# American Institute of Mathematical Sciences

February  2018, 11(1): 103-117. doi: 10.3934/dcdss.2018007

## Finite difference and Legendre spectral method for a time-fractional diffusion-convection equation for image restoration

 AMNEA Group, Department of Mathematics, Faculty of Sciences and Technics, Moulay Ismail University, B.P. 509, Errachidia, Morocco

* Corresponding author: M. R. Sidi Ammi

Received  June 2016 Revised  April 2017 Published  January 2018

In this paper, we consider a time fractional diffusion-convection equation and its application for image processing. A time discretization scheme is introduced and a stability result and error estimates are proved. Practical experiments are then provided showing that the fractional approach is more efficient than the ordinary integer one (Perona-Malik). A fully discrete scheme is proposed by using a Legendre collocation method. The convergence of this method is obtained by proving a priori error estimates.

Citation: Moulay Rchid Sidi Ammi, Ismail Jamiai. Finite difference and Legendre spectral method for a time-fractional diffusion-convection equation for image restoration. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 103-117. doi: 10.3934/dcdss.2018007
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##### References:
PSNR and SNR as functions of α
Numerical example for filtering a noisy image: results after 10 steps. (a) Original image. (b) noisy image. (c), (d) and (e) Results of fractional filter with $\alpha = 0.81$, $\alpha = 0.82$ and $\alpha = 0.83$. (f) Result of PM model
The PSNR and SNR for images in Figure 2
 Noise PM α = 0.81 α = 0.82 α = 0.83 PSNR 17.5181 21.3257 23.3666 23.3814 23.3785 SNR 11.8396 15.6472 17.6881 17.7028 17.7000
 Noise PM α = 0.81 α = 0.82 α = 0.83 PSNR 17.5181 21.3257 23.3666 23.3814 23.3785 SNR 11.8396 15.6472 17.6881 17.7028 17.7000
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