Image Approximation by Rectangular Wavelet Transform

Author: Zavadsky, Vyacheslav

Source: Journal of Mathematical Imaging and Vision, Volume 27, Number 2, February 2007 , pp. 129-138(10)

Publisher: Springer

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Abstract:

We study image approximation by a separable wavelet basis <EquationSource Format="TEX"><![CDATA[$$ {psi(2^{k_1}x-i)psi(2^{k_2}y-j), phi(x-i)psi(2^{k_2}y-j), psi(2^{k_1}(x-i)phi(y-j), phi(x-i)phi(y-i)},$ where $k_1, k_2 in mathbb{Z}_+; i,jinmathbb{Z}; $$]]></EquationSource> 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 <EquationSource Format="TEX"><![CDATA[$$ mathcal{O}(N^{-M}) $$]]></EquationSource> 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 <EquationSource Format="TEX"><![CDATA[$$ mathcal{O}(N^{-M/2}) $$]]></EquationSource> 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.

Keywords: non-linear wavelet compression; rectangular wavelet transform; hyperbolic wavelets; anisotropic Besov space; sparse grid

Document Type: Research article

DOI: http://dx.doi.org/10.1007/s10851-007-0777-z

Publication date: 2007-02-01