B. Alipanahi, A. Delong, M. T. Weirauch, and B. J. Frey, Predicting the sequence specificities of dna-and rna-binding proteins by deep learning, Nature biotechnology, vol.33, issue.8, pp.831-838, 2015.

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.

D. Belanger, B. Yang, and A. Mccallum, End-to-end learning for structured prediction energy networks, Proc. International Conference on Machine Learning (ICML), 2017.

C. Bertocchi, E. Chouzenoux, M. Corbineau, J. Pesquet, and M. Prato, Deep unfolding of a proximal interior point method for image restoration, Inverse Problems, 2019.
URL : https://hal.archives-ouvertes.fr/hal-01943475

D. P. Bertsekas and J. N. Tsitsiklis, Parallel and distributed computation: numerical methods, 1989.

A. Buades, B. Coll, and J. Morel, A non-local algorithm for image denoising, Proc. Conference on Computer Vision and Pattern Recognition (CVPR), 2005.

A. Chambolle, V. Caselles, D. Cremers, M. Novaga, and T. Pock, An introduction to total variation for image analysis. Theoretical foundations and numerical methods for sparse recovery, vol.9, p.227, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00437581

A. Chambolle and J. Darbon, On total variation minimization and surface evolution using parametric maximum flows, International journal of computer vision, vol.84, issue.3, p.288, 2009.

J. Chang and Y. Chen, Pyramid stereo matching network, Proc. Conference on Computer Vision and Pattern Recognition (CVPR), pp.5410-5418, 2018.

Y. Chen and T. Pock, Trainable nonlinear reaction diffusion: A flexible framework for fast and effective image restoration, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.39, issue.6, pp.1256-1272, 2016.

R. Collobert and J. Weston, A unified architecture for natural language processing: Deep neural networks with multitask learning, Proc. International Conference on Machine Learning (ICML), 2008.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, Image denoising by sparse 3-D transform-domain collaborative filtering, IEEE Transactions on Image Processing, vol.16, issue.8, pp.2080-2095, 2007.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, BM3D image denoising with shape-adaptive principal component analysis, Proceedings of SPARS'09, Signal Processing wiht Adaptive Sparse Structured Representations, 2009.
URL : https://hal.archives-ouvertes.fr/inria-00369582

W. Dong, L. Zhang, G. Shi, and X. Li, Nonlocally centralized sparse representation for image restoration, IEEE transactions on Image Processing, vol.22, issue.4, pp.1620-1630, 2012.

E. M. Eksioglu, Decoupled algorithm for mri reconstruction using nonlocal block matching model: Bm3d-mri, Journal of Mathematical Imaging and Vision, vol.56, issue.3, pp.430-440, 2016.

M. Elad and M. Aharon, Image denoising via sparse and redundant representations over learned dictionaries, IEEE Transactions on Image processing, vol.15, issue.12, pp.3736-3745, 2006.

F. Facchinei and J. Pang, Finite-dimensional variational inequalities and complementarity problems, 2007.

A. K. Fletcher, P. Pandit, S. Rangan, S. Sarkar, and P. Schniter, Plug-in estimation in high-dimensional linear inverse problems: A rigorous analysis, Adv. in Neural Information Processing Systems (NeurIPS), 2018.

L. Franceschi, P. Frasconi, M. Donini, and M. Pontil, A bridge between hyperparameter optimization and larning-to-learn, 2017.

A. Geiger, P. Lenz, and R. Urtasun, Are we ready for autonomous driving? the kitti vision benchmark suite, 2012 IEEE Conference on Computer Vision and Pattern Recognition, pp.3354-3361, 2012.

G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Modeling & Simulation, vol.7, issue.3, pp.1005-1028, 2009.

K. Gregor and Y. Lecun, Learning fast approximations of sparse coding, Proc. International Conference on Machine Learning (ICML), 2010.

K. He, X. Zhang, S. Ren, and J. Sun, Deep residual learning for image recognition, Proc. Conference on Computer Vision and Pattern Recognition (CVPR), 2016.

J. Hiriart-urruty and C. , Convex analysis and minimization algorithms I: Fundamentals, Springer science & business media, vol.305, 2013.

A. Juditsky, A. Nemirovski, and C. Tauvel, Solving variational inequalities with stochastic mirror-prox algorithm, Stochastic Systems, vol.1, issue.1, pp.17-58, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00318043

D. P. Kingma and J. Ba, Adam: A method for stochastic optimization, Proc. International Conference on Learning Representations (ICLR), 2013.

G. Korpelevich, The extragradient method for finding saddle points and other problems, Matecon, vol.12, pp.747-756, 1976.

B. Lecouat, J. Ponce, and J. Mairal, Fully trainable and interpretable non-local sparse models for image restoration, 2020.
URL : https://hal.archives-ouvertes.fr/hal-02414291

S. Lefkimmiatis, Non-local color image denoising with convolutional neural networks, Proc. Conference on Computer Vision and Pattern Recognition (CVPR, 2017.

