Generalizations of $ p $-Laplace operator for image enhancement: Part 2

George Baravdish; Yuanji Cheng; Olof Svensson; Freddie Åström

doi:10.3934/cpaa.2020152

Article Contents

2020, Volume 19, Issue 7: 3477-3500. Doi: 10.3934/cpaa.2020152

This issue Previous Article The Lazer-McKenna conjecture for an anisotropic planar elliptic problem with exponential Neumann data Next Article Multiplicity of radial and nonradial solutions to equations with fractional operators

Generalizations of $ p $-Laplace operator for image enhancement: Part 2

1.
Linköping University, Sweden
2.
Malmö University, Sweden
3.
Heidelberg University, Germany

^* Corresponding author
^* Corresponding author

Received: March 2018

Revised: January 2020

Published: April 2020

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Abstract

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.

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Mathematics Subject Classification: Primary: 35K55; Secondary: 35J75.

Citation:

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

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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.)

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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

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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

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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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References

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