A direct D-bar reconstruction algorithm for recovering a complex conductivity in 2-D

A direct reconstruction algorithm for complex conductivities in $W^{2,\infty}(\Omega)$, where $\Omega$ is a bounded, simply connected Lipschitz domain in $\mathbb{R}^2$, is presented. The framework is based on the uniqueness proof by Francini [Inverse Problems 20 2000], but equations relating the Dirichlet-to-Neumann to the scattering transform and the exponentially growing solutions are not present in that work, and are derived here. The algorithm constitutes the first D-bar method for the reconstruction of conductivities and permittivities in two dimensions. Reconstructions of numerically simulated chest phantoms with discontinuities at the organ boundaries are included.