Mat-1.218 Inversio-ongelmien seminaari - Seminar on inverse problems

Numerical methods in statistical EIT

Sampsa Pursiainen, 20.-27.10.2003

Electrical Impedance Tomography (EIT) is an imaging method that provides information about the electromagnetic properties within a 2D- or 3D-body based on voltage measurements on the boundary. In this work, the sought quantity is a scalar-valued conductivity distribution. Voltage measurements refer to a finite set of potential values that are measured by an array of contact electrodes attached on the boundary. The voltage data is generated by injecting currents into the domain through the electrodes.

The issue of this work is to discuss numerical methods and computational techniques that can be applied to the discretized mathematical model of the EIT problem and also to use them in connection with some demonstrative numerical simulations. The computational work that has to be performed before resulting in a proper solution is usually large and can often be diminished remarkably by optimizing the efficiency of the applied numerical methods.

The major interest is concentrated on solving the inverse problem in a statistical sense by treating the conductivity as a random variable with some posterior probability distribution and by employing Markov chain Monte Carlo (MCMC) sampling methods for estimating the properties of the posterior distribution. The purpose is to develop such a Monte Carlo algorithm that finding a proper approximative solution would necessitate as small sample enesembles as possible. Drawing a sample from the posterior distribution demands for solving one or more forward problems, i.e. linear systems. Consequently, another important issue is to discover an effective linear algebraic method of solving the forward problem. Statistical solutions are measured against least-squares solutions which appear more frequently in literature.

In the simulations, we restrict ourselves to cases where the conductivity distribution is given in most parts of the body and only a relatively small anomaly is sought. The need for a method of locating small perturbations arises in connection with various real world applications of EIT such as detecting and classifying tumors from breast tissue.

Summarizing the findings, due to the strong ill-conditioned nature and non-linearity of the inverse problem it is often difficult to obtain any appropriate numerical solutions.Workability of the least-squares approach depends on the applied regularization method. It is difficult to construct regularization method favoring arbitrary structures, e.g. strongly discontinuous conductivities. The statistical model is preferable to the least-squares approach only if there is accurate enough a priori knowledge available.

Kenrick Bingham 16.10.2003