Continuous limits of discrete perimeters
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 2, p. 207-230

We consider a class of discrete convex functionals which satisfy a (generalized) coarea formula. These functionals, based on submodular interactions, arise in discrete optimization and are known as a large class of problems which can be solved in polynomial time. In particular, some of them can be solved very efficiently by maximal flow algorithms and are quite popular in the image processing community. We study the limit in the continuum of these functionals, show that they always converge to some “crystalline” perimeter/total variation, and provide an almost explicit formula for the limiting functional.

DOI : https://doi.org/10.1051/m2an/2009044
Classification:  49Q20,  65K10
Keywords: generalized coarea formula, total variation, anisotropic perimeter
@article{M2AN_2010__44_2_207_0,
author = {Chambolle, Antonin and Giacomini, Alessandro and Lussardi, Luca},
title = {Continuous limits of discrete perimeters},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {44},
number = {2},
year = {2010},
pages = {207-230},
doi = {10.1051/m2an/2009044},
zbl = {1185.94008},
mrnumber = {2655948},
language = {en},
url = {http://www.numdam.org/item/M2AN_2010__44_2_207_0}
}

Chambolle, Antonin; Giacomini, Alessandro; Lussardi, Luca. Continuous limits of discrete perimeters. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 2, pp. 207-230. doi : 10.1051/m2an/2009044. http://www.numdam.org/item/M2AN_2010__44_2_207_0/

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