S. Lefkimmiatis, Universal denoising networks: a novel cnn architecture for image denoising, Proc. Conference on Computer Vision and Pattern Recognition (CVPR), 2018.

C. Lemaréchal and C. Sagastizábal, Practical aspects of the moreau-yosida regularization: Theoretical preliminaries, SIAM Journal on Optimization, vol.7, issue.2, pp.367-385, 1997.

Y. Li, M. Tofighi, J. Geng, V. Monga, and Y. C. Eldar, Efficient and interpretable deep blind image deblurring via algorithm unrolling, IEEE Transactions on Computational Imaging, vol.6, pp.666-681, 2020.

D. Liu, B. Wen, Y. Fan, C. C. Loy, and T. S. Huang, Non-local recurrent network for image restoration, Adv. in Neural Information Processing Systems (NeurIPS), 2018.

M. Lustig, D. Donoho, and J. M. Pauly, Sparse mri: The application of compressed sensing for rapid mr imaging, Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine, vol.58, issue.6, pp.1182-1195, 2007.

D. Maclaurin, D. Duvenaud, and R. Adams, Gradient-based hyperparameter optimization through reversible learning, Proc. International Conference on Machine Learning (ICML), 2015.

J. Mairal, F. Bach, and J. Ponce, Task-driven dictionary learning, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.34, issue.4, pp.791-804, 2011.
URL : https://hal.archives-ouvertes.fr/inria-00521534

J. Mairal, F. Bach, and J. Ponce, Sparse modeling for image and vision processing, Foundations and Trends in Computer Graphics and Vision, vol.8, issue.2-3, pp.85-283, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01081139

J. Mairal, F. R. Bach, J. Ponce, G. Sapiro, and A. Zisserman, Non-local sparse models for image restoration, Proc. International Conference on Computer Vision (ICCV), 2009.
URL : https://hal.archives-ouvertes.fr/hal-02414291

D. Martin, C. Fowlkes, D. Tal, and J. Malik, A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics, 2001.

N. Mayer, E. Ilg, P. Häusser, P. Fischer, D. Cremers et al., A large dataset to train convolutional networks for disparity, optical flow, and scene flow estimation, IEEE International Conference on Computer Vision and Pattern Recognition (CVPR), 2016.

P. Mertikopoulos, B. Lecouat, H. Zenati, C. Foo, V. Chandrasekhar et al., Optimistic mirror descent in saddle-point problems: Going the extra (gradient) mile, Proc. International Conference on Learning Representations (ICLR), 2018.
URL : https://hal.archives-ouvertes.fr/hal-01891551

L. Metz, B. Poole, D. Pfau, and J. Sohl-dickstein, Unrolled generative adversarial networks, Proc. International Conference on Learning Representations (ICLR, 2017.

T. Miyato, T. Kataoka, M. Koyama, and Y. Yoshida, Spectral normalization for generative adversarial networks, Proc. International Conference on Learning Representations (ICLR), 2018.

V. Monga, Y. Li, and Y. C. Eldar, Algorithm unrolling: Interpretable, efficient deep learning for signal and image processing, 2019.

G. Montavon, W. Samek, and K. Müller, Methods for interpreting and understanding deep neural networks, Digital Signal Processing, vol.73, pp.1-15, 2018.

J. J. Moreau, Fonctions convexes duales et points proximaux dans un espace Hilbertien, CR Acad. Sci. Paris Sér. A Math, vol.255, pp.2897-2899, 1962.
URL : https://hal.archives-ouvertes.fr/hal-01867195

H. Nikaidô and K. Isoda, Note on non-cooperative convex games, Pacific Journal of Mathematics, vol.5, pp.807-815, 1955.

A. V. Oord, S. Dieleman, H. Zen, K. Simonyan, O. Vinyals et al., Wavenet: A generative model for raw audio, 2016.

A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury et al., Pytorch: An imperative style, high-performance deep learning library, Adv. in Neural Information Processing Systems (NeurIPS), 2019.

P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence (PAMI), vol.12, issue.7, pp.629-639, 1990.

T. Plötz and S. Roth, Neural nearest neighbors networks, Adv. in Neural Information Processing Systems (NeurIPS), 2018.

X. Qu, Y. Hou, F. Lam, D. Guo, J. Zhong et al., Magnetic resonance image reconstruction from undersampled measurements using a patch-based nonlocal operator, Medical image analysis, vol.18, issue.6, pp.843-856, 2014.

S. Ravishankar and Y. Bresler, Mr image reconstruction from highly undersampled k-space data by dictionary learning, IEEE transactions on medical imaging, vol.30, issue.5, pp.1028-1041, 2010.

