Free access
Volume 38, Number 2, March-April 2004
Page(s) 291 - 320
Published online 15 March 2004
  1. R.A. Adams, Sobolev Spaces. Academic Press, New York (1975).
  2. 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. [CrossRef] [MathSciNet]
  3. L. Ambrosio and V.M. Tortorelli, On the approximation of functionals depending on jumps by quadratic, elliptic functionals. Boll. Un. Mat. Ital. 6-B (1992) 105–123.
  4. L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press (2000).
  5. J.W. Barrett, X. Feng and A. Prohl, Convergence of a fully discrete finite element method for a degenerate parabolic system modeling nematic liquid crystals with variable degree of orientation, preprint.
  6. G. Bellettini and A. Coscia, Discrete approximation of a free discontinuity problem. Numer. Funct. Anal. Optimiz. 15 (1994) 201–224. [CrossRef]
  7. A. Blake and A. Zisserman, Visual reconstruction. MIT Press, Cambridge, MA (1987).
  8. B. Bourdin, Image segmentation with a finite element method. ESAIM: M2AN 33 (1999) 229–244. [CrossRef] [EDP Sciences]
  9. A. Braides, Approximation of free-discontinuity problems. Lect. Notes Math. 1694, Springer-Verlag (1998).
  10. A. Braides and G. Dal Maso, Nonlocal approximation of the Mumford-Shah functional. Calc. Var. Partial Differential Equations 5 (1997) 293–322. [CrossRef] [MathSciNet]
  11. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, Second Edition, Springer-Verlag, New York (2001).
  12. J.W. Cahn and J.E. Hilliard, Free energy of a nonuniform system I, Interfacial free energy. J. Chem. Phys. 28 (1958) 258–267. [CrossRef]
  13. A. Chambolle, Image segmentation by variational methods: Mumford-Shah functional and the discrete approximation. SIAM J. Appl. Math. 55 (1995) 827–863. [CrossRef] [MathSciNet]
  14. A. Chambolle and G. Dal Maso, Discrete approximation of the Mumford-Shah functional in dimension two. ESAIM: M2AN 33 (1999) 651–672. [CrossRef] [EDP Sciences]
  15. P.G. Ciarlet, Basic error estimates for elliptic problems, in Handbook of Numer. Anal. II, Elsevier Sciences Publishers (1991).
  16. G. Dal Maso, An introduction to Γ-convergence, Birkhäuser Boston, Boston, MA (1993).
  17. E. De Giorgi, M. Carriero and A. Leaci, Existence theorem for a minimum problem with discontinuity set. Arch. Rat. Mech. Anal. 108 (1989) 195–218. [CrossRef] [MathSciNet]
  18. 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. [MathSciNet]
  19. F. Dibos and E. Séré, An approximation result for the minimizers of the Mumford-Shah functional. Boll. Un. Mat. Ital. A 11 (1997).
  20. C.M. Elliott, D.A. French and F.A. Milner, A second order splitting method for the Cahn-Hilliard equation. Numer. Math. 54 (1989) 575–590. [CrossRef] [MathSciNet]
  21. S. Esedoglu and J. Shen, Digital inpainting based on the Mumford-Shah-Euler image model. European J. Appl. Math. 13 (2002) 353–370. [MathSciNet]
  22. X. Feng and A. Prohl, Analysis of total variation flow and its finite element approximations. ESAIM: M2AN 37 (2003) 533–556. [CrossRef] [EDP Sciences]
  23. X. Feng and A. Prohl, On gradient flow of the Mumford-Shah functional. (in preparation).
  24. D. Geman and S. Geman, Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Trans. Patten Anal. Mach. Intell. 6 (1984) 721–741. [CrossRef] [PubMed]
  25. R. Glowinski, J.L. Lions and R. Trémoliéres, Numerical analysis of variational inequalities. North-Holland, New York. Stud. Math. Appl. 8 (1981).
  26. M. Gobbino, Gradient flow for the one-dimensional Mumford-Shah strategies. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 27 (1998) 145–193. [MathSciNet]
  27. J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod (1969).
  28. R. March and M. Dozio, A variational method for the recovery of smooth boundaries. Im. Vis. Comp. 15 (1997) 705–712. [CrossRef]
  29. L. Modica, The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal. 98 (1987) 123–142. [CrossRef] [MathSciNet]
  30. L. Modica and S. Mortola, Un esempio di Γ-convergenza. Boll. Un. Mat. Ital. B 14 (1977) 285–299. [MathSciNet]
  31. J.-M. Morel and S. Solimini, Variational Methods in Image Segmentation, Birkhäuser (1995).
  32. D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42 (1989) 577–685. [CrossRef] [MathSciNet]
  33. R.H. Nochetto and C. Verdi, Convergence past singularities for a fully discrete approximation of curvature-driven interfaces. SIAM J. Numer. Anal. 34 (1997) 490–512. [CrossRef] [MathSciNet]
  34. J. Simon, Compact sets in the space Lp(0,T;B). Ann. Mat. Pura Appl. 146 (1987) 65–96. [CrossRef] [MathSciNet]
  35. M. Struwe, Geometric evolution problems. IAS/Park City Math. Series 2 (1996) 259–339.

Recommended for you