English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We consider the hybrid problem of reconstructing the isotropic electricconductivity of a body $\Omega$ from interior Current Density Imaging dataobtainable 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 analternating split Bregman algorithm for treating such least gradient problems,for $a\in L^2(\Omega)$ non-negative and $f\in H^{1/2}(\partial \Omega)$. Wegive a detailed convergence proof by focusing to a large extent on the dualproblem. This leads naturally to the alternating split Bregman algorithm. Thedual problem also turns out to yield a novel method to recover the full vectorfield $J$ from knowledge of its magnitude, and of the voltage $f$ on theboundary. We then present several numerical experiments that illustrate theconvergence behavior of the proposed algorithm.

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