Uncertainty principles and optimally sparse wavelet transforms

TL;DR

This paper develops a new framework for analyzing wavelet transform localization, leading to the design of uncertainty-minimizing windows that produce optimally sparse representations.

Contribution

It introduces a novel localization framework distinguishing group generators from observables, and defines transform uncertainty based on the entire window, enhancing sparsity in wavelet transforms.

Findings

Uncertainty-minimizing windows lead to sparser wavelet representations.

The new framework relates window uncertainty to the localization of the wavelet kernel.

The approach diverges from traditional localization theories by focusing on the whole window's effect.

Abstract

In this paper we introduce a new localization framework for wavelet transforms, such as the 1D wavelet transform and the Shearlet transform. Our goal is to design nonadaptive window functions that promote sparsity in some sense. For that, we introduce a framework for analyzing localization aspects of window functions. Our localization theory diverges from the conventional theory in two ways. First, we distinguish between the group generators, and the operators that measure localization (called observables). Second, we define the uncertainty of a signal transform based on a window as a whole, instead of defining the uncertainty of an individual window. We show that the uncertainty of a window function, in the signal space, is closely related to the localization of the reproducing kernel of the wavelet transform, in phase space. As a result, we show that using uncertainty minimizing window functions, results in representations which are optimally sparse in some sense.