A compactness result for a second-order variational discrete model

Andrea Braides; Anneliese Defranceschi; Enrico Vitali

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2012)

  • Volume: 46, Issue: 2, page 389-410
  • ISSN: 0764-583X

Abstract

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We analyze a nonlinear discrete scheme depending on second-order finite differences. This is the two-dimensional analog of a scheme which in one dimension approximates a free-discontinuity energy proposed by Blake and Zisserman as a higher-order correction of the Mumford and Shah functional. In two dimension we give a compactness result showing that the continuous problem approximating this difference scheme is still defined on special functions with bounded hessian, and we give an upper and a lower bound in terms of the Blake and Zisserman energy. We prove a sharp bound by exhibiting the discrete-to-continuous Γ-limit for a special class of functions, showing the appearance new ‘shear’ terms in the energy, which are a genuinely two-dimensional effect.

How to cite

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Braides, Andrea, Defranceschi, Anneliese, and Vitali, Enrico. "A compactness result for a second-order variational discrete model." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 46.2 (2012): 389-410. <http://eudml.org/doc/273165>.

@article{Braides2012,
abstract = {We analyze a nonlinear discrete scheme depending on second-order finite differences. This is the two-dimensional analog of a scheme which in one dimension approximates a free-discontinuity energy proposed by Blake and Zisserman as a higher-order correction of the Mumford and Shah functional. In two dimension we give a compactness result showing that the continuous problem approximating this difference scheme is still defined on special functions with bounded hessian, and we give an upper and a lower bound in terms of the Blake and Zisserman energy. We prove a sharp bound by exhibiting the discrete-to-continuous Γ-limit for a special class of functions, showing the appearance new ‘shear’ terms in the energy, which are a genuinely two-dimensional effect.},
author = {Braides, Andrea, Defranceschi, Anneliese, Vitali, Enrico},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {computer vision; finite-difference schemes; gamma-convergence; free-discontinuity problems; -convergence; Mumford-Shah functional},
language = {eng},
number = {2},
pages = {389-410},
publisher = {EDP-Sciences},
title = {A compactness result for a second-order variational discrete model},
url = {http://eudml.org/doc/273165},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Braides, Andrea
AU - Defranceschi, Anneliese
AU - Vitali, Enrico
TI - A compactness result for a second-order variational discrete model
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2012
PB - EDP-Sciences
VL - 46
IS - 2
SP - 389
EP - 410
AB - We analyze a nonlinear discrete scheme depending on second-order finite differences. This is the two-dimensional analog of a scheme which in one dimension approximates a free-discontinuity energy proposed by Blake and Zisserman as a higher-order correction of the Mumford and Shah functional. In two dimension we give a compactness result showing that the continuous problem approximating this difference scheme is still defined on special functions with bounded hessian, and we give an upper and a lower bound in terms of the Blake and Zisserman energy. We prove a sharp bound by exhibiting the discrete-to-continuous Γ-limit for a special class of functions, showing the appearance new ‘shear’ terms in the energy, which are a genuinely two-dimensional effect.
LA - eng
KW - computer vision; finite-difference schemes; gamma-convergence; free-discontinuity problems; -convergence; Mumford-Shah functional
UR - http://eudml.org/doc/273165
ER -

References

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