Image compression by rectangular wavelet transform

We study image compression by a separable wavelet basis { ψ(2k1x − i)ψ(2k2y − j), φ(x − i)ψ(2k2y − j), ψ(2k1(x − i)φ(y − j), φ(x − i)φ(y − i) } , where k1, k2 ∈ Z+; i, j ∈ Z; and φ, ψ are elements of a standard biorthogonal wavelet basis in L2(R). Because k1 6= k2, 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 log 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.