Global calibrations for the non-homogeneous Mumford-Shah functional

Massimiliano Morini

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2002)

  • Volume: 1, Issue: 3, page 603-648
  • ISSN: 0391-173X

Abstract

top
Using a calibration method we prove that, if Γ ⊂ Ω is a closed regular hypersurface and if the function g is discontinuous along Γ and regular outside, then the function u β which solves Δ u β = β ( u β - g ) in Ω ∖ Γ ∂ ν u β = 0 on ∂ Ω ∪ Γ is in turn discontinuous along Γ and it is the unique absolute minimizer of the non-homogeneous Mumford-Shah functional ∫ Ω ∖ S u | ∇ u | 2 d x + ℋ n - 1 ( S u ) + β ∫ Ω ∖ S u ( u - g ) 2 d x , over S B V ( Ω ) , for β large enough. Applications of the result to the study of the gradient flow by the method of minimizing movements are shown.

How to cite

top

Morini, Massimiliano. "Global calibrations for the non-homogeneous Mumford-Shah functional." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 1.3 (2002): 603-648. <http://eudml.org/doc/84482>.

@article{Morini2002,
abstract = {Using a calibration method we prove that, if $\Gamma \subset \Omega $ is a closed regular hypersurface and if the function $g$ is discontinuous along $\Gamma $ and regular outside, then the function $u_\{\beta \}$ which solves\[ \left\lbrace \begin\{array\}\{ll\}\Delta u\_\{\beta \}=\beta (u\_\{\beta \}-g)& \text\{in $\Omega \setminus \Gamma $\}\\ \partial \_\{\nu \} u\_\{\beta \}=0 & \text\{on $\partial \Omega \cup \Gamma $\} \end\{array\}\right.\]is in turn discontinuous along $\Gamma $ and it is the unique absolute minimizer of the non-homogeneous Mumford-Shah functional\[ \int \_\{\Omega \setminus S\_u\}|\nabla u|^2\, dx +\{\mathcal \{H\}\}^\{n-1\}(S\_u)+\beta \int \_\{\Omega \setminus S\_u\}(u-g)^2\, dx, \]over $SBV(\Omega )$, for $\beta $ large enough. Applications of the result to the study of the gradient flow by the method of minimizing movements are shown.},
author = {Morini, Massimiliano},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {3},
pages = {603-648},
publisher = {Scuola normale superiore},
title = {Global calibrations for the non-homogeneous Mumford-Shah functional},
url = {http://eudml.org/doc/84482},
volume = {1},
year = {2002},
}

TY - JOUR
AU - Morini, Massimiliano
TI - Global calibrations for the non-homogeneous Mumford-Shah functional
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2002
PB - Scuola normale superiore
VL - 1
IS - 3
SP - 603
EP - 648
AB - Using a calibration method we prove that, if $\Gamma \subset \Omega $ is a closed regular hypersurface and if the function $g$ is discontinuous along $\Gamma $ and regular outside, then the function $u_{\beta }$ which solves\[ \left\lbrace \begin{array}{ll}\Delta u_{\beta }=\beta (u_{\beta }-g)& \text{in $\Omega \setminus \Gamma $}\\ \partial _{\nu } u_{\beta }=0 & \text{on $\partial \Omega \cup \Gamma $} \end{array}\right.\]is in turn discontinuous along $\Gamma $ and it is the unique absolute minimizer of the non-homogeneous Mumford-Shah functional\[ \int _{\Omega \setminus S_u}|\nabla u|^2\, dx +{\mathcal {H}}^{n-1}(S_u)+\beta \int _{\Omega \setminus S_u}(u-g)^2\, dx, \]over $SBV(\Omega )$, for $\beta $ large enough. Applications of the result to the study of the gradient flow by the method of minimizing movements are shown.
LA - eng
UR - http://eudml.org/doc/84482
ER -

