A geometric approach to three-dimensional discrete electrical impedance tomography

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Abstract

Electrical impedance tomography (EIT) is an imaging modality with many possiblepractical applications. It is mainly used for geophysical applications, for which it iscalled electrical resistivity tomography. There have also been many proposed medicalapplications such as respiratory monitoring and breast tumour screening.Although there have been many uniqueness and stability results published over thelast few decades, most of the results are in the context of the theoretical continuousproblem. In practice however, we almost always have to solve a discretised problemfor which very few theoretical results exist. In this thesis we aim to bridge the gapbetween the continuous and discrete problems.The first problem we solve is the three-dimensional triangulation problem of uniquelyembedding a tetrahedral mesh in R3. We parameterise the problem in terms of dihedral angles and we provide a constructive procedure for identifying the independentangles 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 locallyunique. We also present a numerical example to illustrate that the result works inpractice. Without the understanding of the geometric constraints involved in em-bedding a three-dimensional triangulation, we cannot solve more complex problemsinvolving embeddings of finite element meshes.We next investigate the discrete EIT problem for anisotropic conductivity. It iswell known that the entries of the finite element system matrix for piecewise linearpotential and piecewise constant conductivity are equivalent to conductance values ofresistors defined on the edges of the finite element mesh. We attempt to tackle theproblem of embedding a finite element mesh in R3, such that it is consistent with someknown edge conductance values.It is a well known result that for the anisotropic conductivity problem, the bound-ary data is invariant under diffeomorphisms that fix the boundary. Before investigatingthis effect on the discrete case, we define the linear map from conductivities to edgeconductances and investigate the injectivity of this map for a simplistic example. Thisprovides an illustrative example of how a poor choice of finite element mesh can resultin a non-unique solution to the discrete inverse problem of EIT. We then extend theinvestigation to finding interior vertex positions and conductivity distributions thatare consistent with the known edge conductances. The results show that if the totalnumber of interior vertex coordinates and anisotropic conductivity variables is largerthan the number of edges in the mesh, then there exist discrete diffeomorphisms thatperturb the vertices and conductivities such that no change in the edge conductancesis observed. We also show that the non-uniqueness caused by the non-injectivity ofthe linear map has a larger effect than the non-uniqueness caused by diffeomorphisminvariance.

Keyword(s)

discrete electrical impedance tomography

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