQU13## Therefore the D matrix in Eqs. (3) or (4) can be split into two lower order DCT matrices as ##EQU14## The realization of Eq. (4) for the Nth order DCT with the matrix D given by Eq. (21) is depicted in FIG. 1. The schematic block diagram depicted in FIG. 1 is to exhibit the architectural or functional organization of performing a one-dimensional forward split-radix DCT of order-N. The signals are flowing from left to right. The N pixel samples were input parallelly to an adder bank which consists of N units two-input adders. The top half of the adder bank output is used to perform a (N/2)th order forward DCT, in which N/8 constants in the form of real decimal numbers are required. However, these constants may be calculated beforehand and stored in the read-only memory (ROM) prior to the actual processing. The bottom half of the adder bank output, after passing through another stage of arithmetic operations consisting of N/2-2 additions, 2 scaler multiplications, and permutation, is split again into two halves. Each half is to perform a (N/4)th order inverse DCT (IDCT). The N/4 output of the first IDCT and the permutated N/4 output of the second IDCT are both fed into a butterfly bank, which consists of N/2 simultaneous multiplications with constant factors and followd by N/2 simultaneous additions. The N/2 multiplication factors used in the butterfly bank may again be calculated and stored in the ROM prior to the actual processing. Shown in two places in FIG. 1, the N output are the transform coefficients of a one-dimensional forward split-radix DCT of order-N. In other words, in the derived split-radix DCT algorithm just presented, the Nth order DCT consists of an (N/2)th order DCT and two (N/4)th order IDCT. 
     Due to the fact that the DCT matrix is unitary i.e., T -1  =T t , the inverse DCT may be obtained from the transpose of T(N) given in Eq. (4) as ##EQU15## where in Eq. (22) D t  (N/2) can be obtained from Eq. (21) in the split-radix algorithm as ##EQU16## The realization of Eq. (22) for the Nth order IDCT with matrix D t  given by Eq. (23) is depicted in FIG. 2. The schematic block diagram depicted in FIG. 2 is to exhibit the architectural or functional organization of performing a one-dimensional inverse split-radix DCT of order-N. Shown in two places in FIG. 2, the input are the DCT transform coefficients while the output are the pixel samples. The signals in FIG. 2 are flowing from left to right. Functionally, the inverse DCT (IDCT) is to perform the exactly inverse operation of the forward DCT. Hence, the block diagram shown in FIG. 2 is a mirror image of FIG. 1. Notice that the DCT and IDCT blocks in FIG. 1 have been replaced by corresponding IDCT and DCT blocks in FIG. 2, while the other blocks remain unchanged functionally. Clearly it can be seen that it consists of an (N/2)th order IDCT and two (N/4)th order DCT. 
     3. THE FORWARD SPLIT-RADIX SCALED DCT 
     Based on the recursive DCT relation given in Eq. (4) and the split-radix relation in Eq. (21), we can derive the forward split-radix algorithm for the scaled DCT to be implemented by fused multiply and add operations in the following steps: 
     (a) In reference to the previous scaled DCT articles, the DCT matrix may be factored out as ##EQU17## where F(N) is a diagonal matrix and Y(N) is the so-called scaled DCT matrix. 
     (b) We observe that scalar factors may also be factored out from the RT(N/2) in Eq. (13) as ##EQU18## where in Eq. (25) ##EQU19## for c i  =cos (in/2N) and ##EQU20## for i=1, 3, 5, . . . , N/2-1. Thus the scaled D matrix in Eq. (21) becomes ##EQU21## Now the recursive formula for the scaled DCT becomes ##EQU22## where in Eq. (27) D(N/2) is given in Eq. (26). 
     To be more illustrative, we consider the following special cases for the scaled DCT to be implemented by fused multiply and add operations: 
     Case 1. The 4th order forward split-radix scaled DCT: The 4th order DCT is known as given by ##EQU23## where in Eq. (28) α=1/√2, β=cos (π/8), δ=sin (π/8), and P(4) is a permutation matrix given by ##EQU24## Based on the above steps for factorization of T(4) we have ##EQU25## where t 2  =tan (π/8). Therefore the 4th order scaled DCT is ##EQU26## 
     The signal flow-graph of the 4th order scaled DCT sketched according to Eq. (31) is shown in FIG. 3. It requires 8 fused multiply and add operations that are designated by circles in FIG. 3. 
