A projected gradient algorithm for color image decomposition

In this paper, we are interested in color image processing, and in particular color image decomposition. The problem of image decomposition consists in splitting an original image f into two components u and v. u should contain the geometric information of the original image, while v should be made of the oscillating patterns of f , such as textures. We propose here a scheme based on a projected gradient algorithm to compute the solution of various decomposition models for color images or vector-valued images. We provide a direct convergence proof of the scheme, and we give some analysis on color texture modeling. Key-words: Color image decomposition, projected gradient algorithm, color texture modeling.

[1]  I. Ekeland,et al.  Convex analysis and variational problems , 1976 .

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

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

[4]  C. Vogel.  Computational Methods for Inverse Problems , 1987 .

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

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

[7]  Guillermo Sapiro,et al.  Anisotropic diffusion of multivalued images with applications to color filtering , 1996, IEEE Trans. Image Process..

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

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

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

[11]  Ron Kimmel,et al.  A general framework for low level vision , 1998, IEEE Trans. Image Process..

[12]  Tony F. Chan,et al.  Color TV: total variation methods for restoration of vector-valued images , 1998, IEEE Trans. Image Process..

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

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

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

[16]  Ronald F. Gariepy.  FUNCTIONS OF BOUNDED VARIATION AND FREE DISCONTINUITY PROBLEMS (Oxford Mathematical Monographs) , 2001 .

[17]  S. Osher,et al.  IMAGE DECOMPOSITION AND RESTORATION USING TOTAL VARIATION MINIMIZATION AND THE H−1 NORM∗ , 2002 .

[18]  Stanley Osher,et al.  Modeling Textures with Total Variation Minimization and Oscillating Patterns in Image Processing , 2003, J. Sci. Comput..

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

[20]  S. Osher,et al.  Geometric Level Set Methods in Imaging, Vision, and Graphics , 2011, Springer New York.

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

[22]  Jean-François Aujol.  Contribution à l'analyse de textures en traitement d'images par méthodes variationnelles et équations aux dérivées partielles , 2004 .

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

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

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

[26]  Luminita A. Vese,et al.  Image Decomposition Using Total Variation and div(BMO) , 2005, Multiscale Model. Simul..

[27]  Antonin Chambolle,et al.  Dual Norms and Image Decomposition Models , 2005, International Journal of Computer Vision.

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

[29]  I. Daubechiesa,et al.  Variational image restoration by means of wavelets : Simultaneous decomposition , deblurring , and denoising , 2005 .

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

[31]  G. Aubert,et al.  Modeling Very Oscillating Signals. Application to Image Processing , 2005 .

[32]  Michael Elad,et al.  Submitted to Ieee Transactions on Image Processing Image Decomposition via the Combination of Sparse Representations and a Variational Approach , 2022 .

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

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

[35]  J. Darbon.  Total variation minimization with L/sup 1/ data fidelity as a contrast invariant filter , 2005, ISPA 2005. Proceedings of the 4th International Symposium on Image and Signal Processing and Analysis, 2005..

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

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

[38]  A. Haddad.  Méthodes variationnelles en traitement d'image , 2005 .

[39]  Wotao Yin,et al.  Image Cartoon-Texture Decomposition and Feature Selection Using the Total Variation Regularized L1 Functional , 2005, VLSM.

[40]  Tony F. Chan,et al.  Aspects of Total Variation Regularized L[sup 1] Function Approximation , 2005, SIAM J. Appl. Math..

[41]  Stanley Osher,et al.  Color Texture Modeling and Color Image Decomposition in a Variational-PDE Approach , 2006, 2006 Eighth International Symposium on Symbolic and Numeric Algorithms for Scientific Computing.

[42]  Tony F. Chan,et al.  Structure-Texture Image Decomposition—Modeling, Algorithms, and Parameter Selection , 2006, International Journal of Computer Vision.

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

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

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

[46]  Tony F. Chan,et al.  Combining geometrical and textured information to perform image classification , 2006, J. Vis. Commun. Image Represent..

[47]  Jean-François Aujol,et al.  Color image decomposition and restoration , 2006, J. Vis. Commun. Image Represent..

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

[49]  Jérôme Gilles,et al.  Noisy Image Decomposition: A New Structure, Texture and Noise Model Based on Local Adaptivity , 2007, Journal of Mathematical Imaging and Vision.

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

[51]  Wotao Yin,et al.  A comparison of three total variation based texture extraction models , 2007, J. Vis. Commun. Image Represent..

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

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

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

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

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