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Publication 19-CNA-017

Mumford-Shah functionals on graphs and their asymptotics

Marco Caroccia
Scuola Normale Superiore
Piazza dei Cavalieri, 7, 56126 Pisa PI, Italy
and
Universita degli studi di Firenze
Dipartimento di Matematica e Informatica "Ulisse Dini"
Viale Giovanni Battista Morgagni, 67/A, 50134 Firenze FI, Italy
marco.caroccia@sns.it; marco.caroccia@unifi.it

Antonin Chambolle
CMAP, CNRS and École Polytechnique
91128 Palaiseau, France
chambolle.antonin@polytechnique.fr

Dejan Slepčev
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
slepcev@andrew.cmu.edu

Abstract: We consider adaptations of the Mumford-Shah functional to graphs. These are based on discretizations of nonlocal approximations to the Mumford-Shah functional.Motivated by applications in machine learning we study the random geometric graphs associated to random samples of a measure. We establish the conditions on the graph constructions under which the minimizers of graph Mumford-Shah functionals converge to a minimizer of a continuum Mumford-Shah functional. Furthermore we explicitly identify the limiting functional. Moreover we describe an efficient algorithm for computing the approximate minimizers of the graph Mumford-Shah functional.

Get the paper in its entirety as 19-CNA-017.pdf

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