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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

* Corresponding author

Received  March 2018 Revised  January 2020 Published  April 2020

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.

Citation: George Baravdish, Yuanji Cheng, Olof Svensson, Freddie Åström. Generalizations of $ p $-Laplace operator for image enhancement: Part 2. Communications on Pure & 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. Google Scholar

[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.  Google Scholar

[3]

G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal., 4 (1991), 271-283.   Google Scholar

[4]

M. G. CrandallH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

Y. Giga, Surface evolution equations, Monographs in Mathematics, vol. 99, Birkhäuser Verlag, Basel, 2006.  Google Scholar

[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.  Google Scholar

[10]

B. Kawohl, Variational versus pde-based approaches in mathematical image processing, in CRM Proceedings and Lecture Notes, vol. 44, (2008), 113–126.  Google Scholar

[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.  Google Scholar

[12]

G. Lieberman, Second Order Parabolic Differential Equations, World Scientific, 1996. doi: 10.1142/3302.  Google Scholar

[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.  Google Scholar

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. Google Scholar

[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.  Google Scholar

[3]

G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal., 4 (1991), 271-283.   Google Scholar

[4]

M. G. CrandallH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

Y. Giga, Surface evolution equations, Monographs in Mathematics, vol. 99, Birkhäuser Verlag, Basel, 2006.  Google Scholar

[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.  Google Scholar

[10]

B. Kawohl, Variational versus pde-based approaches in mathematical image processing, in CRM Proceedings and Lecture Notes, vol. 44, (2008), 113–126.  Google Scholar

[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.  Google Scholar

[12]

G. Lieberman, Second Order Parabolic Differential Equations, World Scientific, 1996. doi: 10.1142/3302.  Google Scholar

[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.  Google Scholar

Figure 1.  The angle $ d\theta $ and the grid points
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
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