Abstract
In this paper, we are interested in texture modeling with functional analysis spaces. We focus on the case of 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.
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Acar, R., Vogel, C.: Analysis of total variation penalty methods for ill-posed problems. Inverse Probl. 10, 1217–1229 (1994)
Adams, R.A.: Sobolev Spaces. Mathematics, vol. 65. Academic Press, San Diego (1975)
Allard, W.K.: Total variation regularization for image denoising II: Examples. SIAM J. Imaging Sci. 1, 400–417 (2008)
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. Oxford University Press, Oxford (2000)
Andreu-Vaillo, F., Caselles, V., Mazon, J.M.: Parabolic Quasilinear Equations Minimizing Linear Growth Functionals. Progress in Mathematics, vol. 223. Birkhauser, Basel (2002)
Aubert, G., Aujol, J.-F.: Modeling very oscillating signals. Application to image processing. Appl. Math. Optim. 51(2), 163–182 (2005)
Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations. Applied Mathematical Sciences, vol. 147. Springer, Berlin (2001)
Aujol, J.-F.: Some first-order algorithms for total variation based image restoration. J. Math. Imaging Vis. 34(3), 307–327 (2009)
Aujol, J.-F., Chambolle, A.: Dual norms and image decomposition models. Int. J. Comput. Vis. 63(1), 85–104 (2005)
Aujol, J.-F., Chan, T.F.: Combining geometrical and textured information to perform image classification. J. Vis. Commun. Image Represent. 17(5), 1004–1023 (2006)
Aujol, J.-F., Gilboa, G.: Constrained and SNR-based solutions for TV-Hilbert space image denoising. J. Math. Imaging Vis. 26(1–2), 217–237 (2006)
Aujol, J.-F., Kang, S.H.: Color image decomposition and restoration. J. Vis. Commun. Image Represent. 17(4), 916–928 (2006)
Aujol, J.-F., Aubert, G., Blanc-Fíraud, L., Chambolle, A.: Image decomposition into a bounded variation component and an oscillating component. J. Math. Imaging Vis. 22(1), 71–88 (2005)
Aujol, J.-F., Gilboa, G., Chan, T., Osher, S.: Structure-texture image decomposition—modeling, algorithms, and parameter selection. Int. J. Comput. Vis. 67(1), 111–136 (2006)
Beck, A., Teboule, M.: Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems. IEEE Trans. Image Process. 18(11), 2419–2434 (2009)
Bect, J., Blanc-Féraud, L., Aubert, G., Chambolle, A.: A l1-unified variational framework for image restoration. In: ECCV 04. Lecture Notes in Computer Sciences, vol. 3024, pp. 1–13. Springer, Berlin (2004)
Bermudez, A., Moreno, C.: Duality methods for solving variational inequalities. Comput. Maths. Appl. 7(1), 43–58 (1981)
Bioucas-Dias, J., Figueiredo, M.: Thresholding algorithms for image restoration. IEEE Trans. Image Process. 16(12), 2980–2991 (2007)
Blomgren, P., Chan, T.: Color TV: Total variation methods for restoration of vector valued images. IEEE Trans. Image Process. 7(3), 304–309 (1998)
Bresson, X., Chan, T.: Fast dual minimization of the vectorial total variation norm and applications to color image processing. Inverse Probl. Imaging 2(4), 455–484 (2008)
Chambolle, A.: An algorithm for total variation minimization and its applications. J. Math. Imaging Vis. 20, 89–97 (2004)
Chambolle, A.: Total variation minimization and a class of binary MRF models. In: EMMCVPR. Lecture Notes in Computer Sciences, vol. 3757, pp. 136–152. Springer, Berlin (2005)
Chambolle, A., Lions, P.L.: Image recovery via total variation minimization and related problems. Numer. Math. 76(3), 167–188 (1997)
Chambolle, A., Levine, S.E., Lucier, B.J.: Some variations on total variation-based image smoothing (2009)
Chan, T., Esedoglu, S.: Aspects of total variation regularized L 1 function approximation. SIAM J. Appl. Math. 65(5), 1817–1837 (2005)
Chan, T., Shen, J.: Image Processing and Analysis—Variational, PDE, Wavelet, and Stochastic Methods. SIAM, Philadelphia (2005)
Chan, T., Golub, G., Mulet, P.: A nonlinear primal-dual method for total variation-based image restoration. SIAM J. Sci. Comput. 20(6), 1964–1977 (1999)
