Fractional $ 1 $-Laplacian evolution equations to remove multiplicative noise

Tianling Gao; Qiang Liu; Zhiguang Zhang

doi:10.3934/dcdsb.2021254

Article Contents

2022, Volume 27, Issue 9: 4837-4854. Doi: 10.3934/dcdsb.2021254

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

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
^* Corresponding author: Qiang Liu

Received: April 30, 2021

Revised: July 31, 2021

Early access: October 2021

Published: September 2022

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)

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Abstract

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.

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Mathematics Subject Classification: Primary: 35R11, 35K65; Secondary: 68U10.

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Figure 1. (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 $

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Figure 2. (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 $

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