# American Institute of Mathematical Sciences

• Previous Article
Boundedness and asymptotic behavior in a predator-prey model with indirect pursuit-evasion interaction
• DCDS-B Home
• This Issue
• Next Article
The existence of time-dependent attractor for wave equation with fractional damping and lower regular forcing term
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

* Corresponding author: Qiang Liu

Received  May 2021 Revised  August 2021 Published  September 2022 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.

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.
(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, $\lambda = 0.01$
(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, $\lambda = 0.01$
 [1] Niklas Sapountzoglou, Aleksandra Zimmermann. Renormalized solutions for stochastic $p$-Laplace equations with $L^1$-initial data: The case of multiplicative noise. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 3979-4002. doi: 10.3934/dcds.2022041 [2] Yichen Zhang, Meiqiang Feng. A coupled $p$-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075 [3] Kaixuan Zhu, Ji Li, Yongqin Xie, Mingji Zhang. Dynamics of non-autonomous fractional reaction-diffusion equations on $\mathbb{R}^{N}$ driven by multiplicative noise. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5681-5705. doi: 10.3934/dcdsb.2020376 [4] Xiaohui Zhang, Xuping Zhang. Upper semi-continuity of non-autonomous fractional stochastic $p$-Laplacian equation driven by additive noise on $\mathbb{R}^n$. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022081 [5] Claudianor O. Alves, Vincenzo Ambrosio, Teresa Isernia. Existence, multiplicity and concentration for a class of fractional $p \& q$ Laplacian problems in $\mathbb{R} ^{N}$. Communications on Pure and Applied Analysis, 2019, 18 (4) : 2009-2045. doi: 10.3934/cpaa.2019091 [6] Umberto De Maio, Peter I. Kogut, Gabriella Zecca. On optimal $L^1$-control in coefficients for quasi-linear Dirichlet boundary value problems with $BMO$-anisotropic $p$-Laplacian. Mathematical Control and Related Fields, 2020, 10 (4) : 827-854. doi: 10.3934/mcrf.2020021 [7] Yupeng Li, Wuchen Li, Guo Cao. Image segmentation via $L_1$ Monge-Kantorovich problem. Inverse Problems and Imaging, 2019, 13 (4) : 805-826. doi: 10.3934/ipi.2019037 [8] Phuong Le. Symmetry of singular solutions for a weighted Choquard equation involving the fractional $p$-Laplacian. Communications on Pure and Applied Analysis, 2020, 19 (1) : 527-539. doi: 10.3934/cpaa.2020026 [9] Zaizheng Li, Qidi Zhang. Sub-solutions and a point-wise Hopf's lemma for fractional $p$-Laplacian. Communications on Pure and Applied Analysis, 2021, 20 (2) : 835-865. doi: 10.3934/cpaa.2020293 [10] Elhoussine Azroul, Abdelmoujib Benkirane, and Mohammed Shimi. On a nonlocal problem involving the fractional $p(x,.)$-Laplacian satisfying Cerami condition. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3479-3495. doi: 10.3934/dcdss.2020425 [11] Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $p$-Laplacian. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3851-3863. doi: 10.3934/dcdss.2020445 [12] Simone Creo, Maria Rosaria Lancia, Paola Vernole. Transmission problems for the fractional $p$-Laplacian across fractal interfaces. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022047 [13] Pablo Amster, Mariel Paula Kuna, Dionicio Santos. Stability, existence and non-existence of $T$-periodic solutions of nonlinear delayed differential equations with $\varphi$-Laplacian. Communications on Pure and Applied Analysis, 2022, 21 (8) : 2723-2737. doi: 10.3934/cpaa.2022070 [14] Changchun Liu, Pingping Li. Global existence for a chemotaxis-haptotaxis model with $p$-Laplacian. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1399-1419. doi: 10.3934/cpaa.2020070 [15] Nicholas J. Kass, Mohammad A. Rammaha. Local and global existence of solutions to a strongly damped wave equation of the $p$-Laplacian type. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1449-1478. doi: 10.3934/cpaa.2018070 [16] Nikolay Dimitrov, Stepan Tersian. Existence of homoclinic solutions for a nonlinear fourth order $p$-Laplacian difference equation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 555-567. doi: 10.3934/dcdsb.2019254 [17] Jiayi Han, Changchun Liu. Global existence for a two-species chemotaxis-Navier-Stokes system with $p$-Laplacian. Electronic Research Archive, 2021, 29 (5) : 3509-3533. doi: 10.3934/era.2021050 [18] Mohamed Karim Hamdani, Lamine Mbarki, Mostafa Allaoui, Omar Darhouche, Dušan D. Repovš. Existence and multiplicity of solutions involving the $p(x)$-Laplacian equations: On the effect of two nonlocal terms. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022129 [19] Yong Zhou, Jia Wei He. New results on controllability of fractional evolution systems with order $\alpha\in (1,2)$. Evolution Equations and Control Theory, 2021, 10 (3) : 491-509. doi: 10.3934/eect.2020077 [20] Abdolrahman Razani, Giovany M. Figueiredo. A positive solution for an anisotropic $(p,q)$-Laplacian. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022147

2021 Impact Factor: 1.497

## Tools

Article outline

Figures and Tables