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Advanced convex analysis for improved variational image reconstruction. (English) Zbl 1414.49001

Münster: Univ. Münster, Mathematisch-Naturwissenschaftliche Fakultät, Fachbereich Mathematik und Informatik (Diss.). 252 p. (2018).
This more than 250-page long work represents the PhD thesis of the author, successfully defended at the University of Münster in the field of Inverse Problems. It consists of nine chapters grouped in four parts, followed by a conclusion and the usual lists of bibliographical references and figures.
The first part consists of two chapters and is dedicated to an introduction to variational methods in inverse problems and image processing, where topics like ill-posed operator equations, Tikhonov regularization, total variation regularization, Bayesian modeling and sparse signal recovery are presented together with corresponding results.
The next two chapters present some exact and inexact first order optimization methods for solving convex optimization problems, respectively. After some preliminaries on convex analysis in Hilbert spaces, the reader finds some facts about gradient descent and forward-backward splitting methods, the latter with some acceleration results, followed by the alternating direction method of multipliers and some primal-dual methods, again with accelerations, as well as some convergence criteria. Some of these algorithms are extended to the inexact case, where additional hypotheses are also considered in order to improve the convergence speed or results, various types of errors being tested in the iterative steps of the algorithms and the corresponding convergence rates analyzed. Computational results of numerical experiments performed on deblurring problems with TV-regularization are provided, too, while some technical results close the chapter.
The second main part of the thesis consists of three chapters and deals with bias (defined as unavoidable, systematic errors, which occur while using variational methods) reduction in variational regularization by means of Bregman distances, more precisely by employing their infimal convolution. The analysis is presented for some model manifolds (differential, absolutely one-homogeneous), in the presence of clean and noisy data, and for the established regularization functionals such as isotropic and anisotropic total variation, and polyhedral regularization. Besides the theoretical results and investigations on connections to other methods some experimental results on \(\ell^1\) deconvolution, anisotropic and isotropic TV-denoising are also provided in order to stress the achieved improvements. Again, some additional technical results are exiled in appendices at the end of the chapters.
The last part comprises two chapters and presents a variational method for joint reconstruction by means of Bregman distances. The concept of Bregman iterations is extended to multiple channels and it is shown that the resulting method is able to couple the edge information of the respective channel, in the context of total variation, leading to a similar structure in all images, that proves to be useful in applications, for instance in medical imaging in the context of positron emission tomography and magnetic resonance imaging. The benefits of channel weighting are stressed and a discussion about parallel and anti-parallel gradients is also provided. Again, computational results for PET-MRI models are provided, showing the viability of the theoretical results.
Worth noticing is also that all the parts end with short sections dedicated to both conclusions and ideas for future research. All in one this is a very interesting PhD thesis, written on a modern and very actual topic that has offered the author both theoretical challenges and nice applications. I am not very sure that the first part of the title was really necessary, but on the other hand it provides a good image to the reader, who is thus warned that there is some solid theory in this work besides the expected applications.

MSC:

49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
49J40 Variational inequalities
49N45 Inverse problems in optimal control
49N60 Regularity of solutions in optimal control
68U10 Computing methodologies for image processing
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