Discrete Cosine Transform (DCT)

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32 In part M, a simplification can be made: So define ) ( ) ( ) ( ) ( ) ( 6 2 6 6 4 6 6 6 6 6 2 6 6 4 2 6 6 4 2 6 2 4 6 4 6 6 4 6 4 6 6 2 4 6 6 2 4 4 C C a a a C C a C a C a C a C a C a b C C a a a C C a C a C a C a C a C a b + + + − = + − + − = + − = − − + − = + − − − = + − = 4 4 ) ( 4 and 6 6 4 4 6 4 6 6 2 6 2 temp Ra b temp Qa b a a C temp C C R C C Q − = − − = + = + = − =
If you count the operations you will get 29 additions and 5 multiplications. Because an 8x8 2-D DCT can be computed by applying 8-point 1-D DCT to each row and each column of the 2D data. Therefore, 2x8x29 = 464 additions and 2x8x5 = 80 multiplications are required in total. A more efficient DCT algorithm has been proposed by Feig and Winograd in IEEE Trans. on ASSP, pp. 2174-2193, 1992. 33
Page 1 and 2: Discrete Cosine Transform (DCT) 1
Page 3 and 4: Two related transform are the discr
Page 5 and 6: Formulation: DCT is a linear and i
Page 7 and 8: DCT-II: f j = ∑ − n 1 k = 0 x k
Page 9 and 10: DCT-IV: f j = ∑ − n 1 k = 0 x k
Page 11 and 12: Inverse Transforms: IDCT-I is DCT-
Page 13 and 14: References: 1. Rao and Yip, Discre
Page 15 and 16: Then the 1-D DCT is applied to the
Page 17 and 18: 4 C u πu cos S 16 8 ( u) = 7 ∑ x
Page 19 and 20: We have the following DCT v.s. DFT
Page 21 and 22: Let X M = (f(0), f(1), f(2), f(3),
Page 23 and 24: F M = X M x (1/8)T M Because cosα
Page 25 and 26: 25 X M = F M x (8T M -1 ) = FM x L
Page 27 and 28: 27 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
Page 29 and 30: 29 So the IDFT can be performed by
Page 31: 31 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪

Discrete Cosine Transform (DCT)

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