     Case 2. The 8th order forward split-radix scaled DCT: According to Eq. (21) and the factorization step (b) for D(4), we have from Eq. (21) for N=8 that ##EQU27## where λ=cos (π/16), γ=cos (3π/16), t 1  =tan (π/16), and t 3  =tan (3π/16). Thus the scaled D(4) is ##EQU28## Now the 8th order forward scaled DCT is given by ##EQU29## where in Eq. (33) P(8) is given in Eq. (6), Y(4) in Eq. (31), and D(4) in Eq. (32). The signal flow-graph sketched according to Eq. (33) is shown in FIG. 4. It takes 26 fused multiply and add operations to implement the one-dimensional 8th order forward scaled DCT. As one can clearly see from both FIG. 3 and FIG. 4, the 4th order scaled DCT is included in the 8th order scaled DCT. This recursive property offers user flexibility of processing different sizes of input and design modularity. 
     Repeatedly making use of the split-radix recursive formulas, we can design the 16-point and 32-point split-radix scaled DCT in the same fashion as demonstrated above for the 8th order scaled DCT. The signal flow-graphs for the 16-point and the 32-point forward split-radix scaled DCT are shown in FIGS. 5 and 6, respectively. Each circle in FIGS. 4, 5, and 6 represents a fused multiply and add operation. The fused operation at each circle location in FIGS. 5 and 6 is listed in Tables 1 and 2, respectively, where f denotes the first input (at the top) and g the second input (at the bottom) of the fused arithmetic processing unit. Moreover, in Tables 1 through 4 t i  =tan(iπ/64), α=cos(π/4), β=cos (π/8), λ=cos (π/16), γ=cos (3π/16), r=γ/λ, and ρ=αλ. Again, one can clearly see from FIG. 6, the 32-point forward split-radix scaled DCT contains the 4-point, the 8-point, and the 16-point forward scaled DCT. 
     4. THE INVERSE SPLIT-RADIX SCALED DCT 
     The inverse split-radix scaled DCT can be obtained by taking the transpose of Y(N) in Eq. (27). Thus ##EQU30## where in Eq. (34), G(N) is the inverse split-radix scaled DCT of order N, and D t  (N/2) is given by taking the transpose of D(N/2) in Eq. (26), i.e., ##EQU31## The multiplication factors of T(N/4) in Eq. (35) can be transferred to the left-hand side and finally be absorbed in the left most multiplying matrix B t  (N/2) on the right-hand side of Eq. (35). This procedure becomes evident in the following special cases. 
     Case 3. The 4th order inverse split-radix scaled DCT: Taking the transpose of Y(4) in Eq. (31) we have ##EQU32## The signal flow-graph for the 4th order inverse split-radix scaled DCT sketched according to Eq. (36) is shown in FIG. 7. It again requires 8 fused multiply and add operations in implementation. 
     Case 4. The 8th order inverse split-radix scaled DCT: According to the recursive formula in Eq. (34) for N=8, we take the transpose of D(4) in Eq. (32) to obtain ##EQU33## Substituting G(4) in Eq. (36) and D t  (4) in Eq. (37) to Eq. (34) for N=8, we have ##EQU34## where in Eq. (38) P(8) is a permutation matrix defined by ##EQU35## The signal flow-graph for the split-radix inverse 8th order scaled DCT sketched according to Eq. (38) is shown in FIG. 8. Again, it takes 26 fused multiply and add operations and contains the 4th order inverse scaled DCT. 
     Similarly as in the forward split-radix scaled DCT cases, we can use the recursive formulas given above to derive the signal flow-graphs for the 16-point and 32-point split-radix inverse scaled DCT. These are shown in FIGS. 9 and 10, respectively. Again, each circle in FIGS. 9 and 10 denotes a fused multiply and add operation. The fused operation at each circle location in FIGS. 9 and 10 is listed in Tables 3 and 4 respectively. As expected, the 32-point split-radix inverse scaled DCT contains the 4-point, the 8-point, and the 16-point inverse scaled DCT. 
     Although the invention has been described in terms of a preferred embodiment, it will be obvious to those skilled in the art that alterations and modifications may be made without departing from the invention. Accordingly, it is intended that all such alterations and modifications be included within the spirit and scope of the invention as defined by file appended claims. 
