. Si-en-un-point-x, ?u change de signe et si ?u est non nul, alors on considère qu'en x l'image u possède un bord

. Perona-malik, Sur le même principe que l'EDP de la chaleur, écrire le schéma numérique pour (2.5) (une discrétisation possible de la divergence est donnée en (2.6))

+. Modèle-u, Programmer l'algorithme (4.57)?(4.60) Le tester en débruitage d'image, et en décomposition d'image

]. D. Références1, J. A. Adalsteinsson, and . Sethian, The fast constructions of extension velocities in level set methods, Journal of Computational Physics, vol.148, pp.2-22, 1999.

G. Aubert and J. F. , Signed distance functions and viscosity solutions of discontinuous hamilton-jacobi equations, 2002.
URL : https://hal.archives-ouvertes.fr/inria-00072081

G. Aubert and J. F. , Modeling Very Oscillating Signals. Application to Image Processing, Applied Mathematics and Optimization, vol.51, issue.2, 2005.
DOI : 10.1007/s00245-004-0812-z

URL : https://hal.archives-ouvertes.fr/hal-00202000

G. Aubert and J. F. , Optimal partitions, regularized solutions, and application to image classification, Applicable Analysis, vol.420, issue.1, pp.15-35, 2005.
DOI : 10.1023/A:1020874308076

URL : https://hal.archives-ouvertes.fr/hal-00202002

G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing, Applied Mathematical Sciences, vol.147, 2002.

J. F. Aujol and G. Aubert, Signed distance functions and discontinuous Hamilton-Jacobi equations, TMR Viscosity Solutions and their Applications, 2002.
URL : https://hal.archives-ouvertes.fr/inria-00072081

J. F. Aujol, G. Aubert, and L. Blanc-féraud, Wavelet-based level set evolution for classification of textured images, ICIP '03, 2003.
URL : https://hal.archives-ouvertes.fr/hal-00202004

J. F. Aujol, G. Aubert, L. Blanc-féraud, and A. Chambolle, Decomposing an image : Application to SAR images, Scale-Space '03, 2003.

J. F. Aujol, G. Aubert, L. Blanc-féraud, and A. Chambolle, Image Decomposition into a Bounded Variation Component and an Oscillating Component, Journal of Mathematical Imaging and Vision, vol.15, issue.3, pp.71-88, 2005.
DOI : 10.1007/s10851-005-4783-8

URL : https://hal.archives-ouvertes.fr/hal-00202001

J. F. Aujol, G. Aubert, and L. Blanc-féraud, Wavelet-based level set evolution for classification of textured images, IEEE Transactions on Image Processing, vol.12, issue.12, pp.1634-1641, 2003.
DOI : 10.1109/TIP.2003.819309

URL : https://hal.archives-ouvertes.fr/hal-00202004

J. F. Aujol and A. Chambolle, Dual Norms and Image Decomposition Models, International Journal of Computer Vision, vol.19, issue.3, pp.85-104, 2005.
DOI : 10.1007/s11263-005-4948-3

URL : https://hal.archives-ouvertes.fr/inria-00071453

G. Barles, Solutions de Viscosité des équations de Hamilton-Jacobi, 1987.

F. Cao, Geometric curve evolution and image processing, Lecture Notes in Applied Mathematics, vol.1805, 2003.
DOI : 10.1007/b10404

V. Caselles, F. Catte, T. Coll, and F. Dibos, Geometric models for active contours, Proceedings., International Conference on Image Processing, pp.1-31, 1993.
DOI : 10.1109/ICIP.1995.537567

A. Chambolle, An algorithm for total variation minimization and applications, JMIV, vol.20, pp.89-97, 2004.

A. Chambolle and P. L. Lions, Image recovery via total variation minimization and related problems, Numerische Mathematik, vol.76, issue.2, pp.167-188, 1997.
DOI : 10.1007/s002110050258

T. F. Chan and L. A. Vese, Active contours without edges, IEEE Transactions on Image Processing, vol.10, issue.2, pp.266-77, 2001.
DOI : 10.1109/83.902291

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.2.1828

A. Cohen, Wavelet methods in numerical analysis, 2003.
DOI : 10.1016/S1570-8659(00)07004-6

P. L. Combettes and J. C. Pesquet, Image Restoration Subject to a Total Variation Constraint, IEEE Transactions on Image Processing, vol.13, issue.9, 2004.
DOI : 10.1109/TIP.2004.832922

URL : https://hal.archives-ouvertes.fr/hal-00017934

M. G. Crandall, Viscosity solutions: A primer, Viscosity Solutions and Applications, 1997.
DOI : 10.1017/S0308210500026512

M. G. Crandall, H. Ishii, and P. L. Lions, user's guide to viscosity solutions\\ of second order\\ partial differential equations, Bulletin of the American Mathematical Society, vol.27, issue.1, pp.1-67, 1992.
DOI : 10.1090/S0273-0979-1992-00266-5

D. L. Donoho and M. Johnstone, Adapting to Unknown Smoothness via Wavelet Shrinkage, Journal of the American Statistical Association, vol.31, issue.432, pp.1200-1224, 1995.
DOI : 10.1080/01621459.1979.10481038

I. Ekeland and R. Temam, Analyse convexe et problèmes variationnels, 1983.

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, vol.19, 1991.

