Series reversion for electrical impedance tomography with modeling errors

TL;DR

This paper develops and analyzes series reversion methods for electrical impedance tomography that account for modeling errors and unknown contact admittivities, providing new numerical algorithms and demonstrating their effectiveness with simulated 3D data.

Contribution

It extends series reversion techniques to realistic electrode measurements with unknown internal and contact admittivities, including regularization for modeling errors.

Findings

Methods are proven to be convergent under certain injectivity conditions.

Regularized approaches improve robustness against modeling errors.

Numerical examples demonstrate effectiveness in 3D simulations.

Abstract

This work extends the results of [Garde and Hyv\"onen, Math. Comp. 91:1925-1953] on series reversion for Calder\'on's problem to the case of realistic electrode measurements, with both the internal admittivity of the investigated body and the contact admittivity at the electrode-object interfaces treated as unknowns. The forward operator, sending the internal and contact admittivities to the linear electrode current-to-potential map, is first proven to be analytic. A reversion of the corresponding Taylor series yields a family of numerical methods of different orders for solving the inverse problem of electrical impedance tomography, with the possibility to employ different parametrizations for the unknown internal and boundary admittivities. The functionality and convergence of the methods is established only if the employed finite-dimensional parametrization of the unknowns allows the Fr\'echet derivative of the forward map to be injective, but we also heuristically extend the methods to more general settings by resorting to regularization motivated by Bayesian inversion. The performance of this regularized approach is tested via three-dimensional numerical examples based on simulated data. The effect of modeling errors related to electrode shapes and contact admittances is a focal point of the numerical studies.