Imaging of Anisotropic Conductivities from Current Densities in Two Dimensions

We consider the imaging of anisotropic conductivity tensors gamma - (gamma ij) 1 <= i, j <= 2 from knowledge of several internal current densities J=gamma del u, where u satisfies a second-order elliptic equation. del (gamma del u) = 0 on a bounded domain X subset of R-2 with prescribed boundary conditions on partial derivative X. We show that gamma can be uniquely reconstructed from four well-chosen functionals J and that noise in the data is differentiated once during the reconstruction. The inversion procedure is local in the sense that ( most of) the tensor gamma(x) can be reconstructed from knowledge of the functionals J in the vicinity of x. We obtain the existence of an open set of boundary conditions on partial derivative X that guarantee stable reconstructions by using the technique of complex geometrical optics solutions. The explicit inversion procedure is presented in several numerical simulations, which demonstrate the influence of the choice of boundary conditions on the stability of the reconstruction. This problem finds applications in the medical imaging modality called current density imaging or magnetic resonance electrical impedance tomography.