Discrete Cosine Transform (DCT)

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DCT is often used in signal and image processing, especially for lossy data compression, because it has a strong “Energy Compaction” property: most of the signal information tends to be concentrated in a few low-frequency components of the DCT, approaching the KLT for signals based on certain limits of Markov processes. DCT is used in JPEG image compression, MJPEG and MPEG video compression. MDCT is used in MP3, AAC, etc. audio compression. 4
Formulation: DCT is a linear and invertible function F: R n →R n (where R denotes the set of real numbers.), or equivalently on nxn matrix. DCT-I: 1 f j = ( x 2 0 + ( −1) j x n−1 ) + ∑ − n 2 k = 1 x k π cos[ n −1 jk] 5
Page 1 and 2: Discrete Cosine Transform (DCT) 1
Page 3: Two related transform are the discr
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 and 32: 31 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪
Page 33: If you count the operations you wil

Discrete Cosine Transform (DCT)

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