| Summary: | <p>This thesis is concerned with the forward and inverse problems associated with Electrical Capacitance Tomography (ECT) and Electrical Impedance Tomography (EIT). We collectively refer to these paradigms, and other associated modalities, as Electric Field Sensing (EFS). Common to all EFS modalities is the aim to determine the conductivity and permittivity of the material inside a volume from measurements of the electric potential and electric flux taken on the boundary of the volume. EFS modalities have many applications in industrial processes and medical diagnostics due mainly to the non-invasive, non-ionising, and non-destructive property of electric field measurements.</p>
<p>For each application the choice of electrode configuration and sensor input are critical: the shape of the generated electric field and the measurement sensitivity and material specificity are dependent on these choices. We are particularly motivated to study planar configurations of electrodes with time-dependent voltage impulses applied sequentially to different subsets of the electrodes.</p>
<p>We derive five PDE models that describe the electric field in either the frequency or the time domain. In particular, we propose a novel choice of boundary conditions, determined by a coupled ODE, to model additional circuit components and parasitic capacitances inherent in all EFS sensors. For each model, we prove existence and uniqueness results. We also discuss the numerical solution of each model using the Finite Element Method (FEM).</p>
<p>We then consider the inverse problems associated to each of the five models. We detail four, practically inspired, measurement paradigms and prove equivalence by explicitly constructing bijections between them. Our unified approach of combining sequential measurements can be applied to each of the five models. Most notably, we prove the equivalence between sequential measurements taken on two electrodes with measurements taken concurrently on all electrodes. We also discuss the partial data case where not all electrode pairs are sampled.</p>
<p>Finally, in the case of the partial data, we discuss novel methods for estimating the missing data using known measurements of a closely related parameter distribution: the benefit being reconstruction algorithms developed for the full data problem can then be applied to the partial data case.</p>
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