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Fractional derivative based nonlinear diffusion model for image denoising

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Abstract

In this article, a new conformable fractional anisotropic diffusion model for image denoising is presented, which contains the spatial derivative along with the time-fractional derivative. This model is a generalization of the diffusion model (Welk et al. in Scale space. Springer, Berlin, pp 585–597, 2005) with forward–backward diffusivities. The proposed model is very efficient for noise removal of the noisy images in comparison to the classical anisotropic diffusion model. The numerical experiments are performed using an explicit scheme for different-different values of fractional order derivative \(\alpha \). The experimental results are obtained in terms of peak signal to noise ratio (PSNR) as a metric.

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References

  1. Abu Hammad, M., Khalil, R.: Conformable heat differential equation. Int. J. Pure Appl. Math. 94(2), 215–221 (2014)

    Article  MATH  Google Scholar 

  2. Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Avct, D., Eroglue, B.R.I., Ozdemir, N.: Conformable heat equation on a radial symmetric plate. Therm Sci. 21(2), 819–826 (2017)

    Article  Google Scholar 

  4. Ahmad, A.A., José, A., Machado, T.: A critical analysis of the conformable derivative. Nonlinear Dyn. 95, 3063–3073 (2019)

    Article  MATH  Google Scholar 

  5. Alvarez, L., Lions, P.L., Morel, J.M.: Image selective smoothing and edge detection by nonlinear diffusion \(\text{ II}^{\ast }\). SIAM J. Numer. Anal. 29(3), 845–866 (1992)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Chen, D., Sun, S., Zhang, C., Chen, Y., Xue, D.: Fractional-order TV-\(\text{ L}^{2}\) model for image denoising. Cent. Eur. J. Phys. 11(10), 1414–1422 (2013)

    Google Scholar 

  8. Catte, F., Lions, P.L., Morel, J.M., Coll, T.: Image selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numer. Anal. 29, 182–193 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chan, T.F., Golub, G.H., Mulet, P.: A nonlinear primal-dual method for total variation-based image restoration. SIAM J. Sci. Comput. 20(6), 1964–1977 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chang, Q., Chern, I.-L.: Acceleration methods for total variation-based image denoising. SIAM J. Sci. Comput. 25(3), 982–994 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Charbonnier, P., Blanc-Feraud, L., Aubert, G., Barlaud, M.: Two deterministic half-quadratic regularization algorithms for computed imaging. In: Proceedings of 1st International Conference on Image Processing, vol. 2, pp. 168–172 (1994)

  12. Cuesta, E., Finat, J.: Image processing by means of linear integro-differential equation. In: Proceedings of the Third IASTED International Conference on Visualization, Imaging and Image Processing, Benalmadena, Spain (2003)

  13. Çenesiz, Y., Kurt, A.: The solutions of time and space conformable fractional heat equations with conformable Fourier transform. Acta Univ. Sapientiae Math. 7(2), 130–140 (2015)

    MathSciNet  MATH  Google Scholar 

  14. Evans, L.C., Spruck, J.: Motion of level sets by mean curvature. I. J. Differ. Geom. 33, 635–681 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gorenflo, R., Mainardi, F.: Fractional calculus: integral and differential equations of fractional order. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics. Springer, NewYork (1997)

    MATH  Google Scholar 

  16. Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)

    Book  MATH  Google Scholar 

  17. Hermann, R.: Fractional Calculus, An Introduction for Physicists, 2nd edn. World Scientific Publishing Co. Pvt. Ltd, Singapore (2014)

    Book  Google Scholar 

  18. Hammad, A., Khalil, R.: Fractional Fourier series with applications. Am. J. Comput. Appl. Math. 4(6), 187–191 (2014)

    Google Scholar 

  19. Hristov, J.: Derivatives with non-singular kernels from the Caputo–Fabrizio definition and beyond: appraising analysis with emphasis on diffusion models. Frontiers 1, 270–342 (2017)

    Google Scholar 

  20. Khalil, R., Al Horani, M., Yousef, A., Sabebah, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kumar, A., Kumar, S., Yan, S.P.: Residual power series method for fractional diffusion equations. Fund. Inform. 151(14), 213–230 (2017)

    MathSciNet  MATH  Google Scholar 

  22. Kumar, S., Sarfaraz, M., Ahmad, M.K.: Denoising method based on wavelet coefficients via diffusion equation. Iran. J. Sci. Technol. Trans. A Sci. 42, 721–726 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lapidus, L., Pinder, G.F.: Numerical solution of partial differential equations in science and engineering. SIAM Rev. 25(4), 581–582 (1983)

    Article  Google Scholar 

  24. Li, Z., Liu, L., Dehghan, S., Chen, Y., Xue, D.: A review and evaluation of numerical tools for fractional calculus and fractional order controls. Int. J. Control 90(6), 1165–1181 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  25. Mondal, S., Bairagi, N., Lahiri, A.: A fractional calculus approach to Rosenzweig–MacArthur predator prey model and its solution. J. Mod. Methods Numer. Math. 8(12), 66–76 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  26. Munkhamar, J.: Riemann–Liouville fractional derivatives and the Taylor–Riemann series. Uppsala University Department of Mathematics, Project Report, 7 (2004)

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

    Article  Google Scholar 

  28. Rosen, J.G.: The gradient projection method for nonlinear programming, part II, nonlinear constraints. J. Soc. Ind. Appl. Math. 9(4), 514–532 (1961)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  30. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives-Theory and Applications. Gordon and Breach Science, Amsterdam (1993)

    MATH  Google Scholar 

  31. Strong, D.: Adaptive total variation minimizing image restoration. Ph.D. Thesis, UCLA Mathematics Department, USA (1997)

  32. Weickert, J.: Anisotropic Diffusion in Image Processing. Teubner, Stuttgart (1998)

    MATH  Google Scholar 

  33. Weickert, J.: A review of nonlinear diffusion filtering. In: ter Haar Romeny, B., Florack, L., Koenderink, J., Viergever, M. (eds.) Scale-Space Theory in Computer Vision. Scale-Space 1997. Lecture Notes in Computer Science, vol. 1252. Springer, Berlin, Heidelberg (1997)

  34. Welk, M., Theis, D., Brox, T., Weickert, J.: PDE-Based deconvolution with forward-backward diffusivities and diffusion tensors. In: Kimmel, R., Sochen, N.A., Weickert, J. (eds.) Scale Space and PDE Methods in Computer Vision. Scale-Space 2005. Lecture Notes in Computer Science, vol. 3459. Springer, Berlin, Heidelberg (2005)

  35. Yang, X.J., Gao, F., Srivastava, H.M.: New rheological models within local fractional derivative. Romanian Rep. Phys. 69(3), 1–12 (2017)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, School of Basic Sciences and Research, Sharda University, Greater Noida, UP, 201306, India

    Santosh Kumar, Khursheed Alam & Alka Chauhan

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  1. Santosh Kumar
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  2. Khursheed Alam
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  3. Alka Chauhan
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Correspondence to Santosh Kumar.

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Kumar, S., Alam, K. & Chauhan, A. Fractional derivative based nonlinear diffusion model for image denoising. SeMA 79, 355–364 (2022). https://doi.org/10.1007/s40324-021-00255-0

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