Wavelet-vaguelette decompositions and homogeneous equations

Namyong Lee, Purdue University

Abstract

We describe the wavelet-vaguelette decomposition (WVD) for solving a homogeneous equation $Y=Af+Z,$ where A satisfies $\widehat{A\sp{\*}Af}(\xi)=\vert\xi\vert\sp{-2\alpha}\ f(\xi)$ for some $\alpha\ge0.$ We find a sufficient condition on functions to have a WVD. This result generalizes Daubechies's work on the discrete wavelet transform. We examine the relation between the WVD-based method and variational problems for solving a homogeneous equation. Algorithms are derived as exact minimizers of variational problems of the form; given observed function Y, minimize over all g in the Besov space $B\sbsp{1,1}{\beta\sb0}(R\sp{d})$ the functional $\Vert Y-Ag\Vert\sbsp{\cal Y}{2}+2\gamma\vert g\vert\sb{B\sbsp{1,1}{\beta\sb0}},$ where $\cal Y$ is a separable Hilbert space. We use the theory of nonlinear wavelet approximation in $L\sp2(R\sp{d})$ to derive accurate error bounds for recovering f through wavelet shrinkage applied to observed data Y corrupted with independent and identically distributed mean zero Gaussian noise Z. We give a new proof of the rate of convergence of wavelet shrinkage that allows us to estimate rather sharply the best shrinkage parameter. We conduct tomographic reconstruction computations that support the hypothesis that near-optimal shrinkage parameters can be derived if one knows (or can estimate) only two parameters about a phantom image f: the largest $\beta$ for which $f\in B\sbsp{p,p}{\beta}(R\sp2),\ p={3\over\beta+3/2},$ and the seminorm $\vert f\vert\sb{B\sbsp{p,p}{\beta}}.$ Both theoretical and experimental results indicate that our choice of shrinkage parameters yields uniformly better results than Kolaczyk's procedure and classical filtered backprojection method.

Degree

Ph.D.

Advisors

Lucier, Purdue University.

Subject Area

Mathematics

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