Broken ray tomography in the disk

TL;DR

This paper investigates the problem of reconstructing functions inside a disk using integrals over broken rays starting and ending on a boundary subset, providing positive results under certain geometric and regularity conditions.

Contribution

It offers new positive results for broken ray tomography in a disk when the function is quasianalytic in the angular variable and the boundary set of tomography is open.

Findings

Unique reconstruction when the domain is a disk and the function is quasianalytic.

Reconstruction is possible with an open boundary subset of tomography.

Analysis of the singleton boundary case.

Abstract

Given a bounded $C^1$ domain $\Omega\subset\R^n$ and a nonempty subset $E$ of its boundary (set of tomography), we consider broken rays which start and end at points of $E$. We ask: If the integrals of a function over all such broken rays are known, can the function be reconstructed? We give positive answers when $\Omega$ is a ball and the unknown function is required to be uniformly quasianalytic in the angular variable and the set of tomography is open. We also analyze the situation when the set of tomography is a singleton.