Conformal mapping and impedance tomography

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.