Conformal mapping and impedance tomography

Abstract

Akduman and Kress (2002 Inverse Problems 18 1659–1672), Haddar and Kress (2005 Inverse Problems 21 935–953), and Kress (2004 Math. Comput. Simul. 66 255–265) have employed a conformal mapping technique for the inverse problem to recover a perfectly conducting or a non-conducting inclusion in a homogeneous background medium from the Cauchy data on the accessible exterior boundary. We propose an extension of this approach to two-dimensional inverse electrical impedance tomography with piecewise constant conductivities. A main ingredient of our method is the incorporation of the transmission condition on the unknown interior boundary via a nonlocal boundary condition in terms of an integral equation. We present the foundations of the method, a local convergence result and exhibit the feasibility of the method via numerical examples.