     
                                           TABLE 1__________________________________________________________________________Fused Operations in 16-Point Forward ScaledDCTColumn #Row #1   2      3     4    5     6__________________________________________________________________________1    f + g    f + g        f + g                      f + g2    f + g    f + g        f + g                      f - g3    f + g    f + g        f - g                      t.sub.8 f + g4    f + g    f + g        f - g                      -f + t.sub.8 g5    f + g    f - g        f + αg                      f + t.sub.4 g6    f + g    f - g   f + g                 f - αg                      f + t.sub.12 g7    f + g    f - g  -f - g                 αf - g                      t.sub.12 f - g8    f + f - g        αf + g                      t.sub.4 f-99    f - g            f + αg                      f + βg                            f + t.sub.2 g10   f - g    f + g        f - αg                      f + βg                            f + t.sub.6 g11   f - g    -f + g       t.sub.8 f-g                      f - βg                            f + t.sub.10 g12   f - g    f + g        f + t.sub.8 g                      f - βg                            f + t.sub.14 g13   f - g    -f + g       f + αg                      f + βg                            t.sub.14 f - g14   f - g    f + g        f - αg                      f + βg                            t.sub.10 f - g15   f - g    -f + g       t.sub.8 f - g                      f - βg                            t.sub.6 f - g16   f - g            f + t.sub.8 g                      f - βg                            t.sub.2 f - g__________________________________________________________________________ 
    
     
                                           TABLE 2__________________________________________________________________________Fused Operations in 32-Point Forward Scaled DCTColumn #Row #    1   2   3     4    5    6    7   8    9__________________________________________________________________________1   f + g   f + g       f + g      f + g                       f + g2   f + g   f + g       f + g      f + g                       f - g3   f + g   f + g       f + g      f - g                       t.sub.8 f + g4   f + g   f + g       f + g      f - g                       -f + t.sub.8 g5   f + g   f + g       f - g      f + αg                       f + t.sub.4 g6   f + g   f + g       f - g  f + g                  f - αg                       f + t.sub.12 g7   f + g   f + g       f - g -f + g                  αf - g                       t.sub.12 f - g8   f + g   f + g       f - g      αf + g                       t.sub.4 f - g9   f + g   f - g          f + αg                       f + βg                                f + t.sub.2 g10  f + g   f - g       f + g      f - αg                       f + βg                                f + t.sub.6 g11  f + g   f - g       -f + g     t.sub.8 f - g                       f - βg                                f + t.sub.10 g12  f + g   f - g       f + g      f + t.sub.8 g                       f - βg                                f + t.sub.14 g13  f + g   f - g       -f + g     f + αg                       f + βg                                t.sub.14 f - g14  f + g   f - g       f + g      f - αg                       f + βg                                t.sub.10 f - g15  f + g   f - g       -f + g     t.sub.8 f - g                       f - βg                                t.sub.6 f - g16  f + g   f - g          f + t.sub.8 g                       f - βg                                t.sub.2 f - g17  f - g              f + αg                       f + βg                                f + λg                                     f + t.sub.1 g18  f - g   f + g      f - αg                       f + βg                                f + ρg                                     f + t.sub.3 g19  f - g   -f + g     t.sub.8 f - g                       f - βg                                f + ρg                                     f + t.sub.5 g20  f - g   f + g      f + t.sub.8 g                       f - βg                                f + λg                                     f + t.sub.7 g21  f - g   -f + g     f + t.sub.4 g                       f + rg   f - λg                                     f + t.sub.9 g22  f - g   f + g      f + t.sub.12 g                       f - rg                            f - g                                f - ρg                                     f + t.sub.11 g23  f - g   -f + g     t.sub.12 f - g                       rf + g                            f + g                                f - ρg                                     f + t.sub.13 g24  f - g   f + g      t.sub.4 f - g                       -rf + g  f - λg                                     f + t.sub.15 g25  f - g   -f + g     f + αg                       f + βg                                f + λg                                     t.sub.15 f - g26  f - g   f + g      f - αg                       f + βg                                f + ρg                                     t.sub.13 f - g27  f - g   -f + g     t.sub.8 f - g                       f - βg                                f + ρg                                     t.sub.11 f - g28  f - g   f + g      f + t.sub.8 g                       f - βg                                f + λg                                     t.sub.9 f - g29  f - g   -f + g     f + t.sub.4 g                       f + rg   f - λg                                     t.sub.7 f - g30  f - g   f + g      f + t.sub.12 g                       f - rg                            f - f - ρg                                     t.sub.5 f - g31  f - g   -f + g     t.sub.12 f - g                       rf + g                            f + g                                f - ρg                                     t.sub.3 f - g32  f - g              t.sub.4 f - g                       -rf + g  f - λg                                     t.sub.1 f - g__________________________________________________________________________ 
    
     
