Figure 1.
The angle $ d\theta $ and the grid points
-
Previous Article
Multiplicity of radial and nonradial solutions to equations with fractional operators
- CPAA Home
- This Issue
-
Next Article
The Lazer-McKenna conjecture for an anisotropic planar elliptic problem with exponential Neumann data
July
2020, 19(7): 3477-3500.
doi: 10.3934/cpaa.2020152
Generalizations of $ p $-Laplace operator for image enhancement: Part 2
| 1. | Linköping University, Sweden |
| 2. | Malmö University, Sweden |
| 3. | Heidelberg University, Germany |
We have in a previous study introduced a novel elliptic operator $ \Delta_{(p, q)} u = |\nabla u|^q\Delta_1 u +(p-1)|\nabla u|^{p-2} \Delta_{\infty} u $, $ p \ge 1 $, $ q\ge 0, $ as a generalization of the $ p $-Laplace operator. In this paper, we establish the well-posedness of the parabolic equation $ u_{t} = |\nabla u|^{1-q}\Delta_{(1+q, q)}, $ where $ q = q(|\nabla u|) $ is continuous and has range in $ [0, 1], $ in the framework of viscosity solutions. We prove the consistency and convergence of the numerical scheme of finite differences of this parabolic equation. Numerical simulations shows the advantage of this operator applied to image enhancement.
Keywords:
p-Laplace operator,
parabolic equations,
viscosity solutions,
image denoising,
inpainting,
Perona-Malik equations,
inverse problems.
Mathematics Subject Classification:
Primary: 35K55; Secondary: 35J75.
Citation:
George Baravdish, Yuanji Cheng, Olof Svensson, Freddie Åström. Generalizations of $ p $-Laplace operator for image enhancement: Part 2. Communications on Pure and Applied Analysis,
2020, 19
(7)
: 3477-3500.
doi: 10.3934/cpaa.2020152
![]()
References:
| [1] |
G. Baravdish, Y. Cheng, O. Svensson and F. Åström, Extension of $p$-Laplace operator for image denoising, in IFIP Conference on System Modeling and Optimization, Springer, (2015), 107–116. |
| [2] |
G. Barles,
Nonlinear Neumann boundary conditions for quasilinear degenerate elliptic equations and applications, J. Differ. Equ., 154 (1999), 191-224.
doi: 10.1006/jdeq.1998.3568. |
| [3] |
G. Barles and P. E. Souganidis,
Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal., 4 (1991), 271-283.
|
| [4] |
M. G. Crandall, H. Ishii and P. L. Lions,
User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.
doi: 10.1090/S0273-0979-1992-00266-5. |
| [5] |
K. Does, An Evolution Equation Involping the Normalized p-Laplacian, Ph.D thesis, Universität zu Köln, 2009.
doi: 10.3934/cpaa.2011.10.361. |
| [6] |
K. Does,
An evolution equation involving the normalized $p$-laplacian, Commun. Pure Appl. Math., 10 (2011), 361-396.
doi: 10.3934/cpaa.2011.10.361. |
| [7] |
P. Dupuis and H. Ishii,
On oblique derivative problems for fully nonlinear second-order elliptic pde's on domains with corners, Hokkaido Math. J., 20 (1991), 135-164.
doi: 10.14492/hokmj/1381413798. |
| [8] |
Y. Giga, Surface evolution equations, Monographs in Mathematics, vol. 99, Birkhäuser Verlag, Basel, 2006. |
| [9] |
H. Ishii and P. L. Lions,
Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Differ. Equ., 83 (1990), 26-78.
doi: 10.1016/0022-0396(90)90068-Z. |
| [10] |
B. Kawohl, Variational versus pde-based approaches in mathematical image processing, in CRM Proceedings and Lecture Notes, vol. 44, (2008), 113–126. |
| [11] |
A. Kuijper,
Geometrical pdes based on second-order derivatives of gauge coordinates in image processing, Image Vision Comput., 27 (2009), 1023-1034.
doi: 10.1016/j.imavis.2008.09.003. |
| [12] |
G. Lieberman, Second Order Parabolic Differential Equations, World Scientific, 1996.
doi: 10.1142/3302. |
| [13] |
A. M. Oberman,
Finite difference methods for the infinity Laplace and $p$-Laplace equations, J. Comput. Appl. Math., 254 (2013), 65-80.
