Discrete approximation of the Mumford-Shah functional in dimension two

Antonin Chambolle; Gianni Dal Maso

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (1999)

  • Volume: 33, Issue: 4, page 651-672
  • ISSN: 0764-583X

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Chambolle, Antonin, and Dal Maso, Gianni. "Discrete approximation of the Mumford-Shah functional in dimension two." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 33.4 (1999): 651-672. <http://eudml.org/doc/193939>.

@article{Chambolle1999,
author = {Chambolle, Antonin, Dal Maso, Gianni},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {classes SBV and GSBV of functions of bounded variation; Mumford-Shah functional; variational approximation; -convergence; special functions of bounded variation; integral functionals},
language = {eng},
number = {4},
pages = {651-672},
publisher = {Dunod},
title = {Discrete approximation of the Mumford-Shah functional in dimension two},
url = {http://eudml.org/doc/193939},
volume = {33},
year = {1999},
}

TY - JOUR
AU - Chambolle, Antonin
AU - Dal Maso, Gianni
TI - Discrete approximation of the Mumford-Shah functional in dimension two
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 1999
PB - Dunod
VL - 33
IS - 4
SP - 651
EP - 672
LA - eng
KW - classes SBV and GSBV of functions of bounded variation; Mumford-Shah functional; variational approximation; -convergence; special functions of bounded variation; integral functionals
UR - http://eudml.org/doc/193939
ER -

References

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  1. [1] L. Ambrosio, A compactness theorem for a new class of functions of bounded variation. Boll. Un. Mat. Ital. B 3 (1989) 857-881. Zbl0767.49001MR1032614
  2. [2] L. Ambrosio, Variational problems in SBV and image segmentation. Acta Appl. Math. 17 (1989) 1-40. Zbl0697.49004MR1029833
  3. [3] L. Ambrosio, Existence theory for a new class of variational problems. Arch. Rational Mech. Anal. 111 (1990) 291-322. Zbl0711.49064MR1068374
  4. [4] 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. Zbl0722.49020MR1075076
  5. [5] L. Ambrosio and V.M. Tortorelli, On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. B 6 (1992) 105-123. Zbl0776.49029MR1164940
  6. [6] H. Attouch, Variational Convergence for Functions and Operators. Pitman, London (1984). Zbl0561.49012MR773850
  7. [7] G. Bellettini and A. Coscia, Discrete approximation of a free discontinuity problem. Numer. Funct. Anal. Optim. 15 (1994) 201-224. Zbl0806.49002MR1272202
  8. [8] G. Bouchitté, A. Braides and G. Buttazzo, Relaxation results for some free discontinuity problems. J. Reine Angew. Math. 458 (1995) 1-18. Zbl0817.49015MR1310950
  9. [9] B. Bourdin and A. Chambolle, Implementation of an adaptive finite-elements approximation of the Mumford-Shah functional. Preprint LPMTM, Université Paris-Nord/CEREMADE, Université de Paris-Dauphine (1998); Numer. Math. (to appear). Zbl0961.65062MR1771782
  10. [10] A. Braides and G. Dal Maso, Non-Local Approximation of the Mumford-Shah Functional. Calc. Var. Partial Differential Equations 5 (1997) 293-322. Zbl0873.49009MR1450713
  11. [11] A. Chambolle, Image segmentation by variational methods: Mumford and Shah functional and the discrete approximations. SIAM J. Appl. Math. 55 (1995) 827-863. Zbl0830.49015MR1331589
  12. [12] G. Cortesani, Strong approximation of GSBV functions by piecewise smooth functions. Ann. Univ. Ferrara Sez. VII (N.S.) (to appear). Zbl0916.49002MR1686747
  13. [13] G. Dal Maso, An Introduction to Γ-Convergence. Birkhäuser, Boston (1993). Zbl0816.49001MR1201152
  14. [14] E. De Giorgi and L. Ambrosio, Un nuovo funzionale del calcolo delie variazioni. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 82 (1988) 199-210. Zbl0715.49014MR1152641
  15. [15] E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 58 (1975) 842-850. Zbl0339.49005MR448194
  16. [16] F. Dibos and E. Séré, An approximation result for the minimizers of Mumford-Shah functional. Boll. Un, Mat. Ital. A 11 (1997) 149-162. Zbl0873.49008MR1438364
  17. [17] L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992). Zbl0804.28001MR1158660
  18. [18] H. Federer, Geometric Measure Theory. Springer-Verlag, New York (1969). Zbl0176.00801MR257325
  19. [19] J. Frehse, Capacity methods in the theory of partial differential equations. Jahresber. Deutsch. Math.-Verein. 84 (1982) 1-44. Zbl0486.35002MR644068
  20. [20] E. Giusti, Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Basel (1983). Zbl0545.49018MR775682
  21. [21] J.M. Morel and S. Solimini, Variational Models in Image Segmentation. Birkhäuser, Boston, (1995). Zbl0827.68111MR1321598
  22. [22] D. Mumford and J. Shah, Boundary detection by minimizing functionals, I, in Proc. IEEE Conf. on Computer Vision and Pattern Recognition, San Francisco (1985). 
  23. [23] D. Mumford and J. Shah, Optimal approximation by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42 (1989) 577-685. Zbl0691.49036MR997568
  24. [24] P.A. Raviard and J.M. Thomas, Introduction à l'analyse numérique des équations aux dérivées partielles. Masson, Paris (1983). Zbl0561.65069
  25. [25] W.P. Ziemer, Weakly Differentiable Functions. Springer-Verlag, Berlin (1989). Zbl0692.46022MR1014685

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