J. Gomes and O. Faugeras, Reconciling Distance Functions and Level Sets, Journal of Visual Communication and Image Representation, vol.11, issue.2, pp.209-223, 1999.
DOI : 10.1006/jvci.1999.0439

URL : https://hal.archives-ouvertes.fr/inria-00073006

J. B. Hiriart-urruty and C. Lemarechal, Convex Analysis ans Minimisation Algorithms I, volume 305 of Grundlehren der mathematischen Wissenschaften, 1993.

F. Malgouyres, Minimizing the total variation under a general convex constraint for image restoration, IEEE Transactions on Image Processing, vol.11, issue.12, pp.1450-1456, 2002.
DOI : 10.1109/TIP.2002.806241

R. Malladi, J. A. Sethian, and B. C. Vemuri, Shape modeling with front propagation: a level set approach, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.17, issue.2, 1995.
DOI : 10.1109/34.368173

S. G. Mallat, A Wavelet Tour of Signal Processing, 1998.

Y. Meyer, Oscillating patterns in image processing and in some nonlinear evolution equations The Fifteenth Dean Jacquelines B, 2001.

J. M. Morel and S. Solimini, Variational Methods in Image Segmentation, Progress in Nonlinear Differential Equations and Their Applications. Birkhauser, 1995.
DOI : 10.1007/978-1-4684-0567-5

S. Osher and R. P. Fedkiw, Level Set Methods: An Overview and Some Recent Results, Journal of Computational Physics, vol.169, issue.2, pp.463-502, 2001.
DOI : 10.1006/jcph.2000.6636

S. J. Osher, A. Sole, and L. A. Vese, Image decomposition and restoration using total variation minimization and the H ?1 norm. Multiscale Modeling and Simulation : A, SIAM Interdisciplinary Journal, vol.1, issue.3, pp.349-370, 2003.

N. Paragios and R. Deriche, Geodesic active regions and level set methods for supervised texture segmentation, International Journal of Computer Vision, vol.46, issue.3, 2002.

D. Peng, B. Merriman, S. Osher, H. Zhao, and M. Kang, A PDE-Based Fast Local Level Set Method, Journal of Computational Physics, vol.155, issue.2, 1998.
DOI : 10.1006/jcph.1999.6345

T. Rockafellar, Convex Analysis, volume 224 of Grundlehren der mathematischen Wissenschaften, 1983.

E. Rouy and A. Tourin, A Viscosity Solutions Approach to Shape-From-Shading, SIAM Journal on Numerical Analysis, vol.29, issue.3, pp.867-884, 1992.
DOI : 10.1137/0729053

L. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena, vol.60, issue.1-4, pp.259-268, 1992.
DOI : 10.1016/0167-2789(92)90242-F

C. Samson, Contribution à la classification d'images satellitaires par approche variationnelle et équations aux dérivées partielles, 2000.

C. Samson, L. Blanc-féraud, G. Aubert, and J. Zerubia, A level set method for image classification, International Journal of Computer Vision, vol.40, issue.3, pp.187-197, 2000.
DOI : 10.1023/A:1008183109594

C. Samson, L. Blanc-féraud, G. Aubert, and J. Zerubia, A variational model for image classification and restoration, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.22, issue.5, pp.460-472, 2000.
DOI : 10.1109/34.857003

M. Sussman, P. Smereka, and S. Osher, A Level Set Approach for Computing Solutions to Incompressible Two-Phase Flow, Journal of Computational Physics, vol.114, issue.1, pp.146-159, 1994.
DOI : 10.1006/jcph.1994.1155

G. Unal, A. Yezzi, and H. Krim, Information-Theoretic Active Polygons for Unsupervised Texture Segmentation, International Journal of Computer Vision, vol.27, issue.2, 2002.
DOI : 10.1007/s11263-005-4880-6

L. A. Vese and T. F. Chan, A multiphase level set framework for image segmentation using the Mumford and Shah model, International Journal of Computer Vision, vol.50, issue.3, pp.271-293, 2002.
DOI : 10.1023/A:1020874308076

L. A. Vese and S. J. Osher, Modeling textures with total variation minimization and oscillating patterns in image processing, Journal of Scientific Computing, vol.19, issue.1/3, pp.553-572, 2003.
DOI : 10.1023/A:1025384832106