Alternating minimization method for total variation based wavelet shrinkage model

Communications in Computational Physics

Volumn 8, Issue 5, 2010, Pages 976-994

  Zeng, Tieyong ^a   Li, Xiaolong ^b   Ng, Michael ^a  

^a HONG KONG BAPTIST UNIVERSITY   (Hong Kong)

^b PEKING UNIVERSITY   (China)

Author keywords

Alternating minimization; Convergence; Gibbs oscillation; Total variation; Wavelet shrinkage

Indexed keywords

EID: 77958555462     PISSN: 18152406     EISSN: 19917120     Source Type: Journal    
DOI: 10.4208/cicp.210709.180310a     Document Type: Article

References (36)

1. Luigi Ambrosio, Nicola Fusco and Diego Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, 2000.
2. Gilles Aubert and Pierre Kornprobst,Mathematical Problems in Image Processing, Springer, 2002.
3. J. F. Aujol, G. Aubert, L. Blanc-Ferraud and A. Chambolle, Image decomposition into a bounded variation component and an oscillating component, J. Math. Imaging. Vision., 22 (1) (2005), 71-88.
4. Heinz Bauer and Robert B. Burckel (Translator), Measure and Integration Theory, Volume 26 of De Gruyter Studies in Mathematics,Walter de Gruyter, Berlin, New York, 2001.
5. E. Candes and D. L. Donoho, Curvelets-a surprisingly effective nonadaptive respresentation for objects with edges, In Curves and surfaces, pages 105-120, Vanderbilt University Press, Nashville, TN, US, 2000.
6. A. Chambolle, An algorithm for total variation minimization and applications, J. Math. Imaging. Vision., 20 (1-2) (2004), 89-97.
7. A. Chambolle and P.-L. Lions, Image recovery via total minimization and related problems, Numer. Math., 76 (1997), 167-188.
8. Antonin Chambolle, Total variation minimization and a class of binary MRF models, in Energy Minimization Methods in Computer Vision and Pattern Recognition, Lecture Notes in Computer Science, 3757 (2005), 136-152. Springer Berlin/Heidelberg.
9. T. Chan and H. M. Zhou, Total variation improved wavelet thresholding in image compression, In Proceedings to the International Conference on Image Processing, Vancouver, BC, Canada, Sept. 10-13, 2000.
10. T. F. Chan and H.M. Zhou, Optimal construction of wavelet coefficients using total variation regularization in image compression, Technical Report CAM 00-27, University of California Los Angeles, July 2000.
11. Tony F. Chan and Ke Chen, An optimization-based multilevel algorithm for total variation image denoising, Multiscale. Model. Sim., 5 (2) (2006), 615-645.
12. S. Chang, B. Yu and M. Vetterli, Spatially adaptive wavelet thresholding with context modeling forimage denoising, IEEE. Trans. Image. Process., 9 (9) (2000), 1522-1531.
13. Patrick L. Combettes and Valérie R. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale. Model. Sim., 4 (4) (2005), 1168-1200.
14. I. Daubechies, M. Defrise and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure. Appl. Math., 57 (11) (2004), 1413-1457.
15. D. L. Donoho, De-noising by soft-thresholding, IEEE. Trans. Inf. Theor., 41 (3) (1995), 613-627.
16. D. L. Donoho and I. M. Johnstone, Ideal spatial adaptation by wavelet shrinkage, Biometrika., 81 (3) (1994), 425-455.
17. E. J. Candes, L. Demanet, D. L. Donoho and L. Ying, Fast discrete curvelet transforms, Multiscale. Model. Sim., 5 (2005), 861-899.
18. F. Malgouyres, A noise selection approach of image restoration, SPIE, Internatinal Conference on Wavelets IX, Unser Laine and Aldroubi, Eds., San Diego, 4473 (2001), 34-41.
19. A. Jalobeanu, L. Blanc-féraud and J. Zerubia, Satellite image deblurring using complex wavelet packets, Int. J. Comput. Vision., 51 (3) (2003), 205-217.
20. F. Malgouyres, Minimizing the total variation under a general convex constraint for image restoration, IEEE, Trans. Image. Process., 11 (12) (2002), 1450-1456.
21. M. Amiri, Z. Azimifar and P. Fieguth, Correlated non-linear wavelet shrinkage, Proceedings to the International Conference on Image Processing, 2008, 2348-2351.
22. M. Ng, L. Qi, Y. Yang and Y. Huang, On semismooth Newton's methods for total variation minimization, J. Math. Imaging. Vision., 27 (2007), 265-276.
23. R. R. Coifman and A. Sowa, Combining the calculus of variations and wavelets for image enhancement, Appl. Comput. Harmon. Anal., 9 (2000), 1-18.
24. L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D., 60 (1992), 259-268.
25. S. Durand and J. Froment, Reconstruction of wavelet coefficients using total varaition minimization, SIAM. J. Sci. Comput., 24 (2003), 1754-1767.
26. S. Lintner and F.Malgouyres, Solving a variational image restoration model which involves l contraints, Inverse. Probl., 20 (3) (2004), 815-831.
27. S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, 1999 (2nd ed.).
28. Yves Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, University Lecture Series (Vol. 22), American Mathematical Society, Providence, RI, 2001. Note: The fifteenth Dean Jacqueline B. Lewis memorial lectures.
29. J.-L. Starck, M. Elad and D. L. Donoho, Image decomposition via the combination of sparse representation and a variational approach, IEEE Trans. Image Process., 14 (10) (2005), 1570-1582.
30. Gabriele Steidl, Joachim Weickert, Thomas Brox, Pavel Mráze and Martin Welk, On the equivalence of soft wavelet shrinkage, total variation diffusion, total variation regularization, and sides, SIAMJ. Numer. Anal., 42 (2) (2004), 686-713.
31. Yang Wang and Haomin Zhou, Modeling textures with total variation minimization and oscillating patterns in image processing, J. Sci. Comput., 19 (1-3) (2003), 553-572.
32. Yang Wang and Haomin Zhou, Total variation wavelet-basedmedical image denoising, Int. J. Biomedical. Imaging., 2006 (2006), 1-6.
33. You-Wei Wen, Michael. Ng and Wai-Ki Ching, Iterative algorithms based on decoupling of deblurring and denoising for image restoration, SIAM J. Sci. Comput., 30 (5) (2008), 2655-2674.
34. L. Woog, Adapted Waveform Algorithms for Denoising, PhD thesis, Yale University, April 1996. Available at http://www.cs.yale.edu/homes/woog.html.
35. T. Zeng, Incorporating known features into a total variation dictionary model for source separation, Proceedings of IEEE ICIP'08., pp. 577-580, 2008, San Diego, California, U.S.A.
36. T. Zeng and M. Ng, On the total variation dictionary model, IEEE Trans. Image Process., 19 (3) (2010), 821-825.