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  3. Volume 9, Number 3, 2022
  4. A nonlinear fractional partial differential equation for image inpainting

A nonlinear fractional partial differential equation for image inpainting

MMC.
2022;
Volume 9, Number 3
: pp. 536–546
https://doi.org/10.23939/mmc2022.03.536
Received: January 05, 2022
Accepted: June 19, 2022

Mathematical Modeling and Computing, Vol. 9, No. 3, pp. 536–546 (2022)

Authors:
  1. O. Gouasnouane
  2. N. Moussaid
  3. S. Boujena
  4. K. Kabli
1
University Hassan II of Casablanca, FST Mohammedia, Laboratory of Mathematics, Computer Science and Applications; University Hassan II of Casablanca, Ain-Chock Sciences Faculty, Laboratory of Modelisation, Analysis, Control and Statistics (MACS)
2
University Hassan II of Casablanca, FST Mohammedia, Laboratory of Mathematics, Computer Science and Applications (LMCSA)
3
University Hassan II of Casablanca, Ain-Chock Sciences Faculty, Laboratory of Modelisation, Analysis, Control and Statistics (MACS)
4
University Hassan II of Casablanca, Ain-Chock Sciences Faculty, Laboratory of Modelisation, Analysis, Control and Statistics (MACS)

Image inpainting is an important research area in image processing.  Its main purpose is to supplement missing or damaged domains of images using information from surrounding areas.  This step can be performed by using nonlinear diffusive filters requiring a resolution of partial differential evolution equations.  In this paper, we propose a filter defined by a partial differential nonlinear evolution equation with spatial fractional derivatives.  Due to this, we were able to improve the performance obtained by known inpainting models based on partial differential equations and extend certain existing results in image processing.

The discretization of the fractional partial differential equation of the proposed model is carried out using the shifted Grünwald–Letnikov formula, which allows us to build stable numerical schemes. 

The comparative analysis shows that the proposed model produces an improved image quality better or comparable to that obtained by various other efficient models known from the literature.

image processing
image inpainting
fractional calculus
fractional order partial differential equation
nonlinear diffusion
fractional derivative
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