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generalized coarea formula

ESAIM: Mathematical Modelling and Numerical Analysis (1)

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Continuous limits of discrete perimeters
Antonin Chambolle, Alessandro Giacomini, Luca Lussardi
Journal:

ESAIM: Mathematical Modelling and Numerical Analysis / Volume 44 / Issue 2 / March 2010

Published online by Cambridge University Press:

16 December 2009, pp. 207-230

Print publication:

March 2010
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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.