  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.
