1. MMC
  2. All Volumes and Issues
  3. Volume 9, Number 2, 2022
  4. Time-fractional diffusion equation for signal and image smoothing

Time-fractional diffusion equation for signal and image smoothing

MMC.
2022;
Volume 9, Number 2
: pp. 351–364
https://doi.org/10.23939/mmc2022.02.351
Received: October 06, 2020
Revised: May 06, 2022
Accepted: May 07, 2022

Mathematical Modeling and Computing, Vol. 9, No. 2, pp. 351–364 (2022)

Authors:
  1. A. Ben-Loghfyry
  2. A. Hakim
1
LAMAI, University of Cadi Ayyad, Marrakesh, Morocco
2
LAMAI, University of Cadi Ayyad, Marrakesh, Morocco

In this paper, we utilize a time-fractional diffusion equation for image denoising and signal smoothing.  A discretization of our model is provided.  Numerical results show some remarkable results with a great performance, visually and quantitatively, compared to some well known competitive models.

image denoising
fractional derivative
time-fractional order derivative
tensor diffusion
  1. Ben-loghfyry A., Hakim A.  Robust time-fractional diffusion filtering for noise removal.  Mathematical Methods in the Applied Sciences. 1–17 (2022).
  2. El Alaoui El Fels A., Ben-loghfyry A., El Ghorfi M.  Performance of denoising algorithms in the improvement of lithological discrimination.  Modeling Earth Systems and Environment. 1–8 (2022).
  3. Hakim A., Ben-Loghfyry A.  A total variable-order variation model for image denoising.  AIMS Mathematics. 4 (5), 1320–1335 (2019).
  4. Pan H., Wen Y. W., Zhu H. M.  A regularization parameter selection model for total variation based image noise removal.  Applied Mathematical Modelling. 68, 353–367 (2019).
  5. Laghrib A., Ben-Loghfyry A., Hadri A., Hakim A.  A nonconvex fractional order variational model for multi-frame image super-resolution.  Signal Processing: Image Communication. 67, 1–11 (2018).
  6. Hakim M., Ghazdali A., Laghrib A.  A multi-frame super-resolution based on new variational data fidelity term.  Applied Mathematical Modelling. 87, 446–467 (2020).
  7. Baloochian H., Ghaffary H. R., Balochian S.  Enhancing fingerprint image recognition algorithm using fractional derivative filters.  Open Computer Science. 7 (1), 9–16 (2017).
  8. Ferrah I., Chaou A. K., Maadjoudj D., Teguar M.  Novel colour image encoding system combined with ANN for discharges pattern recognition on polluted insulator model.  IET Science, Measurement & Technology. 14, 718–725 (2020).
  9. Nasreddine K., Benzinou A., Fablet R.  Signal and image registration: Application to decrypt marine biological archives.  Traitement du Signal. 26 (4), 255–268 (2009).
  10. Frohn-Schauf C., Henn S., Witsch K.  Multigrid based total variation image registration.  Computing and Visualization in Science. 11 (2), 101–113 (2008).
  11. Alaa H., Alaa N. E., Aqel F., Lefraich H.  A new Lattice Boltzmann method for a Gray–Scott based model applied to image restoration and contrast enhancement.  Mathematical Modeling and Computing. 9 (2), 187–202 (2022).
  12. Alaa K., Atouni M., Zirhem M.  Image restoration and contrast enhancement based on a nonlinear reaction-diffusion mathematical model and divide & conquer technique.  Mathematical Modeling and Computing. 8 (3), 549–559 (2021).
  13. Guichard F., Moisan L., Morel J. M.  A review of PDE models in image processing and image analysis.  Journal de Physique IV. 12 (1), 137–154 (2002).
  14. Chan T. F., Shen J., Vese L.  Variational PDE models in image processing.  Notices of the American Mathematical Society. 50 (1), 14–26 (2003).
  15. Aubert G., Kornprobst P.  Mathematical problems in image processing: partial differential equations and the calculus of variations.  Vol. 147. Springer Science & Business Media (2006).
  16. Perona P., Malik J.  Scale-space and edge detection using anisotropic diffusion.  IEEE Transactions on pattern analysis and machine intelligence. 12 (7), 629–639 (1990).
  17. Catté F., Lions P. L., Morel J. M., Coll T.  Image selective smoothing and edge detection by nonlinear diffusion.  SIAM Journal on Numerical analysis. 29 (1), 182–193 (1992).
  18. Weickert J.  Applications of nonlinear diffusion in image processing and computer vision.  Acta Math. Univ. Comenianae. 70 (1), 33–50 (2001).
  19. Sabatier J., Agrawal O. P., Machado J. T.  Advances in fractional calculus. Vol. 4. Springer (2007).
  20. Machado J. T., Kiryakova V.  The chronicles of fractional calculus.  Fractional Calculus and Applied Analysis. 20 (2), 307–336 (2017).
  21. Yang Q., Chen D., Zhao T., Chen Y.  Fractional calculus in image processing: a review.  Fractional Calculus and Applied Analysis. 19 (5), 1222–1249 (2016).
  22. Uchaikin V. V.  On time-fractional representation of an open system response.  Fractional Calculus and Applied Analysis. 19 (5), 1306 (2016).
  23. Chang A., Sun H.  Time-space fractional derivative models for CO$_2$ transport in heterogeneous media.  Fractional Calculus and Applied Analysis. 21 (1), 151–173 (2018).
  24. Zhao X., Sun Z. Z.  Time-fractional derivatives.  Numerical Methods. 3, 23–48 (2019).
  25. Oliveira D. S., de Oliveira E. C.  On a Caputo-type fractional derivative.  Advances in Pure and Applied Mathematics. 10 (2), 81–91 (2019).
  26. Li C., Qian D., Chen Y.  On Riemann-Liouville and Caputo derivatives.  Discrete Dynamics in Nature and Society. 2011, 562494 (2011).
  27. Weickert J.  Anisotropic diffusion in image processing. Vol. 1.  Teubner Stuttgart (1998).
  28. Zhang J., Chen K.  A total fractional-order variation model for image restoration with nonhomogeneous boundary conditions and its numerical solution.  SIAM Journal on Imaging Sciences. 8 (4), 2487–2518 (2015).
  29. Rudin L. I., Osher S., Fatemi E.  Nonlinear total variation based noise removal algorithms.  Physica D: Nonlinear Phenomena. 60 (1–4), 259–268 (1992).