Image compression by rectangular wavelet transform
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We study image compression by a separable wavelet basis {ψ(2^k_1x-i)ψ(2^k_2y-j), ϕ(x-i)ψ(2^k_2y-j), ψ(2^k_1(x-i)ϕ(y-j), ϕ(x-i)ϕ(y-i)}, where k_1, k_2 ∈Z_+; i,j∈Z; and ϕ,ψ are elements of a standard biorthogonal wavelet basis in L_2(R). Because k_1 k_2, the supports of the basis elements are rectangles, and the corresponding transform is known as the rectangular wavelet transform. We prove that if one-dimensional wavelet basis has M dual vanishing moments then the rate of approximation by N coefficients of rectangular wavelet transform is O(N^-M^C N) for functions with mixed derivative of order M in each direction. The square wavelet transform yields the approximation rate is O(N^-M/2) for functions with all derivatives of the total order M. Thus, the rectangular wavelet transform can outperform the square one if an image has a mixed derivative. We provide experimental comparison of image compression which shows that rectangular wavelet transform outperform the square one.
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