Reciprocity gap functional methods for potentials/sources with small volume support for two elliptic equations

TL;DR

This paper develops and analyzes reciprocity gap functional methods for reconstructing small interior regions in inverse shape problems related to diffuse optical tomography and inverse scattering, providing rigorous and computationally simple algorithms.

Contribution

It introduces asymptotic expansions of reciprocity gap functionals and applies MUSIC-type and direct sampling algorithms for small volume region recovery.

Findings

MUSIC-type algorithm effectively recovers subregions in diffuse optical tomography.

Direct sampling method accurately identifies small regions from limited data.

Method is stable under noisy measurement conditions.

Abstract

In this paper, we consider inverse shape problems coming from diffuse optical tomography and inverse scattering. In both problems, our goal is to reconstruct small volume interior regions from measured data on the exterior surface of an object. In order to achieve this, we will derive an asymptotic expansion of the reciprocity gap functional associated with each problem. The reciprocity gap functional takes in the measured Cauchy data on the exterior surface of the object. In diffuse optical tomography, we prove that a MUSIC-type algorithm can be used to recover the unknown subregions. This gives an analytically rigorous and computationally simple method for recovering the small volume regions. For the problem coming from inverse scattering, we recover the subregions of interest via a direct sampling method. The direct sampling method presented here allows us to accurately recover the small volume region from one pair of Cauchy data. We also prove that the direct sampling method is stable with respect to noisy data. Numerical examples will be presented for both cases in two dimensions where the measurement surface is the unit circle.