By Lotfi Asker Zadeh
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2— Calculation of 2D DCT by Sparse Matrix and Cosine Symmetry In this method, the 1D DCT is expressed as a matrix product, and is factored into 4 sparse matrices, only one of which contains values other than 0, 1, or -1 [CSF77]. Page 48 When extended to the 2D case by forming the tensor product of the 1D DCTs, the multiplication matrix can be optimized using cosine identities [PM93]. 6 The 2D DCT transform is calculated by first performing row-wise 1D DCT. and then column-wise 1D DCT. This factoring is suggested by the dataflow diagram for the 1D DCT.
PN-1), we want to find a set of basis vectors β = (β0, β1, . βN-1), so we can rewrite x in this basis as xk = (k0, k1, . kN-1). We choose β in such way that a truncated representation t of x, given as tk = (k0, k1, . kM-1, 0,0, . 0) has minimum error: It can be shown that this transformation [RY90]: 1. completely decorrelates the signal in transform space, 2. minimizes the mean square error in its truncated representation (guarantees minimum error in compression), 3. concentrates the most energy in the fewest number of coefficients, 4.
We assume a first-order autoregessive model, where r is the onestep correlation coefficient: The variance s2 of en is: and s2 is related to the variance of the pixel distribution sx2: Statistics have been collected for the correlation ρ in the pixel, scan line, and motion directions for various video effects (motion, pan, break). 2. These images show extremely high correlations in all three directions, showing that the energy of the pixel representation can be concentrated in each direction with the DCT.