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Issue ESAIM: M2AN
Volume 42, Number 5, September-October 2008
Page(s) 729 - 748
DOI 10.1051/m2an:2008024
Published online 04 July 2008

ESAIM: M2AN 42 (2008) 729-748
DOI: 10.1051/m2an:2008024

A variational model in image processing with focal points

Andrea Braides1 and Giuseppe Riey2

1  Dipartimento di Matematica, Università di Roma `Tor Vergata', Via della Ricerca scientifica 1, 00133 Roma, Italy. riey@mat.unical.it
2  Dipartimento di Matematica, Università della Calabria, Via P. Bucci, 87036 Arcavacata di Rende (CS), Italy.

Received June 6, 2007. Revised January 12, 2008. Published online July 4, 2008.

Abstract
We propose a model for segmentation problems involving an energy concentrated on the vertices of an unknown polyhedral set, where the contours of the images to be recovered have preferred directions and focal points. We prove that such an energy is obtained as a $\Gamma$-limit of functionals defined on sets with smooth boundary that involve curvature terms of the boundary. The minimizers of the limit functional are polygons with edges either parallel to some prescribed directions or pointing to some fixed points, that can also be taken as unknown of the problem.


Mathematics Subject Classification. 68U10, 94A08, 49J45.

Key words: $\Gamma$-convergence, curvature functionals, segmentation problems, image processing.


© EDP Sciences, SMAI 2008


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