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A new cartoon + texture image decomposition model based on the Sobolev space

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Abstract

Image decomposition aims to decompose a given image into one structural component and another oscillatory component. In most variational decomposition models, the structural component is often measured by the total variation norm and the oscillatory component is measured by its dual norm or others. In this paper, we let the structural component belong to the bounded variation space, the oscillatory texture be in the Sobolev space \(W^{ - 1,\;1}\), and the \(H^{ - 1}\) norm model the residual part. The new model combines the advantages of total variation regularization and weaker norm oscillatory component modeling, and it can well decompose the cartoon and texture while preserving some edges and contours. To solve this optimal problem, an effective numerical algorithm based on the splitting versions of augmented Lagrangian method is discussed in detail. Experimental results are reported to show the visual qualities compared with some state-of-the-art methods.

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References

  1. Nunes, J., Guyot, S., Delechelle, E.: Texture analysis based on local analysis of the bidimensional empirical mode decomposition. Mach. Vision Appl. 16, 177–188 (2005)

    Article  Google Scholar 

  2. Daubechies, I., Teschke, G.: Variational image restoration by means of wavelets: simultaneous decomposition, deblurring, and denoising. Appl. Comp. Harmon. Anal. 19(1), 1–16 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. 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 

  4. Cho, H., Lee, H., Kang, H., Lee, S.: Bilateral texture filtering. ACM Trans. Graphics 33(4), 128 (2014)

    Article  Google Scholar 

  5. Chen, Q., Xu, J., Koltun, V.: Fast image processing with fully-convolutional networks. Proc. IEEE Int. Conf. Computer Vision, 2497–2506 (2017)

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Meyer, Y.: Oscillating patterns in image processing and nonlinear evolution equations. American Mathematical Society, Boston, MA, USA (2001). The Fifteenth Dean Jacqueline B.

  8. 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 

  9. 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 

  10. 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 

  11. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  14. Aujol, J., Gilboa, G., Chan, T., et al.: Structure-texture image decomposition—modeling, algorithms, and parameter selection. Int. J. Computer Vision 67(1), 111–136 (2006)

    Article  MATH  Google Scholar 

  15. Wen, Y., Sun, H., Ng, M.: A primal-dual method for the Meyer model of cartoon and texture decomposition. Numer. Linear Algebra Appl. 26(2), 1–17 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  16. Tang, L., Zhang, H., He, C., et al.: Non-convex and non-smooth variational decomposition for image restoration. Appl. Math. Model. 69, 355–377 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hao, Y., Xu, J., Bai, J., et al.: Image decomposition combining a total variational filter and a Tikhonov quadratic filter. Multidim. Syst. Signal Process. 26(3), 739–751 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Belyaev, A., Fayolle, P.: Adaptive curvature-guided image filtering for structuretexture imagedecomposition. IEEE Trans. Image Process. 27(10), 5192–5203 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kim, Y., Ham, B., Do, M., et al.: Structure-texture image decomposition using deep variational priors. IEEE Trans. Image Process. 28(6), 2692–2704 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  20. Chan, T., Esedoglu, S., Parky, F.: Image decomposition combining staircasing reduction and texture extraction. J. Visual Comm. Image R. 18, 464–486 (2007)

    Article  Google Scholar 

  21. Xu, J., Feng, X., Hao, Y., et al.: Image decomposition and staircase effect reduction based on total generalized variation. J. Syst. Eng. Electron. 25(1), 168–174 (2014)

    Article  Google Scholar 

  22. Liu, X.: A new TGV-Gabor model for cartoon-texture image decomposition. IEEE Signal Process. Lett. 25(8), 1221–1225 (2018)

    Article  Google Scholar 

  23. Xu, J., Hao, Y., Li, M., et al.: A novel variational model for image decomposition. Signal Image Video Process 13(5), 413–447 (2019)

    Article  Google Scholar 

  24. 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)

    Article  MathSciNet  MATH  Google Scholar 

  25. Li, Y., Feng, X.: Coupled dictionary learning method for image decomposition. Sci. China-Inf. Sci. 56(3), 1–12 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Schaeffer, H., Osher, S.: A low patch-rank interpretation of texture. SIAM J. Imag. Sci. 6(1), 226–262 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Han, D., Kong, W., Zhang, W.: A partial splitting augmented Lagrangian method for low patch-rank image decomposition. J. Math. Imag. Vision 51(1), 145–160 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ono, S., Miyata, T., Yamada, 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 

  29. Zhu, Z., Yin, H., Chai, Y., et al.: A novel multi-modality image fusion method based on image decomposition and sparse representation. Inf. Sci. 432, 516–529 (2018)

    Article  MathSciNet  Google Scholar 

  30. Wang, W., Zhao, X., Ng, M.: A cartoon-plus-texture image decomposition model for blind deconvolution. Mul. Syst. Signal Process. 27(2), 541–562 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Han, Y., Xu, C., Baciu, G., et al.: Cartoon and Texture decomposition-based color transfer for fabric images. IEEE Trans. Multimedia 19(1), 80–92 (2017)

    Article  Google Scholar 

  32. He, B., Tao, M., Xu, M., et al.: Alternating direction based contraction method for generally separable linearly constrained convex programming problems. Optimization 62, 573–596 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  33. Li, M., Zhang, Y., Xiao, M., et al.: On Schatten-q quasinorm induced matrix decomposition model for salient object detection. Pattern Recogn. 96, 1–12 (2019)

    Google Scholar 

  34. Liu, Z., Wali, S., Duan, Y., et al.: Proximal ADMM for Euler’s elastica based image decomposition model. Numer. Math. Theor. Meth. Appl. 12(2), 370–402 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  35. Xu, J., Hao, Y., Zhang, X., et al.: A cartoon+texture image decomposition variational model based on preserving the local geometric characteristics. IEEE Access 8, 46574–46584 (2020)

    Article  Google Scholar 

Download references

Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. U1504603), Key Scientific Research Project of Colleges and Universities in Henan Province (Nos.18A120002,19A110014), Youth Science Foundation of Henan University of Science and Technology (2015QN021).

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

    Jianlou Xu, Wanqing Shang & Yan Hao

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  1. Jianlou Xu
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  2. Wanqing Shang
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  3. Yan Hao
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Correspondence to Jianlou Xu.

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Xu, J., Shang, W. & Hao, Y. A new cartoon + texture image decomposition model based on the Sobolev space. SIViP 16, 1569–1576 (2022). https://doi.org/10.1007/s11760-021-02111-0

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