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 and Continuous Dynamical Systems - S, 2018, 11 (1) : 103-117. doi: 10.3934/dcdss.2018007
References:
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E. Bansch and K. Mikula, A coarsening finite element strategy in image selective smoothing, Computing and Visualization in Science, 1 (1997), 53-61.  doi: 10.1007/s007910050005.

[2]

M. BenoitM. PierreO. Alain and C. Ceyral, Fractional differentiation for edge detection, Signal Processing, 83 (2003), 2421-2432. 

[3]

C. Bernardi and Y. Maday, Approximations Spectrales de Probléms Aux Limites Elliptiques Springer-Verlag, Berlin, 1992.

[4]

A. Brook and T. Hughes, Streamline Upwind/Petrov-Galerkin formulation for convection dominated flow with particular emphasis on the incompressible Navier-Stokes equations, Computational Methods in Applied Mechanics and Engineering, 32 (1982), 199-259.  doi: 10.1016/0045-7825(82)90071-8.

[5]

F. CatteP. L. LionsJ. M. Morel and T. Coll, Image selective smoothing and edge detection by nonlinear diffusion, SIAM Journal of Numerical Analysis, 29 (1992), 182-193.  doi: 10.1137/0729012.

[6]

M. Cui, Compact finite difference method for the fractional diffusion equation, J. Comput. Phys., 228 (2009), 7792-7804.  doi: 10.1016/j.jcp.2009.07.021.

[7]

K. DabovA. FoiV. Katkovnik and K. Egiazarian, Color image denoising via sparse 3d collaborative filtering with grouping constraint in luminance-chrominance space, IEEE Int. Conf. Image Process, 16 (2007), 2080-2095.  doi: 10.1109/TIP.2007.901238.

[8]

A. M. A. El-SayedS. H. Behiry and W. E. Raslan, Adomian's decomposition method for solving an intermediate fractional advection-dispersion equation, Comput. Math. Appl., 59 (2010), 1759-1765.  doi: 10.1016/j.camwa.2009.08.065.

[9]

A. M. A. El-Sayed and M. Gaber, The Adomian decomposition method for solving partial differential equations of fractal order in finite domains, Phys. Lett. A, 359 (2006), 175-182.  doi: 10.1016/j.physleta.2006.06.024.

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L. C. Evans and J. Spruck, Motion of level sets by mean curvatures, Journal of Differential Geometry, 33 (1991), 635-681.  doi: 10.4310/jdg/1214446559.

[11]

N. J. FordJ. Xiao and Y. Yan, A finite element method for time fractional partial differential equations, Fractional Calculus and Applied Analysis, 14 (2011), 454-474.  doi: 10.2478/s13540-011-0028-2.

[12]

S. Guo and L. Mei, The fractional variational iteration method using He's polynomials, Phys. Lett. A, 375 (2011), 309-313.  doi: 10.1016/j.physleta.2010.11.047.

[13]

S. M. GuoL. Q. MeiY. Li and Y. F. Sun, The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics, Phys. Lett. A, 376 (2012), 407-411.  doi: 10.1016/j.physleta.2011.10.056.

[14]

J. H. He, A new approach to nonlinear partial differential equations, Commun. Nonlinear Sci. Numer. Simul., 2 (1997), 230-235.  doi: 10.1016/S1007-5704(97)90007-1.

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J. H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Engrg., 178 (1999), 257-262.  doi: 10.1016/S0045-7825(99)00018-3.

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J. H. He, A coupling method of homotopy technique and a perturbation technique for non-linear problems, Inter. J. Non-Linear Mech., 35 (2000), 37-43.  doi: 10.1016/S0020-7462(98)00085-7.

[17]

Q. HuangG. Huang and H. Zhan, A finite element solution for the fractional advection-dispersion equation, Adv. Water Resour., 31 (2008), 1578-1589.  doi: 10.1016/j.advwatres.2008.07.002.

