Iterative Potts and Blake-Zisserman Minimization for the Recovery of Functions with Discontinuities from Indirect Measurements

A. Weinmann, M. Storath

Proceedings of The Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 471, no. 2176, paper no. 20140638, April 2015.

Signals with discontinuities appear in many problems in the applied sciences ranging from mechanics, electrical engineering to biology and medicine. The concrete data acquired are typically discrete, indirect and noisy measurements of some quantities describing the signal under consideration. The task is to restore the signal and, in particular, the discontinuities. In this respect, classical methods perform rather poor, whereas non-convex non-smooth variational methods seem to be the correct choice. Examples are methods based on Mumford-Shah and piecewise constant Mumford-Shah functionals and discretized versions which are known as Blake-Zisserman and Potts functionals. Owing to their non-convexity, minimization of such functionals is challenging. In this paper, we propose a new iterative minimization strategy for Blake-Zisserman as well as Potts functionals and a related jump-sparsity problem dealing with indirect, noisy measurements. We provide a convergence analysis and underpin our findings with numerical experiments.

@ARTICLE(http://bigwww.epfl.ch/publications/weinmann1501.html,
AUTHOR="Weinmann, A. and Storath, M.",
TITLE="Iterative {P}otts and {B}lake-{Z}isserman Minimization for the
        Recovery of Functions with Discontinuities from Indirect
        Measurements",
JOURNAL="Proceedings of {T}he {R}oyal {S}ociety of {L}ondon {A}:
        {M}athematical, Physical and Engineering Sciences",
YEAR="2015",
volume="471",
number="2176",
pages="",
month="April",
note="paper no.\ 20140638")

© 2015 The Royal Society of London. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from The Royal Society of London.
This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.