Equation 1016 in .NET

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Equation 1016
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Optimizing the DCT This matrix can be factored out even more if we take advantage of these relations, which can be found in any book of standard mathematical formulas
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Equation 1017
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Using Equation 1017 we find that Equation 1018
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A derivation from Equation 1018 The other values are derived in a similar manner
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Equation 711 Equation 720
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Factoring the DCT Matrix
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Substituting Equation 1018 into Equation 1016 gives the results in Equation 1019 Equation 1019
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Equation 1019 can be factored into Equation 1020 Equation 1020 S =
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Equation 1021 S =
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Optimizing the DCT Putting Equation 1021 into Equation 1015 gives the matrix shown in Equation 1022
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Equation 1022
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Factoring the DCT Matrix
Equation 1022 appears to be much more complicated than the original in Equation 102, but it actually requires fewer steps to calculate Multiplying a row vector and M requires 64 multiplication and 56 addition operations In the factored matrices any nonzero value that is not equal to +/-1 represents a multiplication and all but one nonzero value in each column represents an addition A zero in a factor represents no operation and almost all of the array elements are zeros Table 101 shows the operations required for each matrix multiplication with the factored matrices
Table 101 Operations Required After Factorization
Matrix
Addition
Multiplication
1 2 3 4 5 6 7 Total
0 8 4 2 0 4 8 26
0 12 0 0 2 0 0 14
Most processors take longer to execute multiplication instructions than additions For example, on my system the speed advantage is about 4:1 in favor of addition Therefore, it is usually advantageous to replace multiplication operations with addition In the factored matrix most of the multiplication operations are in groups of the form Equation 1023
X = A cos Y = A sin
+ B sin - B cos
This form is well known in computer graphics because it represents the rotation of the point (A, B) by an angle A rotation uses four multiplication operations and two additions, but if it is calculated as shown in Equation 1023, it requires 3 multiplications and 3 additions If this method is used for multiplying the matrix, the total number of operations required to multiply a row by the DCT matrix becomes 11 multiplications and 29 additions
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Optimizing the DCT
Equation 1024
T = cos (A + B) X = T - (cos - sin )B Y = -T + (cos + sin )A
The following code example is an implementation of an IDCT function that slavishly follows the factorization in Equation 1022 so that you can clearly see how the factored matrix multiplications are implemented in code typedef double MATRIX [8][8] ; const double C1 = (sqrt (20) * cos (M_PI/160)) ; const double C2 = (sqrt (20) * cos (20*M_PI/160)) const double C3 = (sqrt (20) * cos (30*M_PI/160)) const double S1 = (sqrt (20) * sin (M_PI/160)) ; const double S2 = (sqrt (20) * sin (20*M_PI/160)) const double S3 = (sqrt (20) * sin (30*M_PI/160)) const double SQRT2 = (10 / sqrt(20)) ; unsigned int Limit (double input) { double value = input + 1285 ; if (value < 0) return 0 ; else if (value > 255) return 255 ; else return (unsigned int) value ; } void InverseDCT (MATRIX input, MATRIX output) { double tmp[SampleWidth][SampleWidth] ; for (int row = 0 ; row < 8 ; ++ row) { double a0 = input[row][0] double a1 = input[row][4] double a2 = input[row][2] double a3 = input[row][6] double a4 = input[row][1] double a5 = input[row][5] double a6 = input[row][3] double a7 = input[row][7] double b0 double b1 b2 = S2 b3 = C2 double r0 double b2 double b3 b4 = S1 b7 = C1 = = * * = = = * * (a0 + a1) ; (a0 - a1) ; 2 - C2 * a3 ; a2 + S2 * a3 ; S2 * (a2 + a3) ; r0 - (S2+C2) * a3 ; r0 - (S2-C2) * a2 ; a4 - C1 * a7 ; a4 + S1 * a7 ;
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; ; ; ;
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Factoring the DCT Matrix double r1 = S1 * (a4+a7) ; double b4 = r1 - (S1+C1) * a7 double b7 = r1 - (S1-C1) * a4 b5 = C3 * a5 - S3 * a6 ; b6 = S3 *a5 + C3 * a6 ; double r2 = C3 * (a5 + a6) ; double b5 = r2 - (C3+S3) * a6 double b6 = r2 - (C3-S3) * a5 double double double double double double double double double double double double double double double double double double double double double double double double double double double double double double double double c0 c1 c2 c3 c4 c5 c6 c7 d0 d1 d2 d3 d4 d5 d6 d7 e0 e1 e2 e3 e4 e5 e6 e7 f0 f1 f2 f3 f4 f5 f6 f7 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = b0 b1 b2 b3 b4 b5 b7 b7 c0 c1 c2 c3 c4 c6 c6 c7 ; ; ; ; + +
// //