A Wavelet Plancherel Theory with Application to Multipliers and Sparse Approximations

TL;DR

This paper develops a wavelet-Plancherel theory that extends continuous wavelet analysis, enabling efficient implementation of multiplicative operators and sparse approximations in signal processing.

Contribution

It introduces a new wavelet-Plancherel framework that allows operations in coefficient space to be efficiently performed in the window-signal space, reducing computational complexity.

Findings

Efficient implementation of continuous wavelet multipliers with polynomial symbols.

Linear complexity in the resolution for Calderón-Toeplitz operators.

Framework for greedy sparse approximations using wavelet systems.

Abstract

We introduce an extension of continuous wavelet theory that enables an efficient implementation of multiplicative operators in the coefficient space. In the new theory, the signal space is embedded in a larger abstract signal space -- the so called window-signal space. There is a canonical extension of the wavelet transform to an isometric isomorphism between the window-signal space and the coefficient space. Hence, the new framework is called a wavelet-Plancherel theory, and the extended wavelet transform is called the wavelet-Plancherel transform. Since the wavelet-Plancherel transform is an isometric isomorphism, any operation in the coefficient space can be pulled-back to an operation in the window-signal space. It is then possible to improve the computational complexity of methods that involve a multiplicative operator in the coefficient space, by performing all computations directly in the window-signal space. As one example application, we show how continuous wavelet multipliers (also called Calder\'{o}n-Toeplitz Operators), with polynomial symbols, can be implemented with linear complexity in the resolution of the 1D signal. As another example, we develop a framework for efficiently computing greedy sparse approximations to signals based on elements of continuous wavelet systems.