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Signal, Image and Video Processing

, Volume 8, Issue 1, pp 39–47 | Cite as

Image decomposition using adaptive second-order total generalized variation

  • Jianlou XuEmail author
  • Xiangchu Feng
  • Yan Hao
  • Yu Han
Original Paper

Abstract

This paper proposes a new model for image decomposition which separates an image into a cartoon, consisting only of geometric objects, and an oscillatory component, consisting of textures or noise. The proposed model is given in a variational formulation with adaptive second-order total generalized variation (TGV). The adaptive behavior preserves the key features such as object boundaries and textures while avoiding staircasing effect. To speed up the computation, the split Bregman method is used to solve the proposed model. Experimental results and comparisons demonstrate the proposed model is more effective for image decomposition than the methods of the state-of-the-art image decomposition models.

Keywords

Image decomposition Cartoon Texture TGV  Split Bregman Staircasing effect 

Notes

Acknowledgments

This work is supported by the National Science Foundation of China (No. 60872138, 61105011, 61271294). Meanwhile, the authors also thank Editor-in-Chief Prof. Murat Kunt and anonymous reviewers for their constructive comments that greatly improve this paper.

References

  1. 1.
    Meyer, Y.: Oscillating patterns in image processing and nonlinear evolution equations. University Lecture Series, American Mathematical Society, Boston (2001)Google Scholar
  2. 2.
    Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica. D 60, 259–268 (1992)Google Scholar
  3. 3.
    Vese, L., Osher, S.: Modeling textures with total variation minimization and oscillating patterns in image processing. J. Math. Imaging Vis. 20, 7–18 (2004)Google Scholar
  4. 4.
    Osher, S., Sole, A., Vese, L.: Image decomposition and restoration using total variation minimization and the \(\text{ H}^{-1}\) norm. Multiscale Model. Simul. 1(3), 349–370 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Jiang, L., Yin, H., Feng, X.: Adaptive variational models for image decomposition combining staircase reduction and texture extraction. J. Syst. Eng. Electron. 20(2), 254–259 (2009)Google Scholar
  6. 6.
    Chan, T.F., Esedoglu, S.F., Parky, E.: Image decomposition combining staircase reduction and texture extraction. J. Vis. Commun. Image Represent. 18(6), 464–486 (2007)CrossRefGoogle Scholar
  7. 7.
    Aujol, J.F., Chambolle, A.: Dual norms and image decomposition models. Intl. J. Comp. Vis. 63(1), 85–104 (2005)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Duval, V., Aujol, J.F., Vese, L.: Mathematical modeling of textures: Application to color image decomposition with a projected gradient algorithm. J. Math. Imaging Vis. 37(3), 232–248 (2010)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Bai, J., Feng, X.: Image denoising and decomposition using non-convex functional. Chinese J. Electron. 21(1), 102–106 (2012)MathSciNetGoogle Scholar
  10. 10.
    Jiang, L., Feng, X., Yin, H.: Image decomposition using optimally sparse representations and a variational approach. Signal, Image and Video Processing 22(4), 1287–1292 (2007)Google Scholar
  11. 11.
    Starck, J., Elad, M., Donoho, D.: Image decomposition via the combination of sparse representations and a variational approach. IEEE Trans. Image Process. 14(10), 1570–1582 (2005)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Imaging Sci. 3(3), 492–526 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Bredies, K., Valkonen, T.: Inverse problems with second-order total generalized variation constraints. In: Proceedings of SampTA 2011–9th International Conference on Sampling Theory and Applications, Singapore, pp. 1–4 (2011)Google Scholar
  14. 14.
    Bredies, K., Dong, Y., Hintermüller M.: Spatially dependent regularization parameter selection in total generalized variation models for image restoration. International Journal of Computer Mathematics, pp. 1–15 (2012), online, doi: 10.1080/00207160.2012.700400
  15. 15.
    Knoll, F., Bredies, K., Pock, T., Stollberger, R.: Second order total generalized variation (TGV) for MRI. Magn. Res. Med. 65(2), 480–491 (2011)CrossRefGoogle Scholar
  16. 16.
    Dong, F., Liu, Z., Kong, D., Liu, K.: An improved LOT model for image restoration. J. Math. Imaging Vis. 34(1), 89–97 (2009)Google Scholar
  17. 17.
    Goldstein, T., Osher, S.: The split Bregman method for L1 regularized problems. SIAM J. Imaging Sci. 2(2), 323–343 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Han, Y., Wang, W., Feng, X.: A new fast multiphase image segmentation algorithm based on nonconvex regularizer. Pattern Recongnit. 3(4), 363–372 (2012)CrossRefGoogle Scholar
  19. 19.
    Goldstein, T., Bresson, X., Osher, S.: Geometric applications of the split Bregman method: Segmentation and surface reconstruction. J. Sci. Comput. 45(1–3), 272–293 (2010) Google Scholar
  20. 20.
    Afonso, M., Bioucas-Dias, J., Figueiredo, M.: An augmented lagrangian approach to the constrained optimization formulation of imaging inverse problems. IEEE Trans. Image Process. 20(3), 681–695 (2011)Google Scholar
  21. 21.
    Wu, C., Tai, X.C.: Augmented Lagrangian method, dual methods and split-Bregman iterations for ROF, vectorial TV and higher order models. SIAM J. Imaging Sci. 3(3), 300–339 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Setzer, S.: Operator splittings, Bregman methods and frame shrinkage in image processing. Int. J. Comput. Vision 92(3), 265–280 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Hao, Y., Feng, X.C., Xu, J.L.: Multiplicative noise removal via sparse and redundant representations over learned dictionaries and total variation. Signal Process. 92(6), 1536–1549 (2012)CrossRefGoogle Scholar
  25. 25.
    Wang, Z., Bovik, A., Sheikh, H., Simoncelli, E.: Image quality assessment: From error visibility to structural similarity. IEEE Trans. Image Process. 13(4), 1–14 (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2012

Authors and Affiliations

  1. 1.School of SciencesXidian UniversityXi’anPeople’s Republic of China
  2. 2.School of Mathematics and StatisticsHenan University of Science and TechnologyLuoyangPeople’s Republic of China

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