Image compression by rectangular wavelet transform

Original · EN

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; and ϕ,ψ are elements of a standard biorthogonal wavelet basis in L₂(R). Because k₁≠ 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⁻ᵐᶜ 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.