H. Attouch, J. Bolte, and B. F. Svaiter, Convergence of descent methods for semialgebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods, Math. Program, vol.137, issue.1, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00790042

C. Barbanson, A. Almansa, Y. Ferrec, and P. Monasse, Relief Computation from Images of a Fourier Transform Spectrometer for Interferogram Correction, Light, Energy and the Environment, 2016.
DOI : 10.1364/FTS.2016.FM3E.6
URL : https://hal.archives-ouvertes.fr/hal-01405856

C. Barnes, E. Shechtman, A. Finkelstein, and D. B. Goldman, Patchmatch: A randomized correspondence algorithm for structural image editing, ACM Transactions on Graphics, vol.28, issue.3, p.24, 2009.
DOI : 10.1145/2018396.2018421
URL : http://www.connellybarnes.com/work/publications/2011_patchmatch_cacm.pdf

A. Beck and M. Teboulle, Mirror descent and nonlinear projected subgradient methods for convex optimization, Operations Research Letters, vol.31, issue.3, pp.167-175, 2003.
DOI : 10.1016/S0167-6377(02)00231-6

A. Beck and M. Teboulle, A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems, SIAM Journal on Imaging Sciences, vol.2, issue.1, pp.183-202, 2009.
DOI : 10.1137/080716542
URL : http://ie.technion.ac.il/%7Ebecka/papers/finalicassp2009.pdf

D. P. Bertsekas, Nonlinear programming, Athena Scientific, 1995.

J. Bolte, A. Daniilidis, A. Lewis, and M. Shiota, Clarke Subgradients of Stratifiable Functions, SIAM Journal on Optimization, vol.18, issue.2, pp.556-572, 2007.
DOI : 10.1137/060670080
URL : http://arxiv.org/pdf/math/0601530

J. Bolte, S. Sabach, and M. Teboulle, Proximal alternating linearized minimization for nonconvex and nonsmooth problems, Mathematical Programming, vol.4, issue.1-2, p.2014
DOI : 10.1007/BF01584660
URL : https://hal.archives-ouvertes.fr/hal-00916090

J. Bolte, A. Daniilidis, and A. Lewis, The ??ojasiewicz Inequality for Nonsmooth Subanalytic Functions with Applications to Subgradient Dynamical Systems, SIAM Journal on Optimization, vol.17, issue.4, pp.1205-1223, 2007.
DOI : 10.1137/050644641

M. Lev and . Bregman, The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR computational mathematics and mathematical physics, pp.200-217, 1967.

A. Bugeau and V. Ta, Patch-based image colorization, International Conference on Pattern Recognition, pp.3058-3061, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00708969

A. Bugeau, V. Ta, and N. Papadakis, Variational Exemplar-Based Image Colorization, IEEE Transactions on Image Processing, vol.23, issue.1, pp.298-307, 2014.
DOI : 10.1109/TIP.2013.2288929
URL : https://hal.archives-ouvertes.fr/hal-00803219

M. Burger, H. Dirks, and C. Schonlieb, A Variational Model for Joint Motion Estimation and Image Reconstruction, SIAM Journal on Imaging Sciences, vol.11, issue.1, pp.94-128, 2018.
DOI : 10.1137/16M1084183

Y. Censor and S. A. Zenios, Proximal minimization algorithm withD-functions, Journal of Optimization Theory and Applications, vol.44, issue.3, pp.451-464, 1992.
DOI : 10.1007/BF00940051

A. Chambolle, P. Tan, and S. Vaiter, Accelerated Alternating Descent Methods for Dykstra-Like Problems, Journal of Mathematical Imaging and Vision, vol.23, issue.3, 2017.
DOI : 10.1007/978-1-4612-1015-3
URL : https://hal.archives-ouvertes.fr/hal-01346532

G. Charpiat, M. Hofmann, and B. Schölkopf, Automatic Image Colorization Via Multimodal Predictions, European Conference on Computer Vision, pp.126-139, 2008.
DOI : 10.1007/11744047_2
URL : https://hal.archives-ouvertes.fr/inria-00481552

T. Chen, Y. Wang, V. Schillings, and C. Meinel, Grayscale image matting and colorization, Proceedings of Asian Conference on Computer Vision, 2004.

