Image Approximation by Rectangular Wavelet Transform

ISSN： 0924-9907

Source： Journal of Mathematical Imaging and Vision, Vol.27, Iss.2, 2007-02, pp. : 129-138

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Abstract

We study image approximation by a separable wavelet basis and &PHgr;,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 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 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.

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