doi: 10.1016/j.cam.2012.11.023. |
show all references
References:
| [1] |
G. Baravdish, Y. Cheng, O. Svensson and F. Åström, Extension of $p$-Laplace operator for image denoising, in IFIP Conference on System Modeling and Optimization, Springer, (2015), 107–116. |
| [2] |
G. Barles,
Nonlinear Neumann boundary conditions for quasilinear degenerate elliptic equations and applications, J. Differ. Equ., 154 (1999), 191-224.
doi: 10.1006/jdeq.1998.3568. |
| [3] |
G. Barles and P. E. Souganidis,
Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal., 4 (1991), 271-283.
|
| [4] |
M. G. Crandall, H. Ishii and P. L. Lions,
User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.
doi: 10.1090/S0273-0979-1992-00266-5. |
| [5] |
K. Does, An Evolution Equation Involping the Normalized p-Laplacian, Ph.D thesis, Universität zu Köln, 2009.
doi: 10.3934/cpaa.2011.10.361. |
| [6] |
K. Does,
An evolution equation involving the normalized $p$-laplacian, Commun. Pure Appl. Math., 10 (2011), 361-396.
doi: 10.3934/cpaa.2011.10.361. |
| [7] |
P. Dupuis and H. Ishii,
On oblique derivative problems for fully nonlinear second-order elliptic pde's on domains with corners, Hokkaido Math. J., 20 (1991), 135-164.
doi: 10.14492/hokmj/1381413798. |
| [8] |
Y. Giga, Surface evolution equations, Monographs in Mathematics, vol. 99, Birkhäuser Verlag, Basel, 2006. |
| [9] |
H. Ishii and P. L. Lions,
Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Differ. Equ., 83 (1990), 26-78.
doi: 10.1016/0022-0396(90)90068-Z. |
| [10] |
B. Kawohl, Variational versus pde-based approaches in mathematical image processing, in CRM Proceedings and Lecture Notes, vol. 44, (2008), 113–126. |
| [11] |
A. Kuijper,
Geometrical pdes based on second-order derivatives of gauge coordinates in image processing, Image Vision Comput., 27 (2009), 1023-1034.
doi: 10.1016/j.imavis.2008.09.003. |
| [12] |
G. Lieberman, Second Order Parabolic Differential Equations, World Scientific, 1996.
doi: 10.1142/3302. |
| [13] |
A. M. Oberman,
Finite difference methods for the infinity Laplace and $p$-Laplace equations, J. Comput. Appl. Math., 254 (2013), 65-80.
doi: 10.1016/j.cam.2012.11.023. |
Figure 2.
Removal of Gaussian noise with standard deviation 20 in picewise affine image regions poses particular problems for total variation based models as staircasing effects commonly occur in these regions. Increasing the $ q $ value towards 1, the model resembles isotropic diffusion with a smooth symmetric kernel, and consequently blurs image edges. The adaptive selection of $ q $ , shown in the upper right panel, illustrates the trade-off between edge retention and smoothing of affine regions. The adaptive approach is perceived as producing a visually similar results to $ q = 0 $ . The error values reflect the best PSNR value for each method. (In case of reference to color we refer to the online version.)
Figure 3.
A comparison of denoising results for different $ q $ -values and the adaptive strategy is shown. All results reflects the best PSNR value of each method and, as seen, the proposed adaptive method shows improvement over using some fixed $ q $ -values
Figure 4.
In the first row, the missing values (to be inpainted) are the diagonal lines, in the second row it is the circles and in the third row it is the pattern. In addition, the image is corrupted by normally distributed Gaussian noise with standard deviation 10. All comparison of inpainting results reflect the PSNR value of each method after 2000 iterations. For each of these patterns, we set $ \lambda_{1} = 0.05 $ , $ \lambda_2 = 0.1 $ , $ \tau = 0.2 $ and $ k_1 = k_2 = 0.1 $ in $ q(s) $ . As the figure shows, the proposed method performs improvement over some fixed $ q $ -values. Gaussian smoothing, i.e., $ q = 1 $ , produces oversmoothed results for each test pattern. The case $ q = 0 $ and the adaptive strategy show similar performance. Although, adaptively controlled $ q $ gives better PSNR values
Figure 5.
