Geodesic
A compactness result for a second-order variational discrete model
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 2, pp. 389-410

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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.

DOI : 10.1051/m2an/2011043
Classification : 49J45, 49Q20, 68U10, 65D19, 65M06
Keywords: computer vision, finite-difference schemes, gamma-convergence, free-discontinuity problems 
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     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://geodesic.mathdoc.fr/articles/10.1051/m2an/2011043/}
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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

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