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Time-fractional diffusion equation-based image denoising model

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Abstract

In the last two decades, due to the use of total variation (TV) denoising, variation models for image reconstruction have achieved great success. Compare to some known variation models which have just adopted the space-fractional diffusion equation, the time-fractional diffusion equation has another adjustable time-fractional derivative order to control the diffusion process. Therefore, in this paper, a new denoising model based on time-fractional diffusion equation is proposed, where the discretization in space is the application of classical difference scheme while the discretization in time is the approximation of the Caputo differential. Furthermore, both the explicit and implicit numerical scheme named TFEIS are built and its stability and convergence are rigorously discussed. We prove the full-discretization is stable under some conditions and the numerical solution converges to the exact one with order \(O(\tau^{2 - \alpha } + h^{2} )\), where τ and \(h\) are the time and space step size, respectively. Finally, various evaluation indexes such as histogram recovery degree, signal-to-noise ratio, and edge texture retention index are applied to compare new and original models comprehensively in lots of images. Computer simulation results show our models

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References

  1. Aubert, G., Kornprobct, P.: Mathematical problems in image processing, Ser. Applied Mathematics Sciences. Springer-Verlag, New York (2002)

    Book  Google Scholar 

  2. Zeidler, E.: Nonlinear functional analysis and its applications III: variational methods and optimization. Springer-Verlag, New York (1985)

    Book  Google Scholar 

  3. Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D 60, 259–268 (1991)

    Article  MathSciNet  Google Scholar 

  4. 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. Image Process. 12(12), 1579–1590 (2003)

    Article  Google Scholar 

  5. You, Y.L., Kaveh, M.: Fourth-order partial differential equations for noise removal. IEEE Trans. Image Process. 9(10), 1723–1730 (2000)

    Article  MathSciNet  Google Scholar 

  6. Chan, T.F., Marquina, A., Mulet, P.: High-order total variation-based image restoration. SIAM J. Sci. Comput. 22(2), 503–516 (2000)

    Article  MathSciNet  Google Scholar 

  7. Chan, T.F., Esedoglu, S., Park, F.E.: A fourth order dual method for staircase reduction in texture extraction and image restoration problems. UCLA CAM Report, 05–28 (2005)

  8. Osher, S., Sole, A., Vese, L.: Image decomposition and restoration using total variation minimization and the H1 norm. Multiscale Model. Simul. 1, 349–370 (2003)

    Article  MathSciNet  Google Scholar 

  9. Yang, J., Zhang, Y., Yin, W.W.: An efficient TV-L1 algorithm for deblurring multichannel images corrupted by impulsive noise. SIAM J. Sci. Comput. 31(4), 2842–2865 (2009)

    Article  MathSciNet  Google Scholar 

  10. Tang, L.M., Huang, D.R.: Multiscale image restoration and reconstruction in the framework of variation. Acta. Electtonica Sin. 41(12), 2353–2360 (2013)

    Google Scholar 

  11. Bai, J., Feng, X.C.: Imageddenoising using generalized anisotropic diffusion. J. Math. Imageing Vis. 60, 994–1007 (2018)

    Article  Google Scholar 

  12. Bai, J., Feng, X.-C.: Fractional-order anisotropic diffusion for image denoising. IEEE Trans. Image Process. 16, 2492–2502 (2007)

    Article  MathSciNet  Google Scholar 

  13. Abirami, A., Prakash, P., Thangavel, K.: Fractional diffusion equation based image denoising model using CN–GL scheme. Int. J. Comput. Math. 95(6–7), 1222–1239 (2018)

    Article  MathSciNet  Google Scholar 

  14. Podlubny, I.: Fractional differential equations: an introduction to fractional derivative, fractional differential equations, to methods of their solution and some of their applications, mathematics in science and engineering, Elsevier Science, (1999)

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

    Article  MathSciNet  Google Scholar 

  16. Zhang, J., Wei, Z., Xiao, L.: Adaptive fractional-order multi-scale method for image denoising. J. Math. Imaging Vis. 43, 39–49 (2012)

    Article  MathSciNet  Google Scholar 

  17. Liu, S.-C., Chang, S.: Dimension estimation of discrete-time fractional Brownian motion with applications to image texture classification. IEEE Trans. Image Process. 6(8), 1176–1184 (1997)

    Article  Google Scholar 

  18. Li, B., Liao, X.F., Jiang, Y.: A novel image encryption scheme based on improved random number generator and its implementation. Nonlinear Dyn. 95, 1781–1805 (2019)

    Article  Google Scholar 

  19. Wu, G.C., Luo, M.K., Huang, L.L., Banerjee, S.: Short memory fractional differential equation for new memristor and neural network design. Nonlinear Dyn. 100, 3611–3623 (2020)

    Article  Google Scholar 

  20. Unser, M., Blu, T.: Fractional splines and wavelets. SIAM Rev. 40(1), 43–67 (2000)

    Article  MathSciNet  Google Scholar 

  21. Unser, M.: Splines: a perfect fit for signal and image processing. IEEEE Signal Process. Image 16(6), 22–38 (1999)

    Article  Google Scholar 

  22. Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intel. 12(4), 629–639 (1990)

    Article  Google Scholar 

  23. Chen, D., Chen, Y., Xue, D.: Three-fractional-order TV-L2 models for image denoising. J. Comput. Inf. Syst. 9, 4774–4780 (2002)

    Google Scholar 

  24. Ahkhanov, A.A.: A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280, 424–438 (2015)

    Article  MathSciNet  Google Scholar 

  25. Richtmyer, R.D., Morton, K.W.: Difference methods for initial value problem 2nd edition. Krieger, Malabar, FL (1994)

    MATH  Google Scholar 

  26. Cao, K., Feng, W., Shang, Q.W.: A novel image denoising algorithm based on Crank–Nicholson semi-implicit difference scheme. Proc. Eng. 23, 647–652 (2011)

    Article  Google Scholar 

  27. Yang, Z.Z., Yan, Y.X., Zhou, J.M.: Edge detection based on fractional differential. J. Sichuan Univ. (Eng. Sci. Ed.) 40, 152 (2008)

    Google Scholar 

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 11971337).

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Authors and Affiliations

  1. School of Mathematics, Sichuan University, Chengdu, 610065, People’s Republic of China

    Xingran Liao & Minfu Feng

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  1. Xingran Liao
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  2. Minfu Feng
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Correspondence to Xingran Liao.

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Appendix

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Liao, X., Feng, M. Time-fractional diffusion equation-based image denoising model. Nonlinear Dyn 103, 1999–2017 (2021). https://doi.org/10.1007/s11071-020-06136-x

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