Tomograms and other transforms. A unified view

TL;DR

This paper introduces a unified framework for wavelet-like, quasidistribution, and tomographic transforms, providing explicit formulas and detailed analysis of transforms related to symplectic and affine groups, with a focus on scale-time and scale-frequency tomograms.

Contribution

It presents a comprehensive unification of various signal transforms, deriving explicit relations and analyzing properties of tomograms within a common theoretical framework.

Findings

Explicit formulas relating different transforms are derived.

The properties of scale-time and scale-frequency tomograms are thoroughly analyzed.

Transform interpretations as sampling tools and projectors are provided.

Abstract

A general framework is presented which unifies the treatment of wavelet-like, quasidistribution, and tomographic transforms. Explicit formulas relating the three types of transforms are obtained. The case of transforms associated to the symplectic and affine groups is treated in some detail. Special emphasis is given to the properties of the scale-time and scale-frequency tomograms. Tomograms are interpreted as a tool to sample the signal space by a family of curves or as the matrix element of a projector.