The Identification Problem for the attenuated X-ray transform

TL;DR

This paper investigates the conditions under which the attenuation and source in the attenuated X-ray transform can be uniquely recovered, highlighting the role of Hamiltonian flow and support conditions for stability.

Contribution

It provides new insights into the uniqueness, non-uniqueness, and stability of recovering attenuation and source functions, especially under non-trapping and radial conditions.

Findings

Unique recovery when perturbations are supported in non-trapping regions.

Non-uniqueness results for radial attenuation and source.

Hölder stability estimates for the inverse problem.

Abstract

We study the problem of recovery both the attenuation $a$ and the source $f$ in the attenuated X-ray transform in the plane. We study the linearization as well. It turns out that there are natural Hamiltonian flow that determines which singularities we can recover. If the perturbations $\delta a$, $\delta f$ are supported in a compact set that is non-trapping for that flow, then the problem is well posed. Otherwise, it may not be, and least in the case of radial $a$, $f$, it is not. We present uniqueness and non-uniqueness results for both the linearized and the non-linear problem; as well as a H\"older stability estimate.