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. © EDP Sciences, SMAI, 2011.

A compactness result for a second-order variational discrete model / Braides, A.; Defranceschi, A.; Vitali, E.. - In: ESAIM. MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS. - ISSN 2822-7840. - 46:2(2012), pp. 389-410. [10.1051/m2an/2011043]

A compactness result for a second-order variational discrete model

Braides A.;
2012-01-01

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. © EDP Sciences, SMAI, 2011.
2012
46
2
389
410
Braides, A.; Defranceschi, A.; Vitali, E.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/139503
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