A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition

Abdelghafour Atlas; Mostafa Bendahmane; Fahd Karami; Driss Meskine; Omar Oubbih

doi:10.3934/dcdsb.2020321

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September 2021, 26(9): 4963-4998. doi: 10.3934/dcdsb.2020321

A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition

Abdelghafour Atlas ^1, , Mostafa Bendahmane ^2, , Fahd Karami ^3,, , Driss Meskine ^3, and Omar Oubbih ^3,

1.	Ecole Nationale des Sciences Appliquées de Marrakech, Université Cadi Ayyad, B.P. 575 Avenue Abdelkrim Al Khattabi Marrakech, Morocco
2.	Institut de Mathématiques de Bordeaux and INRIA-Carmen Bordeaux Sud-Ouest, Université de Bordeaux, 33076 Bordeaux Cedex, France
3.	Ecole Supérieure de Technologie d'Essaouira, Université Cadi Ayyad, B.P. 383 Essaouira El Jadida, Essaouira, Morocco

^* Corresponding author: Fahd Karami

Received May 2020 Revised July 2020 Published September 2021 Early access November 2020

This paper is devoted to the mathematical and numerical study of a new proposed model based on a fractional diffusion equation coupled with a nonlinear regularization of the Total Variation operator. This model is primarily intended to introduce a weak norm in the fidelity term, where this norm is considered more appropriate for capturing very oscillatory characteristics interpreted as a texture. Furthermore, our proposed model profits from the benefits of a variable exponent used to distinguish the features of the image. By using Faedo-Galerkin method, we prove the well-posedness (existence and uniqueness) of the weak solution for the proposed model. Based on the alternating direction implicit method of Peaceman-Rachford and the approximations of the Gr$ \ddot{u} $nwald-Letnikov operators, we develop the numerical discretization of our fractional diffusion equation. Experimental results claim that our model provides high-quality results in cartoon-texture-edges decomposition and image denoising. In particular, our model can successfully reduce the staircase phenomenon during the image denoising. Furthermore, small details, texture and fine structures still maintained in the restored image. Finally, we compare our numerical results with the existing models in the literature.

Keywords: Image denoising, image decomposition, texture, TV operator, staircase effect, fractional calculus, nonlinear diffusion, exponent variable, Orlicz-space, Faedo-Galerkin method, Peaceman-Rachford ADI-method.

Mathematics Subject Classification: 26A33, 35K57, 94A08, 46E3x.

Citation: Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4963-4998. doi: 10.3934/dcdsb.2020321

References:

[1]	R. Aboulaich, D. Meskine and A. Souissi, New diffusion models in image processing, Comput. Math. Appl., 56 (2008), 874-882. doi: 10.1016/j.camwa.2008.01.017.
[2]	R. A. Adams, Sobolev Spaces, Ac. Press, New york, 1975.
[3]	L. Afraites, A. Atlas, F. Karami and D. Meskine, Some class of parabolic systems applied to image processing, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1671-1687. doi: 10.3934/dcdsb.2016017.
[4]	L. Alvarez, P.-L. Lions and J.-M. Morel, Image selective smoothing and edge detection by nonlinear diffusion. Ⅱ, SIAM J. Numer. Anal., 29 (1992), 845-866. doi: 10.1137/0729052.
[5]	F. Andreu, C. Ballester, V. Caselles and J. M. Mazón, Minimizing total variation flow, Differential Integral Equations, 14 (2001), 321-360.
[6]	N. Aronszajn, Boundary values of functions with finite Dirichlet integral, Techn. Report of Univ. of Kansas, 14 (1955), 77-94.
[7]	G. Aubert and J.-F. Aujol, Modeling very oscillating signals. Application to image processing, Appl. Math. Optim., 51 (2005), 163-182. doi: 10.1007/s00245-004-0812-z.
[8]	J.-F. Aujol, G. Aubert, L. Blanc-Féraud and A. Chambolle, Image decomposition into a bounded variation component and an oscillating component, J. Math. Imaging Vision, 22 (2005), 71-88. doi: 10.1007/s10851-005-4783-8.
[9]	A. Buades, B. Coll and J. M. Morel, A non-local algorithm for image denoising, IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2 (2005), 60-65.
[10]	A. Chambolle, R. A. DeVore, N.-Y. Lee and B. J. Lucier, Nonlinear wavelet image processing: Variational problems, compression, and noise removal through wavelet shrinkage, IEEE Trans. Image Process., 7 (1998), 319–335. doi: 10.1109/83.661182.
[11]	E. A. Coddington and N. Levinson, Theory of ordinary differential equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, (1955).
[12]	E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.
[13]	K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, Springer-Verlag, Berlin, (2010). doi: 10.1007/978-3-642-14574-2.
[14]	S. Dipierro, X. Ros-Oton and E. Valdinoci, Nonlocal problems with Neumann boundary conditions, Rev. Mat. Iberoam., 33 (2017), 377-416. doi: 10.4171/RMI/942.
[15]	D. L. Donoho, De-noising by soft-thresholding, IEEE Transactions on Information Theory, 41 (1995), 613-627. doi: 10.1109/18.382009.
[16]	A. Elmahi and D. Meskine, Parabolic equations in Orlicz spaces, J. London Math. Soc. (2), 72 (2005), 410-428. doi: 10.1112/S0024610705006630.
[17]	A. Elmoataz, X. Desquesnes and O. Lézoray, Non-Local Morphological PDEs and $p$-Laplacian Equation on Graphs With Applications in Image Processing and Machine Learning, IEEE Journal of Selected Topics in Signal Processing, 6 (2012), 764-779. doi: 10.1109/JSTSP.2012.2216504.
[18]	E. Gagliardo, Ulteriori proprietà di alcune classi di funzioni in più variabili, Ricerche Mat., 8 (1959), 24-51.
[19]	J. B. Garnett, P. W. Jones, T. M. Le and L. A Vese, Modeling oscillatory components with the homogeneous spaces $B\dot MO^{-\alpha}$ and $\dot W{}^{-\alpha,p}$, Pure Appl. Math. Q., 7 (2011), 275-318. doi: 10.4310/PAMQ.2011.v7.n2.a2.
[20]	J. B. Garnett, T. M. Le, Y. Meyer and L. A Vese, Image decompositions using bounded variation and generalized homogeneous Besov spaces, Appl. Comput. Harmon. Anal., 23 (2007), 25-56. doi: 10.1016/j.acha.2007.01.005.
[21]	Y. Giga, M. Muszkieta and P. Rybka, A duality based approach to the minimizing total variation flow in the space $H^{-s}$, Jpn. J. Ind. Appl. Math., 36 (2019), 261-286. doi: 10.1007/s13160-018-00340-4.
[22]	G. Gilboa and S. Osher, Nonlocal linear image regularization and supervised segmentation, Multiscale Model. Simul., 6 (2007), 595-630. doi: 10.1137/060669358.
[23]	G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028. doi: 10.1137/070698592.
[24]	J.-P. Gossez, Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Trans. Amer. Math. Soc., 190 (1974), 163-205. doi: 10.1090/S0002-9947-1974-0342854-2.
[25]	Z. Guo, J. Yin and Q. Liu, On a reaction-diffusion system applied to image decomposition and restoration, Math. Comput. Modelling, 53 (2011), 1336-1350. doi: 10.1016/j.mcm.2010.12.031.
[26]	Y. Jin, J. Jost and G. Wang, A new nonlocal variational setting for image processing, Inverse Probl. Imaging, 9 (2015), 415-430. doi: 10.3934/ipi.2015.9.415.
[27]	Y. Kim and L. A. Vese, Image recovery using functions of bounded variation and Sobolev spaces of negative differentiability, Inverse Probl. Imaging, 3 (2009), 43-68. doi: 10.3934/ipi.2009.3.43.
[28]	S. Kindermann, S. Osher and P. W. Jones, Deblurring and denoising of images by nonlocal functionals, Multiscale Model. Simul., 4 (2005), 1091-1115. doi: 10.1137/050622249.
[29]	T. M. Le and L. A. Vese, Image decomposition using total variation and $ \rm{div} $($ \rm{BMO} $), Multiscale Model. Simul., 4 (2005), 390-423. doi: 10.1137/040610052.
[30]	L. H. Lieu and L. A. Vese, Image restoration and decomposition via bounded total variation and negative Hilbert-Sobolev spaces, Appl. Math. Optim., 58 (2008), 167-193. doi: 10.1007/s00245-008-9047-8.
[31]	X. Liu and L. Huang, A new nonlocal total variation regularization algorithm for image denoising, Math. Comput. Simulation, 97 (2014), 224-233. doi: 10.1016/j.matcom.2013.10.001.
[32]	Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, American Mathematical Society, Providence, RI, (2001). doi: 10.1090/ulect/022.
[33]	K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, Inc., New York, (1993).
[34]	S. Osher, A. Solé and L. Vese, Image decomposition and restoration using total variation minimization and the $H^{-1}$ norm, Multiscale Model. Simul., 1 (2003), 349-370. doi: 10.1137/S1540345902416247.
[35]	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.
[36]	L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D., 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F.
[37]	L. N. Slobodeckiĭ, Generalized Sobolev spaces and their application to boundary problems for partial differential equations, Leningrad. Gos. Ped. Inst. Učen. Zap., 197 (1958), 54-112.
[38]	N. N. Tarkhanov, The Analysis of Solutions of Elliptic Equations, Kluwer Academic Publishers Group, Dordrecht, (1997). doi: 10.1007/978-94-015-8804-1.
[39]	L. A. Vese and S. J. Osher, Modeling textures with total variation minimization and oscillating patterns in image processing, J. Sci. Comput., 19 (2003), 553-572. doi: 10.1023/A:1025384832106.
[40]	Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: from error visibility to structural similarity, IEEE Transactions on Image Processing, 13 (2004), 600-612. doi: 10.1109/TIP.2003.819861.
[41]	L. P. Yaroslavsky, Digital Picture Processing, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-642-81929-2.

