R. Acar and C. Vogel, Analysis of bounded variation penalty methods for ill-posed problems, Inverse Problems, vol.10, issue.6, pp.1217-1229, 1994.
DOI : 10.1088/0266-5611/10/6/003

L. Ambrosio, N. Fusco, and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford mathematical monographs, 2000.

H. Attouch, . Buttazzo, and G. Michaille, Variational analysis in Sobolev and BV spaces : applications to PDEs and optimization. MPS-SIAM series on op- timization, 2006.
DOI : 10.1137/1.9781611973488

G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing, Partial Differential Equations and the Calculus of Variations, Applied Mathematical Sciences, vol.147, 2006.

G. Aubert and J. Aujol, Modeling Very Oscillating Signals. Application to Image Processing, Applied Mathematics and Optimization, vol.51, issue.2, pp.163-182, 2005.
DOI : 10.1007/s00245-004-0812-z

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

G. Aubert, J. Aujol, L. Blanc-feraud, and A. Chambolle, Image decomposition into a bounded variation component and an oscillating component, Journal of Mathematical Imaging and Vision, vol.22, pp.71-88, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00202001

M. Bergounioux and L. Piffet, A BV 2 (?) model for image denoising and/or texture extraction , submitted, Set Valued Analysis, 2010.

A. Chambolle, An algorithm for total variation minimization and applications, Journal of Mathematical Imaging and Vision, vol.20, pp.89-97, 2004.

F. Demengel, FonctionsàFonctionsà hessien borné. Annales de l'institut Fourier, pp.155-190, 1984.

Y. Meyer, Oscillating patterns in image processing and nonlinear evolution equations, 2002.
DOI : 10.1090/ulect/022

S. Osher, E. Fatemi, and L. Rudin, Nonlinear total variation based noise removal algorithms, Physica D, vol.60, pp.259-268, 1992.

S. Osher and L. Vese, Image denoising and decomposition with total variation minimization and oscillatory functions. Special issue on mathematics and image analysis, Journal of Mathematical Imaging and Vision, vol.20, pp.7-18, 2004.

L. Piffet, Modèles variationnels pour l'extraction de textures 2D, 2010.

W. Yin, D. Goldfarb, and S. Osher, A comparison of three total variation based texture extraction models, Journal of Visual Communication and Image Representation, vol.18, issue.3, pp.240-252, 2007.
DOI : 10.1016/j.jvcir.2007.01.004

R. Acar and C. R. Vogel, Analysis of bounded variation penalty methods for ill-posed problems, Inverse Problems, vol.10, issue.6, pp.1217-1229, 1994.
DOI : 10.1088/0266-5611/10/6/003

L. Ambrosio, N. Fusco, and D. Pallara, Free Discontinuity Problems and Special Functions with Bounded Variation, Oxford Mathematical Monographs, 2000.
DOI : 10.1007/978-3-0348-8974-2_2

H. Attouch, L. M. Briceño-arias, and P. L. Combettes, A Parallel Splitting Method for Coupled Monotone Inclusions, SIAM Journal on Control and Optimization, vol.48, issue.5, p.3246, 2010.
DOI : 10.1137/090754297

H. Attouch, G. Buttazzo, and G. Michaille, Variational analysis in sobolev and BV spaces: applications to PDEs and optimization. MPS-SIAM series on optimization, 2006.
DOI : 10.1137/1.9781611973488

G. Aubert and J. F. Aujol, Modeling Very Oscillating Signals. Application to Image Processing, Applied Mathematics and Optimization, vol.51, issue.2, pp.163-182, 2005.
DOI : 10.1007/s00245-004-0812-z

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

G. Aubert, J. F. Aujol, L. Blanc-feraud, and A. Chambolle, Image decomposition into a bounded variation component and an oscillating component, J. Math. Imaging Vis, vol.22, issue.1, pp.71-88, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00202001

G. Aubert and P. Kornprobst, Mathematical problems in image processing, partial differential equations and the calculus of variations, Applied Mathematical Sciences, vol.147, 2006.

J. F. Aujol, Some First-Order Algorithms for Total Variation Based Image Restoration, Journal of Mathematical Imaging and Vision, vol.33, issue.2, pp.307-327, 2009.
DOI : 10.1007/s10851-009-0149-y

URL : https://hal.archives-ouvertes.fr/hal-00260494

A. Chambolle, An algorithm for total variation minimization and applications, J. Math. Imaging Vis, vol.20, pp.89-97, 2004.

