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Augmented Lagrangian Method, Dual Methods and Split Bregman Iteration for ROF Model

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Scale Space and Variational Methods in Computer Vision (SSVM 2009)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 5567))

Abstract

In the recent decades the ROF model (total variation (TV) minimization) has made great successes in image restoration due to its good edge-preserving property. However, the non-differentiability of the minimization problem brings computational difficulties. Different techniques have been proposed to overcome this difficulty. Therein methods regarded to be particularly efficient include dual methods of CGM (Chan, Golub, and Mulet) [7] Chambolle [6] and split Bregman iteration [14], as well as splitting-and-penalty based method [28] [29]. In this paper, we show that most of these methods can be classified under the same framework. The dual methods and split Bregman iteration are just different iterative procedures to solve the same system resulted from a Lagrangian and penalty approach. We only show this relationship for the ROF model. However, it provides a uniform framework to understand these methods for other models. In addition, we provide some examples to illustrate the accuracy and efficiency of the proposed algorithm.

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References

  1. Bertsekas, D.P.: Multiplier methods: a survey. Automatica 12, 133–145 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  2. Blomgren, P., Chan, T.F.: Color TV: total variation methods for restoration of vector-valued images. IEEE Trans. Image Process. 7, 304–309 (1998)

    Article  Google Scholar 

  3. Bregman, L.M.: The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Computational Mathematics and Mathematical Physics 7, 200–217 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bresson, X., Chan, T.F.: Fast minimization of the vectorial total variation norm and applications to color image processing. UCLA CAM Report 07-25 (2007)

    Google Scholar 

  5. Carter, J.L.: Dual methods for total variation – based image restoration. Ph.D. thesis, UCLA (2001)

    Google Scholar 

  6. Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20, 89–97 (2004)

    Article  MathSciNet  Google Scholar 

  7. Chan, T.F., Golub, G.H., Mulet, P.: A nonlinear primal-dual method for total variation-based image restoration. SIAM J. Sci. Comput. 20, 1964–1977 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chan, T., Marquina, A., Mulet, P.: High-order total variation-based image restoration. SIAM J. Sci. Comput. 22, 503–516 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chan, T.F., Osher, S., Shen, J.: The digital TV filter and nonlinear denoising. IEEE Trans. Image Process. 10, 231–241 (2001)

    Article  MATH  Google Scholar 

  10. Chan, T.F., Kang, S.H., Shen, J.H.: Total variation denoising and enhancement of color images based on the CB and HSV color models. J. Visual Commun. Image Repres. 12, 422–435 (2001)

    Article  Google Scholar 

  11. Chan, T., Esedoglu, S., Park, F.E., Yip, A.: Recent developments in total variation image restoration. UCLA CAM Report 05-01 (2005)

    Google Scholar 

  12. Chan, T.F., Esedoglu, S., Park, F.E.: A fourth order dual method for staircase reduction in texture extraction and image restoration problems. UCLA CAM Report 05-28 (2005)

    Google Scholar 

  13. Glowinski, R., Le Tallec, P.: Augmented Lagrangians and operator-splitting methods in nonlinear mechanics. SIAM, Philadelphia (1989)

    Book  MATH  Google Scholar 

  14. Goldstein, T., Osher, S.: The split Bregman method for L1 regularized problems. UCLA CAM Report 08-29 (2008)

    Google Scholar 

  15. Hestenes, M.R.: Multiplier and gradient methods. Journal of Optimization Theory and Applications 4, 303–320 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hinterberger, W., Scherzer, O.: Variational methods on the space of functions of bounded Hessian for convexification and denoising. Computing 76, 109–133 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kimmel, R., Malladi, R., Sochen, N.: Images as embedded maps and minimal surfaces: movies, color, texture, and volumetric medical images. Int’l J. Computer Vision 39, 111–129 (2000)

    Article  MATH  Google Scholar 

  18. Lysaker, M., Lundervold, A., Tai, X.-C.: Noise removal using fourth-order partial differential equation with applications to medical Magnetic Resonance Images in space and time. IEEE Trans. Image Process. 12, 1579–1590 (2003)

