@MASTERSTHESIS\{IMM2005-03849,
    author       = "J. Pedersen",
    title        = "Modular Algorithms for Large Scale Total Variation Image Deblurring",
    year         = "2005",
    keywords     = "Total Variation, Tikhonov, ill posed problems, regularization, steep gradients, MOORe Tools, large scale inversion, iterative methods, object oriented implementation, non linear optimization, Newton's method, augmented system",
    school       = "Informatics and Mathematical Modelling, Technical University of Denmark, {DTU}",
    address      = "Richard Petersens Plads, Building 321, {DK-}2800 Kgs. Lyngby",
    type         = "",
    note         = "Supervised by Professor Per Christian Hansen and Associate Professor Hans Bruun Nielsen",
    url          = "http://www2.compute.dtu.dk/pubdb/pubs/3849-full.html",
    abstract     = "We present the theory behind inverse problems and illustrate that regularization is needed to obtain useful solutions. The most frequently used solution techniques for solving ill posed problems are based on the use of 2 norms, which are known not to be suitable when edges are desired in the regularized solution. The thesis covers the development of an algorithm for solving ill posed problems, where edges and steep gradients are allowed. The key idea is to make use of Total Variation, thus the algorithm solves a Tikhonov based optimization problem with the Total Variation functional as penalty term. The algorithm is based on Newton s method with a line search procedure.

The Total Variation functional is the essential subject for allowing edges in the solutions. We explain the theory connected with Total Variation, and we discuss how important it is to use an appropriate discretization, and propose a way of doing this. Furthermore, we discuss how to deal with the non linearity of the functional by introducing an extra set of variables. Introducing these variables we obtain an augmented system, which is globally  more  linear and thus easier to solve. The purpose of this is to obtain a faster convergence.

The implementation of the algorithm is carried out with focus on efficiency. It is implemented in the framework of the modular Matlab toolbox MOORe Tools, which is designed for solving large scale inverse problems. This results in a modular and object oriented structure, which together with the interior use of iterative methods makes the implementation suitable for solving large scale problems. All implementation aspects are explained in detail. Finally, some experiments using the algorithm are carried out."
}