F. Demengel, Fonctions à hessien borné Annales de l'institut Fourier, pp.155-190, 1984.
DOI : 10.5802/aif.969

URL : http://archive.numdam.org/article/AIF_1984__34_2_155_0.pdf

R. Echegut and L. Piffet, A variational model for image texture identification (preprint)

I. Ekeland and R. Temam, Convex Analysis and Variational problems, SIAM Classic in Applied Mathematics, vol.28, 1999.
DOI : 10.1137/1.9781611971088

J. Fadili and G. Peyré, Total variation projection with first order schemes (preprint)

W. Hinterberger and O. Scherzer, Variational Methods on the Space of Functions of Bounded Hessian for Convexification and Denoising, Computing, vol.80, issue.3, pp.109-133, 2006.
DOI : 10.1007/s00607-005-0119-1

Y. Meyer, Oscillating patterns in image processing and nonlinear evolution equations, AMS, vol.22, 2002.
DOI : 10.1090/ulect/022

S. Osher, E. Fatemi, and L. Rudin, Nonlinear total variation based noise removal algorithms, Physica D, vol.60, pp.259-268, 1992.

S. Osher and L. Vese, Modeling textures with total variation minimization and oscillating patterns in image processing, J. Sci. Comput, vol.19, pp.1-3, 2003.

S. J. Osher and L. A. Vese, Image denoising and decomposition with total variation minimization and oscillatory functions. Special issue on mathematics and image analysis, J. Math. Imaging Vis, vol.20, issue.12, pp.7-18, 2004.

L. Piffet, Modèles variationnels pour l'extraction de textures 2D, 2010.

P. Weiss, L. Blanc-féraud, and G. Aubert, Efficient Schemes for Total Variation Minimization Under Constraints in Image Processing, SIAM Journal on Scientific Computing, vol.31, issue.3, pp.31-2047, 2009.
DOI : 10.1137/070696143

URL : https://hal.archives-ouvertes.fr/inria-00166096

W. Yin, D. Goldfarb, and S. Osher, A comparison of three total variation based texture extraction models, Journal of Visual Communication and Image Representation, vol.18, issue.3, pp.240-252, 2007.
DOI : 10.1016/j.jvcir.2007.01.004

R. Acar and C. R. Vogel, Analysis of bounded variation penalty methods for ill-posed problems, Inverse Problems, vol.10, issue.6, pp.1217-1229, 1994.
DOI : 10.1088/0266-5611/10/6/003

R. A. Adams, Sobolev spaces, 1978.

S. Aliney, A property of the minimum vectors of a regularizing functional defined by means of the absolute norm, IEEE Transactions on Signal Processing, vol.45, issue.4, pp.913-917, 1997.
DOI : 10.1109/78.564179

L. Ambrosio, N. Fusco, and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford mathematical monographs, 2000.

H. Attouch, L. M. Briceño-arias, and P. L. Combettes, A Parallel Splitting Method for Coupled Monotone Inclusions, SIAM Journal on Control and Optimization, vol.48, issue.5, p.3246, 2010.
DOI : 10.1137/090754297

H. Attouch, . Buttazzo, and G. Michaille, Variational analysis in Sobolev and BV spaces : applications to PDEs and optimization. MPS-SIAM series on optimization, 2006.
DOI : 10.1137/1.9781611973488

G. Aubert and J. F. Aujol, Modeling Very Oscillating Signals. Application to Image Processing, Applied Mathematics and Optimization, vol.51, issue.2, pp.163-182, 2005.
DOI : 10.1007/s00245-004-0812-z

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

G. Aubert, J. F. Aujol, L. Blanc-feraud, and A. Chambolle, Image decomposition into a bounded variation component and an oscillating component, Journal of Mathematical Imaging and Vision, vol.22, issue.1, pp.71-88, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00202001

G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing , Partial Differential Equations and the Calculus of Variations, Applied Mathematical Sciences, vol.147, 2006.

