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doi: 10.3934/dcdsb.2021254
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Fractional $ 1 $-Laplacian evolution equations to remove multiplicative noise

Tianling Gao 1, , Qiang Liu 1,, and  Zhiguang Zhang 2,

1. 

College of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, China
2. 

College of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing 404100, China

* Corresponding author: Qiang Liu

Received  May 2021 Revised  August 2021 Early access October 2021

Fund Project: The first author is supported by NSF grant of Guandong(2018A030310454), The second author is supported by NSF grant of Guandong(2020A1515010554), The third author is supported by Young Foundation of Three Gorges University(19QN09)

In this paper, we propose a new image denosing model to remove the multiplicative noise by a maximum a posteriori estimation and an inhomogeneous fractional $ 1 $-Laplace evolution equation. The main difficulty of the problem is the equation will become very singular when $ u(x) = u(y) $. The existence and uniqueness of the weak positive solution are proved. Numerical examples demonstrate the better capability of our model on some heavy multiplicative noised images.

Keywords: Fractional $ 1 $-Laplacian, existence and uniqueness, multiplicative noise, image restoration.
Mathematics Subject Classification: Primary: 35R11, 35K65; Secondary: 68U10.
Citation: Tianling Gao, Qiang Liu, Zhiguang Zhang. Fractional $ 1 $-Laplacian evolution equations to remove multiplicative noise. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021254
References:
[1]

B. Abdellaoui, A. Attar, R. Bentifour and I. Peral, On fractional $p$-Laplacian parabolic problem with general data, Ann. Mat. Pura Appl., 197 (2018), 329-356.  doi: 10.1007/s10231-017-0682-z.  Google Scholar

[2]

F. Andreu, J. M. Mzaón, J. D. Rossi and J. Toledo, A nonlocal $p$-Laplacian evolution equation with Neumann boundary conditions, J. Math. Pures Appl., 90 (2008), 201-227.  doi: 10.1016/j.matpur.2008.04.003.  Google Scholar

[3]

G. Aubert and J.-F. Aujol, A variational approach to removing multiplicative noise, SIAM J. Appl. Math., 68 (2008), 925-946.  doi: 10.1137/060671814.  Google Scholar

[4]

L. Brasco, E. Lindgren and E. Parini, The fractional Cheeger problem, Interfaces Free Bound., 16 (2004), 419-458.  doi: 10.4171/IFB/325.  Google Scholar

[5]

A. Buades, B. Coll and J. M. Morel, A review of image denoising algorithms, with a new one, Multiscale Model. Simul., 4 (2005), 490-530.  doi: 10.1137/040616024.  Google Scholar

[6]

B. Chen, J.-L. Cai, W.-S. Chen and Y. Li, A multiplicative noise removal approach based on partial differential equation model, Math. Probl. Eng., 2012 (2012), Art. ID 242043, 14 pp. doi: 10.1155/2012/242043.  Google Scholar

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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.  Google Scholar

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F. Dong, H. Zhang and D.-X. Kong, Nonlocal total variation models for multiplicative noise removal using split Bregman iteration, Math. Comput. Modelling, 55 (2012), 939-954.  doi: 10.1016/j.mcm.2011.09.021.  Google Scholar

[9]

W. Feng, H. Lei and Y. Gao, Speckle reduction via higher order total variation approach, IEEE Trans. Image Process., 23 (2014), 1831-1843.  doi: 10.1109/TIP.2014.2308432.  Google Scholar

[10]

G. Gilboa and S. Osher, Nonlocal linear image regularization and supervised segmentation, Multiscale Model. Simul., 6 (2007), 595-630.  doi: 10.1137/060669358.  Google Scholar

[11]

G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028.  doi: 10.1137/070698592.  Google Scholar

[12]

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.  Google Scholar

[13]

Y.-M. Huang, M. K. Ng and Y.-W. Wen, A new total variation method for multiplicative noise removal, SIAM J. Imaging Sci., 2 (2009), 20-40.  doi: 10.1137/080712593.  Google Scholar

[14]

Z. Jin and X. Yang, Analysis of a new variational model for multiplicative noise removal, J. Math. Anal. Appl., 362 (2010), 415-426.  doi: 10.1016/j.jmaa.2009.08.036.  Google Scholar

[15]

F. Li, M. K. Ng and C. Shen, Multiplicative noise removal with spatially varying regularization parameters, SIAM J. Imaging Sci., 3 (2010), 1-20.  doi: 10.1137/090748421.  Google Scholar

[16]

