A convergent algorithm for the hybrid problem of reconstructing conductivity from minimal interior data
This work provides a convergent algorithm for a hybrid imaging problem that uses minimal interior data, which is important for medical imaging applications like MRI-based conductivity reconstruction.
The paper presents an alternating split Bregman algorithm for reconstructing isotropic electric conductivity from interior current density imaging data, requiring only the magnitude of one current and boundary voltage. The algorithm is proven convergent via dual problem analysis, and numerical experiments demonstrate its behavior.
We consider the hybrid problem of reconstructing the isotropic electric conductivity of a body $Ω$ 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 $\partialΩ$. As previously shown, the corresponding voltage potential u in $Ω$ 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 $a\in L^2(Ω)$ non-negative and $f\in H^{1/2}(\partial Ω)$. 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.