Artifacts in the inversion of the broken ray transform in the plane

TL;DR

This paper investigates the limitations of reconstructing singularities in the broken ray transform in the plane, showing that conjugate points cause artifacts and analyzing both theoretical and numerical aspects of these artifacts.

Contribution

It provides a detailed analysis of how conjugate points in broken ray transforms lead to artifacts, with specific insights into the V-line and parallel ray transforms.

Findings

Conjugate points prevent local recovery of singularities.

Artifacts arise in the presence of conjugate points during reconstruction.

Numerical experiments illustrate the theoretical limitations.

Abstract

We study the integral transform over a general family of broken rays in $\mathbb{R}^2$. It is natural for broken rays to have conjugate points, for example, when they are reflected from a curved boundary. If there are conjugate points, we show that the singularities cannot be recovered from local data and therefore artifacts arise in the reconstruction. We apply these conclusions to two examples, the V-line Radon transform and the parallel ray transform. In each example, a detailed discussion of the local and global recovery of singularities is given and we perform numerical experiments to illustrate the results.