Skip to main content
SpringerLink
Log in

Image decomposition based on the adaptive direction total variation and \(\mathbb {G}\)-norm regularization

  • Original Paper
  • Published:
Signal, Image and Video Processing Aims and scope Submit manuscript

Abstract

To improve the decomposition quality, it is very important to describe the local structure of the image in the proposed model. This fact motivates us to improve the Meyer’s decomposition model via coupling one weighted matrix with one rotation matrix into the total variation norm. In the proposed model, the weighted matrix can be used to enhance the diffusion along with the tangent direction of the edge and the rotation matrix is used to make the difference operator couple with the coordinate system of the normal direction and the tangent direction efficiently. With these operations, our proposed model owns the advantage of the local adaption and also describes the image structure robustly. Since the proposed model has the splitting structure, we can employ the alternating direction method of multipliers to solve it. Furthermore, the convergence of the numerical method can be efficiently kept under the framework of this algorithm. Numerical results are presented to show that the proposed model can decompose better cartoon and texture components than other testing methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Notes

  1. Here we set the W with the size \(16\times 16\). \(\tan ^{-1}\) is the arctangent function with output range of \((-\frac{\pi }{2},\frac{\pi }{2})\)

References

  1. Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60(1–4), 259–268 (1992)

    Article  MathSciNet  Google Scholar 

  2. Kose, K., Cevher, V., Cetin, A.: Filtered variation method for denoising and sparse signal processing. In: IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 3329-3332 (2012)

  3. Meyer, Y.: Oscillating patterns in image processing and nonlinear evolution equations. Univ. Lect. Ser. 22, 1–123 (2001)

    Article  MathSciNet  Google Scholar 

  4. 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  Google Scholar 

  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  Google Scholar 

  6. Aujol, J., Aubert, G., Fraud, L., Chambolle, A.: Image decomposition into a bounded variation component and an oscillating component. J. Math. Imaging Vis. 22(1), 71–88 (2005)

    Article  MathSciNet  Google Scholar 

  7. Liu, X.: A new TGV-Gabor model for cartoon-texture image decomposition. IEEE Signal Process. Lett. 25(8), 1121–1125 (2018)

    Article  Google Scholar 

  8. Gao, Y., Bredies, K.: Infimal convolution of oscillation total generalized variation for the recovery of images with structured texture. SIAM J. Imaging Sci. 11(3), 2021–2063 (2018)

    Article  MathSciNet  Google Scholar 

  9. Duan, J., Qiu, Z., Lu, W., Wang, G., Panc, Z., Bai, L.: An edge-weighted second order variational model for image decomposition. Digit. Signal Process. 49, 162–181 (2016)

    Article  Google Scholar 

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

    Article  Google Scholar 

  11. 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), e2224 (2019)

    Article  MathSciNet  Google Scholar 

  12. Tang, L., Zhang, H., He, C., Fang, Z.: Nonconvex and nonsmooth variational decomposition for image restoration. Appl. Math. Model. 69, 355–377 (2019)

    Article  MathSciNet  Google Scholar 

  13. Aujo, J., Gilboa, G., Chan, T., Osher, S.: Structure-texture image decompositionłmodeling, algorithms, and parameter selection. Int. J. Comput. Vis. 67(1), 111–136 (2006)

    Article  Google Scholar 

  14. Jalalzai, K.: Some remarks on the staircasing phenomenon in total variation-based image denoising. J. Math. Imaging Vis. 54(2), 256–268 (2016)

    Article  MathSciNet  Google Scholar 

  15. Duan, J., Lu, W., Pan, Z., Bai, L.: An edge-weighted second order variational model for image decomposition. Digit. Signal Process. 49, 162–181 (2006)

    Article  Google Scholar 

  16. Bayram, I., Kamasak, M.: Directional total variation. IEEE Signal Process. Lett. 19(12), 781–784 (2012)

    Article  Google Scholar 

  17. Kongskov, R., Dong, Y.: Directional total generalized variation regularization for impulse noise removal. In: Scale Space and Variational Methods in Computer Vision, pp. 221–231 (2017)

  18. Kongskov, R., Dong, Y., Knudsen, K.: Directional total generalized variation regularization. BIT Numer. Math. 59(4), 903–928 (2019)

    Article  MathSciNet  Google Scholar 

  19. Parisotto, S., Calatroni, L., Caliari, M., Schonlieb, C., Weickert, J.: Anisotropic osmosis filtering for shadow removal in images. Inverse Probl. 35(5), 54001 (2019)

