Mathematical modeling of textures: Application to color image decomposition with a projected gradient algorithm

Journal of Mathematical Imaging and Vision

Volumn 37, Issue 3, 2010, Pages 232-248

  Duval, Vincent ^a   Aujol, Jean François ^b,c   Vese, Luminita A ^d  

^a INSTITUT TELECOM   (France)

^b UNIVERSITÉ PARIS SACLAY   (France)

^c CNRS   (France)

^d UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Color image decomposition; Color texture modeling; Projected gradient algorithm; Texture modeling

Indexed keywords

COLOR IMAGES; COLOR TEXTURES; DECOMPOSITION MODEL; GEOMETRIC INFORMATION; IMAGE DECOMPOSITION; MATHEMATICAL MODELING; ORIGINAL IMAGES; OSCILLATING PATTERN; PROJECTED GRADIENT; TEXTURE MODELING; TWO-COMPONENT; VECTOR VALUED;

ALGORITHMS; COLOR IMAGE PROCESSING; TEXTURES;

COLOR;

EID: 77954952357     PISSN: 09249907     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10851-010-0203-9     Document Type: Article

References (74)

1. Acar, R., Vogel, C.: Analysis of total variation penalty methods for ill-posed problems. Inverse Probl. 10, 1217-1229 (1994)
2. Adams, R. A.: Sobolev Spaces. Mathematics, vol. 65. Academic Press, San Diego (1975)
3. Allard, W. K.: Total variation regularization for image denoising II: Examples. SIAM J. Imaging Sci. 1, 400-417 (2008)
4. Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. Oxford University Press, Oxford (2000)
5. Andreu-Vaillo, F., Caselles, V., Mazon, J. M.: Parabolic Quasilinear Equations Minimizing Linear Growth Functionals. Progress in Mathematics, vol. 223. Birkhauser, Basel (2002)
6. Aubert, G., Aujol, J.-F.: Modeling very oscillating signals. Application to image processing. Appl. Math. Optim. 51 (2), 163-182 (2005)
7. 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)
8. Aujol, J.-F.: Some first-order algorithms for total variation based image restoration. J. Math. Imaging Vis. 34 (3), 307-327 (2009)
9. Aujol, J.-F., Chambolle, A.: Dual norms and image decomposition models. Int. J. Comput. Vis. 63 (1), 85-104 (2005)
10. 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)
11. 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)
12. Aujol, J.-F., Kang, S. H.: Color image decomposition and restoration. J. Vis. Commun. Image Represent. 17 (4), 916-928 (2006)
13. 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)
14. 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)
15. 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)
16. 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)
17. Bermudez, A., Moreno, C: Duality methods for solving variational inequalities. Comput. Maths. Appl. 7 (1), 43-58 (1981)
18. Bioucas-Dias, J., Figueiredo, M.: Thresholding algorithms for image restoration. IEEE Trans. Image Process. 16 (12), 2980-2991 (2007)
19. Blomgren, P., Chan, T: Color TV: Total variation methods for restoration of vector valued images. IEEE Trans. Image Process. 7 (3), 304-309 (1998)
20. 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)
21. Chambolle, A.: An algorithm for total variation minimization and its applications. J. Math. Imaging Vis. 20, 89-97 (2004)
22. 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)
23. Chambolle, A., Lions, PL.: Image recovery via total variation minimization and related problems. Numer. Math. 76 (3), 167-188 (1997)
24. Chambolle, A., Levine, S. E., Lucier, B. J.: Some variations on total variation-based image smoothing (2009)
25. 1 function approximation. SIAM J. Appl. Math. 65 (5), 1817-1837 (2005)
26. Chan, T, Shen, J.: Image Processing and Analysis-Variational, PDE, Wavelet, and Stochastic Methods. SIAM, Philadelphia (2005)
27. 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)
28. 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)
29. 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)
30. Combettes, PL., Pesquet, J.: Image restoration subject to a total variation constraint. IEEE Trans. Image Process. 13 (9), 1213-1222 (2004)
31. Combettes, PL., Wajs, V: Signal recovery by proximal forwardbackward splitting. SIAM J. Multiscale Model. Simul. 4 (4), 1168-1200 (2005)
32. Darbon, J.: Total variation minimization with l1 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) (Pubitemid 44459692)
33. 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)
34. Daubechies, I., Teschke, G.: Variational image restoration by means of wavelets: simultaneous decomposition, deblurring and denoising. Appl. Comput. Harmon. Anal. 19, 1-16 (2005)
