Detection of multiple rigid inclusions from ERT data using the complete-electrode model

Abstract

This paper considers inverse shape reconstruction of rigid inclusions from electrical resistance tomography data and presents a novel combination of forward solution using the method of fundamental solutions (MFS) and Bayesian inverse reconstruction through a Markov chain Monte Carlo (MCMC) algorithm. The MFS provides fast and highly accurate forward solution with less computational effort than the more standard boundary element
and finite element methods. This is particularly important as MCMC inverse solution requires the forward problem to be solved thousands of times. However, using the Bayesian approach and MCMC estimation has big advantages in terms of uncertainty and reliability assessment. The mathematical model is governed by the two-dimensional Laplace’s equation in a multiply-connected domain subject to a homogeneous Dirichlet boundary condition on the unknown rigid inclusion and a piecewise Robin boundary condition on the outer boundary containing the electrodes. The measurements are the constant voltage values on the electrodes calculated from multiple current patterns and solving the inverse problem means detecting the location, shape and size of the inner object, as well as the contact impedances for each electrode (if unknown). A series of numerical examples using
simulated data demonstrate the accuracy and stability of the proposed approach.