A compactness result for a second-order variational discrete model
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 2, p. 389-410

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 : https://doi.org/10.1051/m2an/2011043
Classification:  49J45,  49Q20,  68U10,  65D19,  65M06
Keywords: 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 - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {46},
     number = {2},
     year = {2012},
     pages = {389-410},
     doi = {10.1051/m2an/2011043},
     zbl = {1272.49095},
     mrnumber = {2855647},
     language = {en},
     url = {http://www.numdam.org/item/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 - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 2, pp. 389-410. doi : 10.1051/m2an/2011043. http://www.numdam.org/item/M2AN_2012__46_2_389_0/

[1] R. Alicandro and M. Cicalese, A general integral representation result for continuum limits of discrete energies with superlinear growth. SIAM J. Math. Anal. 36 (2004) 1-37. | MR 2083851 | Zbl 1070.49009

[2] R. Alicandro, M. Focardi and M.S. Gelli, Finite difference approximation of energies in fracture mechanics. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 29 (2000) 671-709. | Numdam | MR 1817714 | Zbl 1072.49020

[3] L. Ambrosio and V.M. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence. Comm. Pure Appl. Math. 43 (1990) 999-1036. | MR 1075076 | Zbl 0722.49020

[4] L. Ambrosio and V.M. Tortorelli, On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. B 6 (1992) 105-123. | MR 1164940 | Zbl 0776.49029

[5] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, Oxford (2000). | MR 1857292 | Zbl 0957.49001

[6] L. Ambrosio, L. Faina and R. March, Variational approximation of a second order free discontinuity problem in computer vision. SIAM J. Math. Anal. 32 (2001) 1171-1197. | MR 1856244 | Zbl 0996.46014

[7] G. Bellettini and A. Coscia, Approximation of a functional depending on jumps and corners. Boll. Un. Mat. Ital. B 8 (1994) 151-181. | MR 1274324 | Zbl 0808.49014

[8] A. Blake and A. Zisserman, Visual Reconstruction. MIT Press, Cambridge, MA (1987). | MR 919733

[9] B. Bourdin and A. Chambolle, Implementation of an adaptive finite-element approximation of the Mumford-Shah functional. Numer. Math. 85 (2000) 609-646. | MR 1771782 | Zbl 0961.65062

[10] M. Brady and B.K.P. Horn, Rotationally symmetric operators for surface interpolation. Computer Vision, Graphics, and Image Processing 22 (1983) 70-94. | Zbl 0535.65003

[11] A. Braides, Lower semicontinuity conditions for functionals on jumps and creases. SIAM J. Math Anal. 26 (1995) 1184-1198. | MR 1347416 | Zbl 0832.49009

[12] A. Braides, Approximation of Free-discontinuity Problems. Springer Verlag, Berlin (1998). | MR 1651773 | Zbl 0909.49001

[13] A. Braides, Γ-convergence for Beginners. Oxford University Press, Oxford (2002). | MR 1968440 | Zbl 1198.49001

[14] A. Braides, Discrete approximation of functionals with jumps and creases, in Homogenization, 2001 (Naples) GAKUTO Internat. Ser. Math. Sci. Appl. 18. Tokyo, Gakkōtosho (2003) 147-153. | MR 2022557 | Zbl 1039.49029

[15] A. Braides and M.S. Gelli, Limits of discrete systems with long-range interactions. J. Convex Anal. 9 (2002) 363-399. | MR 1970562 | Zbl 1031.49022

[16] A. Braides and A. Piatnitski, Overall properties of a discrete membrane with randomly distributed defects. Arch. Ration. Mech. Anal. 189 (2008) 301-323. | MR 2413098 | Zbl 1147.74039

[17] A. Braides, A.J. Lew and M. Ortiz, Effective cohesive behavior of layers of interatomic planes. Arch. Ration. Mech. Anal. 180 (2006) 151-182. | MR 2210908 | Zbl 1093.74013

[18] A. Braides, M. Solci and E. Vitali, A derivation of linear elastic energies from pair-interaction atomistic systems. Netw. Heterog. Media 2 (2007) 551-567 | MR 2318845 | Zbl 1183.74017

[19] M. Carriero, A. Leaci and F. Tomarelli, A second order model in image segmentation: Blake and Zisserman functional, in Variational Methods for Discontinuous Structures (Como, 1994), Progr. Nonlin. Diff. Eq. Appl. 25, edited by R. Serapioni and F. Tomarelli. Basel, Birkhäuser (1996) 57-72. | MR 1414488 | Zbl 0915.49004

[20] M. Carriero, A. Leaci and F. Tomarelli, Strong minimizers of Blake and Zisserman functional. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25 (1997) 257-285. | Numdam | MR 1655518 | Zbl 1015.49010

[21] M. Carriero, A. Leaci and F. Tomarelli, Density estimates and further properties of Blake and Zisserman functional, in From Convexity to Nonconvexity, Nonconvex Optim. Appl. 55, edited by R. Gilbert and Pardalos. Kluwer Acad. Publ., Dordrecht (2001) 381-392 | MR 1864879 | Zbl 1043.49003

[22] M. Carriero, A. Leaci and F. Tomarelli, Euler equations for Blake and Zisserman functional. Calc. Var. Partial Diff. Eq. 32 (2008) 81-110. | MR 2377407 | Zbl 1138.49021

[23] M. Carriero, A. Leaci and F. Tomarelli, A Dirichlet problem with free gradient discontinuity. Adv. Mat. Sci. Appl. 20 (2010) 107-141 | MR 2760721 | Zbl 1220.49001

[24] M. Carriero, A. Leaci and F. Tomarelli, A candidate local minimizer of Blake and Zisserman functional. J. Math Pures Appl. 96 (2011) 58-87 | MR 2812712 | Zbl 1218.49029

[25] A. Chambolle, Un théorème de Γ-convergence pour la segmentation des signaux. C. R. Acad. Sci., Paris, Ser. I 314 (1992) 191-196. | MR 1150831 | Zbl 0772.49010

[26] A. Chambolle, Image segmentation by variational methods: Mumford and Shah functional and the discrete approximations. SIAM J. Appl. Math. 55 (1995) 827-863. | MR 1331589 | Zbl 0830.49015

[27] A. Chambolle, Finite-differences approximation of the Mumford-Shah functional. ESAIM: M2AN 33 (1999) 261-288. | Numdam | MR 1700035 | Zbl 0947.65076

[28] A. Chambolle and G. Dal Maso, Discrete approximation of the Mumford-Shah functional in dimension two. ESAIM: M2AN 33 (1999) 651-672. | Numdam | MR 1726478 | Zbl 0943.49011

[29] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Co., Amsterdam (1978). | MR 520174 | Zbl 0511.65078

[30] S. Conti, I. Fonseca and G. Leoni A Γ-convergence result for the two-gradient theory of phase transitions. Comm. Pure Appl. Math. 55 (2002) 857-936. | MR 1894158 | Zbl 1029.49040

[31] S. Geman and D. Geman, Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE PAMI 6 (1984) 721-724. | Zbl 0573.62030

[32] W.E.L. Grimson, From Images to Surfaces. The MIT Press Classic Series. MIT, Cambridge (1981).

[33] D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42 (1989) 577-685. | MR 997568 | Zbl 0691.49036

[34] P. Santos and E. Zappale, Lower Semicontinuity in SBH. Mediterranean J. Math. 5 (2008) 221-235. | MR 2427396 | Zbl 1169.49009

[35] B. Schmidt, On the derivation of linear elasticity from atomistic models. Netw. Heterogen. Media 4 (2009) 789-812. | MR 2552170 | Zbl 1183.74020