Propagation and recovery of singularities in the inverse conductivity problem

Abstract

The ill-posedness of Calder\'on's inverse conductivity problem, responsible for the poor spatial resolution of Electrical Impedance Tomography (EIT), has been an impetus for the development of hybrid imaging techniques, which compensate for this lack of resolution by coupling with a second type of physical wave, typically modeled by a hyperbolic PDE. Here we show how, using EIT data alone, to efficiently detect interior jumps and other singularities of the conductivity. Analysis of the complex geometrical optics solutions of Astala and P\"aiv\"arinta [\emph{Ann. Math.}, {\bf 163} (2006)] in 2D makes it possible to exploit an underlying complex principal type structure of the problem. We show that the leading term in a Neumann series is an invertible nonlinear generalized Radon transform of the conductivity. The wave front set of all higher-order terms can be characterized, and, under a prior, some are smoother than the leading term. Numerics indicate that this approach effectively detects inclusions within inclusions via EIT.