Inverse problem for a nonlocal diffuse optical tomography equation

TL;DR

This paper investigates a nonlocal inverse problem in diffuse optical tomography, proving uniqueness of diffusion and absorption coefficients under certain conditions and demonstrating the optimality of these conditions.

Contribution

It establishes the uniqueness of coefficients in a nonlocal diffuse optical tomography inverse problem and shows the optimality of the measurement condition.

Findings

Uniqueness of coefficients given identical DN data and boundary conditions.

Optimality of the boundary measurement condition.

Construction of counterexamples without the boundary condition.

Abstract

In this article a nonlocal analogue of an inverse problem in diffuse optical tomography is considered. We show that whenever one has given two pairs of diffusion and absorption coefficients $(\gamma_j,q_j)$, $j=1,2$, such that there holds $q_1=q_2$ in the measurement set $W$ and they generate the same DN data, then they are necessarily equal in $\mathbb{R}^n$ and $\Omega$, respectively. Additionally, we show that the condition $q_1|_W=q_2|_W$ is optimal in the sense that without this restriction one can construct two distinct pairs $(\gamma_j,q_j)$, $j=1,2$ generating the same DN data.