J. F. Aujol, Some First-Order Algorithms for Total Variation Based Image Restoration, Journal of Mathematical Imaging and Vision, vol.33, issue.2, pp.307-327, 2009.
DOI : 10.1007/s10851-009-0149-y

URL : https://hal.archives-ouvertes.fr/hal-00260494

J. 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

J. Aujol, G. Gilboa, T. Chan, and S. Osher, Structure-Texture Image Decomposition???Modeling, Algorithms, and Parameter Selection, International Journal of Computer Vision, vol.4, issue.2, pp.111-136, 2005.
DOI : 10.1007/s11263-006-4331-z

URL : https://hal.archives-ouvertes.fr/hal-00201977

D. Azé, Eléments d'analyse convexe et variationnelle, 1997.

M. Bergounioux, L. Piffet, M. Bergounioux, M. Tran, and P. , A second-order model for image denoising, Set-Valued and Variational Analysis A second order model for 3D-texture extraction, p.530816, 2010.

P. Blomgren, T. Chan, P. Mulet, and C. Wong, Total variation image restoration: numerical methods and extensions, Proceedings of International Conference on Image Processing, pp.384-387, 1997.
DOI : 10.1109/ICIP.1997.632128

K. Bredies, K. Kunisch, and T. Pock, Total Generalized Variation, SIAM Journal on Imaging Sciences, vol.3, issue.3, pp.492-526, 2010.
DOI : 10.1137/090769521

C. Bouman and K. Sauer, A generalized Gaussian image model for edge-preserving MAP estimation, IEEE Transactions on Image Processing, vol.2, issue.3, pp.296-310, 1993.
DOI : 10.1109/83.236536

V. Caselles, A. Chambolle, and M. Novaga, The Discontinuity Set of Solutions of the TV Denoising Problem and Some Extensions, Multiscale Modeling & Simulation, vol.6, issue.3, pp.879-894, 2007.
DOI : 10.1137/070683003

A. Chambolle, An algorithm for total variation minimization and applications, Journal of Mathematical Imaging and Vision, vol.20, pp.89-97, 2004.

A. Chambolle and P. Lions, Image recovery via total variation minimization and related problems Numerische Mathematik, pp.167-188, 1997.

T. Chan, A. Marquina, and P. Mulet, High-Order Total Variation-Based Image Restoration, SIAM Journal on Scientific Computing, vol.22, issue.2, pp.503-516, 2000.
DOI : 10.1137/S1064827598344169

T. Chan and S. Esedoglu, Function Approximation, SIAM Journal on Applied Mathematics, vol.65, issue.5, pp.1817-1837, 2005.
DOI : 10.1137/040604297

T. F. Chan, S. Esedoglu, and F. E. Park, Image decomposition combining staircase reduction and texture extraction, Journal of Visual Communication and Image Representation, vol.18, issue.6, pp.464-486, 2007.
DOI : 10.1016/j.jvcir.2006.12.004

J. Darbon, J. Sigelle, and M. , Image Restoration with Discrete Constrained Total Variation Part I: Fast and Exact Optimization, Journal of Mathematical Imaging and Vision, vol.2, issue.4, pp.277-291, 2006.
DOI : 10.1007/s10851-006-8803-0

F. Demengel, FonctionsàFonctionsà hessien borné Annales de l'institut Fourier, pp.155-190, 1984.

D. C. Dobson and F. Santosa, Recovery of Blocky Images from Noisy and Blurred Data, SIAM Journal on Applied Mathematics, vol.56, issue.4, pp.1181-1198, 1996.
DOI : 10.1137/S003613999427560X

V. Duval, J. Aujol, and Y. Gousseau, The TVL1 Model: A Geometric Point of View, Multiscale Modeling & Simulation, vol.8, issue.1, pp.154-189, 2009.
DOI : 10.1137/090757083

URL : https://hal.archives-ouvertes.fr/hal-00380195

R. Echegut and L. Piffet, A Variational Model for Image Texture Identification, pp.sub- mitted
DOI : 10.1007/978-3-642-12598-0_41

URL : https://hal.archives-ouvertes.fr/hal-00439431

I. Ekeland and R. Temam, Convex Analysis and Variational problems, SIAM Classic in Applied Mathematics, vol.28, 1999.
DOI : 10.1137/1.9781611971088

J. Fadili and G. Peyré, Total Variation Projection with First Order Schemes, Preprint

L. C. Evans and R. Gariepy, Measure theory and fine properties of functions, 1992.

W. Hinterberger and O. Scherzer, Variational Methods on the Space of Functions of Bounded Hessian for Convexification and Denoising, Computing, vol.80, issue.3, pp.109-133, 2006.
DOI : 10.1007/s00607-005-0119-1

