Fast dual minimization of the vectorial total variation norm and applications to color image processing

Inverse Problems and Imaging

Volumn 2, Issue 4, 2008, Pages 455-484

  Bresson, Xavier ^a   Chan, Tony F ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

BV space; Chromaticity brightness color representation; Denoising on manifold; Dual formulation; Image deblurring; Image decomposition; Image denoising; Image inpainting; Inverse scale space; ROF model; Vector valued TV norm; Wavelet shrinkage

Indexed keywords

EID: 84929309148     PISSN: 19308337     EISSN: 19308345     Source Type: Journal    
DOI: 10.3934/ipi.2008.2.455     Document Type: Article

References (40)

1. L. Ambrosio, N. Fusco and D. Pallara, “Functions of Bounded Variation and Free Disconti- nuity Problems,” edition, Oxford University Press, 2000.
2. J. P. Aubin, “Applied Functional Analysis,” edition, John Wiley and Sons Inc., New York, 1979.
3. J.-F. Aujol and A. Chambolle, Dual norms and image decomposition models, International Journal of Computer Vision, 63 (2005), 85–104.
4. J. F. Aujol and S. H. Kang, Color image decomposition and restoration, Journal of Visual Communication and Image Representation, 17 (2006), 916–928.
5. J. Bect, L. Blanc-Feraud, G. Aubert and A. Chambolle, A l1-unified variational framework for image restoration, in “European Conference on Computer Vision,” (2004), 1–13.
6. M. Bertalmio, G. Sapiro, V. Caselles and C. Ballester, Image inpainting, in “International Conference on Computer Graphics and Interactive Techniques,” (2000), 417–424.
7. M. Bertalmio, L. Vese, G. Sapiro and S. Osher, Simultaneous structure and texture image inpainting, IEEE Transactions on Image Processing, 12 (2003), 882–889.
8. P. Blomgren and T. F. Chan, Color TV: Total variation methods for restoration of vector- valued images, IEEE Transactions on Image Processing, 7 (1998), 304–309.
9. I. Bogdanova, X. Bresson, J. Thiran and P. Vandergheynst, Scale-Space analysis and active contours for omnidirectional images, IEEE Transactions on Image Processing, 16 (2007), 1888–1901.
10. X. Bresson and J.-P. Thiran, Image segmentation model using active contour and image decomposition, in “IEEE International Conference on Image Processing,” (2006), 1657–1660.
11. H. Brezis, F. Bethuel and F. Helein, Singular limit for the minimization of Ginzburg-Landau functionals, Comptes rendus de l’Academie des Sciences, Serie Mathematique, 314 (1992), 891-895.
12. F. Catt, P. L. Lions, J. M. Morel and T. Coll, Image selective smoothing and edge detection by nonlinear diffusion, SIAM Journal on Numerical Analysis, 29 (1992), 182–193.
13. A. Chambolle, An algorithm for total variation minimization and applications, Journal of Mathematical Imaging and Vision, 20 (2004), 89–97.
14. A. Chambolle, R. DeVore, N. Lee and B. Lucier, Nonlinear wavelet image processing: Vari- ational problems, compression, and noise removal through wavelet shrinkage, IEEE Transac- tions on Image Processing, 7 (1998), 319–333.
15. T. F. Chan, G. H. Golub and P. Mulet, A nonlinear primal-dual method for total variation- based image restoration, SIAM journal on scientific computing, 20 (1999), 1964–1977.
16. T. F. Chan, S. H. Kang and J. Shen, Total variation denoising and enhancement of color images based on the CB and HSV color models, Journal of Visual Communication and Image Representation, 12 (2001), 422–435.
17. T. F. Chan and J. Shen, Variational image inpainting, Communications on Pure and Applied Mathematics, 58 (2005), 579–619.
18. T. F. Chan and C.-K. Wong, Total variation blind deconvolution, IEEE Transactions on Image Processing, 7 (1998), 370–375.
19. J. Darbon and S. Peyronnet, A vectorial self-dual morphological filter based on total variation minimization, in “International Symposium on Visual Computing” (ISVC), 3804 (2005), 388–395.
20. J. Darbon and M. Sigelle, Image restoration with discrete constrained total variation part I: Fast and exact optimization, Journal of Mathematical Imaging and Vision, 26 (2006), 277–291.
21. D. L. Donoho and I. M. Johnstone, Ideal spatial adaptation by wavelet shrinkage, Biometrika, 81 (1994), 425–455.
22. I. Ekeland and R. Temam, “Convex Analysis and Variational Problems,” edition, Society for Industrial and Applied Mathematics, 1976.
23. M. Fornasier and R. March, Restoration of color images by vector valued BV functions and variational calculus, SIAM Journal on Applied Mathematics, 68 (2007), 437–460.
24. D. Graziani, “On the L1-Lower Semicontinuity and Relaxation in BV (RM) for Integral Functionals with Discontinuous Integrand,” Ph.D thesis, 2006.
25. L. He, A. Marquina and S. Osher, Blind deconvolution using TV regularization and bregman iteration, International Journal of Imaging Systems and Technology, 15 (2003), 74–83.
26. B. Horn and B. Schunck, Determining optical flow, Artificial Intelligence, 17 (1981), 185–203.
27. S. H. Kang and R. March, Variational models for image colorization via chromaticity and brightness decomposition, IEEE Transactions on Image Processing, 16 (2007), 2251–2261.
28. E. Kreyszig, “Differential Geometry,” edition, Paperback, 1991.
29. F. Memoli, M. Bertalmio, L. T. Cheng, G. Sapiro and S. Osher, Variational problems and partial differential equations on implicit surfaces: Bye bye triangulated surfaces?, in “Geo- metric Level Set Methods in Imaging, Vision, and Graphics,” Springer, New York, (2003), 381–397.
30. Y. Meyer, “Oscillating Patterns in Image Processing and Nonlinear Evolution Equations,” edition, American Mathematical Society, 22, 2001.
31. S. Osher, M. Burger, D. Goldfarb, J. Xu and W. Yin, An iterative regularization method for total variation-based image restoration, SIAM Multiscale Modeling & Simulation, 4 (2005), 460–489.
32. L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259–268.
33. G. Sapiro and D. L. Ringach, Anisotropic diffusion of multi-valued images with applications to color filtering, IEEE Transactions on Image Processing, 5 (1996), 1582–1586.
34. M. Sion, On general minimax theorems, Pacific Journal of Mathematics, 9 (1958), 171–176.
35. N. Sochen, R. Kimmel and R. Malladi, A general framework for low level vision, IEEE Transactions on Image Processing, 7 (1998), 310–318.
36. E. Tadmor, S. Nezzar and L. Vese, Multiscale hierarchical decomposition of images with applications to deblurring, denoising and segmentation, Communications on Mathematical Sciences, 6 (2008), 281–307.
37. S. Osher, X.-C. Tai and R. Holm, Image inpainting using a TV-stokes equation, in “Image Processing Based on Partial Differential Equations,” Tai, Lie, Chan and Osher eds, Springer, (2007), 3–22.
38. A. N. Tikhonov and V. A. Arsenin, “Solution of Ill-posed Problems,” edition, Winston & Sons, 1977.
39. L. A. Vese and S. J. Osher, Color texture modeling and color image decomposition in a variational-PDE approach, in “Symbolic and Numeric Algorithms for Scientific Computing” (SYNASC), (2006), 103–110.
40. J. Von Neumann, Zur theorie der gesellschaftsspiele, Math. Annalen., 100 (1928), 295–320.