Image Approximation by Rectangular Wavelet Transform

Abstract

We study image approximation by a separable wavelet basis $\{\psi(2^{k_1}x-i)\psi(2^{k_2}y-j), \phi(x-i)\psi(2^{k_2}y-j), \psi(2^{k_1}(x-i)\phi(y-j), \phi(x-i)\phi(y-i)\},$ where $k_1, k_2 \in \mathbb{Z}_+; i,j\in\mathbb{Z};$ 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 provide a self-contained proof that if one-dimensional wavelet basis has M dual vanishing moments then the rate of approximation by N coefficients of rectangular wavelet transform is $\mathcal{O}(N^{-M})$ for functions with mixed derivative of order M in each direction. These results are consistent with optimal approximation rates for such functions. The square wavelet transform yields the approximation rate is $\mathcal{O}(N^{-M/2})$ 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 approximation which shows that rectangular wavelet transform outperform the square one.