B. Hofmann, B. Kaltenbacher, C. Pöschl, and O. Scherzer, A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators, Inverse Problems, vol.23, issue.3, pp.987-1010, 2007.
DOI : 10.1088/0266-5611/23/3/009

K. Ito and K. Kunisch, Lagrange Multiplier Approach to Variational Problems and Applications, Advances in Design and Control, 2008.
DOI : 10.1137/1.9780898718614

T. M. Le and L. A. Vese, ), Multiscale Modeling & Simulation, vol.4, issue.2, pp.390-423, 2005.
DOI : 10.1137/040610052

L. H. Lieu and L. A. Vese, Image Restoration and Decomposition via Bounded Total Variation and Negative Hilbert-Sobolev Spaces, Oscillating patterns in image processing and nonlinear evolution equations, pp.167-193, 2002.
DOI : 10.1007/s00245-008-9047-8

C. Louchet, Modèles variationnels et bayésiens pour le débruitage d'images : de la variation totale vers les moyennes non-locales, 2008.

Y. Nesterov, Gradient methods for minimizing composite objecive function, CORE discussion paper, 2007.

M. Nikolova, Estimées localement fortement homogènes Compterendus de l'académie des sciences, t, pp.665-670, 1997.

M. Nikolova, Local Strong Homogeneity of a Regularized Estimator, SIAM Journal on Applied Mathematics, vol.61, issue.2, pp.633-658, 2000.
DOI : 10.1137/S0036139997327794

M. Nikolova, Weakly Constrained Minimization: Application to the Estimation of Images and Signals Involving Constant Regions, Journal of Mathematical Imaging and Vision, vol.21, issue.2, pp.155-175, 2004.
DOI : 10.1023/B:JMIV.0000035180.40477.bd

M. Nikolova, A Variational Approach to Remove Outliers and Impulse Noise, Journal of Mathematical Imaging and Vision, vol.20, issue.1/2, pp.99-120, 2004.
DOI : 10.1023/B:JMIV.0000011920.58935.9c

M. Nikolova, Analysis of the Recovery of Edges in Images and Signals by Minimizing Nonconvex Regularized Least-Squares, Multiscale Modeling & Simulation, vol.4, issue.3, pp.960-991, 2005.
DOI : 10.1137/040619582

M. Nikolova, Model distortions in Bayesian MAP reconstruction, Inverse Problems and Imaging, vol.1, issue.2, pp.399-422, 2007.
DOI : 10.3934/ipi.2007.1.399

S. Osher, E. Fatemi, and L. Rudin, Nonlinear total variation based noise removal algorithms, Physica D, vol.60, pp.259-268, 1992.

S. Osher, A. Sole, and L. Vese, Image decomposition and restoration using total variation minimization and the H 1 norm, SIAM Journal on Multiscale Modeling and Simulation, pp.1-3, 2003.

S. Osher and L. Vese, Modeling textures with total variation minimization and oscillating patterns in image processing, Journal of Scientific Computing, vol.19, pp.1-3, 2003.

S. J. Osher and L. A. Vese, Image denoising and decomposition with total variation minimization and oscillatory functions. Special issue on mathematics and image analysis, J. Math. Imaging Vision, vol.20, issue.12, pp.7-18, 2004.

L. Piffet, Modèles variationnels pour l'extraction de textures 2D, 2010.

R. Rockafellar and T. , Convex Analysis, 1972.
DOI : 10.1515/9781400873173

W. Ring, Structural properties of solutions of total variation regularization problems. ESSAIM, Math Model ling and Numerical Analysis, pp.799-840, 2000.

P. Weiss, L. Blanc-féraud, and G. Aubert, Efficient Schemes for Total Variation Minimization Under Constraints in Image Processing, SIAM Journal on Scientific Computing, vol.31, issue.3, 2009.
DOI : 10.1137/070696143

URL : https://hal.archives-ouvertes.fr/inria-00166096

W. Yin, D. Goldfarb, and S. Osher, A comparison of three total variation based texture extraction models, Journal of Visual Communication and Image Representation, vol.18, issue.3, pp.240-252, 2007.
DOI : 10.1016/j.jvcir.2007.01.004