Continuous limits of discrete perimeters

Antonin Chambolle; Alessandro Giacomini; Luca Lussardi

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 44, Issue: 2, page 207-230
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Chambolle, Antonin, Giacomini, Alessandro, and Lussardi, Luca. "Continuous limits of discrete perimeters." ESAIM: Mathematical Modelling and Numerical Analysis 44.2 (2010): 207-230. <http://eudml.org/doc/250806>.

@article{Chambolle2010,
abstract = { 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. },
author = {Chambolle, Antonin, Giacomini, Alessandro, Lussardi, Luca},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Generalized coarea formula; total variation; anisotropic perimeter; generalized coarea formula},
language = {eng},
month = {3},
number = {2},
pages = {207-230},
publisher = {EDP Sciences},
title = {Continuous limits of discrete perimeters},
url = {http://eudml.org/doc/250806},
volume = {44},
year = {2010},
}

TY - JOUR
AU - Chambolle, Antonin
AU - Giacomini, Alessandro
AU - Lussardi, Luca
TI - Continuous limits of discrete perimeters
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 44
IS - 2
SP - 207
EP - 230
AB - 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.
LA - eng
KW - Generalized coarea formula; total variation; anisotropic perimeter; generalized coarea formula
UR - http://eudml.org/doc/250806
ER -

References

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