A Bayesian approach and total variation priors in 3D electrical impedance tomography

Abstract

The reconstruction of resistivity distribution in electrical impedance tomography (EIT) is a nonlinear ill-posed inverse problem which necessitates regularization. In this paper the regularized EIT problem is discussed from a Bayesian point of view. The basic idea in the Bayesian approach is to describe the resistivity distribution and voltage measurements as multivariate random variables. The regularization (prior information) is incorporated into the prior density. The solution for the inverse problem is obtained as a point estimate (typically mean or maximum) of the posterior density, which is the product of the prior density and the so-called likelihood density. A class of methods that can be used to compute the posterior mean are the so-called Markov chain Monte Carlo (MCMC) methods. These seem to be especially suitable when the prior information contain inequality constraints and nonsmooth functionals. In this paper the Bayesian approach to three dimensional EIT is examined with an example in which the retrieval of a blocky three dimensional resistivity distribution is carried out by using MCMC methods.