  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.
