Necessary conditions for extremals of Blake & Zisserman functional

Michele Carriero; Antonio Leaci; Franco Tomarelli

doi:10.1016/S1631-073X(02)02231-8

Necessary conditions for extremals of Blake & Zisserman functional
[Conditions nécessaires d'extrémalité pour la fonctionnelle de Blake & Zisserman]

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

Comptes Rendus. Mathématique, Volume 334 (2002) no. 4, pp. 343-348.

Résumé
Abstract

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 : 2001-09-10
Accepté le : 2001-11-26
Publié le : 2002-01-01

DOI : 10.1016/S1631-073X(02)02231-8

Affiliations des auteurs :

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},
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     pages = {343--348},
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     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/

Bibliographie
Cité par

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[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

Alessandro Lanza; Antonio Leaci; Serena Morigi; Franco Tomarelli Symmetrised fractional variation with L1 fidelity for signal denoising via Grünwald-Letnikov scheme, Applied Mathematics and Computation, Volume 500 (2025), p. 129429 | DOI:10.1016/j.amc.2025.129429
Antonio Leaci; Franco Tomarelli Symmetrized fractional total variation for signal and image analysis, Advances in Continuous and Discrete Models, Volume 2023 (2023) no. 1 | DOI:10.1186/s13662-023-03762-8
Danilo Percivale; Franco Tomarelli Smooth and Broken Minimizers of Some Free Discontinuity Problems, Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs, Volume 22 (2017), p. 431 | DOI:10.1007/978-3-319-64489-9_17
Michele Carriero; Antonio Leaci; Franco Tomarelli Corrigendum to “A candidate local minimizer of Blake and Zisserman functional” [J. Math. Pures Appl. 96 (1) (2011) 58–87], Journal de Mathématiques Pures et Appliquées, Volume 104 (2015) no. 5, p. 1005 | DOI:10.1016/j.matpur.2015.03.012
Michele Carriero; Antonio Leaci; Franco Tomarelli A Survey on the Blake–Zisserman Functional, Milan Journal of Mathematics, Volume 83 (2015) no. 2, p. 397 | DOI:10.1007/s00032-015-0246-x
Paolo Facchi; Marilena Ligabò; Sergio Solimini Tomography: mathematical aspects and applications, Physica Scripta, Volume 90 (2015) no. 7, p. 074007 | DOI:10.1088/0031-8949/90/7/074007
Tommaso Boccellari; Franco Tomarelli Generic uniqueness of minimizer for Blake Zisserman functional, Revista Matemática Complutense, Volume 26 (2013) no. 2, p. 361 | DOI:10.1007/s13163-012-0103-1
M. Carriero; A. Leaci; F. Tomarelli Free gradient discontinuity and image inpainting, Journal of Mathematical Sciences, Volume 181 (2012) no. 6, p. 805 | DOI:10.1007/s10958-012-0716-4
Michele Carriero; Antonio Leaci; Franco Tomarelli Uniform density estimates for Blake Zisserman functional, Discrete Continuous Dynamical Systems - A, Volume 31 (2011) no. 4, p. 1129 | DOI:10.3934/dcds.2011.31.1129
Michele Carriero; Antonio Leaci; Franco Tomarelli A candidate local minimizer of Blake and Zisserman functional, Journal de Mathématiques Pures et Appliquées, Volume 96 (2011) no. 1, p. 58 | DOI:10.1016/j.matpur.2011.01.005
Michele Carriero; Antonio Leaci; Franco Tomarelli Euler equations for Blake and Zisserman functional, Calculus of Variations and Partial Differential Equations, Volume 32 (2008) no. 1, p. 81 | DOI:10.1007/s00526-007-0129-2
Michele Carriero; Antonio Leaci; Franco Tomarelli Calculus of variations and image segmentation, Journal of Physiology-Paris, Volume 97 (2003) no. 2-3, p. 343 | DOI:10.1016/j.jphysparis.2003.09.008

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