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Optimizing the DCT going to multiply the rows of T by the columns of M to create a temporary matrix Then we are going to multiply the columns of the temporary matrix by the rows of MT When we multiply T M each output row depends only on the corresponding row in T so we can treat the calculation of each row separately Similarly, when we multiply MT and the temporary matrix, each column in the output depends only on the corresponding column in the temporary matrix The DCT transform matrix as originally presented in 7 is repeated with the substitution in Equation 102 Each row/column dot product requires 8 multiplication operations and 7 additions; therefore, transforming each row requires 64 multiplication operations and 56 additions
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Equation 102
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Notice that there is much symmetry in the matrix We will exploit these symmetries by factoring the DCT transform matrix into the product of several sparse matrices The first transformation we are going to make is to factor out the constant value from each element in the matrix and redefine the IDCT using the equivalent definition shown in Equation 103 Equation 103 V = MTTM
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Factoring the DCT Matrix
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For now we will ignore the in Equation 104
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factor and simply work with the matrix shown
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Equation 104
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The next few simplifications take advantage of the symmetries in the values of the cosine function to reduce the number of unique values in the transform matrix The properties of the cosine function can be found in any mathematical handbook If you refer to Figure 101 you can see that the cosine function is cyclical such that Equation 105
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cos x = cos (x + 2 )
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Using Equation 105 we can replace every occurrence of with , giving Equation 106
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in Equation 104
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Figure 101 Cosine Function
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Optimizing the DCT
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Equation 106
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Again referring to Figure 101, the cosine function is symmetric along the x-axis such that Equation 107 Using Equation 107 we can replace all the matrix elements in Equation 106 with arguments to the cosine function that are greater than , giving Equation 108 Equation 108
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The cosine function is also symmetric along the y-axis such that Equation 109
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Factoring the DCT Matrix
Using Equation 109 we can replace all arguments to the cosine function in Equation 108 that are greater than , giving Equation 1010
Equation 7070
The value of the cosine function at Equation 1011
is well known
Substituting Equation 1011 in to Equation 1010 gives Equation 1012
Equation 1012
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Optimizing the DCT Disregarding the sign, only seven distinct values remain in the transform matrix Now that the values within the transform matrix have been simplified we will factor the matrix into the product of several sparse matrices The primary goal in the factorization of the transform matrix is to create matrix factors with as many zero values as possible The secondary goal is to create matrices with the values +/-1 Zeros are great, ones are so-so, and everything else is bad The process used to factor the matrix is called Gaussian elimination Gaussian elimination is beyond the scope of this book However we have attempted to include enough steps for a reader with a basic knowledge of linear algebra to clearly see how the factors are obtained The following examples of matrix multiplication operations illustrate the principles of row reduction used to factor the DCT matrix
= = = = = =
Column Interchange Column Addition Row Interchange Row Addition
Notice that if the matrix in Equation 1012 is divided in half vertically, the left half of each row is either a mirror image of the right or a negative mirror image We can factor the matrix to group the mirror image rows together and the negative mirror image rows together (Equation 1013) This first factorization is not strictly necessary Its only purpose is to make the remaining factorization steps clearer
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Factoring the DCT Matrix
Equation 7073
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Optimizing the DCT
Equation 1014
In Equation 1014, notice that the nonzero elements at the upper left corner of the center matrix form the same mirror pattern as the rows of the matrix in Equation 1013 We factor again in a similar manner to attack that corner (Equation 1015)
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Factoring the DCT Matrix
Take a look at the 4 4 submatrix at the lower right corner of the second matrix in Equation 1015 and in Equation 1016 Equation 1015