A convergent algorithm for the hybrid problem of reconstructing conductivity from minimal interior 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 \[u=\hbox{argmin} \{\int_Ωa(x)|\nabla u|: u \in H^{1}(Ω), \ \ u|_{\partial Ω}=f\},\] 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.

References in corpus (4)

• General Resolvents for Monotone Operators: Characterization and Extension
• Convergence of the alternating split Bregman algorithm in infinite-dimensional Hilbert spaces
• Conductivity imaging from one interior measurement in the presence of perfectly conducting and insulating inclusions
• New Demiclosedness Principles for (firmly) nonexpansive operators

Cited by in corpus (5)

• Imaging Conductivity from Current Density Magnitude using Neural Networks
• Uniqueness of minimizers of weighted least gradient problems arising in conductivity imaging
• A regularized weighted least gradient problem for conductivity imaging
• Least Gradient Problems with Neumann Boundary Condition
• Electrical Networks with Prescribed Current and Applications to Random Walks on Graphs