[18]

B. Jian and C. F. Xiang, Fractional-order anisotropic diffusion for image denoising, IEEE Transactions on Image Processing, 16 (2007), 2492-2502.  doi: 10.1109/TIP.2007.904971.

[19]

Z. JunZ. Wei and L. Xiao, Adaptive fractional-order multi-scale method for image denoising, Journal of Mathematical Imaging and Vision, 43 (2012), 39-49.  doi: 10.1007/s10851-011-0285-z.

[20]

Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, Journal of Computational Physics, 225 (2007), 1533-1552.  doi: 10.1016/j.jcp.2007.02.001.

[21]

B. Lu, Bäcklund transformation of fractional Riccati equation and its applications to nonlinear fractional partial differential equations, Phys. Lett. A, 376 (2012), 2045-2048.  doi: 10.1016/j.physleta.2012.05.013.

[22]

Z. Odibat and S. Momani, Fractional Green function for linear time-fractional equations of fractional order, Appl. Math. Lett., 21 (2008), 194-199.  doi: 10.1016/j.aml.2007.02.022.

[23]

G. PedramS. C. MicaelA. B. Jon and M. F. F. Nuno, An efficient method for segmentation of images based on fractional calculus and natural selection, Expert Systems with Applications, 39 (2012), 12407-12417. 

[24]

P. Perona and J. Malik, Scale space and edge detection using anisotropic diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence, 12 (1990), 629-639.  doi: 10.1109/34.56205.

[25]

A. G. Radwan, S. K. Abd-El-Hafiz and S. H. AbdElHaleem, Image encryption in the fractional-order domain, in Proceedings of IEEE International Conference on Engineering and Technology (ICET), New Cairo City, IEEE, New York: 2012, 1–6. doi: 10.1109/ICEngTechnol.2012.6396148.

[26]

Y. Shih and H. C. Elman, Modified streamline diffusion schemes for convection-diffusion problems, Computational Methods in Applied Mechanics and Engineering, 174 (1999), 137-151.  doi: 10.1016/S0045-7825(98)00283-7.

[27]

Y. ShihC. Rei and H. Wang, A novel PDE based image restoration: Convection-diffusion equation for image denoising, Journal of Computational and Applied Mathematics, 231 (2009), 771-779.  doi: 10.1016/j.cam.2009.05.001.

[28]

J. L. StarckE. J. Candes and D. L. Donoho, The Curvelet transform for image denoising, IEEE Transactions on Image Processing, 11 (2002), 670-684.  doi: 10.1109/TIP.2002.1014998.

[29]

J. Weickert, Anisotropic Diffusion in Image Processing B. G. Teunbner Stuttgart, 1998.

[30]

C. Wielgus, Perona-Malik Equation and Its Numerical Properties Thesis, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, 2010.

[31]

G. Wu and E. W. M. Lee, Fractional variational iteration method and its application, Phys. Lett. A, 374 (2010), 2506-2509.  doi: 10.1016/j.physleta.2010.04.034.

[32]

Z. YiP. Yi-FeiJ. Hu and Z. Ji-Liu, A class of fractional-order variational image inpainting models, Applied Mathematics and Information Sciences, 6 (2012), 299-306. 

[33]

P. Yi-FeiZ. Ji-Liu and Y. Xiao, Fractional differential mask: A fractional differential-based approach for multiscale texture enhancement, IEEE Transactions on Image Processing, 19 (2010), 491-511.  doi: 10.1109/TIP.2009.2035980.

[34]

X. YinS. Zhou and M. A. Siddique, Fractional nonlinear anisotropic diffusion with p-Laplace variation method for image restoration, Multimedia Tools and Applications, 75 (2016), 4505-4526.  doi: 10.1007/s11042-015-2488-6.

[35]

S. Zhang and H. Q. Zhang, Fractional sub-equation method and its applications to nonlinear fractional PDEs, Phys. Lett. A, 375 (2011), 1069-1073.  doi: 10.1016/j.physleta.2011.01.029.

show all references

References:
[1]

E. Bansch and K. Mikula, A coarsening finite element strategy in image selective smoothing, Computing and Visualization in Science, 1 (1997), 53-61.  doi: 10.1007/s007910050005.

