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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 |
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Vol. 11 (2018), No. 8, 1901–1943
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DOI: 10.2140/apde.2018.11.1901
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Abstract |
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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
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Mathematical Subject Classification 2010
Primary: 35R30, 58J40, 65N21
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Milestones
Received: 12 December 2016
Revised: 21 September 2017
Accepted: 14 November 2017
Published: 6 June 2018
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Authors |
| Allan Greenleaf | |
| Department of Mathematics University of Rochester Rochester, NY United States |
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| Matti Lassas | |
| Department of Mathematics and
Statistics University of Helsinki Helsinki Finland |
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| Matteo Santacesaria | |
| Dipartimento di Matematica Politecnico di Milano Milano Italy |
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| Samuli Siltanen | |
| Department of Mathematics and
Statistics University of Helsinki Helsinki Finland |
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| Gunther Uhlmann | |
| Department of Mathematics University of Washington Seattle, WA United States |
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