@MASTERSTHESIS\{IMM2013-07023,
    author       = "J. Larsen",
    title        = "Sparsity regularization for inverse problems using curvelets",
    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} Supervisors: Kim Knudsen, {DTU} Compute, kiknu@dtu.dk, and Jakob Lemvig, {DTU} Compute",
    url          = "http://www.compute.dtu.dk/English.aspx",
    abstract     = "In inverse problems information about a physical system is achieved from observations or measurements by reversing the effect of a model that acts on the sought information. Often this approach leads to mathematical problems without existence or uniqueness of a solution, or to problems with an unstable solution in the sense that small perturbations in the observations/measurements might cause unbounded changes to the solution. This issue is known as ill-posedness.

The concept of regularization deals with ill-posedness by replacing the original problem with a nearby problem that is not ill-posed. Different regularization methods are useful in different situation. If for instance a sought solution is known in advance to be an edgecontaining image, this a priori information can be included in the regularized model to compensate for some of the ill posedness.

The relatively new concept of curvelets provide a way to decompose L2(R2) functions into frame-coefficients. Much like for wavelets, the curvelet coefficients with high magnitudes indicate a jump discontinuity at a certain translate of the decomposed function. This feature can be included in a sparsity regularization model that promotes a solution to have many zero valued curvelet coefficients. The sparsity promoting feature thus promotes a solution to contain edges.

This thesis reviews theory on curvelet based regularization in comparison with the more well established edge-preserving methods total variation and wavelet-based regularization. Further, two concrete inverse problems are used to demonstrate inversions using the three different regularization methods. Namely the de-blurring of a digital image and a computed tomography problem are considered.

The curvelet based regularization method shows result with qualities close to the results of total variation regularization in a {2D} tomography problem, and the use of sparse expansions in regularization appears to have a promising future due to the great attention on the subject. The computational costs and extra efforts needed to implement curvelet based regularization compared to total variation does not justify a commercialization of the method in its current version. By construction curvelets capture orientation in more directions than wavelets on {2D} domains and curevelets are therefore better suited than wavelets for {2D} images with singularities along curves."
}