S. Conti, J. Ginster, and M. Rumpf, A BV functional and its relaxation for joint motion estimation and image sequence recovery, ESAIM: Mathematical Modelling and Numerical Analysis, vol.49, issue.5, pp.1463-1487, 2015.
DOI : 10.1007/0-387-28831-7_15
URL : http://arxiv.org/pdf/1504.01998

Z. Denkowska, P. Maciej, and . Denkowski, A long and winding road to definable sets, Journal of Singularities, vol.13, pp.57-86, 2015.
DOI : 10.5427/jsing.2015.13d

D. Fortun, P. Bouthemy, and C. Kervrann, Aggregation of local parametric candidates with exemplar-based occlusion handling for optical flow, Computer Vision and Image Understanding, vol.145, pp.81-94, 2016.
DOI : 10.1016/j.cviu.2015.11.020
URL : https://hal.archives-ouvertes.fr/hal-01001758

D. Fortun, P. Bouthemy, and C. Kervrann, A Variational Aggregation Framework for Patch-Based Optical Flow Estimation, Journal of Mathematical Imaging and Vision, vol.93, issue.3, pp.280-299, 2016.
DOI : 10.1007/s11263-011-0422-6
URL : https://hal.archives-ouvertes.fr/hal-01408771

T. Goldstein, B. O. Donoghue, S. Setzer, and R. Baraniuk, Fast Alternating Direction Optimization Methods, SIAM Journal on Imaging Sciences, vol.7, issue.3, pp.1588-1623, 2014.
DOI : 10.1137/120896219

O. Güler, On the Convergence of the Proximal Point Algorithm for Convex Minimization, SIAM Journal on Control and Optimization, vol.29, issue.2, pp.403-419, 1991.
DOI : 10.1137/0329022

R. Kumar-gupta, A. Yong, -. Chia, and D. Rajan, Ee Sin Ng, and Huang Zhiyong. Image colorization using similar images, ACM International Conference on Multimedia, pp.369-378, 2012.

T. Horiuchi, Colorization algorithm using probabilistic relaxation, Image and Vision Computing, vol.22, issue.3, pp.197-202, 2004.
DOI : 10.1016/j.imavis.2003.08.004

T. Horiuchi and S. Hirano, Colorization algorithm for grayscale image by propagating seed pixels, Proceedings 2003 International Conference on Image Processing (Cat. No.03CH37429), p.457, 2003.
DOI : 10.1109/ICIP.2003.1246997

R. Ironi, D. Cohen-or, and D. Lischinski, Colorization by example, Rendering Techniques, pp.201-210, 2005.

S. H. Kang and R. March, Variational Models for Image Colorization via Chromaticity and Brightness Decomposition, IEEE Transactions on Image Processing, vol.16, issue.9, pp.2251-2261, 2007.
DOI : 10.1109/TIP.2007.903257
URL : http://www.iac.rm.cnr.it/~march/kang-march.pdf

A. Levin, D. Lischinski, and Y. Weiss, Colorization using optimization, ACM Transactions on Graphics, vol.23, issue.3, pp.689-694, 2004.
DOI : 10.1145/1015706.1015780
URL : http://www.cs.huji.ac.il/~yweiss/Colorization/colorization-siggraph04.pdf

J. Moreau, Proximité et dualité dans un espace hilbertien Bulletin de la Société mathématique de France, pp.273-299, 1965.
DOI : 10.24033/bsmf.1625

