American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021254
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 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 & 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 [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

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
(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] 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 [2] 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 & Continuous Dynamical Systems - B, 2021, 26 (10) : 5681-5705. doi: 10.3934/dcdsb.2020376 [3] 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 & Applied Analysis, 2019, 18 (4) : 2009-2045. doi: 10.3934/cpaa.2019091 [4] 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 & Related Fields, 2020, 10 (4) : 827-854. doi: 10.3934/mcrf.2020021 [5] Yupeng Li, Wuchen Li, Guo Cao. Image segmentation via $L_1$ Monge-Kantorovich problem. Inverse Problems & Imaging, 2019, 13 (4) : 805-826. doi: 10.3934/ipi.2019037 [6] Phuong Le. Symmetry of singular solutions for a weighted Choquard equation involving the fractional $p$-Laplacian. Communications on Pure & Applied Analysis, 2020, 19 (1) : 527-539. doi: 10.3934/cpaa.2020026 [7] Zaizheng Li, Qidi Zhang. Sub-solutions and a point-wise Hopf's lemma for fractional $p$-Laplacian. Communications on Pure & Applied Analysis, 2021, 20 (2) : 835-865. doi: 10.3934/cpaa.2020293 [8] Elhoussine Azroul, Abdelmoujib Benkirane, and Mohammed Shimi. On a nonlocal problem involving the fractional $p(x,.)$-Laplacian satisfying Cerami condition. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3479-3495. doi: 10.3934/dcdss.2020425 [9] Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $p$-Laplacian. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3851-3863. doi: 10.3934/dcdss.2020445 [10] Changchun Liu, Pingping Li. Global existence for a chemotaxis-haptotaxis model with $p$-Laplacian. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1399-1419. doi: 10.3934/cpaa.2020070 [11] 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 & Applied Analysis, 2018, 17 (4) : 1449-1478. doi: 10.3934/cpaa.2018070 [12] Nikolay Dimitrov, Stepan Tersian. Existence of homoclinic solutions for a nonlinear fourth order $p$-Laplacian difference equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 555-567. doi: 10.3934/dcdsb.2019254 [13] 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 [14] Yong Zhou, Jia Wei He. New results on controllability of fractional evolution systems with order $\alpha\in (1,2)$. Evolution Equations & Control Theory, 2021, 10 (3) : 491-509. doi: 10.3934/eect.2020077 [15] Mei Yu, Xia Zhang, Binlin Zhang. Property of solutions for elliptic equation involving the higher-order fractional Laplacian in $\mathbb{R}^n_+$. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3597-3612. doi: 10.3934/cpaa.2020157 [16] Jinguo Zhang, Dengyun Yang. Fractional $p$-sub-Laplacian operator problem with concave-convex nonlinearities on homogeneous groups. Electronic Research Archive, 2021, 29 (5) : 3243-3260. doi: 10.3934/era.2021036 [17] Ekta Mittal, Sunil Joshi. Note on a $k$-generalised fractional derivative. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 797-804. doi: 10.3934/dcdss.2020045 [18] Mihai Mihăilescu, Julio D. Rossi. Monotonicity with respect to $p$ of the First Nontrivial Eigenvalue of the $p$-Laplacian with Homogeneous Neumann Boundary Conditions. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4363-4371. doi: 10.3934/cpaa.2020198 [19] Mohan Mallick, R. Shivaji, Byungjae Son, S. Sundar. Bifurcation and multiplicity results for a class of $n\times n$ $p$-Laplacian system. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1295-1304. doi: 10.3934/cpaa.2018062 [20] Jong Yoon Hyun, Yoonjin Lee, Yansheng Wu. Connection of $p$-ary $t$-weight linear codes to Ramanujan Cayley graphs with $t+1$ eigenvalues. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020133

2020 Impact Factor: 1.327

Tools

Article outline

Figures and Tables