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

We present a procedure for recovering the conformal factor of an anisotropic conductivity matrix in a known conformal class in a domain in $\mathbb{R}^n$ with $n\geq 2$. The method requires one internal measurement together with a priori knowledge of the conformal class of the conductivity matrix. This problem arises in the medical imaging modality of current density impedance imaging (CDII), and the interior data needed can be obtained using MRI-based techniques for measuring current densities (CDI) and diffusion tensors (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 measurements. 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.