Advertisement

Truncated Fractional-Order Total Variation Model for Image Restoration

  • Raymond Honfu Chan
  • Hai-Xia LiangEmail author
Article
  • 96 Downloads

Abstract

Fractional-order derivative is attracting more and more interest from researchers working on image processing because it helps to preserve more texture than total variation when noise is removed. In the existing works, the Grunwald–Letnikov fractional-order derivative is usually used, where the Dirichlet homogeneous boundary condition can only be considered and therefore the full lower triangular Toeplitz matrix is generated as the discrete partial fractional-order derivative operator. In this paper, a modified truncation is considered in generating the discrete fractional-order partial derivative operator and a truncated fractional-order total variation (tFoTV) model is proposed for image restoration. Hopefully, first any boundary condition can be used in the numerical experiments. Second, the accuracy of the reconstructed images by the tFoTV model can be improved. The alternating directional method of multiplier is applied to solve the tFoTV model. Its convergence is also analyzed briefly. In the numerical experiments, we apply the tFoTV model to recover images that are corrupted by blur and noise. The numerical results show that the tFoTV model provides better reconstruction in peak signal-to-noise ratio (PSNR) than the full fractional-order variation and total variation models. From the numerical results, we can also see that the tFoTV model is comparable with the total generalized variation (TGV) model in accuracy. In addition, we can roughly fix a fractional order according to the structure of the image, and therefore, there is only one parameter left to determine in the tFoTV model, while there are always two parameters to be fixed in TGV model.

Keywords

Image restoration Fractional-order derivative Truncated fractional-order total variation model Total variation Total generalized variation Alternating directional method of multiplier 