Chan, T.F., Kang, S.H., Shen, J.: Total variation denoising and enhancement of color images based on the cb and hsv color models. J. Vis. Commun. Image Represent. 12, 2001 (2000)
Charbonnier, P., Blanc-Feraud, L., Aubert, G., Barlaud, M.: Deterministic edge-preserving regularization in computed imaging. IEEE Trans. Image Process. 6(2), 298–311 (2007)
Combettes, P.L., Pesquet, J.: Image restoration subject to a total variation constraint. IEEE Trans. Image Process. 13(9), 1213–1222 (2004)
Combettes, P.L., Wajs, V.: Signal recovery by proximal forward-backward splitting. SIAM J. Multiscale Model. Simul. 4(4), 1168–1200 (2005)
Darbon, J.: Total variation minimization with l 1 data fidelity as a contrast invariant filter. In: 4th International Symposium on Image and Signal Processing and Analysis (ISPA 2005), pp. 221–226. September (2005)
Darbon, J., Sigelle, M.: Image restoration with discrete constrained total variation Part I: Fast and exact optimization. J. Math. Imaging Vis. 26(3), 261–276 (2006)
Daubechies, I., Teschke, G.: Variational image restoration by means of wavelets: simultaneous decomposition, deblurring and denoising. Appl. Comput. Harmon. Anal. 19, 1–16 (2005)
Daubechies, I., Defrise, M., De Mol, C.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure Appl. Math. 57, 1413–1457 (2004)
Dobson, D., Vogel, C.: Convergence of an iterative method for total variation denoising. SIAM J. Numer. Anal. 34, 1779–1791 (1997)
Durand, S., Fadili, J., Nikolova, M.: Multiplicative noise removal using L 1 fidelity on frame coefficients. CMLA Report, 08-40 (2008)
Duval, V., Aujol, J.-F., Gousseau, Y.: The TVL1 model: a geometric point of view. SIAM J. Multiscale Model. Simul. 8, 154–189 (2009)
Eckstein, J.: The Lions-Mercier algorithm and the alternating direction method are instances of the proximal point algorithm (1988)
Ekeland, I., Témam, R.: Convex Analysis and Variational Problems. Classics in Applied Mathematics, vol. 28. SIAM, Philadelphia (1999)
Fadili, J., Starck, J.-L.: Monotone operator splitting for optimization problems in sparse recovery. In: IEEE ICIP, Cairo, Egypt (2009)
Fu, H., Ng, M., Nikolova, M., Barlow, J.: Efficient minimization methods of mixed l1-l1 and l2-l1 norms for image restoration. SIAM J. Sci. Comput. 27(6), 1881–1902 (2006)
G Ciarlet, P.: Introduction “l’Analyse Numírique Matricielle et” l’Optimisation. Dunod, Paris (1998)
Garnett, J., Jones, P., Le, T., Vese, L.: Modeling oscillatory components with the homogeneous spaces BMO −α and W −α,p. UCLA CAM Report, 07-21, July (2007)
Gilles, J.: Noisy image decomposition: a new structure, textures and noise model based on local adaptivity. J. Math. Imaging Vis. 28(3) (2007)
Goldfarb, D., Yin, W.: Second-order cone programming methods for total variation based image restoration. SIAM J. Sci. Comput. 27(2), 622–645 (2005)
Haddad, A.: Míthodes variationnelles en traitement d’images. PhD thesis, ENS Cachan (2005)
Julesz, B.: Texton gradients: The texton theory revisited. Biol. Cybern. 54, 245–251 (1986)
Le, T., Vese, L.: Image decomposition using total variation and div(BMO). Multiscale Model. Simul., SIAM Interdiscip. J. 4(2), 390–423 (2005)
Lieu, L.: Contribution to problems in image restoration, decomposition, and segmentation by variational methods and partial differential equations. PhD thesis, UCLA (2006)
Lieu, L.H., Vese, L.A.: Image restoration and decomposition via bounded total variation and negative Hilbert-Sobolev spaces. Appl. Math. Optim. 58, 167–193 (2009)
Lions, P.L., Mercier, B.: Algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16(6), 964–979 (1979)
Meyer, Y.: Oscillating Patterns in Image Processing and Nonlinear Evolution Equations. University Lecture Series, vol. 22. American Mathematical Society, Providence (2001). The fifteenth Dean Jacqueline B. Lewis memorial lectures
