The X-ray transform on a generic family of smooth curves

TL;DR

This paper investigates the limitations of reconstructing singularities from X-ray transform data over smooth curves in a Riemannian plane, highlighting issues caused by conjugate points and demonstrating findings through numerical experiments.

Contribution

It reveals the impact of conjugate points on the recoverability of singularities in the X-ray transform and provides numerical evidence of these phenomena.

Findings

Singularities cannot be recovered locally in the presence of conjugate points.

Artifacts may appear in reconstructions due to these limitations.

Numerical experiments support the theoretical results.

Abstract

We study the X-ray transform over a generic family of smooth curves in $\mathbb{R}^2$ with a Riemannian metric $g$. We show that the singularities cannot be recovered from local data in the presence of conjugate points, and therefore artifacts may arise in the reconstruction. We perform numerical experiments to illustrate the results.