An integral equation method for the inverse conductivity problem

TL;DR

This paper introduces an integral equation-based algorithm for reconstructing electrical conductivity images from boundary measurements, utilizing regularization and analytical solutions for improved inverse problem solving.

Contribution

It develops a novel integral equation approach for the inverse conductivity problem, incorporating regularization and analytical eigenfunction techniques.

Findings

Effective reconstruction of conductivity distributions demonstrated.

Regularization improves stability and accuracy.

Analytical solutions simplify computations.

Abstract

We present an image reconstruction algorithm for the Inverse Conductivity Problem based on reformulating the problem in terms of integral equations. We use as data the values of injected electric currents and of the corresponding induced boundary potentials, as well as the boundary values of the electrical conductivity. We have used a priori information to find a regularized conductivity distribution by first solving a Fredholm integral equation of the second kind for the Laplacian of the potential, and then by solving a first order partial differential equation for the regularized conductivity itself. Many of the calculations involved in the method can be achieved analytically using the eigenfunctions of an integral operator defined in the paper.