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
References:

show all references

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
 [1] Xuefeng Zhang, Hui Yan. Image enhancement algorithm using adaptive fractional differential mask technique. Mathematical Foundations of Computing, 2019, 2 (4) : 347-359. doi: 10.3934/mfc.2019022 [2] Mehdi Bastani, Davod Khojasteh Salkuyeh. On the GSOR iteration method for image restoration. Numerical Algebra, Control & Optimization, 2019, 0 (0) : 0-0. doi: 10.3934/naco.2020013 [3] Zbigniew Gomolka, Boguslaw Twarog, Jacek Bartman. Improvement of image processing by using homogeneous neural networks with fractional derivatives theorem. Conference Publications, 2011, 2011 (Special) : 505-514. doi: 10.3934/proc.2011.2011.505 [4] Valerii Maltsev, Michael Pokojovy. On a parabolic-hyperbolic filter for multicolor image noise reduction. Evolution Equations & Control Theory, 2016, 5 (2) : 251-272. doi: 10.3934/eect.2016004 [5] Jia Li, Zuowei Shen, Rujie Yin, Xiaoqun Zhang. A reweighted $l^2$ method for image restoration with Poisson and mixed Poisson-Gaussian noise. Inverse Problems & Imaging, 2015, 9 (3) : 875-894. doi: 10.3934/ipi.2015.9.875 [6] Michael Hintermüller, Monserrat Rincon-Camacho. An adaptive finite element method in $L^2$-TV-based image denoising. Inverse Problems & Imaging, 2014, 8 (3) : 685-711. doi: 10.3934/ipi.2014.8.685 [7] Jianjun Zhang, Yunyi Hu, James G. Nagy. A scaled gradient method for digital tomographic image reconstruction. Inverse Problems & Imaging, 2018, 12 (1) : 239-259. doi: 10.3934/ipi.2018010 [8] Jianhong (Jackie) Shen, Sung Ha Kang. Quantum TV and applications in image processing. Inverse Problems & Imaging, 2007, 1 (3) : 557-575. doi: 10.3934/ipi.2007.1.557 [9] Wenzhong Zhu, Huanlong Jiang, Erli Wang, Yani Hou, Lidong Xian, Joyati Debnath. X-ray image global enhancement algorithm in medical image classification. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1297-1309. doi: 10.3934/dcdss.2019089 [10] Wei Wang, Na Sun, Michael K. Ng. A variational gamma correction model for image contrast enhancement. Inverse Problems & Imaging, 2019, 13 (3) : 461-478. doi: 10.3934/ipi.2019023 [11] Ömer Oruç, Alaattin Esen, Fatih Bulut. A unified finite difference Chebyshev wavelet method for numerically solving time fractional Burgers' equation. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 533-542. doi: 10.3934/dcdss.2019035 [12] Lekbir Afraites, Abdelghafour Atlas, Fahd Karami, Driss Meskine. Some class of parabolic systems applied to image processing. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1671-1687. doi: 10.3934/dcdsb.2016017 [13] Yan Jin, Jürgen Jost, Guofang Wang. A new nonlocal variational setting for image processing. Inverse Problems & Imaging, 2015, 9 (2) : 415-430. doi: 10.3934/ipi.2015.9.415 [14] Marcus Wagner. A direct method for the solution of an optimal control problem arising from image registration. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 487-510. doi: 10.3934/naco.2012.2.487 [15] Yangang Chen, Justin W. L. Wan. Numerical method for image registration model based on optimal mass transport. Inverse Problems & Imaging, 2018, 12 (2) : 401-432. doi: 10.3934/ipi.2018018 [16] Wei Zhu, Xue-Cheng Tai, Tony Chan. Augmented Lagrangian method for a mean curvature based image denoising model. Inverse Problems & Imaging, 2013, 7 (4) : 1409-1432. doi: 10.3934/ipi.2013.7.1409 [17] Nam-Yong Lee, Bradley J. Lucier. Preconditioned conjugate gradient method for boundary artifact-free image deblurring. Inverse Problems & Imaging, 2016, 10 (1) : 195-225. doi: 10.3934/ipi.2016.10.195 [18] Jing Xu, Xue-Cheng Tai, Li-Lian Wang. A two-level domain decomposition method for image restoration. Inverse Problems & Imaging, 2010, 4 (3) : 523-545. doi: 10.3934/ipi.2010.4.523 [19] Yangyang Xu, Wotao Yin. A fast patch-dictionary method for whole image recovery. Inverse Problems & Imaging, 2016, 10 (2) : 563-583. doi: 10.3934/ipi.2016012 [20] Lijun Yi, Zhongqing Wang. Legendre spectral collocation method for second-order nonlinear ordinary/partial differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 299-322. doi: 10.3934/dcdsb.2014.19.299

2018 Impact Factor: 0.545