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  4. Blind image deblurring using Nash game and the fractional order derivative

Blind image deblurring using Nash game and the fractional order derivative

MMC.
2024;
Volume 11, Number 4
: pp. 923–929
https://doi.org/10.23939/mmc2024.04.923
Received: December 08, 2023
Revised: July 19, 2024
Accepted: July 22, 2024

Semmane F. Z., Moussaid N., Ziani M. Blind image deblurring using Nash game and the fractional order derivative. Mathematical Modeling and Computing. Vol. 11, No. 4, pp. 923–929 (2024)

Authors:
  1. F. Z. Semmane
  2. N. Moussaid
  3. M. Ziani
1
LMCSA laboratory, Hassan II University of Casablanca
2
LMCSA laboratory, Hassan II University of Casablanca
3
LMSA, Department of Mathematics, Faculty of Sciences, Mohammed V University in Rabat

This paper presents an innovative approach to blind image deblurring based on fractional order derivatives and Nash game theory.  The integration of fractional order derivatives enhances the deblurring process, capturing intricate image details beyond the capabilities of traditional integer-order derivatives.  The Nash game framework is employed to model the strategic interaction between the image and the unknown blur kernel, fostering a cooperative optimization process.  Experimental results showcase the proposed method's superiority in terms of both Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity Index (SSIM) when compared to existing methods.  The fractional order derivative enhances image structure preservation, while the Nash game facilitates joint optimization of image restoration and blur kernel estimation.

deconvolution
image deblurring
optimisation
Nash game
total variation regularization
fractional order derivative
  1. 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).
  2. Oldham K., Spanier J.  The Fractional Calculus.  Theory and Applications of Differentiation and Integration to Arbitrary Order.  Academic Press (1974).
  3. Rudin L. I., Osher S., Fatemi E.  Nonlinear total variation based noise removal algorithms.  Physica D: Nonlinear Phenomena.  60 (1–4), 259–268 (1996).
  4. Chan T. F., Wong C.-K.  Total variation blind deconvolution.  IEEE Transactions on Image Processing.  7 (3), 370–375 (1998).
  5. Chan T., Esedoglu S., Park F., Yip A.  Recent developments in total variation image restoration.  Mathematical Models of Computer Vision.  17 (2), 17–31 (2005).
  6. Meskine D., Moussaid N., Berhich S.  Blind image deblurring by game theory.  NISS'19: Proceedings of the 2nd International Conference on Networking, Information Systems & Security.  31, 1–7 (2019).
  7. Aboulaich R., Habbal A., Moussaid N.  Optimisation multicrit\`ere: une approche par partage des variables.  ARIMA.  13, 77–89 (2010).
  8. Elmoumen S., Moussaid N., Aboulaich R.  Image retrieval using Nash equilibrium and Kalai–Smorodinsky solution.  Mathematical Modeling and Computing.  8 (4), 646–657 (2021).
  9. Semmane F. Z., Moussaid N., Ziani M.  Searching for similar images using Nash game and machine learning.  Mathematical Modeling and Computing.  11 (1), 239–249 (2024).
  10. Alaa K., Atounti M., Zirhem M.  Image restoration and contrast enhancement based on a nonlinear reaction-diffusion mathematical model and divide and conquer technique.  Mathematical Modeling and Computing.  8 (3), 549–559 (2021).
  11. Abdelouahab M.-S., Hamri N.-E.  The Grünwald–Letnikov fractional-order derivative with fixed memory length.  Mediterranean Journal of Mathematics.  13 (2), 557–572 (2016).
  12. Salah F.-E., Moussaid N.  Machine learning and similar image-based techniques based on Nash game theory.  Mathematical Modeling and Computing.  11 (1), 120–133 (2024).
  13. Wang Z., Bovik A. C., Sheikh H. R., Simoncelli E. P.  Image quality assessment: from error visibility to structural similarity.  IEEE Transactions on Image Processing.  13 (4), 600–612 (2004).
  14. Nasr N., Moussaid N., Gouasnouane O.  The Kalai Smorodinsky solution for blind deconvolution.  Computational and Applied Mathematics.  41 (5), 222 (2022).