@MASTERSTHESIS\{IMM2013-06780,
    author       = "H. Garde",
    title        = "Sparsity Regularization for Electrical Impedance Tomography",
    year         = "2013",
    school       = "Technical University of Denmark, Department of Applied Mathematics and Computer Science",
    address      = "Richard Petersens Plads, Building 324, {DK-}2800 Kgs. Lyngby, Denmark, compute@compute.dtu.dk",
    type         = "",
    note         = "{DTU} supervisor: Kim Knudsen, kiknu@dtu.dk, {DTU} Compute",
    url          = "http://www.compute.dtu.dk/English.aspx",
    abstract     = "This thesis deals with the inverse problem that arises from electrical impedance tomography (EIT) reconstruction. Here the mathematical foundation is laid for solving the forward {EIT} problem uniquely, along with continuity and differentiability results for the forward problem.
The inverse {EIT} problem is investigated in great theoretical detail with respect to regularization techniques, where sparsity and total variation regularization are used to iteratively give approximate solutions to the problem. The use of multiple datasets and partial data are investigated, along with their respective effect on the solution. The idea of using prior information is applied throughout the thesis, in order to improve these approximate solutions, for instance in terms of the gradient used in the iterative algorithm and the bias that is introduced by the regularization.
Furthermore, I have engineered specific basis functions into the solutions of sparsity regularization, by using different parameters for each basis function. This successfully improves the solution to a degree where it is possible to reconstruct sharp edges and the correct contrast, even for very difficult inclusions, something that is otherwise unheard of for a problem as ill-posed and non-linear as {EIT}. Prior information is also applied in an experiment to have total variation regularization determine an approximation to the support of an inclusion, and use this information to improve the solution from the sparsity regularization.
The iterative methods have been implemented successfully in Python using FEniCS [31], that is based on the finite element method."
}