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Higher-Order TV Methods—Enhancement via Bregman Iteration

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Abstract

In this work we analyze and compare two recent variational models for image denoising and improve their reconstructions by applying a Bregman iteration strategy. One of the standard techniques in image denoising, the ROF-model (cf. Rudin et al. in Physica D 60:259–268, 1992), is well known for recovering sharp edges of a signal or image, but also for producing staircase-like artifacts. In order to overcome these model-dependent deficiencies, total variation modifications that incorporate higher-order derivatives have been proposed (cf. Chambolle and Lions in Numer. Math. 76:167–188, 1997; Bredies et al. in SIAM J. Imaging Sci. 3(3):492–526, 2010). These models reduce staircasing for reasonable parameter choices. However, the combination of derivatives of different order leads to other undesired side effects, which we shall also highlight in several examples.

The goal of this paper is to analyze capabilities and limitations of the different models and to improve their reconstructions in quality by introducing Bregman iterations. Besides general modeling and analysis we discuss efficient numerical realizations of Bregman iterations and modified versions thereof.

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References

  1. Alter, F., Caselles, V., Chambolle, A.: Evolution of characteristic functions of convex sets in the plane by the minimizing total variation flow. Interfaces Free Bound. 7(1), 29–53 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, New York (2011). With a foreword by Hédy Attouch

    Book  MATH  Google Scholar 

  3. Benning, M.: Singular regularization of inverse problems. Ph.D. thesis, University of Münster, Institute for Computational and Applied Mathematics, Einsteinstr. 62, 48149 Münster, May 2011

  4. Bertozzi, A.L., Greer, J.B., Osher, S., Vixie, K.: Nonlinear regularizations of TV based PDEs for image processing. In: Chen, G.-Q., Gasper, G., Jerome, J. (eds.) AMS Series of Contemporary Mathematics, vol. 371, pp. 29–40 (2005)

    Google Scholar 

  5. Bishop, R.L., Goldberg, S.I.: Tensor Analysis on Manifolds. Dover, New York (1980)

    Google Scholar 

  6. Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Imaging Sci. 3(3), 492–526 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bregman, L.M.: The relaxation method for finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. Math. Phys. 7, 200–217 (1967)

    Article  Google Scholar 

  8. Buades, A., Coll, B., Morel, J.M.: A review of image denoising algorithms, with a new one. Multiscale Model. Simul. 4(2), 490–530 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Burger, M., Osher, S.: Convergence rates of convex variational regularization. Inverse Probl. 20, 1411–1421 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Burger, M., Osher, S.: A guide to tv zoo. In: Level Set and PDE-based Reconstruction Methods. Springer, Berlin (2012, to appear)

    Google Scholar 

  11. Burger, M., Gilboa, G., Osher, S., Xu, J.: Nonlinear inverse scale space methods. Commun. Math. Sci. 4, 179–212 (2006)

    MathSciNet  MATH  Google Scholar 

  12. Burger, M., Frick, K., Osher, S., Scherzer, O.: Inverse total variation flow. Multiscale Model. Simul. 6(2), 366–395 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Caselles, V., Chambolle, A., Novaga, M.: The discontinuity set of solutions of the TV denoising problem and some extensions. Multiscale Model. Simul. 6(3), 879–894 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20(1–2), 89–97 (2004). Special issue on mathematics and image analysis

    MathSciNet  Google Scholar 

  15. Chambolle, A., Lions, P.-L.: Image recovery via total variation minimization and related problems. Numer. Math. 76, 167–188 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40(1), 120–145 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Chan, T.F., Shen, J.: Image Processing and Analysis: Variational, PDE, Wavelet and Stochastic Methods. SIAM, Philadelphia (2005)

    Book  MATH  Google Scholar 

  18. Chan, T., Marquina, A., Mulet, P.: High-order total variation-based image restoration. SIAM J. Sci. Comput. 22(2), 503–516 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  19. Esser, E., Zhang, X., Chan, T.F.: A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science. SIAM J. Imaging Sci. 3(4), 1015–1046 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7(3), 1005–1028 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Goldstein, T., Osher, S.: The split Bregman method for l1-regularized problems. SIAM J. Imaging Sci. 2(2), 323–343 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Mathematical Modelling: Theory and Applications, vol. 23. Springer, New York (2009)

    Book  MATH  Google Scholar 

  23. Kiwiel, K.C.: Proximal minimization methods with generalized Bregman functions. SIAM J. Control Optim. 35(4), 1142–1168 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  24. Knoll, F., Bredies, K., Pock, T., Stollberger, R.: Second order total generalized variation (TGV) for MRI. Magn. Reson. Med. 65(2), 480–491 (2011)

    Article  Google Scholar 

  25. Meyer, Y.: Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures. University Lecture Series, vol. 22. Am. Math. Soc., Boston (2001)

    MATH  Google Scholar 

  26. Osher, S., Burger, M., Goldfarb, D., Xu, J., Yin, W.: An iterative regularization method for total variation-based image restoration. Multiscale Model. Simul. 4(2), 460–489 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Setzer, S., Steidl, G.: Variational methods with higher order derivatives in image processing. In: Neamtu, M., Schumaker, L.L. (eds.) Approximation XII. Nashboro Press, Brentwood (2008)

    Google Scholar 

  30. Setzer, S., Steidl, G., Teuber, T.: Infimal convolution regularizations with discrete l1-type functionals. Commun. Math. Sci. 9(3), 797–827 (2011)

    MathSciNet  Google Scholar 

  31. Stefan, W., Renaut, R.A., Gelb, A.: Improved total variation-type regularization using higher order edge detectors. SIAM J. Imaging Sci. 3(2), 232–251 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Strong, D., Chan, T.: Edge-preserving and scale-dependent properties of total variation regularization. Inverse Probl. 19(6), S165–S187 (2003). Special section on imaging

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhang, X., Burger, M., Osher, S.: A unified primal-dual algorithm framework based on Bregman iteration. J. Sci. Comput. 46(1), 20–46 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

Financial support is acknowledged to the German Science Foundation (DFG) via grants SFB 656 (Subproject B2) and BU 2327/1. The third author thanks Stanley Osher for initiating his interest in variational methods for image processing.

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Authors and Affiliations

  1. Institut für Numerische und Angewandte Mathematik, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, 48149, Münster, Germany

    Martin Benning, Martin Burger & Jahn Müller

  2. Department of Mathematics, University of California Los Angeles, 405 Hilgard Avenue, Los Angeles, CA, 90095-1555, USA

    Christoph Brune

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  1. Martin Benning
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  2. Christoph Brune
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  3. Martin Burger
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  4. Jahn Müller
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Correspondence to Martin Burger.

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In honor of Stanley Osher for his 70th birthday.

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Benning, M., Brune, C., Burger, M. et al. Higher-Order TV Methods—Enhancement via Bregman Iteration. J Sci Comput 54, 269–310 (2013). https://doi.org/10.1007/s10915-012-9650-3

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  • DOI: https://doi.org/10.1007/s10915-012-9650-3

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