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.
Mots-clés : computer vision, finite-difference schemes, gamma-convergence, free-discontinuity problems
@article{M2AN_2012__46_2_389_0,
author = {Braides, Andrea and Defranceschi, Anneliese and Vitali, Enrico},
title = {A compactness result for a second-order variational discrete model},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {389--410},
publisher = {EDP-Sciences},
volume = {46},
number = {2},
year = {2012},
doi = {10.1051/m2an/2011043},
mrnumber = {2855647},
zbl = {1272.49095},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2011043/}
}
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 PY - 2012 SP - 389 EP - 410 VL - 46 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2011043/ DO - 10.1051/m2an/2011043 LA - en ID - M2AN_2012__46_2_389_0 ER -
%0 Journal Article %A Braides, Andrea %A Defranceschi, Anneliese %A Vitali, Enrico %T A compactness result for a second-order variational discrete model %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2012 %P 389-410 %V 46 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2011043/ %R 10.1051/m2an/2011043 %G en %F M2AN_2012__46_2_389_0
Braides, Andrea; Defranceschi, Anneliese; Vitali, Enrico. A compactness result for a second-order variational discrete model. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 2, pp. 389-410. doi : 10.1051/m2an/2011043. http://archive.numdam.org/articles/10.1051/m2an/2011043/
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