Total fractional-order variation and bilateral filter for image denoising

2024;
: pp. 642–653
https://doi.org/10.23939/mmc2024.03.642
Received: December 08, 2023
Revised: July 21, 2024
Accepted: July 25, 2024

Addouch R., Moussaid N., Gouasnouane O., Ben-Loghfyry A. Total fractional-order variation and bilateral filter for image denoising.  Mathematical Modeling and Computing. Vol. 11, No. 3, pp. 642–653 (2024)

1
LMCA, FSTM of Mohammedia, Hassan II University of Casablanca, Morocco
2
LMCA, FSTM of Mohammedia, Hassan II University of Casablanca, Morocco
3
LMCA, FSTM of Mohammedia, Hassan II University of Casablanca, Morocco
4
LMCMAN, FSTM of Mohammedia, Hassan II University of Casablanca, Morocco

Image denoising stands out as a primary goal in image processing.  However, many existing methods encounter challenges in preserving features such as corners and edges of an image while deleting the noise.  This study investigates and evaluates a fractional-order derivative based on the total $\alpha$-order variation (TV) model and the bilateral total variation (BTV) model.  This choice is motivated by the proven effectiveness of the TV model in noise removal and edge preservation, with the BTV model further utilized to enhance the restoration of fine and intricate details.  The experimental results affirm the efficacy of the proposed model, supported by objective quantitative metrics and subjective assessments of visual appearance.

  1. Ben-Loghfyry A., Hakim A.  A total variable-order variation model for image denoising.  AIMS Mathematics.  4 (5), 1320–1335 (2019).
  2. Zhu M., Wright S. J., Chan T. F.  Duality-based algorithms for total-variation-regularized image restoration.  Computational Optimization and Applications.  47 (3), 377–400 (2010).
  3. 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).
  4. Laghrib A., Hakim A., Raghay S.  A combined total variation and bilateral filter approach for image robust super resolution.  EURASIP Journal on Image and Video Processing.  2015, 19 (2015).
  5. El Mourabit I., El Rhabi M., Hakim A.  Blind deconvolution using bilateral total variation regularization: a theoretical study and application.  Applicable Analysis.  101 (16),  5660–5673 (2022).
  6. Rudin L. I., Osher S., Fatemi E.  Nonlinear total variation based noise removal algorithms.  Physica D: Nonlinear Phenomena.  60 (1–4), 259–268 (1992).
  7. Ben-Loghfyry A., Hakim A., Laghrib A.  A denoising model based on the fractional Beltrami regularization and its numerical solution.  Journal of Applied Mathematics and Computing.  69 (2), 1431–1463 (2023).
  8. Ben-Loghfyry A., Hakim A.  Time-fractional difusion equation for signal and image smoothing.  Mathematical Modeling and Computing.  9 (2), 351–364 (2022).
  9. Ben-Loghfyry A., Hakim A.  Robust-time-fractional diffusion filtering for noise removal.  Mathematical Methods in the Applied Sciences.  45 (16), 9719–9735 (2022).
  10. Ben-Loghfyry A., Charkaoui A.  Regularized Perona & Malik model involving Caputo time-fractional derivative with application to image denoising.  Chaos, Solitons & Fractals.  175 (1), 113925 (2023).
  11. Gouasnouane O., Moussaid N., Boujena S.  A nonlinear fractional partial equation differential for image denoising.  2021 Third International Conference on Transportation and Smart Technologies (TST), Tangier, Morocco. 59–64 (2021).
  12. Gouasnouane O., Moussaid N., Boujena S., Kabli K.  A nonlinear fractional partial differential equation for image inpainting.  Mathematical Modeling and Computing.  9 (3), 536–546 (2022).
  13. 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).
  14. Podlubny I., Chechkin A., Skovranek T., Chen Y., Vinagre-Jara B.M.  Matrix approach to discrete fractional calculus~II: Partial fractional differential equations.  Journal of Computational Physics.  228 (1), 3137–3153 (2009).
  15. Wang H., Du N.  Fast solution methods for space-fractional diffusion equations.  Journal of Computational and Applied Mathematics.  255, 376–383 (2014).
  16. Sayah A., Moussaid N., Gouasnouane O.  Finite difference method for Perona-Malik model with fractional derivative and its application in image processing.  2021 Third International Conference on Transportation and Smart Technologies (TST), Tangier, Morocco. 101–106 (2021).
  17. Zosso D., Bustin R.  A Primal-Dual Projected Gradient algorithm for efficient Beltrami regularization.  UCLA CAM Report. 14–52 (2014).
  18. Zeidler E.  Nonlinear functional analysis and its Applications. III. Variational methods and optimization.  Springer-Verlag, New York (1985).