Title:

A geometric approach to threedimensional discrete electrical impedance tomography

Electrical impedance tomography (EIT) is an imaging modality with many possible practical applications. It is mainly used for geophysical applications, for which it is called electrical resistivity tomography. There have also been many proposed medical applications such as respiratory monitoring and breast tumour screening. Although there have been many uniqueness and stability results published over the last few decades, most of the results are in the context of the theoretical continuous problem. In practice however, we almost always have to solve a discretised problem for which very few theoretical results exist. In this thesis we aim to bridge the gap between the continuous and discrete problems. The first problem we solve is the threedimensional triangulation problem of uniquely embedding a tetrahedral mesh in R3. We parameterise the problem in terms of dihedral angles and we provide a constructive procedure for identifying the independent angles and the independent set of constraints that the dependent angles must satisfy. We then use the implicit function theorem to prove that the embedding is locally unique. We also present a numerical example to illustrate that the result works in practice. Without the understanding of the geometric constraints involved in embedding a threedimensional triangulation, we cannot solve more complex problems involving embeddings of finite element meshes. We next investigate the discrete EIT problem for anisotropic conductivity. It is well known that the entries of the finite element system matrix for piecewise linear potential and piecewise constant conductivity are equivalent to conductance values of resistors defined on the edges of the finite element mesh. We attempt to tackle the problem of embedding a finite element mesh in R3, such that it is consistent with some known edge conductance values. It is a well known result that for the anisotropic conductivity problem, the boundary data is invariant under diffeomorphisms that fix the boundary. Before investigating this effect on the discrete case, we define the linear map from conductivities to edge conductances and investigate the injectivity of this map for a simplistic example. This provides an illustrative example of how a poor choice of finite element mesh can result in a nonunique solution to the discrete inverse problem of EIT. We then extend the investigation to finding interior vertex positions and conductivity distributions that are consistent with the known edge conductances. The results show that if the total number of interior vertex coordinates and anisotropic conductivity variables is larger than the number of edges in the mesh, then there exist discrete diffeomorphisms that perturb the vertices and conductivities such that no change in the edge conductances is observed. We also show that the nonuniqueness caused by the noninjectivity of the linear map has a larger effect than the nonuniqueness caused by diffeomorphism invariance.
