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A novel variational model for image decomposition

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Abstract

Image decomposition denotes a process by which an image is decomposed into several different scales, such as cartoon, texture (or noise) and edge. In order to better separate the noise and preserve the edges, one coupled variational model for image decomposition is proposed in this paper. In this coupled model, an introduced vector field and the gradient of image are intertwined and the orders of this model can be adjusted by the given parameters. To prevent image from being too smooth and edges from being damaged, one weighted function containing a Gaussian convolution is proposed. Meanwhile, considering the equivalence between the solution of the heat diffusion equation and the Gaussian convolution, we turn the convolution computation into a variational model for the introduced vector field. Different from the existing methods, the proposed model firstly contains first- and second-order regularization terms which can remove the noise better; secondly, the solution for the introduced vector field is just given Gaussian convolution. To solve the variational system, the alternating direction method, primal–dual method and Gauss–Seidel iteration are adopted. In addition, the proximal point method is designed for solving the primal variable and dual variable. Extensive numerical experiments verify that the new method can obtain better results than those by some recent methods.

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References

  1. Mallat, S.: A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans Pattern Anal Mach Intell 11(7), 674–693 (1989)

    Article  MATH  Google Scholar 

  2. Buades, A., Le, T., Morel, J., et al.: Fast cartoon + texture image filters. IEEE Trans Image Process 19, 1978–1986 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Meyer, Y.: Oscillating Patterns in Image Processing and Nonlinear Evolution Equations. American Mathematical Society, Boston, MA, USA (2001). The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures, vol. 22 of University Lecture Series

  5. Osher, S., Sole, A., Vese, L.: Image decomposition and restoration using total variation minimization and the H−1 norm. Multiscale Model. Simul. 1(3), 349–370 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Vese, L., Osher, S.: Modeling textures with total variation minimization and oscillating patterns in image processing. J. Sci. Comput. 19(1–3), 553–572 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Aujol, J., Aubert, G., Blanc-Feraud, L., et al.: Image decomposition into a bounded variation component and an oscillating component. J. Math. Imaging Vis. 22(1), 71–88 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Aujol, J., Chambolle, A.: Dual norms and image decomposition models. Int J. Comput. Vis. 63(1), 85–104 (2005)

    Article  MATH  Google Scholar 

  9. Li, Y., Feng, X.: Coupled dictionary learning method for image decomposition. Sci. China Inf. Sci. 56, 1–10 (2013)

    MathSciNet  Google Scholar 

  10. Hao, Y., Xu, J., Bai, J., et al.: Image decomposition combining a total variational filter and a Tikhonov quadratic filter. Multidimens. Syst. Sign. Process. 26, 739–751 (2015)

    Article  MathSciNet  Google Scholar 

  11. Chan, T., Esedoglu, S., Parky, E.: Image decomposition combining staircase reduction and texture extraction. J. Visual Commun. Image Represent. 18(6), 464–486 (2007)

    Article  Google Scholar 

  12. Xu, J., Feng, X., Hao, Y., et al.: Image decomposition and staircase effect reduction based on total generalized variation. J. Syst. Eng. Electron. 25, 168–174 (2014)

    Article  Google Scholar 

  13. Xu, J., Feng, X., Hao, Y., et al.: Image decomposition using adaptive second order total generalized variation. Signal Image Video Process. 8, 39–47 (2014)

    Article  Google Scholar 

  14. Pang, Z., Yang, Y.: A projected gradient algorithm based on the augmented Lagrangian strategy for image restoration and texture extraction. Image Vis. Comput. 29, 117–126 (2011)

    Article  Google Scholar 

  15. Ng, M., Yuan, X., Zhang, W.: Coupled variational image decomposition and restoration model for blurred cartoon- plus-texture images with missing pixels. IEEE Trans. Image Process. 22(6), 2233–2246 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Wang, W., Zhao, X., Ng, M.: A cartoon-plus-texture image decomposition model for blind deconvolution. Multidimens Syst. Sign. Process. 27, 541–562 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Tang, L., He, C.: Multiscale texture extraction with hierarchical (BV,Gp,L2) decomposition. J. Math Imaging Vis. 45, 148–163 (2013)

    Article  MATH  Google Scholar 

  18. Shunsuke, O., Miyata, T., Yamad, I.: Cartoon-texture image decomposition using blockwise low-rank texture characterization. IEEE Trans. Image Process. 23(3), 1128–1142 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Surya Prasath, V., Vorotnikov, D.: On a system of adaptive coupled PDEs for image restoration. J. Math Imaging Vis. 48, 35–52 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Moreno, J., Prasath, V., Vorotnikov, D., et al.: Adaptive diffusion constrained total variation scheme with application to ‘cartoon + texture + edge’ image decomposition. Nuclear Electron. Detect. Technol. 16(18), 81–122 (2015)

    Google Scholar 

  21. Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12(7), 629–639 (1990)

    Article  Google Scholar 

  22. Catte, V., Lions, P., Morel, J., et al.: Image selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numer. Anal. 29(1), 182–193 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lysaker, M., Lundervold, A., Tai, X.: Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time. IEEE Trans. Image Process. 12(12), 1579–1589 (2003)

    Article  MATH  Google Scholar 

  24. Mohammad Reza Hajiaboli: An anisotropic fourth-order diffusion filter for image noise removal. Int. J. Comput. Vis. 92, 177–191 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Cai, J., Chan, R., Nikolova, M.: Fast two-phase image deblurring under impulse noise. J Math. Imaging Vis. 36, 46–53 (2010)

    Article  MathSciNet  Google Scholar 

  26. Hahn, J., Wu, C., Tai, X.: Augmented Lagrangian method for generalized TV-Stokes model. J. Sci. Comput. 50, 235–264 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Xu, J., Feng, A., Hao, Y., et al.: Image deblurring and denoising by an improved variational model. Int. J. Electron. Commun (AEÜ) 70, 1128–1133 (2016)

    Article  Google Scholar 

  29. Tikhonov, A., Arsenin, V.: Solutions of Ill-posed problems. Winston, Washington, DC (1977)

    MATH  Google Scholar 

  30. 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 

  31. Zhang, B., Zhu, Z., Wang, S.: A simple primal–dual method for total variation image restoration. J. Visual Commun. Image Represent. 38, 814–823 (2016)

    Article  Google Scholar 

  32. Wang, Z., Bovik, A., Sheikh, H., et al.: Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13(4), 1–14 (2004)

    Article  Google Scholar 

Download references

Acknowledgements

This work is supported by the National Science Foundation of China (Nos. U1504603, 61301229, 61401383) and Key Scientific Research Project of Colleges and Universities in Henan Province (Nos. 18A120002, 19A110014).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, 471023, China

    Jianlou Xu & Yan Hao

  2. College of Mathematics and Statistics, Shenzhen University, Shenzhen, 518060, China

    Min Li

  3. Institute of Graphics and Image Processing, Xianyang Normal University, Xianyang, 712000, China

    Xiaobo Zhang

Authors
  1. Jianlou Xu
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  2. Yan Hao
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  3. Min Li
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  4. Xiaobo Zhang
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Correspondence to Jianlou Xu.

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Xu, J., Hao, Y., Li, M. et al. A novel variational model for image decomposition. SIViP 13, 967–974 (2019). https://doi.org/10.1007/s11760-019-01434-3

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