Nesterov, Y.: Smooth minimization of non-smooth functions. Math. Program. (A) 103(1), 127–152 (2005)
Ng, M.K., Qi, L., Yang, Y.F., Huang, Y.: On semismooth Newton methods for total variation minimization. J. Math. Imaging Vis. 27, 265–276 (2007)
Nikolova, M., Chan, R.: The equivalence of half-quadratic minimization and the gradient linearization iteration. IEEE Trans. Image Process. 16(6), 1623–1627 (2007)
Osher, S., Paragios, N.: Geometric Level Set Methods in Imaging, Vision, and Graphic. Springer, Berlin (2003)
Osher, S., Sole, A., Vese, L.: Image decomposition and restoration using total variation minimization and the H −1 norm. SIAM J. Multiscale Model. Simul. 1(3), 349–370 (2003)
Rudin, L., Osher, S., Fatemi, E.: Non linear total variation based noise removal algorithms. Physica D 60, 259–268 (2003)
Sapiro, G., Ringach, D.L.: Anisotropic diffusion of multivalued images with applications to color filtering. IEEE Trans. Image Process. 5(11), 1582–1586 (1996)
Setzer, S.: Split Bregman algorithm, Douglas-Rachford splitting and frame shrinkage. Preprint, University of Mannheim (2008)
Sochen, N., Kimmel, R., Malladi, R.: A general framework for low level vision. IEEE Trans. Image Process. 7(3), 310–318 (1998)
Soille, P.: Morphological Image Analysis: Principles and Applications. Springer, New York (2003)
Starck, J.L., ELad, M., Donoho, D.L.: Image decomposition via the combination of sparse representations and a variational approach (2005)
Vese, L.A., Osher, S.J.: Modeling textures with total variation minimization and oscillating patterns in image processing. J. Sci. Comput. 19(1–3), 553–572 (2003)
Vese, L.A., Osher, S.J.: Color texture modeling and color image decomposition in a variational-PDE approach. In: Proceedings of the Eighth International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC ’06), pp. 103–110. IEEE, New York (2006)
Vogel, C.R.: Computational Methods for Inverse Problems. Frontiers in Applied Mathematics, vol. 23. SIAM, Philadelphia (2002)
Weiss, P., Blanc-Fíraud, L., Aubert, G.: Efficient schemes for total variation minimization under constraints in image processing. SIAM J. Sci. Comput. 31(3), 2047–2080 (2009)
Yin, W., Goldfarb, D.: Second-order cone programming methods for total variation based image restoration. SIAM J. Sci. Comput. 27(2), 622–645 (2005)
Yin, W., Goldfarb, D., Osher, S.: Image cartoon-texture decomposition and feature selection using the total variation regularized l 1. In: Variational, Geometric, and Level Set Methods in Computer Vision. Lecture Notes in Computer Science, vol. 3752, pp. 73–84. Springer, Berlin (2005)
Yin, W., Goldfarb, D., Osher, S.: A comparison of three total variation based texture extraction models. J. Vis. Commun. Image Represent. 18(3), 240–252 (2007)
Yuan, J., Schnörr, C., Steidl, G.: Convex Hodge decomposition and regularization of image flows. J. Math. Imaging Vis. 33(2), 169–177 (2009)
Zhu, M., Chan, T.F.: An efficient primal-dual hybrid gradient algorithm for total variation image restoration, May (2008). UCLA CAM Report 08-34
Zhu, M., Wright, S.J., Chan, T.F.: Duality-based algorithms for total variation image restoration. Comput. Optim. Appl. (2008). doi:10.1007/s10589-008-9225-2
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The first and second authors were supported by the grants FREEDOM (ANR07-JCJC-0048-01), “Movies, restoration and missing data”, and NATIMAGES (ANR-08-EMER-009), “Adaptivity for natural images and texture representations.” The third author was supported by the National Science Foundation grant DMS-0714945.
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Duval, V., Aujol, JF. & Vese, L.A. Mathematical Modeling of Textures: Application to Color Image Decomposition with a Projected Gradient Algorithm. J Math Imaging Vis 37, 232–248 (2010). https://doi.org/10.1007/s10851-010-0203-9
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DOI: https://doi.org/10.1007/s10851-010-0203-9