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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September
2022, 27(9): 4837-4854.
doi: 10.3934/dcdsb.2021254
Fractional $ 1 $-Laplacian evolution equations to remove multiplicative noise
| 1. | College of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, China |
| 2. | College of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing 404100, China |
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 and Continuous Dynamical Systems - B,
2022, 27
(9)
: 4837-4854.
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. |
| [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. |
| [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. |
| [4] |
L. Brasco, E. Lindgren and E. Parini,
The fractional Cheeger problem, Interfaces Free Bound., 16 (2004), 419-458.
doi: 10.4171/IFB/325. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [10] |
G. Gilboa and S. Osher,
Nonlocal linear image regularization and supervised segmentation, Multiscale Model. Simul., 6 (2007), 595-630.
doi: 10.1137/060669358. |
| [11] |
G. Gilboa and S. Osher,
Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028.
doi: 10.1137/070698592. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [18] |
D. Puhst,
On the evolutionary fractional $p$-laplacian, Appl. Math. Res. Express., 2015 (2015), 253-273.
doi: 10.1093/amrx/abv003. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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. |
| [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. |
| [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. |
| [4] |
L. Brasco, E. Lindgren and E. Parini,
The fractional Cheeger problem, Interfaces Free Bound., 16 (2004), 419-458.
doi: 10.4171/IFB/325. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [10] |
G. Gilboa and S. Osher,
Nonlocal linear image regularization and supervised segmentation, Multiscale Model. Simul., 6 (2007), 595-630.
doi: 10.1137/060669358. |
| [11] |
G. Gilboa and S. Osher,
Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028.
doi: 10.1137/070698592. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [18] |
D. Puhst,
On the evolutionary fractional $p$-laplacian, Appl. Math. Res. Express., 2015 (2015), 253-273.
doi: 10.1093/amrx/abv003. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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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