  A new non-linear optimization approach is proposed for the sparse
reconstruction of log-conductivities in current density impedance imaging. This
framework comprises of minimizing an objective functional involving a least
squares fit of the interior electric field data corresponding to two boundary
voltage measurements, where the conductivity and the electric potential are
related through an elliptic PDE arising in electrical impedance tomography.
Further, the objective functional consists of a $L^1$ regularization term that
promotes sparsity patterns in the conductivity and a Perona-Malik anisotropic
diffusion term that enhances the edges to facilitate high contrast and
resolution. This framework is motivated by a similar recent approach to solve
an inverse problem in acousto-electric tomography. Several numerical
experiments and comparison with an existing method demonstrate the
effectiveness of the proposed method for superior image reconstructions of a
wide-variety of log-conductivity patterns.