show all references

References:

[1]	R. Aboulaich, D. Meskine and A. Souissi, New diffusion models in image processing, Comput. Math. Appl., 56 (2008), 874-882. doi: 10.1016/j.camwa.2008.01.017.
[2]	R. A. Adams, Sobolev Spaces, Ac. Press, New york, 1975.
[3]	L. Afraites, A. Atlas, F. Karami and D. Meskine, Some class of parabolic systems applied to image processing, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1671-1687. doi: 10.3934/dcdsb.2016017.
[4]	L. Alvarez, P.-L. Lions and J.-M. Morel, Image selective smoothing and edge detection by nonlinear diffusion. Ⅱ, SIAM J. Numer. Anal., 29 (1992), 845-866. doi: 10.1137/0729052.
[5]	F. Andreu, C. Ballester, V. Caselles and J. M. Mazón, Minimizing total variation flow, Differential Integral Equations, 14 (2001), 321-360.
[6]	N. Aronszajn, Boundary values of functions with finite Dirichlet integral, Techn. Report of Univ. of Kansas, 14 (1955), 77-94.
[7]	G. Aubert and J.-F. Aujol, Modeling very oscillating signals. Application to image processing, Appl. Math. Optim., 51 (2005), 163-182. doi: 10.1007/s00245-004-0812-z.
[8]	J.-F. Aujol, G. Aubert, L. Blanc-Féraud and A. Chambolle, Image decomposition into a bounded variation component and an oscillating component, J. Math. Imaging Vision, 22 (2005), 71-88. doi: 10.1007/s10851-005-4783-8.
[9]	A. Buades, B. Coll and J. M. Morel, A non-local algorithm for image denoising, IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2 (2005), 60-65.
[10]	A. Chambolle, R. A. DeVore, N.-Y. Lee and B. J. Lucier, Nonlinear wavelet image processing: Variational problems, compression, and noise removal through wavelet shrinkage, IEEE Trans. Image Process., 7 (1998), 319–335. doi: 10.1109/83.661182.
[11]	E. A. Coddington and N. Levinson, Theory of ordinary differential equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, (1955).
[12]	E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.
[13]	K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, Springer-Verlag, Berlin, (2010). doi: 10.1007/978-3-642-14574-2.
[14]	S. Dipierro, X. Ros-Oton and E. Valdinoci, Nonlocal problems with Neumann boundary conditions, Rev. Mat. Iberoam., 33 (2017), 377-416. doi: 10.4171/RMI/942.
[15]	D. L. Donoho, De-noising by soft-thresholding, IEEE Transactions on Information Theory, 41 (1995), 613-627. doi: 10.1109/18.382009.
[16]	A. Elmahi and D. Meskine, Parabolic equations in Orlicz spaces, J. London Math. Soc. (2), 72 (2005), 410-428. doi: 10.1112/S0024610705006630.
[17]	A. Elmoataz, X. Desquesnes and O. Lézoray, Non-Local Morphological PDEs and $p$-Laplacian Equation on Graphs With Applications in Image Processing and Machine Learning, IEEE Journal of Selected Topics in Signal Processing, 6 (2012), 764-779. doi: 10.1109/JSTSP.2012.2216504.
[18]	E. Gagliardo, Ulteriori proprietà di alcune classi di funzioni in più variabili, Ricerche Mat., 8 (1959), 24-51.
[19]	J. B. Garnett, P. W. Jones, T. M. Le and L. A Vese, Modeling oscillatory components with the homogeneous spaces $B\dot MO^{-\alpha}$ and $\dot W{}^{-\alpha,p}$, Pure Appl. Math. Q., 7 (2011), 275-318. doi: 10.4310/PAMQ.2011.v7.n2.a2.
[20]	J. B. Garnett, T. M. Le, Y. Meyer and L. A Vese, Image decompositions using bounded variation and generalized homogeneous Besov spaces, Appl. Comput. Harmon. Anal., 23 (2007), 25-56. doi: 10.1016/j.acha.2007.01.005.
