Some First-Order Algorithms for Total Variation Based Image Restoration

This paper deals with first-order numerical schemes for image restoration. These schemes rely on a duality-based algorithm proposed in 1979 by Bermùdez and Moreno. This is an old and forgotten algorithm that is revealed wider than recent schemes (such as the Chambolle projection algorithm) and able to improve contemporary schemes. Total variation regularization and smoothed total variation regularization are investigated. Algorithms are presented for such regularizations in image restoration. We prove the convergence of all the proposed schemes. We illustrate our study with numerous numerical examples. We make some comparisons with a class of efficient algorithms (proved to be optimal among first-order numerical schemes) recently introduced by Y. Nesterov.

[1]  A. Bermúdez,et al.  Duality methods for solving variational inequalities , 1981 .

[2]  Tom Goldstein,et al.  The Split Bregman Method for L1-Regularized Problems , 2009, SIAM J. Imaging Sci..

[3]  Yves Meyer,et al.  Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures , 2001 .

[4]  Michael K. Ng,et al.  On Semismooth Newton’s Methods for Total Variation Minimization , 2007, Journal of Mathematical Imaging and Vision.

[5]  I. Daubechies,et al.  An iterative thresholding algorithm for linear inverse problems with a sparsity constraint , 2003, math/0307152.

[6]  Pierre Weiss,et al.  Algorithmes rapides d'optimisation convexe. Applications à la reconstruction d'images et à la détection de changements. (Fast algorithms for convex optimization. Applications to image reconstruction and change detection) , 2008 .

[7]  J. Craggs.  Applied Mathematical Sciences , 1973 .

[8]  T. Chan,et al.  On the Convergence of the Lagged Diffusivity Fixed Point Method in Total Variation Image Restoration , 1999 .

[9]  Jérôme Darbon,et al.  Image Restoration with Discrete Constrained Total Variation Part II: Levelable Functions, Convex Priors and Non-Convex Cases , 2006, Journal of Mathematical Imaging and Vision.

[10]  Rachid Deriche,et al.  Regularizing Flows for Constrained Matrix-Valued Images , 2004 .

[11]  乔花玲,et al.  关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解 , 2003 .

[12]  H. Brezis.  Analyse fonctionnelle : théorie et applications , 1983 .

[13]  Guy Gilboa,et al.  Constrained and SNR-Based Solutions for TV-Hilbert Space Image Denoising , 2006, Journal of Mathematical Imaging and Vision.

[14]  Mohamed-Jalal Fadili,et al.  Multiplicative Noise Removal Using L1 Fidelity on Frame Coefficients , 2008, Journal of Mathematical Imaging and Vision.

[15]  Gilles Aubert,et al.  Efficient Schemes for Total Variation Minimization Under Constraints in Image Processing , 2009, SIAM J. Sci. Comput..

[16]  Mingqiang Zhu,et al.  An Efficient Primal-Dual Hybrid Gradient Algorithm For Total Variation Image Restoration , 2008 .

[17]  Y. Meyer.  Oscillating Patterns in Some Nonlinear Evolution Equations , 2006 .

[18]  Jean-François Aujol,et al.  Irregular to Regular Sampling, Denoising, and Deconvolution , 2009, Multiscale Model. Simul..

[19]  Patrick L. Combettes,et al.  Image restoration subject to a total variation constraint , 2004, IEEE Transactions on Image Processing.

[20]  Mila Nikolova,et al.  Efficient Minimization Methods of Mixed l2-l1 and l1-l1 Norms for Image Restoration , 2005, SIAM J. Sci. Comput..

[21]  Tony F. Chan,et al.  Image processing and analysis - variational, PDE, wavelet, and stochastic methods , 2005 .

[22]  A. Chambolle.  Practical, Unified, Motion and Missing Data Treatment in Degraded Video , 2004, Journal of Mathematical Imaging and Vision.

[23]  Dirk A. Lorenz,et al.  A generalized conditional gradient method for nonlinear operator equations with sparsity constraints , 2007 .

[24]  M. Shirosaki.  Another proof of the defect relation for moving targets , 1991 .

[25]  Stephen J. Wright,et al.  Duality-based algorithms for total-variation-regularized image restoration , 2010, Comput. Optim. Appl..

[26]  Gjlles Aubert,et al.  Mathematical problems in image processing , 2001 .

[27]  Andrés Almansa,et al.  A TV Based Restoration Model with Local Constraints , 2008, J. Sci. Comput..

[28]  Y. Nesterov.  Gradient methods for minimizing composite objective function , 2007 .

