Orthogonal Matched Wavelets with Vanishing Moments: A Sparsity Design Approach

Abstract

This paper presents a novel approach to design orthogonal wavelets matched to a signal with compact support and vanishing moments. It provides a systematic and versatile framework for matching an orthogonal wavelet to a specific signal or application. The central idea is to select a wavelet by optimizing a criterion which promotes sparsity of the wavelet representation of a prototype signal. Optimization is performed over the space of orthogonal wavelet functions with compact support, coming from filter banks with a given order. Parametrizations of this space are accompanied by explicit conditions for extra vanishing moments, to be used for constrained optimization. The approach is developed first for critically sampled wavelet transforms, then for undecimated wavelet transforms. Two different optimization criteria are presented: to achieve sparsity by L1-minimization or by L4-norm maximization. Masking can be employed by introducing weighting factors. Examples are given to demonstrate and evaluate the approach.