Q. Liu, X. Li and T. Gao, A nondivergence $p$-Laplace equation in a removing multiplicative noise model, Nonlinear Anal. RWA, 14 (2013), 2046-2058.  doi: 10.1016/j.nonrwa.2013.02.008.  Google Scholar

[17]

J. M. Mazón, J. D. Rossi and J. Toledo, Fractional $p$-Laplacian evolution equations, J. Math. Pures Appl., 105 (2016), 810-844.  doi: 10.1016/j.matpur.2016.02.004.  Google Scholar

[18]

D. Puhst, On the evolutionary fractional $p$-laplacian, Appl. Math. Res. Express., 2015 (2015), 253-273.  doi: 10.1093/amrx/abv003.  Google Scholar

[19]

L. Rudin, P.-L. Lions and S. Osher, Multiplicative denoising and deblurring: Theory and algorithms, in Geometric Level Set Methods in Imaging, Vision, and Graphics, S. Osher and N. Paragios, eds., Springer, New York, (2003) 103–119. doi: 10.1007/0-387-21810-6_6.  Google Scholar

[20]

L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D: Nonlinear Phenomena, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[21]

S. Segura de León and and C. M. Webler, Global existence and uniqueness for the inhomogeneous $1$-Laplace evolution equation, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1213-1246.  doi: 10.1007/s00030-015-0320-7.  Google Scholar

[22]

J. Shi and S. Osher, A nonlinear inverse scale space method for a convex multiplicative noise models, SIAM J. Img. Sci., 1 (2008), 294-321.  doi: 10.1137/070689954.  Google Scholar

[23]

J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[24]

J. Sun, J. Li and Q. Liu, Cauchy problem of a nonlocal $p$-Laplacian evolution equation with nonlocal convection, Nonlinear Anal. TMA, 95 (2014), 691-702.  doi: 10.1016/j.na.2013.09.023.  Google Scholar

[25]

J. L. Vazquez, The Dirichlet problem for the fractional $p$-Laplacian evolution equation, J. Differential Equations, 260 (2016), 6038-6056.  doi: 10.1016/j.jde.2015.12.033.  Google Scholar

[26]

Z. Zhou, Z. Guo, G. Dong, J. Sun, D. Zhang and B. Wu, A doubly degenerate diffusion model based on the gray level indicator for multiplicative noise removal, IEEE Trans. Image Process., 24 (2015), 249-260.  doi: 10.1109/TIP.2014.2376185.  Google Scholar

[27]

Z. Zhou, Z. Guo and B. Y. Wu, A doubly degenerate diffusion equation in multiplicative noise removal models, J. Math. Anal. Appl., 458 (2018), 58-70.  doi: 10.1016/j.jmaa.2017.08.049.  Google Scholar

show all references

References:
[1]

B. Abdellaoui, A. Attar, R. Bentifour and I. Peral, On fractional $p$-Laplacian parabolic problem with general data, Ann. Mat. Pura Appl., 197 (2018), 329-356.  doi: 10.1007/s10231-017-0682-z.  Google Scholar

[2]

F. Andreu, J. M. Mzaón, J. D. Rossi and J. Toledo, A nonlocal $p$-Laplacian evolution equation with Neumann boundary conditions, J. Math. Pures Appl., 90 (2008), 201-227.  doi: 10.1016/j.matpur.2008.04.003.  Google Scholar

[3]

G. Aubert and J.-F. Aujol, A variational approach to removing multiplicative noise, SIAM J. Appl. Math., 68 (2008), 925-946.  doi: 10.1137/060671814.  Google Scholar

[4]

L. Brasco, E. Lindgren and E. Parini, The fractional Cheeger problem, Interfaces Free Bound., 16 (2004), 419-458.  doi: 10.4171/IFB/325.  Google Scholar

[5]

A. Buades, B. Coll and J. M. Morel, A review of image denoising algorithms, with a new one, Multiscale Model. Simul., 4 (2005), 490-530.  doi: 10.1137/040616024.  Google Scholar

[6]

B. Chen, J.-L. Cai, W.-S. Chen and Y. Li, A multiplicative noise removal approach based on partial differential equation model, Math. Probl. Eng., 2012 (2012), Art. ID 242043, 14 pp. doi: 10.1155/2012/242043.  Google Scholar

[7]

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.  Google Scholar

[8]

F. Dong, H. Zhang and D.-X. Kong, Nonlocal total variation models for multiplicative noise removal using split Bregman iteration, Math. Comput. Modelling, 55 (2012), 939-954.  doi: 10.1016/j.mcm.2011.09.021.  Google Scholar

