Wavelet Design with Optimally Localized Ambiguity Function: a Variational Approach

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.

Existence of Uncertainty Minimizers for the Continuous Wavelet Transform

“…An alternative approach for defining a wavelet uncertainty, based on the concept of observables, was proposed and investigated in [6,8,9]. Observables are localization operators that enable us to define uncertainty functionals that measure the localization of mother wavelets f in time and scale.…”

“…Two observable-based uncertainty functionals were proposed in [8] and [9]. However, the existence of minimizers of these uncertainty functionals was not proved.…”

“…In this section, we recall the observables approach to wavelet uncertainty functionals, and the two wavelet uncertainties introduced in [6,9].…”

“…The phase space uncertainty, introduced in [9], is a way to model the spread of the 2D ambiguity function…”

“…A main result in [9] is a pull-back of the calculation of the phase space uncertainty to the window function, based on the wavelet-Plancherel theory [13], which makes it considerably easier to work with.…”