All the Groups of Signal Analysis from the (1+1)-affine Galilei Group

TL;DR

This paper explores the connections between the (1+1)-affine Galilei group and key groups in signal analysis, revealing how they are related through subgroups and extensions, and analyzing their unitary representations.

Contribution

It uncovers the structural relationships between the affine Galilei group and major signal analysis groups, providing a unified framework and representation analysis.

Findings

All key groups can be derived as subgroups or extensions of the affine Galilei group.

The unitary representations of these groups are studied in detail.

A unified perspective on signal analysis groups is established.

Abstract

We study the relationship between the (1+1)-affine Galilei group and four groups of interest in signal analysis and image processing, viz., the wavelet or the affine group of the line, the Weyl-Heisenberg, the shearlet and the Stockwell groups. We show how all these groups can be obtained either directly as subgroups, or as subgroups of central extensions of the affine Galilei group. We also study this at the level of unitary representations of the groups on Hilbert spaces.