@MASTERSTHESIS\{IMM2014-07020,
    author       = "M. F. Schmidt",
    title        = "A Non-Linear Diffusion Filtering Method for Reconstruction Problems",
    year         = "2014",
    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     = "The purpose of this thesis is to investigate two different approaches for solving reconstruction problems. Reconstruction problems arise in, for example, medical imaging, satellite imaging, data compression, fingerprint analysis, and much more. A reconstruction problem may be seen as an inverse problem for which regularization methods are applied in order to obtain a reasonable and satisfactory reconstruction. This approach leads to the formulation of a minimization problem for which existence and uniqueness results are proven. The minimization problem turns out to be associated with an Euler-Lagrange equation in distributional sense for the minimizer. This Euler-Lagrange equation is turned into a non-linear diffusion problem for which the existence of a solution to the problem in its strong formulation is studied.

The thesis gives an introduction to regularization methods and minimization problems, followed by a study of the related non-linear diffusion problems and their solutions. The proofs of the existence results for the minimization problem and the non-linear diffusion problem are based on [1] and [2], respectively.

In order to investigate the diffusion problems in greater detail, numerical experiments are performed. The discretized iteration schemes for the diffusion problems are implemented in MatLab. The experiments show that one needs an optimal way of choosing the time step length for the discretized diffusion problem and an optimal stopping criteria for the iteration process."
}