[21]	Y. Giga, M. Muszkieta and P. Rybka, A duality based approach to the minimizing total variation flow in the space $H^{-s}$, Jpn. J. Ind. Appl. Math., 36 (2019), 261-286. doi: 10.1007/s13160-018-00340-4.
[22]	G. Gilboa and S. Osher, Nonlocal linear image regularization and supervised segmentation, Multiscale Model. Simul., 6 (2007), 595-630. doi: 10.1137/060669358.
[23]	G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028. doi: 10.1137/070698592.
[24]	J.-P. Gossez, Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Trans. Amer. Math. Soc., 190 (1974), 163-205. doi: 10.1090/S0002-9947-1974-0342854-2.
[25]	Z. Guo, J. Yin and Q. Liu, On a reaction-diffusion system applied to image decomposition and restoration, Math. Comput. Modelling, 53 (2011), 1336-1350. doi: 10.1016/j.mcm.2010.12.031.
[26]	Y. Jin, J. Jost and G. Wang, A new nonlocal variational setting for image processing, Inverse Probl. Imaging, 9 (2015), 415-430. doi: 10.3934/ipi.2015.9.415.
[27]	Y. Kim and L. A. Vese, Image recovery using functions of bounded variation and Sobolev spaces of negative differentiability, Inverse Probl. Imaging, 3 (2009), 43-68. doi: 10.3934/ipi.2009.3.43.
[28]	S. Kindermann, S. Osher and P. W. Jones, Deblurring and denoising of images by nonlocal functionals, Multiscale Model. Simul., 4 (2005), 1091-1115. doi: 10.1137/050622249.
[29]	T. M. Le and L. A. Vese, Image decomposition using total variation and $ \rm{div} $($ \rm{BMO} $), Multiscale Model. Simul., 4 (2005), 390-423. doi: 10.1137/040610052.
[30]	L. H. Lieu and L. A. Vese, Image restoration and decomposition via bounded total variation and negative Hilbert-Sobolev spaces, Appl. Math. Optim., 58 (2008), 167-193. doi: 10.1007/s00245-008-9047-8.
[31]	X. Liu and L. Huang, A new nonlocal total variation regularization algorithm for image denoising, Math. Comput. Simulation, 97 (2014), 224-233. doi: 10.1016/j.matcom.2013.10.001.
[32]	Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, American Mathematical Society, Providence, RI, (2001). doi: 10.1090/ulect/022.
[33]	K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, Inc., New York, (1993).
[34]	S. Osher, A. Solé and L. Vese, Image decomposition and restoration using total variation minimization and the $H^{-1}$ norm, Multiscale Model. Simul., 1 (2003), 349-370. doi: 10.1137/S1540345902416247.
[35]	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.
[36]	L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D., 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F.
[37]	L. N. Slobodeckiĭ, Generalized Sobolev spaces and their application to boundary problems for partial differential equations, Leningrad. Gos. Ped. Inst. Učen. Zap., 197 (1958), 54-112.
[38]	N. N. Tarkhanov, The Analysis of Solutions of Elliptic Equations, Kluwer Academic Publishers Group, Dordrecht, (1997). doi: 10.1007/978-94-015-8804-1.
[39]	L. A. Vese and S. J. Osher, Modeling textures with total variation minimization and oscillating patterns in image processing, J. Sci. Comput., 19 (2003), 553-572. doi: 10.1023/A:1025384832106.
[40]	Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: from error visibility to structural similarity, IEEE Transactions on Image Processing, 13 (2004), 600-612. doi: 10.1109/TIP.2003.819861.
[41]	L. P. Yaroslavsky, Digital Picture Processing, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-642-81929-2.

