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.
DOI : 10.1016/0167-2789(92)90242-F

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

H. Attouch, G. Buttazzo, and G. Michaille, Variational analysis in Sobolev and BV spaces, MPS/SIAM Series on Optimization. Society for Industrial and Applied Mathematics (SIAM), vol.6, 2006.
DOI : 10.1137/1.9781611973488

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

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

J. 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

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.1-3349, 2003.

S. Osher and L. Vese, 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.

S. Osher and L. 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.

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

M. Bergounioux and L. Piffet, A second-order model for image denoising. Set-Valued and Variational Analysis, pp.277-306, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00440872

M. Bergounioux and L. Piffet, A full second order variational model for multiscale texture analysis, Computational Optimization and Applications, vol.18, issue.1???2, pp.215-237, 2013.
DOI : 10.1007/s10589-012-9484-9

M. Bergounioux, On poincaré-wirtinger inequalities in bv -spaces, Control & Cybernetics, vol.4, issue.40, pp.921-929, 2011.

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
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.378.6330

K. Bredies, K. Kunisch, and T. Valkonen, Properties of <mml:math altimg="si2.gif" display="inline" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mtext>-</mml:mtext><mml:msup><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>TGV</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>: The one-dimensional case, Journal of Mathematical Analysis and Applications, vol.398, issue.1, pp.438-454, 2013.
DOI : 10.1016/j.jmaa.2012.08.053

K. Bredies and M. Holler, Abstract, Journal of Inverse and Ill-posed Problems, 2014.
DOI : 10.1515/jip-2013-0068

L. Ambrosio, L. Faina, and R. March, Variational Approximation of a Second Order Free Discontinuity Problem in Computer Vision, SIAM Journal on Mathematical Analysis, vol.32, issue.6, pp.1171-1197, 2001.
DOI : 10.1137/S0036141000368326

M. Carriero, A. Leaci, and F. Tomarelli, Uniform density estimates for the blake & zisserman functional. Discrete and Continuous Dynamical Systems, pp.1129-1150, 2011.

M. Carriero, A. Leaci, and F. Tomarelli, Free gradient discontinuity and image inpainting, Journal of Mathematical Sciences, vol.XLII, issue.6, pp.805-819, 2012.
DOI : 10.1007/s10958-012-0716-4

Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, 2001.
DOI : 10.1090/ulect/022

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

G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing, Partial Differential Equations and the Calculus of Variations Weakly Differentiable Functions -Sobolev Space and Functions of Bounded Variation, Applied Mathematical Sciences, vol.147, 1980.

M. Carriero, A. Leaci, and F. Tomarelli, Special bounded hessian and elatsic-plastic plate, Rend. Ac. Naz delle Scienze, XVI, issue.13, pp.223-258, 1992.

J. Moreau, Inf-convolution des fonctions numériques sur un espace vectoriel, Comptes Rendus de lAcadémie des Sciences de Paris, p.50475049, 1963.

J. Hiriart-urruty and R. R. Phelps, Subdifferential calculus using ? -subdifferentials, Journal of Functional Analysis, vol.118, pp.150-166, 1993.

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

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

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

M. Bergounioux, Inf-convolution model : numerical experiment, 2014.

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