Mat-1.600 Laskennallisen tieteen ja
tekniikan seminaari
15.3.2004 14.15
U322
Sampsa Pursiainen, TKK Matematiikan
laitos
Numerical Methods in
Statistical EIT
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 case,
the sought quantity is a scalar-valued conductivity distribution within
the domain. 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 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. One of the central aims of this thesis is to
introduce methods that can rather commonly be told to be suitable for
solving the EIT problem.
The major interest is concentrated on solving the inverse problem in
terms of Bayesian statistics 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 regularized least-squares solutions which appear more
frequently in literature.
In the simulations, we restrict ourselves to cases where $\sigma$ known
in most parts of the domain 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. The statistical model is
preferable to the least-squares approach only if there is accurate
enough a priori knowledge available. Especially in cases where the
nature of the conductivity distribution is strongly discontinuous it is
advantageous to use the statistical formulation.