Sparsity regularization for inverse problems using curvelets | Jacob Larsen
| | 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. | | Type | Master's thesis [Academic thesis] | | Year | 2013 | | Publisher | 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 | | Series | DTU Compute M.Sc.-2013 | | Note | DTU Supervisors: Kim Knudsen, DTU Compute, kiknu@dtu.dk, and Jakob Lemvig, DTU Compute | | Electronic version(s) | [pdf] | | Publication link | http://www.compute.dtu.dk/English.aspx | | BibTeX data | [bibtex] | | IMM Group(s) | Scientific Computing |
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