Geodesic
Free gradient discontinuity and image inpainting
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XX, Tome 390 (2011), pp. 92-116 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

We introduce and study a formulation of inpainting problem for 2-dimensional images which are locally damaged. This formulation is based on the regularization of the solution of a second order variational problem with Dirichlet boundary condition. A variational approximation algorithm is proposed. 
@article{ZNSL_2011_390_a3,
     author = {M. Carriero and A. Leaci and F. Tomarelli},
     title = {Free gradient discontinuity and image inpainting},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {92--116},
     year = {2011},
     volume = {390},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_390_a3/}
}
TY  - JOUR
AU  - M. Carriero
AU  - A. Leaci
AU  - F. Tomarelli
TI  - Free gradient discontinuity and image inpainting
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2011
SP  - 92
EP  - 116
VL  - 390
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2011_390_a3/
LA  - en
ID  - ZNSL_2011_390_a3
ER  - 
%0 Journal Article
%A M. Carriero
%A A. Leaci
%A F. Tomarelli
%T Free gradient discontinuity and image inpainting
%J Zapiski Nauchnykh Seminarov POMI
%D 2011
%P 92-116
%V 390
%U http://geodesic.mathdoc.fr/item/ZNSL_2011_390_a3/
%G en
%F ZNSL_2011_390_a3
M. Carriero; A. Leaci; F. Tomarelli. Free gradient discontinuity and image inpainting. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XX, Tome 390 (2011), pp. 92-116. http://geodesic.mathdoc.fr/item/ZNSL_2011_390_a3/

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

[2] L. Ambrosio, L. Faina, R. March, “Variational approximation of a second order free discontinuity problem in computer vision”, SIAM J. Math. Anal., 32 (2001), 1171–1197 | DOI | MR | Zbl

[3] G. Aubert, P. Kornprobst, Mathematical problems in image processing, Partial Differential Equations and the Calculus of Variations, Appl. Math. Sci., 147, 2nd ed., Springer, New York, 2006 | MR | Zbl

[4] M. Bertalmío, A. Bertozzi, G. Sapiro, “Navier–Stokes, fluid dynamics, and image and video inpainting”, Proc. IEEE Int. Conf. on Comp. Vision and Pattern Recog., Hawai, 2001

[5] M. Bertalmío, V. Caselles, S. Masnou, G. Sapiro, “Inpainting”, Encyclopedia of Computer Vision, Springer, 2011

[6] M. Bertalmío, L. Vese, G. Sapiro, S. Osher, “Simultaneous structure and texture image inpainting”, IEEE Transactions on Image Processing, 12:8 (2003), 882–889 | DOI

[7] T. Boccellari, F. Tomarelli, “About well-posedness of optimal segmentation for Blake and Zisserman functional”, Istituto Lombardo (Rend. Scienze), 142 (2008), 237–266 | MR

[8] T. Boccellari, F. Tomarelli, Generic uniqueness of minimizer for Blake and Zisserman functional, QDD 66, Dip. Matematica, Politecnico di Milano, 2010, 73 pp. http://www1.mate.polimi.it/biblioteca/qddview.php?id=1390& L=i

[9] A. Blake, A. Zisserman, Visual Reconstruction, The MIT Press, Cambridge, 1987 | MR

[10] A. Braides, A. Defranceschi, E. Vitali, “A compactness result for a second-order variational discrete model”, ESAIM Math. Model. Numer. Anal. (to appear)

[11] G. Carboni, M. Carriero, A. Leaci, I. Sgura, F. Tomarelli, Variational Segmentation and Inpainting via the Blake and Zisserman Functional (to appear)

[12] G. Carboni, I. Sgura, Finite difference approximation of 4th order BVPs for detecting discontinuities in digital data (to appear)

[13] M. Carriero, A. Farina, I. Sgura, “Image segmentation in the framework of free discontinuity problems”, Calculus of Variations: Topics from the Mathematical Heritage of E. De Giorgi, Quad. Mat., 14, Dept. Math., Seconda Univ. Napoli, Caserta, 2004, 86–133 | MR

[14] M. Carriero, A. Leaci, “Existence theorem for a Dirichlet problem with free discontinuity set”, Nonlinear Analysis Th. Meth. Appl., 15:7 (1990), 661–677 | DOI | MR | Zbl

[15] M. Carriero, A. Leaci, F. Tomarelli, “Free gradient discontinuities”, Calculus of Variations, Homogeneization and Continuum Mechanics (Marseille, 1993), Ser. Adv. Math. Appl. Sci., 18, World Sci. Publishing, River Edge, NJ, 1994, 131–147 | MR | Zbl

[16] M. Carriero, A. Leaci, F. Tomarelli, “A second order model in image segmentation: Blake and Zisserman functional”, Variational Methods for Discontinuous Structures (Como, 1994), Progr. Nonlinear Differential Equations Appl., 25, Birkhäuser, Basel, 1996, 57–72 | MR | Zbl

[17] M. Carriero, A. Leaci, F. Tomarelli, “Strong minimizers of Blake Zisserman functional”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 25:1–2 (1997), 257–285 | MR | Zbl

[18] M. Carriero, A. Leaci, F. Tomarelli, “Density estimates and further properties of Blake and Zisserman functional”, From Convexity to Nonconvexity, Nonconvex Optim. Appl., 55, eds. R. Gilbert, Pardalos, Kluwer Acad. Publ., Dordrecht, 2001, 381–392 | DOI | MR | Zbl

