Image decomposition combining a total variational filter and a Tikhonov quadratic filter



In this paper, we propose a new variational model for image decomposition which separates an image into a cartoon, consisting only of geometric objects and an oscillatory component, consisting of texture or noise. In the new model, the \(\hbox {H}^{-1}\)-norm is considered as the data fitting term and the regularization term is composed of a total variational filter and a Tikhonov quadratic filter. These two filters can be automatically selected by a soft threshold function. When the pixels belong to the cartoon area, the total variational filter is adopted, which can preserve the geometric structures of image such as the edges, well. When the pixels belong to texture region, the Tikhonov quadratic filter is chosen,which can extract the texture of image well. To solve the proposed model effectively, the split Bregman method is employed. Experimental results demonstrate that the proposed model and algorithm can obtain better decomposition results than those of classical models.


Total variation Image decomposition Split Bregman Image denoising 



This work is supported by the National Natural Science Fund of China (No: 61301229, 61105011) and the doctoral research fund of Henan University of Science and Technology (No: 09001708, 09001751). In addition, we would like to thank the Associate Editor and anonymous reviewers for their constructive comments that greatly improve this paper.


  1. Afonso, M., Bioucas-Dias, J., & Figueiredo, M. (2011). An augmented lagrangian approach to the constrained optimization formulation of imaging inverse problems. IEEE Transactions on Image Processing, 20(3), 681–695.MathSciNetCrossRefGoogle Scholar
  2. Aujol, J. F., Aubert, G., Blanc-Feraud, L., & Chambolle, A. (2005). Image decomposition into a bounded variation component and an oscillating component. Journal of Mathematical Imaging and Vision, 22, 71–88.MathSciNetCrossRefGoogle Scholar
  3. Aujol, J. F., & Chambolle, A. (2005). Dual norms and image decomposition models. International Journal of Computer Vision, 63, 85–104.CrossRefGoogle Scholar
  4. Bertalmio, M., Vese, L., Sapiro, G., & Osher, S. (2003). Simultaneous structure and texture image inpainting. IEEE Transactions on Image Processing, 12(8), 882–889.CrossRefGoogle Scholar
  5. Buades, A., Le, T. M., Morel, J. M., & Vese, L. A. (2010). Fast cartoon + texture image filters. IEEE Transaction on Image Processing, 19, 1978–1986.MathSciNetCrossRefGoogle Scholar
  6. Chan, T. F., Esedoglu, S., & Parky, F. (2007). Image decomposition combining staircasing reduction and texture extraction. Journal of Visual Communication and Image Representation, 18, 464–486.CrossRefGoogle Scholar
  7. Fadili, M. J., & Starck, J.-L. (2010). MCALab: Reproducible research in signal and image decomposition and inpainting. IEEE Computing in Science and Engineering, 12(1), 44–62.MathSciNetCrossRefGoogle Scholar
  8. Garnett, J. B., & Le, T. M. (2007). Image decompositions using bounded variation and generalized homogeneous Besov space. Applied and Computational Harmonic Analysis, 23, 25–56.MATHMathSciNetCrossRefGoogle Scholar
  9. Goldstein, T., & Osher, S. (2009). The split Bregman method for \(L_1\)-regularized problems. SIAM Journal on Imaging Sciences, 2(2), 323–343.MATHMathSciNetCrossRefGoogle Scholar
  10. Han, Y., Wang, W. W., & Feng, X. C. (2012). A new fast multiphase image segmentation algorithm based on nonconvex regularizer. Pattern Recognition, 45(1), 363–372.MATHCrossRefGoogle Scholar
  11. Han, Y., Feng, X. C., Baciu, G., & Wang, W. W. (2013). Nonconvex sparse regularizer based speckle noise removal. Pattern Recognition, 46(3), 989–1001.CrossRefGoogle Scholar
  12. Jiang, L. L., & Yin, H. Q. (2012). Bregman iteration algorithm for sparse nonnegative matrix factorizations via alternating \(l_1 \)-norm minimization. Multidimensional Systems and Signal Processing, 23(3), 315–328.MATHMathSciNetCrossRefGoogle Scholar
  13. Li, Y. F., & Feng, X. C. (2012). Image decomposition via learning the morphological diversity. Pattern Recognition Letter, 33(2), 111–120.CrossRefGoogle Scholar
  14. Meyer, Y. (2001). Oscillating patterns in image processing and nonlinear evolution equations. Vol. 22 of University Lecture Series. Providence, RI: American Mathematical SocietyGoogle Scholar
  15. Nikolova, M., Ng, M., & Tam, C.-P. (2010). Fast nonconvex nonsmooth minimization methods for image restoration and reconstruction. IEEE Transactions on Image Processing, 19(12), 3073–3088.MathSciNetCrossRefGoogle Scholar
  16. Osher, S., Sole, A., & Vese, L. (2003). Image decomposition and restoration using total variation minimization and the \(\text{ H }^{-1}\) Norm. Multiscale Modeling and Simulation, 1(3), 349–370.MATHMathSciNetCrossRefGoogle Scholar
  17. Rudin, L., Osher, S., & Fatemi, E. (1992). Nonlinear total variation based noise removal algorithms. Physica D, 60, 259–268.MATHCrossRefGoogle Scholar
  18. Setzer, S. (2011). Operator splittings. Bregman methods and frame shrinkage in image processing. International Journal of Computer Vision, 92(3), 265–280.MATHMathSciNetCrossRefGoogle Scholar
  19. Shen, J. (2005). Piecewise H\(^{-1}\) + H\(^{0}\) + H\(^{1 }\) images and the Mumford–Shah–Sobolev model for segmented image decomposition. Applied Mathematics Research Express, 4, 143–167.Google Scholar
  20. Starck, J.-L., Elad, M., & Donoho, D. L. (2005). Image decomposition via the combination of sparse representations and a variational approach. IEEE Transactions on Image Processing, 14(10), 1570–1582.MATHMathSciNetCrossRefGoogle Scholar
  21. Vese, L., & Osher, S. (2003). Modeling textures with total variation minimization and oscillating patterns in image processing. Journal of Scientific Computing, 19(1–3), 553–572.MATHMathSciNetCrossRefGoogle Scholar
  22. Weickert, J., & ter Haar Romeny, B. M. (1998). Efficient and reliable schemes for nonlinear diffusion filtering. IEEE Transactions on Image Processing, 7(3), 398–410.CrossRefGoogle Scholar
  23. Wu, C., & Tai, X. C. (2010). Augmented Lagrangian method, dual methods and split Bregman iterations for ROF, vectorial TV and higher order models. SIAM Journal on Imaging Science, 3(3), 300–339.MATHMathSciNetCrossRefGoogle Scholar
  24. Xu, J. L., Feng, X. C., & Hao, Y. (2012). A coupled variational model for image denoising using a duality strategy and split Bregman. Multidimensional Systems and Signal Processing, doi: 10.1007/s11045-012-0190-7.
  25. Yin, H. Q., & Liu, H. W. (2011). A primal-dual gradient method for image decomposition based on (BV, H\(^{-1}\)). Multidimensional Systems and Signal Processing, 22(4), 335–348.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsHenan University of Science and TechnologyLuoyangChina
  2. 2.School of SciencesXidian UniversityXi’anChina
  3. 3.College of Mathematic and Computational ScienceShenzhen UniversityShenzhen China

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