35. 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)
36. Dobson, D., Vogel, C: Convergence of an iterative method for total variation denoising. SIAM J. Numer. Anal. 34, 1779-1791 (1997)
37. 1 fidelity on frame coefficients. CMLA Report, 08-40 (2008)
38. Duval, V, Aujol, J.-F, Gousseau, Y.: The TVL1 model: a geometric point of view. SIAM J. Multiscale Model. Simul. 8, 154-189 (2009)
39. Eckstein, J.: The Lions-Mercier algorithm and the alternating direction method are instances of the proximal point algorithm (1988)
40. Ekeland, I., Témam, R.: Convex Analysis and Variational Problems. Classics in Applied Mathematics, vol. 28. SIAM, Philadelphia (1999)
41. Fadili, J., Starck, J.-L.: Monotone operator splitting for optimization problems in sparse recovery. In: IEEE ICIP, Cairo, Egypt (2009)
42. 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)
43. G Ciarlet, P.: Introduction "l'Analyse Numírique Matricielle et" l'Optimisation. Dunod, Paris (1998)
44. -a, p. UCLA CAM Report, 07-21, July (2007)
45. Gilles, J.: Noisy image decomposition: a new structure, textures and noise model based on local adaptivity. J. Math. Imaging Vis. 28 (3) (2007)
46. Goldfarb, D., Yin, W.: Second-order cone programming methods for total variation based image restoration. SIAM J. Sci. Comput. 27 (2), 622-645 (2005)
47. Haddad, A.: Míthodes variationnelles en traitement d'images. PhD thesis, ENS Cachan (2005)
48. Julesz, B.: Texton gradients: The texton theory revisited. Biol. Cybern. 54, 245-251 (1986)
49. Le, T, Vese, L.: Image decomposition using total variation and div (BMO). Multiscale Model. Simul., SIAM Interdiscip. J. 4 (2), 390-423 (2005)
50. Lieu, L.: Contribution to problems in image restoration, decomposition, and segmentation by variational methods and partial differential equations. PhD thesis, UCLA (2006)
51. 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)
52. Lions, PL., Mercier, B.: Algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16 (6), 964-979 (1979)
53. 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
54. Nesterov, Y: Smooth minimization of non-smooth functions. Math. Program. (A) 103 (1), 127-152 (2005)
55. 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)
56. Nikolova, M., Chan, R.: The equivalence of half-quadratic minimization and the gradient linearization iteration. IEEE Trans. Image Process. 16 (6), 1623-1627 (2007)
57. Osher, S., Paragios, N.: Geometric Level Set Methods in Imaging, Vision, and Graphic. Springer, Berlin (2003)
58. -1 norm. SIAM J. Multiscale Model. Simul. 1 (3), 349-370 (2003)
59. Rudin, L., Osher, S., Fatemi, E.: Non linear total variation based noise removal algorithms. Physica D 60, 259-268 (2003)
60. Sapiro, G., Ringach, D. L.: Anisotropic diffusion of multivalued images with applications to color filtering. IEEE Trans. Image Process. 5 (11), 1582-1586 (1996)
61. Setzer, S.: Split Bregman algorithm, Douglas-Rachford splitting and frame shrinkage. Preprint, University of Mannheim (2008)
62. Sochen, N., Kimmel, R., Malladi, R.: A general framework for low level vision. IEEE Trans. Image Process. 7 (3), 310-318 (1998)
63. Soille, P.: Morphological Image Analysis: Principles and Applications. Springer, New York (2003)
64. Starck, J. L., Elad, M., Donoho, D. L.: Image decomposition via the combination of sparse representations and a variational approach (2005)
65. 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)
66. 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)
67. Vogel, C. R.: Computational Methods for Inverse Problems. Frontiers in Applied Mathematics, vol. 23. SIAM, Philadelphia (2002)
68. 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)
69. Yin, W., Goldfarb, D.: Second-order cone programming methods for total variation based image restoration. SIAM J. Sci. Comput. 27 (2), 622-645 (2005)
70. 1. In: Variational, Geometric, and Level Set Methods in Computer Vision. Lecture Notes in Computer Science, vol. 3752, pp. 73-84. Springer, Berlin (2005)
71. 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)
72. Yuan, J., Schnörr, C., Steidl, G.: Convex Hodge decomposition and regularization of image flows. J. Math. Imaging Vis. 33 (2), 169-177 (2009)
73. Zhu, M., Chan, T. F.: An efficient primal-dual hybrid gradient algorithm for total variation image restoration, May (2008). UCLA CAM Report 08-34
74. 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