Figure 1. Comparison between the regularization terms of the function $ f(x) = 1/|x| $ associated to the TV operator, which is weak in the numerical simulations, (the lines associated to this function are illustrated by the red curves in both figures: case $ \epsilon = 0 $ and case $ \gamma = 0 $). The first figure illustrates the curves obtained by the function $ f(x) = 1/\sqrt{|x|^{2}+\epsilon} $ and by different values of $ \epsilon $. The second figure illustrates the curves obtained by $ f(x) = ( \mathop{{\rm{log}}}(1+|x|))^{\gamma}/|x| $ and by different values of $ \gamma $. It is clear that the previously proposed converges rapidly to the regularization term associated with the TV operator from $ \gamma = 10^{-1} $. On the other hand, the other function converges from $ \epsilon = 10^{-4} $

Noise level	$\underline{Fig.3(A)}$	$\underline{Fig.3(D)}$	$\underline{Fig.3(B)}$	$\underline{Fig.3(E)}$	$\underline{Fig.3(C)}$	$\underline{Fig.3(F)}$
$ \sigma=10 $	$\mbox{Noisy}$	$\mbox{Restored}$	$\mbox{Noisy}$	$\mbox{Restored}$	$\mbox{Noisy}$	$\mbox{Restored}$
$\mbox{SNR}$	12.54	20.23	13.57	14.84	12.66	15.11
$\mbox{PSNR}$	28.14	35.82	27.22	29.16	28.17	30.60
$\mbox{SSIM}$	0.782	0.971	0.865	0.933	0.590	0.949

$\mbox{Noise level}$	$\underline{Fig.4(A)}$	$\underline{Fig.4(D)}$	$\underline{Fig.4(B)}$	$\underline{Fig.4(E)}$	$\underline{Fig.4(C)}$	$\underline{Fig.4(F)}$
$ \sigma=20 $	$\mbox{Noisy}$	$\mbox{Restored}$	$\mbox{Noisy}$	$\mbox{Restored}$	$\mbox{Noisy}$	$\mbox{Restored}$
$\mbox{SNR}$	06.50	17.49	09.04	12.10	06.62	13.41
$\mbox{PSNR}$	22.10	33.09	22.02	25.38	22.13	28.90
$\mbox{SSIM}$	0.580	0.925	0.743	0.896	0.422	0.837

$\mbox{Noise level}$	$\underline{Fig.5(A)}$	$\underline{Fig.5(D)}$	$\underline{Fig.5(B)}$	$\underline{Fig.5(E)}$	$\underline{Fig.5(C)}$	$\underline{Fig.5(F)}$
$ \sigma=30$	$\mbox{Noisy}$	$\mbox{Restored}$	$\mbox{Noisy}$	$\mbox{Restored}$	$\mbox{Noisy}$	$\mbox{Restored}$
$\mbox{SNR}$	03.01	16.12	05.51	10.07	03.06	12.44
$\mbox{PSNR}$	18.58	31.73	18.56	23.39	18.54	27.92
$\mbox{SSIM}$	0.501	0.884	0.650	0.885	0.325	0.780

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A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition

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