    Article  MathSciNet  Google Scholar 

  20. Aujol, J., Kang, S.: Color image decomposition and restoration. J. Vis. Commun. Image Represent. 17(4), 916–928 (2006)

    Article  Google Scholar 

  21. Rockafellar, R.: Convex Anaysis. Princeton University Press, Princeton (2015)

    Google Scholar 

  22. Wu, C., Tai, X.: Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models. SIAM J. Imaging Sci. 3(3), 300–339 (2010)

    Article  MathSciNet  Google Scholar 

  23. Lu, W., Duan, J., Qiu, Z., n Pan, Z., Liu, W., Bai, L.: Implementation of high order variational models made easy for image processing. Math. Methods Appl. Sci. 39(14), 4208–4233 (2016)

    Article  MathSciNet  Google Scholar 

  24. Moura, C., Kubrusly, C.: The Courant–Friedrichs–Lewy (CFL) Condition: 80 Years After Its Discovery. Birkhauser, Basel (2013)

    Book  Google Scholar 

  25. 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  Google Scholar 

  26. Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20(1–2), 89–97 (2004)

    MathSciNet  MATH  Google Scholar 

  27. Aujol, J.: Some first-order algorithms for total variation based image restoration. J. Math. Imaging Vis. 34(3), 307–327 (2007)

    Article  MathSciNet  Google Scholar 

  28. Bresson, X., Chan, T.: Fast dual minimization of the vectorial total variation norm and applications to color image processing. Inverse Probl. Imaging 2(4), 455–484 (2008)

    Article  MathSciNet  Google Scholar 

  29. Steidl, G., Teuber, T.: Removing multiplicative noise by Douglas–Rachford splitting methods. J. Math. Imaging Vis. 36(2), 168–184 (2010)

    Article  MathSciNet  Google Scholar 

  30. Wu, C., Zhang, J., Tai, X.: Augmented Lagrangian method for total variation restoration with non-quadratic fidelity. Inverse Probl. Imaging 5(1), 237–261 (2011)

    Article  MathSciNet  Google Scholar 

  31. Zhang, J., Chen, R., Deng, C., Wang, S.: Fast linearized augmented method for Euler’s elastica model. Numer. Math.:Theory Methods Appl. 10(1), 98–115 (2017)

    MathSciNet  MATH  Google Scholar 

  32. Gabay, D., Mercier, B.: A dual algorithm for the solution of nonlinear variational problems via finite element approximation. Comput. Math. Appl. 2(1), 17–40 (1976)

    Article  Google Scholar 

  33. Glowinski, R., Marroco, A.: Sur 1’approximation,par elements finis dordre un, et la resolution, par enalisation-dualite, dune classe de problemes de Dirichlet non lineares. Rev. Francaise d’Autom. Inf. Recherche Operationelle 9(R–2), 41–76 (1975)

    MATH  Google Scholar 

  34. Gilles, J., Osher, S.: Bregman implementation of Meyer’s G-norm for cartoon+textures decomposition. UCLA cam, 11–73 (2011)

  35. Yang, G., Burger, P., Firmin, D., Underwood, S.: Structure adaptive anisotropic image filtering. Image Vis. Comput. 14(2), 135–145 (1996)

    Article  Google Scholar 

  36. Hong, L., Wan, Y., Jain, A.: Fingerprint image enhancement: algorithm and performance evaluation. IEEE Trans. Pattern Anal. Mach. Intell. 20(8), 777–789 (1998)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Statistics, Henan Univesity, Kaifeng, 475004, Henan Province, China

    Baoli Shi & Zhi-Feng Pang

  2. Zhongyuan institute of statistics, Zhengzhou institute of finance and economics, Zhengzhou, 450000, Henan Province, China

    Ge Meng

  3. Department of Radiology, Luoyang Orthopedic Hospital of Henan Province, Zhengzhou, 450018, Henan Province, China

    Zhenjiang Zhao

Authors
  1. Baoli Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ge Meng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhenjiang Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zhi-Feng Pang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhi-Feng Pang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was partially supported Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (Changsha University of Science and Technology, China)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shi, B., Meng, G., Zhao, Z. et al. Image decomposition based on the adaptive direction total variation and \(\mathbb {G}\)-norm regularization. SIViP 15, 155–163 (2021). https://doi.org/10.1007/s11760-020-01734-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11760-020-01734-z

Keywords

Search

Navigation