On the identification of piecewise constant coefficients in optical diffusion tomography by level set

TL;DR

This paper introduces a level set regularization method with a split strategy for the challenging inverse problem of simultaneously identifying piecewise constant diffusion and absorption coefficients in optical tomography, supported by theoretical guarantees and numerical tests.

Contribution

It develops a novel combined level set and split strategy approach for joint coefficient identification, with proven regularization properties and demonstrated numerical effectiveness.

Findings

The method guarantees regularization under certain conditions.

Numerical tests show effective simultaneous coefficient reconstruction.

The approach handles the nonlinear, ill-posed nature of the problem.

Abstract

In this paper, we propose a level set regularization approach combined with a split strategy for the simultaneous identification of piecewise constant diffusion and absorption coefficients from a finite set of optical tomography data (Neumann-to-Dirichlet data). This problem is a high nonlinear inverse problem combining together the exponential and mildly ill-posedness of diffusion and absorption coefficients, respectively. We prove that the parameter-to-measurement map satisfies sufficient conditions (continuity in the $L^1$ topology) to guarantee regularization properties of the proposed level set approach. On the other hand, numerical tests considering different configurations bring new ideas on how to propose a convergent split strategy for the simultaneous identification of the coefficients. The behavior and performance of the proposed numerical strategy is illustrated with some numerical examples.