References

top
  1. [1] G. Alberti – G. Bouchitté – G. Dal Maso, The calibration method for the Mumford-Shah functional, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), 249-254. Zbl0948.49005MR1711069
  2. [2] G. Alberti – G. Bouchitté – G. Dal MasoG., The calibration method for the Mumford-Shah functional and free discontinuity problems, Preprint SISSA, Trieste, 2001. Zbl1015.49008MR2001706
  3. [3] L. Ambrosio, Movimenti minimizzanti, Rend. Accad. Naz. Sci. XL Mem. Mat. Sci. Fis. Natur. 113 (1995), 191-246. Zbl0957.49029MR1387558
  4. [4] L. Ambrosio, A compactness theorem for a new class of variational problems, Boll. Un. Mat. It. 3-B (1989), 857-881. Zbl0767.49001MR1032614
  5. [5] L. Ambrosio – N. Fusco – D. Pallara, “Functions of Bounded Variation and Free-Discontinuity Problems”, Oxford University Press, Oxford, 2000. Zbl0957.49001MR1857292
  6. [6] A. Bonnet, On the regularity of edges in image segmentation, Ann. Inst. H. Poincaré, Anal. Non Linéaire. 13 (1996), 485-528. Zbl0883.49004MR1404319
  7. [7] A. Chambolle – F. Doveri, Minimizing movements of the Mumford-Shah energy, Discrete Contin. Dynam. Systems 3 (1997), 153-174. Zbl0948.35073MR1432071
  8. [8] G. Dal Maso – M. G. Mora – M. Morini, Local calibrations for minimizers of the Mumford-Shah functional with rectilinear discontinuity set, J. Math. Pures Appl. 79 (2000), 141-162. Zbl0962.49013MR1749156
  9. [9] E. De Giorgi – L. Ambrosio, Un nuovo funzionale del calcolo delle variazioni, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 82 (1988), 199-210. Zbl0715.49014MR1152641
  10. [10] M. C. Delfour – J. P. Zolésio, Shape Analysis via oriented distance functions, J. Funct. Anal. 123 (1994), 129-201. Zbl0814.49032MR1279299
  11. [11] T. De Pauw – D. Smets, On explicit solutions for the problem of Mumford and Shah, Comm. Contemp. Math. 1 (1999), 201-212. Zbl0953.49022MR1696099
  12. [12] M. Gobbino, Gradient flow for the one-dimensional Mumford-Shah functional, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 27 (1998), 145-193. Zbl0931.49010MR1658873
  13. [13] P. Grisvard, “Majorations en norme du maximum de la résolvante du laplacien dans un polygone. Nonlinear partial differential equations and their applications”, Collège de France Seminar, Vol. XII (Paris, 1991-1993), 87-96, Pitman Res. Zbl0811.35022MR1291845
  14. [14] P. Grisvard, “Elliptic Problems in Nonsmooth Domains”, Monographs and Studies in Mathematics 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. Zbl0695.35060MR775683
  15. [15] A. Lunardi, “Analytic Semigroups and Optimal Regularity in Parabolic Problems”, Progress in Nonlinear Differential Equations and Their Applications, vol. 16, Birkhäuser Verlag, Basel, 1995. Zbl0816.35001MR1329547
  16. [16] M. G. Mora, Local calibrations for minimizers of the Mumford-Shah functional with a triple junction, Preprint SISSA, Trieste, 2001. Zbl1017.49018MR1901148
  17. [17] M. G. Mora – M. Morini, Functional depending on curvatures with constraints, Rend. Sem. Mat. Univ. Padova 104 (2000), 173-199. Zbl1017.49019MR1809356
  18. [18] M. G. Mora – M. Morini, Local calibrations for minimizers of the Mumford-Shah functional with a regular discontinuity set, To appear on Ann. Inst. H. Poincaré, Anal. non linéaire. Zbl1052.49018MR1841127
  19. [19] D. Mumford – J. Shah, Boundary detection by minimizing functionals, I, Proc. IEEE Conf. on Computer Vision and Pattern Recognition (San Francisco, 1985). 
  20. [20] D. Mumford – J. Shah, Optimal approximation by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math. 42 (1989), 577-685. Zbl0691.49036MR997568
  21. [21] T. J. Richardson, Limit theorems for a variational problem arising in computer vision, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19 (1992), 1-49. Zbl0757.49027MR1183756

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.