Current Density Impedance Imaging of an Anisotropic Conductivity in a Known Conformal Class

TL;DR

This paper introduces a method to recover the anisotropic conductivity's conformal factor in a known class using a single internal measurement, relevant for advanced medical imaging techniques like CDII and DTI.

Contribution

It provides a novel reconstruction procedure for anisotropic conductivities in a known conformal class using minimal internal data and a variational approach, extending to inclusions.

Findings

Unique solution for the electric potential via a constrained minimization problem.

Equipotential surfaces are area minimizing in a Riemannian metric derived from data.

Method accommodates perfectly conducting and insulating inclusions.

Abstract

We present a procedure for recovering the conformal factor of an anisotropic conductivity matrix in a known conformal class in a domain in Euclidean space of dimension greater than or equal to 2. The method requires one internal measurement, together with a priori knowledge of the conformal class (local orientation) of the conductivity matrix. This problem arises in the coupled-physics medical imaging modality of Current Density Impedance Imaging (CDII) and the assumptions on the data are suitable for measurements determinable from cross-property based couplings of the two imaging modalities CDII and Diffusion Tensor Imaging (DTI). We show that the corresponding electric potential is the unique solution of a constrained minimization problem with respect to a weighted total variation functional defined in terms of the physical data. Further, we show that the associated equipotential surfaces are area minimizing with respect to a Riemannian metric obtained from the data. The results are also extended to allow the presence of perfectly conducting and/or insulating inclusions.