Propagation and recovery of singularities in the inverse conductivity problem

Abstract

The ill-posedness of Calderón’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. We show in two dimensions how, using EIT data alone, to use propagation of singularities for complex principal-type PDEs to efficiently detect interior jumps and other singularities of the conductivity. Analysis of variants of the CGO solutions of Astala and Päivärinta (Ann. Math. (2) 163:1 (2006), 265–299) allows us to exploit a complex principal-type geometry underlying the problem and show that the leading term in a Born 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 refined descriptions are possible. We present numerics to show that this approach is effective for detecting inclusions within inclusions.