[29]  Antonin Chambolle,et al.  A l1-Unified Variational Framework for Image Restoration , 2004, ECCV.

[30]  Raymond H. Chan,et al.  The Equivalence of Half-Quadratic Minimization and the Gradient Linearization Iteration , 2007, IEEE Transactions on Image Processing.

[31]  Ronny Ramlau,et al.  A Tikhonov-based projection iteration for nonlinear Ill-posed problems with sparsity constraints , 2006, Numerische Mathematik.

[32]  Jean-François Aujol,et al.  Projected Gradient Based Color Image Decomposition , 2009, SSVM.

[33]  Gene H. Golub,et al.  A Nonlinear Primal-Dual Method for Total Variation-Based Image Restoration , 1999, SIAM J. Sci. Comput..

[34]  Michel Barlaud,et al.  Deterministic edge-preserving regularization in computed imaging , 1997, IEEE Trans. Image Process..

[35]  Mohamed-Jalal Fadili,et al.  Monotone operator splitting for optimization problems in sparse recovery , 2009, 2009 16th IEEE International Conference on Image Processing (ICIP).

[36]  Jérôme Darbon,et al.  Image Restoration with Discrete Constrained Total Variation Part I: Fast and Exact Optimization , 2006, Journal of Mathematical Imaging and Vision.

[37]  Jing Yuan,et al.  Convex Hodge Decomposition and Regularization of Image Flows , 2009, Journal of Mathematical Imaging and Vision.

[38]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[39]  P. Lions,et al.  Image recovery via total variation minimization and related problems , 1997 .

[40]  Simon Setzer,et al.  Split Bregman Algorithm, Douglas-Rachford Splitting and Frame Shrinkage , 2009, SSVM.

[41]  Yurii Nesterov,et al.  Smooth minimization of non-smooth functions , 2005, Math. Program..

[42]  J. Aujol,et al.  Some algorithms for total variation based image restoration , 2008 .

[43]  José M. Bioucas-Dias,et al.  A New TwIST: Two-Step Iterative Shrinkage/Thresholding Algorithms for Image Restoration , 2007, IEEE Transactions on Image Processing.

[44]  ANTONIN CHAMBOLLE,et al.  An Algorithm for Total Variation Minimization and Applications , 2004, Journal of Mathematical Imaging and Vision.

[45]  V. Caselles,et al.  Parabolic Quasilinear Equations Min-imizing Linear Growth Functionals , 2004 .

[46]  C. Vogel,et al.  Analysis of bounded variation penalty methods for ill-posed problems , 1994 .

[47]  Robert D. Nowak,et al.  Majorization–Minimization Algorithms for Wavelet-Based Image Restoration , 2007, IEEE Transactions on Image Processing.

[48]  R. Temam,et al.  Analyse convexe et problèmes variationnels , 1974 .

[49]  Wotao Yin,et al.  Second-order Cone Programming Methods for Total Variation-Based Image Restoration , 2005, SIAM J. Sci. Comput..

[50]  O. Nelles,et al.  An Introduction to Optimization , 1996, IEEE Antennas and Propagation Magazine.

[51]  P. G. Ciarlet,et al.  Exercices d'analyse numérique matricielle et d'optimisation , 1982 .

[52]  Antonin Chambolle,et al.  Image Decomposition into a Bounded Variation Component and an Oscillating Component , 2005, Journal of Mathematical Imaging and Vision.

[53]  H. Brezis.  Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert , 1973 .

[54]  Yurii Nesterov,et al.  Introductory Lectures on Convex Optimization - A Basic Course , 2014, Applied Optimization.

[55]  A. Pazy,et al.  On the asymptotic behavior of iterates of nonexpansive mappings in Hilbert space , 1977 .

[56]  D. Dobson,et al.  Convergence of an Iterative Method for Total Variation Denoising , 1997 .

[57]  T. Chan,et al.  Fast dual minimization of the vectorial total variation norm and applications to color image processing , 2008 .

[58]  Antonin Chambolle,et al.  Total Variation Minimization and a Class of Binary MRF Models , 2005, EMMCVPR.

[59]  L. Ambrosio,et al.  Functions of Bounded Variation and Free Discontinuity Problems , 2000 .

[60]  P. G. Ciarlet,et al.  Introduction a l'analyse numerique matricielle et a l'optimisation , 1984 .

[61]  Patrick L. Combettes,et al.  Signal Recovery by Proximal Forward-Backward Splitting , 2005, Multiscale Model. Simul..