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:
[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.  Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[30]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[30]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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
[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, 2020  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]

Yones Esmaeelzade Aghdam, Hamid Safdari, Yaqub Azari, Hossein Jafari, Dumitru Baleanu. Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020402

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (79)
  • HTML views (240)
  • Cited by (7)

Other articles
by authors

[Back to Top]