[9]

W. Feng, H. Lei and Y. Gao, Speckle reduction via higher order total variation approach, IEEE Trans. Image Process., 23 (2014), 1831-1843.  doi: 10.1109/TIP.2014.2308432.  Google Scholar

[10]

G. Gilboa and S. Osher, Nonlocal linear image regularization and supervised segmentation, Multiscale Model. Simul., 6 (2007), 595-630.  doi: 10.1137/060669358.  Google Scholar

[11]

G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028.  doi: 10.1137/070698592.  Google Scholar

[12]

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.  Google Scholar

[13]

Y.-M. Huang, M. K. Ng and Y.-W. Wen, A new total variation method for multiplicative noise removal, SIAM J. Imaging Sci., 2 (2009), 20-40.  doi: 10.1137/080712593.  Google Scholar

[14]

Z. Jin and X. Yang, Analysis of a new variational model for multiplicative noise removal, J. Math. Anal. Appl., 362 (2010), 415-426.  doi: 10.1016/j.jmaa.2009.08.036.  Google Scholar

[15]

F. Li, M. K. Ng and C. Shen, Multiplicative noise removal with spatially varying regularization parameters, SIAM J. Imaging Sci., 3 (2010), 1-20.  doi: 10.1137/090748421.  Google Scholar

[16]

Q. Liu, X. Li and T. Gao, A nondivergence $p$-Laplace equation in a removing multiplicative noise model, Nonlinear Anal. RWA, 14 (2013), 2046-2058.  doi: 10.1016/j.nonrwa.2013.02.008.  Google Scholar

[17]

J. M. Mazón, J. D. Rossi and J. Toledo, Fractional $p$-Laplacian evolution equations, J. Math. Pures Appl., 105 (2016), 810-844.  doi: 10.1016/j.matpur.2016.02.004.  Google Scholar

[18]

D. Puhst, On the evolutionary fractional $p$-laplacian, Appl. Math. Res. Express., 2015 (2015), 253-273.  doi: 10.1093/amrx/abv003.  Google Scholar

[19]

L. Rudin, P.-L. Lions and S. Osher, Multiplicative denoising and deblurring: Theory and algorithms, in Geometric Level Set Methods in Imaging, Vision, and Graphics, S. Osher and N. Paragios, eds., Springer, New York, (2003) 103–119. doi: 10.1007/0-387-21810-6_6.  Google Scholar

[20]

L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D: Nonlinear Phenomena, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[21]

S. Segura de León and and C. M. Webler, Global existence and uniqueness for the inhomogeneous $1$-Laplace evolution equation, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1213-1246.  doi: 10.1007/s00030-015-0320-7.  Google Scholar

[22]

J. Shi and S. Osher, A nonlinear inverse scale space method for a convex multiplicative noise models, SIAM J. Img. Sci., 1 (2008), 294-321.  doi: 10.1137/070689954.  Google Scholar

[23]

J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[24]

J. Sun, J. Li and Q. Liu, Cauchy problem of a nonlocal $p$-Laplacian evolution equation with nonlocal convection, Nonlinear Anal. TMA, 95 (2014), 691-702.  doi: 10.1016/j.na.2013.09.023.  Google Scholar

[25]

J. L. Vazquez, The Dirichlet problem for the fractional $p$-Laplacian evolution equation, J. Differential Equations, 260 (2016), 6038-6056.  doi: 10.1016/j.jde.2015.12.033.  Google Scholar

[26]

Z. Zhou, Z. Guo, G. Dong, J. Sun, D. Zhang and B. Wu, A doubly degenerate diffusion model based on the gray level indicator for multiplicative noise removal, IEEE Trans. Image Process., 24 (2015), 249-260.  doi: 10.1109/TIP.2014.2376185.  Google Scholar

[27]

Z. Zhou, Z. Guo and B. Y. Wu, A doubly degenerate diffusion equation in multiplicative noise removal models, J. Math. Anal. Appl., 458 (2018), 58-70.  doi: 10.1016/j.jmaa.2017.08.049.  Google Scholar

Figure 1.  (a) Original image. (b) the noisy image with PSNR = 12.49. (c)Image restored by AA model, $ \lambda = 0.01 $. (d) Image restored by our model, $ \lambda = 0.01 $
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Figure 2.  (a) Original image. (b) the noisy image with PSNR = 12.01. (c)Image restored by AA model, $ \lambda = 0.01 $. (d) Image restored by our model, $ \lambda = 0.01 $
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