Comptes Rendus
no. 4
Necessary conditions for extremals of Blake & Zisserman functional
[Conditions nécessaires d'extrémalité pour la fonctionnelle de Blake & Zisserman]
Michele Carriero1 ; Antonio Leaci1 ; Franco Tomarelli2
1 Dipartimento di Matematica “Ennio De Giorgi”, Via Arnesano, I-73100 Lecce, Italy
2 Dipartimento di Matematica “Francesco Brioschi”, Politecnico, Piazza Leonardo da Vinci 32, I-20133, Milano, Italy
Comptes Rendus. Mathématique, Volume 334 (2002) no. 4, pp. 343-348.

On donne des conditions nécessaires de minimisation d'une fonctionnelle dépendant de discontinuités libres et de dérivées secondes, reliée à la segmentation d'images. On exhibe un candidat explicite vérifiant toutes les conditions d'extrémalité.

We show some necessary conditions for minimizers of a functional depending on free discontinuities, free gradient discontinuities and second derivatives, which is related to image segmentation. A candidate for minimality of main part of the functional is explicitly exhibited

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)02231-8
Michele Carriero 1 ; Antonio Leaci 1 ; Franco Tomarelli 2

1 Dipartimento di Matematica “Ennio De Giorgi”, Via Arnesano, I-73100 Lecce, Italy
2 Dipartimento di Matematica “Francesco Brioschi”, Politecnico, Piazza Leonardo da Vinci 32, I-20133, Milano, Italy 
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     title = {Necessary conditions for extremals of {Blake} & {Zisserman} functional},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {343--348},
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     year = {2002},
     doi = {10.1016/S1631-073X(02)02231-8},
     language = {en},
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Michele Carriero; Antonio Leaci; Franco Tomarelli. Necessary conditions for extremals of Blake & Zisserman functional. Comptes Rendus. Mathématique, Volume 334 (2002) no. 4, pp. 343-348. doi : 10.1016/S1631-073X(02)02231-8. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02231-8/

[1] L. Ambrosio; N. Fusco; D. Pallara Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monographs, Oxford University Press, 2000

[2] A. Blake; A. Zisserman Visual Reconstruction, MIT Press, Cambridge, 1987

[3] M. Carriero; A. Leaci; F. Tomarelli Free gradient discontinuities (G. Buttazzo; G. Bouchitté; P. Suquet, eds.), Calculus of Variations, Homogenization and Continuum Mechanics, World Scientific, Singapore, 1994, pp. 131-147

[4] M. Carriero; A. Leaci; F. Tomarelli A second order model in image segmentation: Blake & Zisserman functional (R. Serapioni; F. Tomarelli, eds.), Variational Methods for Discontinuous Structures, Birkäuser, 1996, pp. 57-72

[5] M. Carriero; A. Leaci; F. Tomarelli Strong minimizers of Blake & Zisserman functional, Ann. Scuola Norm. Sup. Pisa, Cl. Sci. (4), Volume 25 (1997), pp. 257-285

[6] M. Carriero; A. Leaci; F. Tomarelli Density estimates and further properties of Blake & Zisserman functional (P.D. Panagiotopoulos; R. Gilbert; P.M. Pardalos, eds.), From Convexity to Nonconvexity, Kluwer Academic, 2001, pp. 381-392

[7] E. De Giorgi Free discontinuity problems in calculus of variations (R. Dautray, ed.), Frontiers Pure Appl. Math., North–Holland, Amsterdam, 1991, pp. 55-61

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