M. Nikolova and P. Tan, Alternating proximal gradient descent for nonconvex nonsmooth block-regularised problems, 2018.

P. Ochs, Y. Chen, T. Brox, and T. Pock, iPiano: Inertial Proximal Algorithm for Nonconvex Optimization, SIAM Journal on Imaging Sciences, vol.7, issue.2, pp.1388-1419, 2014.
DOI : 10.1137/130942954
URL : http://arxiv.org/pdf/1404.4805

J. Persch, F. Pierre, and G. Steidl, Exemplar-Based Face Colorization Using Image Morphing, Journal of Imaging, vol.15, issue.4, p.48, 2017.
DOI : 10.1109/83.855431
URL : https://hal.archives-ouvertes.fr/hal-01539058

F. Pierre, J. Aujol, A. Bugeau, N. Papadakis, and V. Ta, Luminance-Chrominance Model for Image Colorization, SIAM Journal on Imaging Sciences, vol.8, issue.1, pp.536-563, 2015.
DOI : 10.1137/140979368
URL : https://hal.archives-ouvertes.fr/hal-01166919

F. Pierre, J. Aujol, A. Bugeau, and V. Ta, Interactive Video Colorization Within a Variational Framework, SIAM Journal on Imaging Sciences, vol.10, issue.4, pp.2293-2325, 2017.
DOI : 10.1137/16M1108625
URL : https://hal.archives-ouvertes.fr/hal-01379702

F. Pierre, J. Aujol, A. Bugeau, and V. Ta, Combinaison linéaire optimale de métriques pour la colorisation d'images, XXVème colloque GRETSI, pp.1-4, 2015.

F. Pierre, J. Aujol, A. Bugeau, and V. Ta, Dépôt Agence de Protection des Programmes No IDDN, 2016.

F. Pierre, J. Aujol, and A. Bugeau, Vinh-Thong Ta, and Nicolas Papadakis. Exemplar-based colorization in RGB color space, International Conference on Image Processing, pp.1-5, 2014.

T. Pock and S. Sabach, Inertial Proximal Alternating Linearized Minimization (iPALM) for Nonconvex and Nonsmooth Problems, SIAM Journal on Imaging Sciences, vol.9, issue.4, pp.1756-1787, 2016.
DOI : 10.1137/16M1064064
URL : http://arxiv.org/pdf/1702.02505

T. Boris and . Polyak, Some methods of speeding up the convergence of iteration methods, USSR Computational Mathematics and Mathematical Physics, vol.4, issue.5, pp.1-17, 1964.

R. T. Rockafellar and J. B. Wets, Variational analysis, 1998.
DOI : 10.1007/978-3-642-02431-3

M. Shiota, Geometry of subanalytic and semialgebraic sets, 2012.
DOI : 10.1007/978-1-4612-2008-4

P. Tan, Y. Ferrec, and L. Roussetrouvì-ere, Correction par méthode variationnelle des non uniformités des détecteurs d'un interféromètre imageur, XXVIe colloque GRETSI 2017, 2017.

T. Welsh, M. Ashikhmin, and K. Mueller, Transferring color to greyscale images, In ACM Transactions on Graphics, vol.21, pp.277-280, 2002.
DOI : 10.1145/566654.566576

Y. Xu and W. Yin, A Block Coordinate Descent Method for Regularized Multiconvex Optimization with Applications to Nonnegative Tensor Factorization and Completion, SIAM Journal on Imaging Sciences, vol.6, issue.3, pp.1758-1789, 2013.
DOI : 10.1137/120887795

L. Yatziv and G. Sapiro, Fast image and video colorization using chrominance blending, IEEE Transactions on Image Processing, vol.15, issue.5, pp.1120-1129, 2006.
DOI : 10.1109/TIP.2005.864231
URL : http://conservancy.umn.edu/bitstream/4079/1/2010.pdf

R. Zhang, P. Isola, and A. A. Efros, Colorful Image Colorization, European Conference on Computer Vision, pp.649-666, 2016.
DOI : 10.1109/CVPR.2015.7298965
URL : http://arxiv.org/pdf/1603.08511