Vol. 11, No. 8, 2018

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Propagation and recovery of singularities in the inverse conductivity problem

Allan Greenleaf, Matti Lassas, Matteo Santacesaria, Samuli Siltanen and Gunther Uhlmann

Vol. 11 (2018), No. 8, 1901–1943
DOI: 10.2140/apde.2018.11.1901
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.

Keywords
electrical impedance tomography, propagation of singularities, Calderón's problem, tomography, Radon transform
Mathematical Subject Classification 2010
Primary: 35R30, 58J40, 65N21
Milestones
Received: 12 December 2016
Revised: 21 September 2017
Accepted: 14 November 2017
Published: 6 June 2018
Authors
Allan Greenleaf
Department of Mathematics
University of Rochester
Rochester, NY
United States
Matti Lassas
Department of Mathematics and Statistics
University of Helsinki
Helsinki
Finland
Matteo Santacesaria
Dipartimento di Matematica
Politecnico di Milano
Milano
Italy
Samuli Siltanen
Department of Mathematics and Statistics
University of Helsinki
Helsinki
Finland
Gunther Uhlmann
Department of Mathematics
University of Washington
Seattle, WA
United States