Mathematics Subject Classification

65K10 

References

  1. 1.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Lysaker, M., Lundervold, A., Tai, X.-C.: Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time. IEEE Trans. Imaging Process. 12, 1579–1590 (2003)CrossRefGoogle Scholar
  3. 3.
    Chan, R.H., Liang, H., Wei, S., Nikolova, M., Tai, X.-C.: High-order total variation regularization approach for axially symmetric object tomography from a single radiograph. Inverse Probl. Imaging 9, 55–77 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Lysaker, M., Tai, X.-C.: Iterative image restoration combining total variation minimization and a second-order functional. Int. J. Comput. Vis. 66, 5–18 (2005)CrossRefGoogle Scholar
  5. 5.
    Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Imaging Sci. 3, 492–526 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Knoll, F., Bredies, K., Pock, T., Stollberger, R.: Second order total generalized variation (TGV) for MRI. Magn. Reson. Med. 65, 480–491 (2011)CrossRefGoogle Scholar
  7. 7.
    Lavoie, J.L., Osler, T.J., Tremblay, R.: Fractional derivatives and special functions. SIAM Rev. 18, 240–268 (1976)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)zbMATHGoogle Scholar
  9. 9.
    Podlubny, I., Chechkin, A., Skovranek, T., Chen, Y., Jara, B.M.V.: Matrix approach to discrete fractional calculus II: partial fractional differential equations. J. Comput. Phys. 228(8), 3137–3153 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Mathieu, B., Melchior, P., Oustaloup, A., Ceyral, C.: Fractional differentiation for edge detection. Signal Process. 83, 2421–2432 (2002)CrossRefGoogle Scholar
  11. 11.
    Bai, J., Feng, X.C.: Fractional-order anisotropic diffusion for image denoising. IEEE Trans. Image Process. 16(10), 2492–2502 (2007)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chan, R.H., Lanza, A., Morigi, S., Sgallari, F.: An adaptive strategy for restoration of textured images using fractional order regularization. Numer. Math. Theory Methods Appl. 6, 276–296 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Cuesta, E., Kirane, M., Malik, S.: Image structure preserving denoising using genelized fractional time integrals. Signal Process. 92, 553–563 (2012)CrossRefGoogle Scholar
  14. 14.
    Hu, X., Li, Y.: A new variational model for image denoising based in fractional-order derivative. In: 2012 International Conference on Systems and Informatics (ICSAI), pp. 1820–1824 (2012)Google Scholar
  15. 15.
    Larnier, S., Mecca, R.: Fractional-order diffusion for image reconstruction. In: 2012 IEEE International Conference on Acoustics, Speech and Signal Processing, pp. 1057–1060 (2012)Google Scholar
  16. 16.
    Pu, Y., Zhou, J., Siarry, P., Zhang, N., Liu, Y.: Fractional partial differential equation: fractional total variation and fractional steepest decent approach-based multiscale denoising model for texture image. Abstr Appl Anal. (2013).  https://doi.org/10.1155/2013/483791 CrossRefzbMATHGoogle Scholar
  17. 17.
    Xu, J., Feng, X., Hao, Y.: A coupled variational model for image denoising using a duality strategy and split Bregman. Multidimens. Syst. Signal Process. 25, 83–94 (2014)CrossRefGoogle Scholar
  18. 18.
    Zhang, J., Chen, K.: A Total fractional-order variation model for image restoration with non-homogeneous boundary conditions and its numerical solution, Computer Vision and Pattern Recognition (2015). arXiv:1509.04237.pdf
  19. 19.
    Zhang, J., Wei, Z., Xiao, L.: Adaptive fractional-order multi-scale method for image denoising. J. Math. Imaging Vis. 43, 39–49 (2011)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Zhang, J., Wei, Z.: A class of fractional-order multi-scale variational models and alternating projection algorithm for image denoising. Appl. Math. Model. 35(5), 2516–2528 (2011)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Wang, W., Lu, P.: A new image deblurring method based on fractional differential. In: Audio, Language and Image Processing, pp. 497–501 (2012)Google Scholar
  22. 22.
    Tian, D., Xue, D., Chen, D., Sun, S.: A fractional-order regulatory CV model for brain MR image segmentation. In: 2013 Chinese Control and Decision Conference, pp. 37–40 (2013)Google Scholar
  23. 23.
    Zhang, Y., Pu, Y.-F., Hu, J.-R., Zhou, J.-L.: A class of fractional-order variational image inpainting models. Appl. Math. Inf. Sci. 6(2), 299–306 (2012)MathSciNetGoogle Scholar
  24. 24.
    Zhang, J., Wei, Z.: Fractional variational model and algorithm for image denoising. In: Proceedings of the Fourth International Conference on Natural Computation., vol 5, pp. 524–528. IEEE, Washington (2008)Google Scholar
  25. 25.
    Glowinski, R., Marrocco, A.: Sur L’approximation, par elements finis d’ordre un, et la resolution, par penalisationdualite, dune classe de problems de Direchlet non linaries. R. A. I. O. R2 9(R–2), 41–76 (1975)Google Scholar
  26. 26.
    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)CrossRefGoogle Scholar
  27. 27.
    Wu, C.L., Tai, X.C.: Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models. SIAM J. Imaging Sci. 3, 300–339 (2010)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Wu, C.L., Zhang, J.Y., Tai, X.C.: Augmented Lagrangian method for total variation restoration with non-quadratic fidelity. Inverse Probl. Imaging 5, 237–261 (2010)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Tai, X.C., Wu, C.L.: Augmented Lagrangian method, dual methods and split Bregman iteration for ROF model. In: Scale Space and Variational Methods in Computer Vision, Second International Conference, SSVM 2009, Voss, Norway, June 1–5, 2009. Proceedings. Lecture Notes in Computer Science 5567, pp. 502–513. Springer, Heidelberg (2009)Google Scholar
  30. 30.
    Zhang, X., Burger, M., Osher, S.: A unified primal-dual algorithm framework based on Bregman iteration. J. Sci. Comput. 46(1), 20–46 (2011)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Goldstein, T., Osher, S.: The split Bregman method for l1-regularized problems. SIAM J. Imaging Sci. 2, 323–343 (2009)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Deng, W., Yin, W.: On the global and linear convergence of generalized alternating direction method of multipliers. J. Sci. Comput. 66(3), 889–916 (2016)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Eckstein, J., Bertsekas, D.: On the Douglas-Rackford Splitting Method and Proximal Point Algorithm for Maximal Monotone Operators, Mathematical Programming, vol. 55. North-Holland, Amsterdam (1992)zbMATHGoogle Scholar
  34. 34.
    Guo, W., Qin, J., Yin, W.: A new detail-preserving regularity scheme. SIAM J. Imaging Sci. 7(2), 1309–1334 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.College of ScienceCity University of Hong KongHong KongChina
  2. 2.Department of Mathematical SciencesXi’an Jiaotong-Liverpool UniversitySuzhouChina

Personalised recommendations