Stochastic Boundary Operators and Model Uncertainties in Electrical Impedance Tomography

Abstract

Electrical impedance tomography (EIT) is an imaging modality which can be applied to conductive targets. In EIT electrodes are placed on the boundary, or surface, of an object and weak electrical currents are injected into the target through the electrodes. The resulting voltages are measured and an estimate for the internal conductivity distribution of the target is computed. There are numerous applications of EIT, from monitoring a patient's pulmonary or brain function to imaging of multi-phase flows or subsurface geophysical structures. The difficulty in EIT, as with all di use tomography modalities, is that the estimation of the internal conductivity distribution is an ill-posed inverse problem. This implies that the estimates are unstable and often non-unique. As a consequence, attention must be paid to the mathematical modelling of the measurements as well as to the estimation methods. Furthermore, in any practical situation where a mathematical model is applied there are parameters which are unknown or uncertain. In order to handle these uncertainties simplifying assumptions or approximations are usually made. These approximations for the modelling uncertainties are typically made since the information is impossible or infeasible to measure. Failing to account for these modelling uncertainties leads to systematic errors, which destroy estimates. Moreover, in practical situations there is limited computational power and time so there is pressure to compute the estimates with a approximative model. A typical way of reducing the computational cost is to shrink the computational domain in which estimates are computed. For example, in geophysical EIT the region of interest may only be very small in comparison to the area where the current ows resulting from the EIT current injections can stretch. If these current ows are not correctly modelled, estimates of the conductivity distribution may have severe artefacts rendering estimate meaningless. In this thesis, we develop a computationally feasible model for practical absolute EIT imaging. Such a model will have to account for the modelling uncertainties which are present in all practical measurements. Also it will have to be suffsaintlyy low dimensional so that the online computation of the conductivity estimates is feasible. In order to achieve this, we develop a model which accounts for the current ows when the computational domain is truncated. The resulting domain truncation model involves the stochastic Dirichlet-to-Neuman operator over the truncation boundary. The Karhunen- Lo eve decomposition is then adapted to give a low-dimensional model for the stochastic Dirichlet-to-Neumann operator. In order to take the modelling errors into account, we use the recently developed Bayesian approximation error approach. In particular, we use the Bayesian approximation error approach to compensate for errors induced from unknown contact impedances, model reduction and the use of an approximative domain shape.