A convergent algorithm for the hybrid problem of reconstructing conductivity from minimal interior data

TL;DR

This paper introduces a convergent alternating split Bregman algorithm for reconstructing electrical conductivity from minimal interior data, specifically the magnitude of current density, and demonstrates its effectiveness through numerical experiments.

Contribution

It develops a novel split Bregman method for solving the weighted least gradient problem in conductivity reconstruction, with a detailed convergence proof and a new approach to recover the full current vector field.

Findings

The algorithm converges reliably in numerical tests.

It effectively reconstructs conductivity from minimal interior data.

A new method for recovering the full current vector field from magnitude and boundary data.

Abstract

We consider the hybrid problem of reconstructing the isotropic electric conductivity of a body $\Omega$ from interior Current Density Imaging data obtainable using MRI measurements. We only require knowledge of the magnitude $|J|$ of one current generated by a given voltage $f$ on the boundary $\partial\Omega$. As previously shown, the corresponding voltage potential u in $\Omega$ is a minimizer of the weighted least gradient problem \[u=\hbox{argmin} \{\int_{\Omega}a(x)|\nabla u|: u \in H^{1}(\Omega), \ \ u|_{\partial \Omega}=f\},\] with $a(x)= |J(x)|$. In this paper we present an alternating split Bregman algorithm for treating such least gradient problems, for $a\in L^2(\Omega)$ non-negative and $f\in H^{1/2}(\partial \Omega)$. We give a detailed convergence proof by focusing to a large extent on the dual problem. This leads naturally to the alternating split Bregman algorithm. The dual problem also turns out to yield a novel method to recover the full vector field $J$ from knowledge of its magnitude, and of the voltage $f$ on the boundary. We then present several numerical experiments that illustrate the convergence behavior of the proposed algorithm.