                                           TABLE 3__________________________________________________________________________Fused Operations in 16-Point Inverse ScaledDCTColumn #Row #    1     2     3     4   5     6__________________________________________________________________________1         f + g f + g     f + g f + g2         f - g f + g     f + αg                           f + βg3         t.sub.8 f - g           f - g     f + αg                           f + βg4         f + t.sub.8 g           f - g     f + g f + αg5         f + t.sub.4 g           f + g     f - g f + αg6         f + t.sub.12 g           f - g f - g                     f - αg                           f + βg7         t.sub.12 f - g           f + g t + g                     f -  αg                           f + βg8         t.sub.14 f - g           -f + g    f - g f + g9   f + t.sub.2 g     f + g f + g           f - g10  f + t.sub.6 g     f + g f - g     f - g f - βg11  f + t.sub.10 g     f - g t.sub.8 f + g                     f + g f - βg12  f + t.sub.14 g     f - g -f + t.sub.8 g                     f - g f - αg13  t.sub.14 f - g     f + g f + g     f + g f - αg14  t.sub.10 f - g     f + g f - g     f - g f - βg15  t.sub.6 f - g     f - g t.sub.8 f + g                     f + g f - βg16  t.sub.2 f - g     f - g -f + t.sub.8 g  f - g__________________________________________________________________________ 
    
     
                                           TABLE 4__________________________________________________________________________Fused Operations in 32-Point Inverse Scaled DCTColumn #Row #    1   2   3     4    5    6    7   8    9__________________________________________________________________________1                 f + g                  f + g     f + g                                f + g                                     f + g2                 f - g                  f + g     f + αg                                f + βg                                     f + λg3                 t.sub.8 f - g                  f - g     f + αg                                f + βg                                     f + λg4                 f + t.sub.8 g                  f - g     f + g                                f + αg                                     f + βg5                 f + t.sub.4 g                  f + g     f - g                                f + αg                                     f + βg6                 f + t.sub. 12 g                  f - g                       f - g                            f - αg                                f + βg                                     f + λg7                 t.sub.12 f - g                  f + g                       t + g                            f - αg                                f + βg                                     f + λg8                 t.sub.4 f - g                  -f + g    f - g                                f + g                                     f + αg9       f + t.sub.2 g             f + g                  f + g         f - g                                     f + αg10      f + t.sub.6 g             f + g                  f - g     f - g                                f - βg                                     f + λg11      f + t.sub.10 g             f - g                  t.sub.8 f + g                            f + g                                f - βg                                     f + λg12      f + t.sub.14 g             f - g                  -f +  t.sub.8 g                            f - g                                f - αg                                     f + βg13      t.sub.14 f - g             f + g                  f + g     f + g                                f - αg                                     f + βg14      t.sub.10 f - g             f + g                  f - g     f - g                                f - βg                                     f + λg15      t.sub.6 f - g             f - g                  t.sub.8 f + g                            f + g                                f - βg                                     f + λg16      t.sub.2 f - g             f - g                  -f + t.sub.8 g                                f - g                                     f + g17  f + t.sub.1 g   f + g     f + g                  f + g              f - g18  f + t.sub.3 g   f + g     f + g                  t - g     f - g    f - λg19  f + t.sub.5 g   f + g     f - g                  t.sub.8 f + g                            f + g    f - λg20  f + t.sub.7 g   f + g     f - g                  -f + t.sub.8 g                            f - g    f - βg21  f + t.sub.9 g   f - g     f + αg                  f + t.sub.4 g                            f + g    f - βg22  f + t.sub.11 g   f - g        f + g             f - αg                  f + t.sub.12 g                            f - g    f - λg23  f + t.sub.13 g   f - g       -f + g             αf - g                  t.sub.12 f - g                            f + g    f - λg24  f + t.sub.15 g   f - g     αf + g                  t.sub.4 f - g                            f - g    f - αg25  t.sub.15 f - g   f + g     f + g                  f + g     f + g    f - αg26  t.sub.13 f - g   f + g     f + g                  f - g     f - g    f - λg27  t.sub.11 f - g   f + g     f - g                  t.sub.8 f + g                            f + g    f - λg28  t.sub.9 f - g   f + g     f - g                  -f + t.sub.8 g                            f - g    f - βg29  t.sub.7 f - g   f - g     f + αg                  f + t.sub.4 g                            f + g    f - βg30  t.sub.5 f - g   f - g       f + g f - αg                  f + t.sub.12 g                            f - g    f - λg31  t.sub.3 f - g   f - g       -f + g             αf - g                  t.sub.12 f - g                            f + g    f - λg32  t.sub.1 f - g   f - g     αf + g                  t.sub.4 f - g      f - g__________________________________________________________________________