Photoacoustic tomography in attenuating media with partial data

TL;DR

This paper addresses the challenge of reconstructing initial acoustic sources in photoacoustic tomography within attenuating media, proving uniqueness and stability for partial data, and introducing a new reconstruction method with numerical validation.

Contribution

It establishes the first proof of uniqueness and stability estimates for partial data inverse problems in attenuating media, and proposes a novel Neumann series reconstruction formula.

Findings

Proved uniqueness of the inverse problem with partial data.

Derived stability estimates for the reconstruction.

Developed a new Neumann series reconstruction method.

Abstract

The attenuation of ultrasound waves in photoacoustic and thermoacoustic imaging presents an important drawback in the applicability of these modalities. This issue has been addressed previously in the applied and theoretical literature, and some advances have been made on the topic. In particular, stability inequalities have been proposed for the inverse problem of initial source recovery with partial observations under the assumption of unique determination of the initial pressure. The main goal of this work is to fill this gap, this is, we prove the uniqueness property for the inverse problem and establish the associated stability estimates as well. The problem of reconstructing the initial condition of acoustic waves in the complete-data setting is revisited and a new Neumann series reconstruction formula is obtained for the case of partial observations in a semi-bounded geometry. A numerical simulation is also included to test the method.