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

TL;DR

This paper introduces the first D-bar method for directly reconstructing complex conductivities and permittivities in 2D, using a novel framework based on a proven uniqueness theorem, demonstrated with simulated chest phantom images.

Contribution

It develops a new direct D-bar reconstruction algorithm for complex conductivities in 2D, extending previous theoretical results to practical imaging applications.

Findings

Successful reconstruction of simulated chest phantoms with boundary discontinuities

First implementation of D-bar method for complex conductivity and permittivity in 2D

Demonstrates feasibility of direct nonlinear inversion in medical imaging

Abstract

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.