Uniqueness, Lipschitz stability and reconstruction for the inverse optical tomography problem

TL;DR

This paper establishes uniqueness and stability results for recovering optical properties in steady-state tomography, and proposes a numerical iterative method for reconstruction with demonstrated numerical examples.

Contribution

It provides the first Lipschitz stability estimates for simultaneous recovery of diffusion and absorption coefficients in optical tomography.

Findings

Proved global uniqueness and Lipschitz stability for absorption with known diffusion.

Established Lipschitz stability for joint diffusion and absorption recovery.

Developed a quasi-Newton iterative algorithm for numerical reconstruction.

Abstract

In this paper, we consider the inverse problem of recovering a diffusion and absorption coefficients in steady-state optical tomography problem from the Neumann-to-Dirichlet map. We first prove a Global uniqueness and Lipschitz stability estimate for the absorption parameter provided that the diffusion is known. Then, we prove a Lipschitz stability result for simultaneous recovery of diffusion and absorption. In both cases the parameters belong to a known finite subspace with a priori known bounds. The proofs relies on a monotonicity result combined with the techniques of localized potentials. To numerically solve the inverse problem, we propose a Kohn-Vogeliustype cost functional over a class of admissible parameters subject to two boundary value problems. The reformulation of the minimization problem via the Neumann-toDirichlet operator allows us to obtain the optimality conditions by using the Frechet differentiability of this operator and its inverse. The reconstruction is then performed by means of an iterative algorithm based on a quasi-Newton method. Finally, we illustrate some numerical results.