Wavelet Design with Optimally Localized Ambiguity Function: a Variational Approach

TL;DR

This paper introduces a variational method for designing mother wavelets with minimal ambiguity function spread, resulting in sharper phase space representations and reduced coefficient correlation in continuous wavelet transforms.

Contribution

It proposes a novel optimization approach using wavelet-Plancharel theory to directly design mother wavelets with optimal localization properties in phase space.

Findings

Wavelet ambiguity functions can be minimized using gradient descent.

Optimized wavelets lead to sparser and sharper phase space representations.

The method simplifies the design process by avoiding complex 2D constraints.

Abstract

In this paper, we design mother wavelets for the 1D continuous wavelet transform with some optimality properties. An optimal mother wavelet here is one that has an ambiguity function with minimal spread in the continuous coefficient space (also called phase space). Since the ambiguity function is the reproducing kernel of the coefficient space, optimal windows lead to phase space representations which are "optimally sharp." Namely, the wavelet coefficients have minimal correlations with each other. Such a construction also promotes sparsity in phase space. The spread of the ambiguity function is modeled as the sum of variances along the axes in phase space. In order to optimize the mother wavelet directly as a 1D signal, we pull-back the variances, defined on the 2D phase space, to the so called window-signal space. This is done using the recently developed wavelet-Plancharel theory. The approach allows formulating the optimization problem of the 2D ambiguity function as a minimization problem of the 1D mother wavelet. The resulting 1D formulation is more efficient and does not involve complicated constraints on the 2D ambiguity function. We optimize the mother wavelet using gradient descent, which yields a locally optimal mother wavelet.