Windowed Radon Transforms, Analytic Signals and the Wave Equation

TL;DR

This paper introduces a windowed Radon transform as a generalization of wavelet transforms, derives an inversion formula, and applies it to the wave equation, leading to wavelet expansions adapted to solutions.

Contribution

It presents a novel windowed Radon transform, derives its inversion, and applies it to develop wavelet solutions for the wave equation, extending analytic signal concepts to higher dimensions.

Findings

Derived a reconstruction formula for the windowed Radon transform.

Introduced the Analytic--Signal transform (AST) as a special case.

Applied the AST to expand solutions of the wave equation.

Abstract

The act of measuring a physical signal or field suggests a generalization of the wavelet transform that turns out to be a windowed version of the Radon transform. A reconstruction formula is derived which inverts this transform. A special choice of window yields the "Analytic--Signal transform" (AST), which gives a partially analytic extension of functions from R^n to C^n. For n =1, this reduces to Gabor's classical definition of "analytic signals." The AST is applied to the wave equation, giving an expansion of solutions in terms of wavelets specifically adapted to that equation and parametrized by real space and imaginary time coordinates (the "Euclidean region").