Monotonicity-based reconstruction of extreme inclusions in electrical impedance tomography

TL;DR

This paper extends the monotonicity-based method in electrical impedance tomography to effectively reconstruct extreme inclusions, such as perfect conductors or insulators, by establishing new convergence principles for boundary maps.

Contribution

It introduces a convergence result for the Neumann-to-Dirichlet map when conductivities approach zero or infinity, enabling the method to handle extreme inclusions.

Findings

Successfully reconstructs perfectly conducting inclusions.

Handles perfectly insulating inclusions.

Generalizes to inclusions with conductivities between zero and infinity.

Abstract

The monotonicity-based approach has become one of the fundamental methods for reconstructing inclusions in the inverse problem of electrical impedance tomography. Thus far the method has not been proven to be able to handle extreme inclusions that correspond to some parts of the studied domain becoming either perfectly conducting or perfectly insulating. The main obstacle has arguably been establishing suitable monotonicity principles for the corresponding Neumann-to-Dirichlet boundary maps. In this work, we tackle this shortcoming by first giving a convergence result in the operator norm for the Neumann-to-Dirichlet map when the conductivity coefficient decays to zero and/or grows to infinity in some given parts of the domain. This allows passing the necessary monotonicity principles to the limiting case. Subsequently, we show how the monotonicity method generalizes to the definite case of reconstructing either perfectly conducting or perfectly insulating inclusions, as well as to the indefinite case where the perturbed conductivity can take any values between, and including, zero and infinity.