[2]

M. BenoitM. PierreO. Alain and C. Ceyral, Fractional differentiation for edge detection, Signal Processing, 83 (2003), 2421-2432. 

[3]

C. Bernardi and Y. Maday, Approximations Spectrales de Probléms Aux Limites Elliptiques Springer-Verlag, Berlin, 1992.

[4]

A. Brook and T. Hughes, Streamline Upwind/Petrov-Galerkin formulation for convection dominated flow with particular emphasis on the incompressible Navier-Stokes equations, Computational Methods in Applied Mechanics and Engineering, 32 (1982), 199-259.  doi: 10.1016/0045-7825(82)90071-8.

[5]

F. CatteP. L. LionsJ. M. Morel and T. Coll, Image selective smoothing and edge detection by nonlinear diffusion, SIAM Journal of Numerical Analysis, 29 (1992), 182-193.  doi: 10.1137/0729012.

[6]

M. Cui, Compact finite difference method for the fractional diffusion equation, J. Comput. Phys., 228 (2009), 7792-7804.  doi: 10.1016/j.jcp.2009.07.021.

[7]

K. DabovA. FoiV. Katkovnik and K. Egiazarian, Color image denoising via sparse 3d collaborative filtering with grouping constraint in luminance-chrominance space, IEEE Int. Conf. Image Process, 16 (2007), 2080-2095.  doi: 10.1109/TIP.2007.901238.

[8]

A. M. A. El-SayedS. H. Behiry and W. E. Raslan, Adomian's decomposition method for solving an intermediate fractional advection-dispersion equation, Comput. Math. Appl., 59 (2010), 1759-1765.  doi: 10.1016/j.camwa.2009.08.065.

[9]

A. M. A. El-Sayed and M. Gaber, The Adomian decomposition method for solving partial differential equations of fractal order in finite domains, Phys. Lett. A, 359 (2006), 175-182.  doi: 10.1016/j.physleta.2006.06.024.

[10]

L. C. Evans and J. Spruck, Motion of level sets by mean curvatures, Journal of Differential Geometry, 33 (1991), 635-681.  doi: 10.4310/jdg/1214446559.

[11]

N. J. FordJ. Xiao and Y. Yan, A finite element method for time fractional partial differential equations, Fractional Calculus and Applied Analysis, 14 (2011), 454-474.  doi: 10.2478/s13540-011-0028-2.

[12]

S. Guo and L. Mei, The fractional variational iteration method using He's polynomials, Phys. Lett. A, 375 (2011), 309-313.  doi: 10.1016/j.physleta.2010.11.047.

[13]

S. M. GuoL. Q. MeiY. Li and Y. F. Sun, The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics, Phys. Lett. A, 376 (2012), 407-411.  doi: 10.1016/j.physleta.2011.10.056.

[14]

J. H. He, A new approach to nonlinear partial differential equations, Commun. Nonlinear Sci. Numer. Simul., 2 (1997), 230-235.  doi: 10.1016/S1007-5704(97)90007-1.

[15]

J. H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Engrg., 178 (1999), 257-262.  doi: 10.1016/S0045-7825(99)00018-3.

[16]

J. H. He, A coupling method of homotopy technique and a perturbation technique for non-linear problems, Inter. J. Non-Linear Mech., 35 (2000), 37-43.  doi: 10.1016/S0020-7462(98)00085-7.

[17]

Q. HuangG. Huang and H. Zhan, A finite element solution for the fractional advection-dispersion equation, Adv. Water Resour., 31 (2008), 1578-1589.  doi: 10.1016/j.advwatres.2008.07.002.

[18]

B. Jian and C. F. Xiang, Fractional-order anisotropic diffusion for image denoising, IEEE Transactions on Image Processing, 16 (2007), 2492-2502.  doi: 10.1109/TIP.2007.904971.