Y. Romano, M. Elad, and P. Milanfar, The little engine that could: Regularization by denoising (red), SIAM Journal on Imaging Sciences, vol.10, issue.4, pp.1804-1844, 2017.

L. I. 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.

M. Scetbon, M. Elad, and P. Milanfar, , 2019.

D. Scharstein and R. Szeliski, A taxonomy and evaluation of dense two-frame stereo correspondence algorithms, International journal of computer vision, vol.47, issue.1-3, pp.7-42, 2002.

D. Simon and M. Elad, Rethinking the CSC model for natural images, Advances in Neural Information Processing Systems (NeurIPS), 2019.

J. Sun, H. Li, and Z. Xu, Deep admm-net for compressive sensing mri, Adv. in Neural Information Processing Systems (NIPS), 2016.

B. Turlach, W. Venables, and S. Wright, Simultaneous variable selection, Technometrics, vol.47, issue.3, p.349, 2005.

Z. Wang, D. Liu, J. Yang, W. Han, and T. Huang, Deep networks for image super-resolution with sparse prior, Proc. Conference on Computer Vision and Pattern Recognition (CVPR), 2015.

S. J. Wright, R. D. Nowak, and M. A. Figueiredo, Sparse reconstruction by separable approximation, IEEE Transactions on Signal Processing, vol.57, issue.7, pp.2479-2493, 2009.

J. Yang, Z. Wang, Z. Lin, S. Cohen, and T. Huang, Coupled dictionary training for image super-resolution, IEEE transactions on image processing, vol.21, issue.8, pp.3467-3478, 2012.

J. Yang, Y. Zhang, and W. Yin, A fast alternating direction method for tvl1-l2 signal reconstruction from partial fourier data, IEEE Journal of Selected Topics in Signal Processing, vol.4, issue.2, pp.288-297, 2010.

K. Yosida, Functional analysis, 1964.

Y. Yu, Better approximation and faster algorithm using the proximal average, Adv. in Neural Information Processing Systems (NIPS), 2013.

F. Zhang, V. Prisacariu, R. Yang, and P. H. Torr, Ga-net: Guided aggregation net for end-to-end stereo matching, Proc. Conference on Computer Vision and Pattern Recognition (CVPR), 2019.

J. Zhang and B. Ghanem, Ista-net: Interpretable optimization-inspired deep network for image compressive sensing, Proc. Conference on Computer Vision and Pattern Recognition (CVPR), 2018.

K. Zhang, W. Zuo, Y. Chen, D. Meng, and L. Zhang, Beyond a gaussian denoiser: Residual learning of deep cnn for image denoising, IEEE Transactions on Image Processing, vol.26, issue.7, pp.3142-3155, 2017.

K. Zhang, W. Zuo, and L. Zhang, Ffdnet: Toward a fast and flexible solution for cnn-based image denoising, IEEE Transactions on Image Processing, vol.27, issue.9, pp.4608-4622, 2018.

Y. Zhang, K. Li, K. Li, B. Zhong, and Y. Fu, Residual non-local attention networks for image restoration, Proc. International Conference on Learning Representations (ICLR), 2019.

S. Zheng, S. Jayasumana, B. Romera-paredes, V. Vineet, Z. Su et al., Conditional random fields as recurrent neural networks, Proc. Conference on Computer Vision and Pattern Recognition (CVPR), 2015.

D. Zoran and Y. Weiss, From learning models of natural image patches to whole image restoration, Proc. Conference on Computer Vision and Pattern Recognition

, Laplacian symmetric -extra-grad 144 33, vol.87

, Laplacian assymetric -extra-grad 480 35, vol.20, p.28

, Non-local TV assymmetric (several ?) 235

, Non-local TV assymmetric -extra-grad 226 37.83 30.98 28, vol.34, p.31

, Non-local TV assymmetric -extra-grad (several ?), vol.307, p.37

, Non-local Laplacian assymmetric (several ?) 235

, Non-local Laplacian assymmetric -extra-grad 226

, Non-local Laplacian assymmetric -extra-grad (several ?), vol.307, p.37

, Performance is measured in terms of average PSNR, Table A5: Patch level grayscale denoising on BSD68, training on BSD400 for all methods

, Sparse Coding + Barzilai-Borwein 68k 37, vol.85, p.31

, Sparse Coding + Variance

, Sparse Coding + TV 68k

, Sparse Coding + TV + Variance 68k 37, vol.84

, Sparse Coding + TV + Variance + Barzilai-Borwein 68k 37, vol.86, pp.29-33

, Non-local group -symmetric 68k 37, vol.94, p.16

, Non-local group -assymetric 68k 37, vol.95, p.29, 2019.

, Non-local group -assymetric + TV 68k

, Non-local group -assymetric + Variance

, Non-local group -assymetric + Variance + TV

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