[19] M. Carriero, A. Leaci, F. Tomarelli, “Necessary conditions for extremals of Blake and Zisserman functional”, C. R. Math. Acad. Sci. Paris, 334:4 (2002), 343–348 | DOI | MR | Zbl

[20] M. Carriero, A. Leaci, F. Tomarelli, “Calculus of Variations and image segmentation”, J. Physiology Paris, 97:2–3 (2003), 343–353 | DOI

[21] M. Carriero, A. Leaci, F. Tomarelli, “Second Order Variational Problems with Free Discontinuity and Free Gradient Discontinuity”, Calculus of Variations: Topics from the Mathematical Heritage of E. De Giorgi, Quad. Mat., 14, Dept. Math., Seconda Univ. Napoli, Caserta, 2004, 135–186 | MR

[22] M. Carriero, A. Leaci, F. Tomarelli, “Euler equations for Blake Zisserman functional”, Calc. Var. Partial Diff. Eqs., 32:1 (2008), 81–110 | DOI | MR | Zbl

[23] M. Carriero, A. Leaci, F. Tomarelli, “Variational approach to image segmentation”, Pure Math. Appl., 20 (2009), 141–156 | MR | Zbl

[24] M. Carriero, A. Leaci, F. Tomarelli, “Uniform density estimates for Blake Zisserman functional”, Discrete Contin. Dyn. Syst. Series A, 31:3 (2011), 1129–1150 | DOI | MR | Zbl

[25] M. Carriero, A. Leaci, F. Tomarelli, “A Dirichlet problem with free gradient discontinuity”, Adv. Math. Sci. Appl., 20:1 (2010), 107–141 | MR | Zbl

[26] M. Carriero, A. Leaci, F. Tomarelli, “A candidate local minimizer of Blake Zisserman functional”, J. Math. Pures Appl., 96 (2011), 58–87 | DOI | MR | Zbl

[27] M. Carriero, A. Leaci, F. Tomarelli, “About Poincaré Inequalities for Functions Lacking Summability”, Note Mat., 31:1 (2011), 67–84 | MR

[28] M. Carriero, A. Leaci, F. Tomarelli, Variational approximation of a second order free discontinuity problem for inpainting (to appear)

[29] M. Carriero, A. Leaci, F. Tomarelli, Segmentation and inpainting of color images (to appear)

[30] V. Caselles, G. Haro, G. Sapiro, J. Verdera, “On geometric variational models for inpainting surface holes”, Computer Vision and Image Understanding, 111 (2008), 351–373 | DOI

[31] T. F. Chan, S. H. Kang, J. Shen, “Euler's elastica and curvature based inpainting”, SIAM J. Appl. Math., 63:2 (2002), 564–592 | MR | Zbl

[32] T. F. Chan, J. Shen, “Variational image inpainting”, Comm. Pure Appl. Math., 58 (2005), 579–619 | DOI | MR | Zbl

[33] E. De Giorgi, “Free discontinuity problems in calculus of variations”, Frontiers in Pure Appl. Math., ed. R. Dautray, North–Holland, Amsterdam, 1991, 55–61 | MR

[34] E. De Giorgi, L. Ambrosio, “Un nuovo tipo di funzionale del calcolo delle variazioni”, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 82 (1988), 199–210 | MR | Zbl

[35] E. De Giorgi, M. Carriero, A. Leaci, “Existence theorem for a minimum problem with free discontinuity set”, Arch. Rational Mech. Anal., 108 (1989), 195–218 | DOI | MR | Zbl

[36] E. De Giorgi, T. Franzoni, “Su un tipo di convergenza variazionale”, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 58:6 (1975), 842–850 | MR | Zbl

[37] R. J. Duffin, “Continuation of biharmonic functions by reflection”, Duke Math. J., 22 (1955), 313–324 | DOI | MR | Zbl

[38] S. Esedoglu, J. H. Shen, “Digital inpainting based on the Mumford–Shah–Euler image model”, Eur. J. Appl. Math., 13:4 (2002), 353–370 | DOI | MR | Zbl

[39] L. C. Evans, R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies Adv. Math., CRC Press, Boca Raton, 1992 | MR | Zbl

[40] H. Federer, Geometric Measure Theory, Springer, Berlin, 1969 | MR | Zbl

[41] F. A. Lops, F. Maddalena, S. Solimini, “Hölder continuity conditions for the solvability of Dirichlet problems involving functionals with free discontinuities”, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18:6 (2001), 639–673 | DOI | MR | Zbl

[42] P. A. Markovich, Applied Partial Differential Equations: a Visual Approach, Springer, New York, 2007 | MR

[43] J. M. Morel, S. Solimini, Variational Models in Image Segmentation, Progr. Nonlinear Differential Equations Appl., 14, Birkhäuser, Basel, 1995 | MR

[44] D. Mumford, J. Shah, “Optimal approximation by piecewise smooth functions and associated variational problems”, Comm. Pure Appl. Math., 42 (1989), 577–685 | DOI | MR | Zbl

[45] J. Verdera, V. Caselles, M. Bertalmio, G. Sapiro, “Inpainting Surface Holes”, International Conference on Image Processing, 2003, 903–906