    Article  MATH  Google Scholar 

  19. Lysaker, M., Tai, X.-C.: Iterative image restoration combining total variation minimization and a second order functional. Int’l J. Computer Vision 66, 5–18 (2006)

    Article  MATH  Google Scholar 

  20. Osher, S., Burger, M., Goldfarb, D., Xu, J.J., Yin, W.T.: An iterative regularization method for total variation-based image restoration. SIAM Multiscale Model. Simul. 4, 460–489 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Powell, M.J.D.: A method for nonlinear constraints in minimization problems. Optimization. In: Fletcher, R. (ed.), pp. 283–298. Academic Press, New York (1972)

    Google Scholar 

  22. Rockafellar, R.T.: A dual approach to solving nonlinear programming problems by unconstrained optimization. Mathematical Programming 5, 354–373 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  23. Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  24. Sapiro, G., Ringach, D.L.: Anisotropic diffusion of multivalued images with applications to color filtering. IEEE Trans. Image Process 5, 1582–1586 (1996)

    Article  MATH  Google Scholar 

  25. Scherer, O.: Denoising with higher order derivatives of bounded variation and an application to parameter estimation. Computing 60, 1–27 (1998)

    Article  MathSciNet  Google Scholar 

  26. Steidl, G.: A note on the dual treatment of higher-order regularization functionals. Computing 76, 135–148 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Tschumperlé, D., Deriche, R.: Vector-valued image regularization with PDEs: a common framework for different applications. IEEE Trans. Pattern Anal. Machine Intell. 27, 506–517 (2005)

    Article  Google Scholar 

  28. Wang, Y.L., Yin, W.T., Zhang, Y.: A fast algorithm for image deblurring with total variation regularization. UCLA CAM Report 07-22 (2007)

    Google Scholar 

  29. Wang, Y., Yang, J., Yin, W., Zhang, Y.: A new alternating minimization algorithm for total variation image reconstruction. SIAM Journal on Imaging Sciences (to appear)

    Google Scholar 

  30. Yin, W.T., Osher, S., Goldfarb, D., Darbon, J.: Bregman iterative algorithms for compressend sensing and related problems. SIAM J. Imaging Sciences 1, 143–168 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  31. You, Y.-L., Kaveh, M.: Fourth-order partial differential equation for noise removal. IEEE Trans. Image Process. 9, 1723–1730 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhu, M., Wright, S.J., Chan, T.F.: Duality-based algorithms for total variation image restoration. UCLA CAM Report 08-33 (2008)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Division of Mathematical Science, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore Department of Mathematics,University of Bergen, Johannes Brunsgate 12, N-5008, Bergen, Norway

    Xue-Cheng Tai

  2. Division of Mathematical Science, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore

    Chunlin Wu

Authors
  1. Xue-Cheng Tai
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  2. Chunlin Wu
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Bergen, Johannes Brunsgate 12, 5008, Bergen, Norway

    Xue-Cheng Tai

  2. Department of Informatics and Centre of Mathematics for Applications, University of Oslo, Oslo, Norway

    Knut Mørken

  3. Simula Research Laboratory, M. Linges v 17, Fornebu, P.O.Box 134, N-1325, Lysaker, Norway

    Marius Lysaker

  4. SINTEF ICT, Oslo, Norway

    Knut-Andreas Lie

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© 2009 Springer-Verlag Berlin Heidelberg

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Tai, XC., Wu, C. (2009). Augmented Lagrangian Method, Dual Methods and Split Bregman Iteration for ROF Model. In: Tai, XC., Mørken, K., Lysaker, M., Lie, KA. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2009. Lecture Notes in Computer Science, vol 5567. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02256-2_42

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  • DOI: https://doi.org/10.1007/978-3-642-02256-2_42

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-02255-5

  • Online ISBN: 978-3-642-02256-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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