A classification of continuous wavelet transforms in dimension three

TL;DR

This paper classifies all matrix groups in three dimensions that produce continuous wavelet transforms with irreducible quasi-regular representations, enabling the development of sparse signal spaces and atomic decompositions.

Contribution

It provides a comprehensive classification of such groups in three dimensions, facilitating the construction of wavelet-based sparse representations.

Findings

Complete catalogue of matrix groups in GL(3,R) for wavelet transforms

Existence results for compactly supported admissible vectors

Framework for atomic decompositions in associated signal spaces

Abstract

This paper presents a full catalogue, up to conjugacy and subgroups of finite index, of all matrix groups $H < {\rm GL}(3,\mathbb{R})$ that give rise to a continuous wavelet transform with associated irreducible quasi-regular representation. For each group in this class, coorbit theory allows to consistently define spaces of sparse signals, and to construct atomic decompositions converging simultaneously in a whole range of these spaces. As an application of the classification, we investigate the existence of compactly supported admissible vectors and atoms for the groups.