Truncated Fractional-Order Total Variation Model for Image Restoration

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  • 1 College of Science, City University of Hong Kong, Hong Kong, China;
    2 Department of Mathematical Sciences, Xi'an Jiaotong-Liverpool University, Suzhou 215123, Jiangsu, China

Received date: 2018-06-07

  Revised date: 2018-11-06

  Online published: 2019-11-28

Supported by

Raymond Honfu Chan's research was supported in part by Hong Kong Research Grants Council (HKRGC) General Research Fund (No. CityU12500915, CityU14306316), HKRGC Collaborative Research Fund (No. C1007-15G) and HKRGC Areas of Excellence (No. AoE/M-05/12). Hai-Xia Liang's research was supported partly by the Natural Science Foundation of Jiangsu Province (No. BK20150373) and partly by Xi'an Jiaotong-Liverpool University Research Enhancement Fund (No.17-01-08).

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.

Cite this article

Raymond Honfu Chan, Hai-Xia Liang . Truncated Fractional-Order Total Variation Model for Image Restoration[J]. Journal of the Operations Research Society of China, 2019 , 7(4) : 561 -578 . DOI: 10.1007/s40305-019-00250-3

References

[1] Rudin, L.I., Osher, S., Fatemi, E.:Nonlinear total variation based noise removal algorithms. Physica D 60, 259-268(1992)
[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)
[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)
[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)
[5] Bredies, K., Kunisch, K., Pock, T.:Total generalized variation. SIAM J. Imaging Sci. 3, 492-526(2010)
[6] Knoll, F., Bredies, K., Pock, T., Stollberger, R.:Second order total generalized variation (TGV) for MRI. Magn. Reson. Med. 65, 480-491(2011)
[7] Lavoie, J.L., Osler, T.J., Tremblay, R.:Fractional derivatives and special functions. SIAM Rev. 18, 240-268(1976)
[8] Podlubny, I.:Fractional Differential Equations. Academic Press, New York (1999)
[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)
[10] Mathieu, B., Melchior, P., Oustaloup, A., Ceyral, C.:Fractional differentiation for edge detection. Signal Process. 83, 2421-2432(2002)
[11] Bai, J., Feng, X.C.:Fractional-order anisotropic diffusion for image denoising. IEEE Trans. Image Process. 16(10), 2492-2502(2007)
[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)
[13] Cuesta, E., Kirane, M., Malik, S.:Image structure preserving denoising using genelized fractional time integrals. Signal Process. 92, 553-563(2012)
[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)
[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)
[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
[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)
[18] Zhang, J., Chen, K.:A Total fractional-order variation model for image restoration with nonhomogeneous boundary conditions and its numerical solution, Computer Vision and Pattern Recognition (2015). arXiv:1509.04237.pdf
[19] Zhang, J., Wei, Z., Xiao, L.:Adaptive fractional-order multi-scale method for image denoising. J. Math. Imaging Vis. 43, 39-49(2011)
[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)
[21] Wang,W.,Lu,P.:Anewimagedeblurringmethodbasedonfractionaldifferential.In:Audio,Language and Image Processing, pp. 497-501(2012)
[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)
[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)
[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)
[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. R29(R-2), 41-76(1975)
[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)
[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)
[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)
[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)
[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)
[31] Goldstein, T., Osher, S.:The split Bregman method for l1-regularized problems. SIAM J. Imaging Sci. 2, 323-343(2009)
[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)
[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)
[34] Guo, W., Qin, J., Yin, W.:A new detail-preserving regularity scheme. SIAM J. Imaging Sci. 7(2), 1309-1334(2014)
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