In comparison of inpainting, the adaptive strategy gives better PSNR value than some fixed $ q $ -values. As expected, $ q = 1 $ blurs image edges and produces a suboptimal result
| [1] |
Yan Chen, Kewei Zhang. Young measure solutions of the two-dimensional Perona-Malik equation in image processing. Communications on Pure and Applied Analysis, 2006, 5 (3) : 617-637. doi: 10.3934/cpaa.2006.5.617 |
| [2] |
Antonio Greco, Giovanni Porru. Optimization problems for the energy integral of p-Laplace equations. Conference Publications, 2013, 2013 (special) : 301-310. doi: 10.3934/proc.2013.2013.301 |
| [3] |
Ryuji Kajikiya. Nonradial least energy solutions of the p-Laplace elliptic equations. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 547-561. doi: 10.3934/dcds.2018024 |
| [4] |
Mikhail D. Surnachev, Vasily V. Zhikov. On existence and uniqueness classes for the Cauchy problem for parabolic equations of the p-Laplace type. Communications on Pure and Applied Analysis, 2013, 12 (4) : 1783-1812. doi: 10.3934/cpaa.2013.12.1783 |
| [5] |
Patrick Guidotti. A family of nonlinear diffusions connecting Perona-Malik to standard diffusion. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 581-590. doi: 10.3934/dcdss.2012.5.581 |
| [6] |
Arrigo Cellina. The regularity of solutions to some variational problems, including the p-Laplace equation for 3≤p < 4. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 4071-4085. doi: 10.3934/dcds.2018177 |
| [7] |
Yangrong Li, Jinyan Yin. Existence, regularity and approximation of global attractors for weakly dissipative p-Laplace equations. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1939-1957. doi: 10.3934/dcdss.2016079 |
| [8] |
Li Song, Yangrong Li, Fengling Wang. Controller and asymptotic autonomy of random attractors for stochastic p-Laplace lattice equations. Evolution Equations and Control Theory, 2022 doi: 10.3934/eect.2022010 |
| [9] |
Joachim Naumann. On the existence of weak solutions of an unsteady p-Laplace thermistor system with strictly monotone electrical conductivities. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 837-852. doi: 10.3934/dcdss.2017042 |
| [10] |
Zheng Zhou. Layered solutions in $R^2$ for a class of $p$-Laplace equations. Communications on Pure and Applied Analysis, 2010, 9 (3) : 819-837. doi: 10.3934/cpaa.2010.9.819 |
| [11] |
Gang Li, Fen Gu, Feida Jiang. Positive viscosity solutions of a third degree homogeneous parabolic infinity Laplace equation. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1449-1462. doi: 10.3934/cpaa.2020071 |
| [12] |
Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Inverse problems for evolution equations with time dependent operator-coefficients. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 737-744. doi: 10.3934/dcdss.2016025 |
| [13] |
Manas Kar, Jenn-Nan Wang. Size estimates for the weighted p-Laplace equation with one measurement. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 2011-2024. doi: 10.3934/dcdsb.2020188 |
| [14] |
Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure and Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395 |
| [15] |
Patrizia Pucci, Raffaella Servadei. Nonexistence for $p$--Laplace equations with singular weights. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1421-1438. doi: 10.3934/cpaa.2010.9.1421 |
| [16] |
Kristian Bredies. Weak solutions of linear degenerate parabolic equations and an application in image processing. Communications on Pure and Applied Analysis, 2009, 8 (4) : 1203-1229. doi: 10.3934/cpaa.2009.8.1203 |
| [17] |
Guoqing Zhang, Jia-yu Shao, Sanyang Liu. Linking solutions for N-laplace elliptic equations with Hardy-Sobolev operator and indefinite weights. Communications on Pure and Applied Analysis, 2011, 10 (2) : 571-581. doi: 10.3934/cpaa.2011.10.571 |
| [18] |
Filippo Gazzola. On the moments of solutions to linear parabolic equations involving the biharmonic operator. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3583-3597. doi: 10.3934/dcds.2013.33.3583 |
| [19] |
Eun Bee Choi, Yun-Ho Kim. Existence of nontrivial solutions for equations of $p(x)$-Laplace type without Ambrosetti and Rabinowitz condition. Conference Publications, 2015, 2015 (special) : 276-286. doi: 10.3934/proc.2015.0276 |
| [20] |
Mohammed Al Horani, Angelo Favini. Inverse problems for singular differential-operator equations with higher order polar singularities. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2159-2168. doi: 10.3934/dcdsb.2014.19.2159 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]