Inversion of circular means and the wave equation on convex planar domains

TL;DR

This paper develops explicit inversion formulas for recovering initial data of the wave equation in convex planar domains from boundary measurements, with exact formulas for circular and elliptical domains, advancing hybrid imaging techniques.

Contribution

It introduces back-projection type inversion formulas for the wave equation in convex domains, including exact formulas for circular and elliptical cases, and extends to mean value recovery over boundary-centered circles.

Findings

Exact inversion formulas for circular and elliptical domains.

The operator al_0m vanishes in circular and elliptical cases.

Applicable to photoacoustic and thermoacoustic tomography.

Abstract

We study the problem of recovering the initial data of the two dimensional wave equation from values of its solution on the boundary $\partial \Om$ of a smooth convex bounded domain $\Om \subset \R^2$. As a main result we establish back-projection type inversion formulas that recover any initial data with support in $\Om$ modulo an explicitly computed smoothing integral operator $\K_\Om$. For circular and elliptical domains the operator $\K_\Om$ is shown to vanish identically and hence we establish exact inversion formulas of the back-projection type in these cases. Similar results are obtained for recovering a function from its mean values over circles with centers on $\partial \Om$. Both reconstruction problems are, amongst others, essential for the hybrid imaging modalities photoacoustic and thermoacoustic tomography.