# A compactness result for a second-order variational discrete model

Andrea Braides; Anneliese Defranceschi; Enrico Vitali

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

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topBraides, 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 -

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