Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-06T09:17:26.420Z Has data issue: false hasContentIssue false
  • English
  • Français

A compactness result for a second-order variational discretemodel

Published online by Cambridge University Press:  23 November 2011

Andrea Braides ,
Anneliese Defranceschi  and
Enrico Vitali
Andrea Braides
Affiliation:
Dipartimento di Matematica, Università di Roma ‘Tor Vergata’, via della Ricerca Scientifica, 00133 Rome, Italy. [email protected]
Anneliese Defranceschi
Affiliation:
Dipartimento di Matematica, Università di Trento, via Sommarive 14, 38123 Povo, Italy
Enrico Vitali
Affiliation:
Dipartimento di Matematica ‘F. Casorati’, Università di Pavia, via Ferrata 1, 27100 Pavia, Italy
Get access

Abstract

We analyze a nonlinear discrete scheme depending on second-order finite differences. Thisis the two-dimensional analog of a scheme which in one dimension approximates afree-discontinuity energy proposed by Blake and Zisserman as a higher-order correction ofthe Mumford and Shah functional. In two dimension we give a compactness result showingthat the continuous problem approximating this difference scheme is still defined onspecial functions with bounded hessian, and we give an upper and a lower bound in terms ofthe Blake and Zisserman energy. We prove a sharp bound by exhibiting thediscrete-to-continuous Γ-limit for a special class of functions, showingthe appearance new ‘shear’ terms in the energy, which are a genuinely two-dimensionaleffect.

Keywords

Computer visionfinite-difference schemesgamma-convergencefree-discontinuity problems
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 2 , March 2012 , pp. 389 - 410
Copyright
© EDP Sciences, SMAI, 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alicandro, R. and Cicalese, M., A general integral representation result for continuum limits of discrete energies with superlinear growth. SIAM J. Math. Anal. 36 (2004) 1–37. Google Scholar
Alicandro, R., Focardi, M. and Gelli, M.S., Finite difference approximation of energies in fracture mechanics. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 29 (2000) 671–709. Google Scholar
Ambrosio, L. and Tortorelli, V.M., Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence. Comm. Pure Appl. Math. 43 (1990) 999–1036. Google Scholar
Ambrosio, L. and Tortorelli, V.M., On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. B 6 (1992) 105–123. Google Scholar
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, Oxford (2000).
Ambrosio, L., Faina, L. and March, R., Variational approximation of a second order free discontinuity problem in computer vision. SIAM J. Math. Anal. 32 (2001) 1171–1197. Google Scholar
Bellettini, G. and Coscia, A., Approximation of a functional depending on jumps and corners. Boll. Un. Mat. Ital. B 8 (1994) 151–181. Google Scholar
A. Blake and A. Zisserman, Visual Reconstruction. MIT Press, Cambridge, MA (1987).
Bourdin, B. and Chambolle, A., Implementation of an adaptive finite-element approximation of the Mumford-Shah functional. Numer. Math. 85 (2000) 609–646. Google Scholar
Brady, M. and Horn, B.K.P., Rotationally symmetric operators for surface interpolation. Computer Vision, Graphics, and Image Processing 22 (1983) 70–94. Google Scholar
Braides, A., Lower semicontinuity conditions for functionals on jumps and creases. SIAM J. Math Anal. 26 (1995) 1184–1198. Google Scholar
A. Braides, Approximation of Free-discontinuity Problems. Springer Verlag, Berlin (1998).
A. Braides, Γ -convergence for Beginners. Oxford University Press, Oxford (2002).
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.
Braides, A. and Gelli, M.S., Limits of discrete systems with long-range interactions. J. Convex Anal. 9 (2002) 363–399. Google Scholar
Braides, A. and Piatnitski, A., Overall properties of a discrete membrane with randomly distributed defects. Arch. Ration. Mech. Anal. 189 (2008) 301–323. Google Scholar
Braides, A., Lew, A.J. and Ortiz, M., Effective cohesive behavior of layers of interatomic planes. Arch. Ration. Mech. Anal. 180 (2006) 151–182. Google Scholar
Braides, A., Solci, M. and Vitali, E., A derivation of linear elastic energies from pair-interaction atomistic systems. Netw. Heterog. Media 2 (2007) 551–567 Google Scholar
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.
Carriero, M., Leaci, A. and Tomarelli, F., Strong minimizers of Blake and Zisserman functional. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25 (1997) 257–285. Google Scholar
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
Carriero, M., Leaci, A. and Tomarelli, F., Euler equations for Blake and Zisserman functional. Calc. Var. Partial Diff. Eq. 32 (2008) 81–110. Google Scholar
Carriero, M., Leaci, A. and Tomarelli, F., A Dirichlet problem with free gradient discontinuity. Adv. Mat. Sci. Appl. 20 (2010) 107–141 Google Scholar
Carriero, M., Leaci, A. and Tomarelli, F., A candidate local minimizer of Blake and Zisserman functional. J. Math Pures Appl. 96 (2011) 58–87 Google Scholar
Chambolle, A., Un théorème de Γ-convergence pour la segmentation des signaux. C. R. Acad. Sci., Paris, Ser. I 314 (1992) 191–196. Google Scholar
Chambolle, A., Image segmentation by variational methods: Mumford and Shah functional and the discrete approximations. SIAM J. Appl. Math. 55 (1995) 827–863. Google Scholar
Chambolle, A., Finite-differences approximation of the Mumford-Shah functional. ESAIM: M2AN 33 (1999) 261–288. Google Scholar
Chambolle, A. and Dal Maso, G., Discrete approximation of the Mumford-Shah functional in dimension two. ESAIM: M2AN 33 (1999) 651–672. Google Scholar
P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Co., Amsterdam (1978).
Conti, S., 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. Google Scholar
Geman, S. and Geman, D., Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE PAMI 6 (1984) 721–724. Google ScholarPubMed
W.E.L. Grimson, From Images to Surfaces. The MIT Press Classic Series. MIT, Cambridge (1981).
Mumford, D. and Shah, J., Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42 (1989) 577–685. Google Scholar
Santos, P. and Zappale, E., Lower Semicontinuity in SBH. Mediterranean J. Math. 5 (2008) 221–235. Google Scholar
Schmidt, B., On the derivation of linear elasticity from atomistic models. Netw. Heterogen. Media 4 (2009) 789–812. Google Scholar