[19]

Z. JunZ. Wei and L. Xiao, Adaptive fractional-order multi-scale method for image denoising, Journal of Mathematical Imaging and Vision, 43 (2012), 39-49.  doi: 10.1007/s10851-011-0285-z.

[20]

Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, Journal of Computational Physics, 225 (2007), 1533-1552.  doi: 10.1016/j.jcp.2007.02.001.

[21]

B. Lu, Bäcklund transformation of fractional Riccati equation and its applications to nonlinear fractional partial differential equations, Phys. Lett. A, 376 (2012), 2045-2048.  doi: 10.1016/j.physleta.2012.05.013.

[22]

Z. Odibat and S. Momani, Fractional Green function for linear time-fractional equations of fractional order, Appl. Math. Lett., 21 (2008), 194-199.  doi: 10.1016/j.aml.2007.02.022.

[23]

G. PedramS. C. MicaelA. B. Jon and M. F. F. Nuno, An efficient method for segmentation of images based on fractional calculus and natural selection, Expert Systems with Applications, 39 (2012), 12407-12417. 

[24]

P. Perona and J. Malik, Scale space and edge detection using anisotropic diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence, 12 (1990), 629-639.  doi: 10.1109/34.56205.

[25]

A. G. Radwan, S. K. Abd-El-Hafiz and S. H. AbdElHaleem, Image encryption in the fractional-order domain, in Proceedings of IEEE International Conference on Engineering and Technology (ICET), New Cairo City, IEEE, New York: 2012, 1–6. doi: 10.1109/ICEngTechnol.2012.6396148.

[26]

Y. Shih and H. C. Elman, Modified streamline diffusion schemes for convection-diffusion problems, Computational Methods in Applied Mechanics and Engineering, 174 (1999), 137-151.  doi: 10.1016/S0045-7825(98)00283-7.

[27]

Y. ShihC. Rei and H. Wang, A novel PDE based image restoration: Convection-diffusion equation for image denoising, Journal of Computational and Applied Mathematics, 231 (2009), 771-779.  doi: 10.1016/j.cam.2009.05.001.

[28]

J. L. StarckE. J. Candes and D. L. Donoho, The Curvelet transform for image denoising, IEEE Transactions on Image Processing, 11 (2002), 670-684.  doi: 10.1109/TIP.2002.1014998.

[29]

J. Weickert, Anisotropic Diffusion in Image Processing B. G. Teunbner Stuttgart, 1998.

[30]

C. Wielgus, Perona-Malik Equation and Its Numerical Properties Thesis, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, 2010.

[31]

G. Wu and E. W. M. Lee, Fractional variational iteration method and its application, Phys. Lett. A, 374 (2010), 2506-2509.  doi: 10.1016/j.physleta.2010.04.034.

[32]

Z. YiP. Yi-FeiJ. Hu and Z. Ji-Liu, A class of fractional-order variational image inpainting models, Applied Mathematics and Information Sciences, 6 (2012), 299-306. 

[33]

P. Yi-FeiZ. Ji-Liu and Y. Xiao, Fractional differential mask: A fractional differential-based approach for multiscale texture enhancement, IEEE Transactions on Image Processing, 19 (2010), 491-511.  doi: 10.1109/TIP.2009.2035980.

[34]

X. YinS. Zhou and M. A. Siddique, Fractional nonlinear anisotropic diffusion with p-Laplace variation method for image restoration, Multimedia Tools and Applications, 75 (2016), 4505-4526.  doi: 10.1007/s11042-015-2488-6.

[35]

S. Zhang and H. Q. Zhang, Fractional sub-equation method and its applications to nonlinear fractional PDEs, Phys. Lett. A, 375 (2011), 1069-1073.  doi: 10.1016/j.physleta.2011.01.029.

Figure 1.  PSNR and SNR as functions of α
Figure 2.  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
Table 1.  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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