Proximity algorithms for the $ {\mathit{L}}^{1}{\mathit{L}}^{2}/{\mathit{T}\mathit{V}}^{\mathit{\alpha }} $ image denoising model

Donghong Zhao; Ruiying Huang; Li Feng; Donghong Zhao; Ruiying Huang; Li Feng

doi:10.3934/math.2024807

AIMS Mathematics

2024, Volume 9, Issue 6: 16643-16665. doi: 10.3934/math.2024807

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Proximity algorithms for the $ {\mathit{L}}^{1}{\mathit{L}}^{2}/{\mathit{T}\mathit{V}}^{\mathit{\alpha }} $ image denoising model

School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, China

Received: 18 January 2024 Revised: 25 April 2024 Accepted: 06 May 2024 Published: 14 May 2024
MSC : 49N45, 65K10, 68U10

Inspired by the ROF model and the $ {L}^{1}/TV $ image denoising model, we propose a combined model to remove Gaussian noise and salt-and-pepper noise simultaneously. This model combines the $ {L}^{1} $ -data fidelity term, $ {L}^{2} $ -data fidelity term and a fractional-order total variation regularization term, and is termed the $ {L}^{1}{L}^{2}/{TV}^{\alpha } $ model. We have used the proximity algorithm to solve the proposed model. Through this method, the non-differentiable term is solved by using the fixed-point equations of the proximity operator. The numerical experiments show that the proposed model can effectively remove Gaussian noise and salt and pepper noise through implementation of the proximity algorithm. As we varied the fractional order $ \alpha $ from 0.8 to 1.9 in increments of 0.1, we observed that different images correspond to different optimal values of α.
- fractional-order total variation,
- double fidelity,
- proximity algorithm,
- convex minimization problem,
- image denoising
Citation: Donghong Zhao, Ruiying Huang, Li Feng. Proximity algorithms for the $ {\mathit{L}}^{1}{\mathit{L}}^{2}/{\mathit{T}\mathit{V}}^{\mathit{\alpha }} $ image denoising model[J]. AIMS Mathematics, 2024, 9(6): 16643-16665. doi: 10.3934/math.2024807

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Abstract

Inspired by the ROF model and the $ {L}^{1}/TV $ image denoising model, we propose a combined model to remove Gaussian noise and salt-and-pepper noise simultaneously. This model combines the $ {L}^{1} $ -data fidelity term, $ {L}^{2} $ -data fidelity term and a fractional-order total variation regularization term, and is termed the $ {L}^{1}{L}^{2}/{TV}^{\alpha } $ model. We have used the proximity algorithm to solve the proposed model. Through this method, the non-differentiable term is solved by using the fixed-point equations of the proximity operator. The numerical experiments show that the proposed model can effectively remove Gaussian noise and salt and pepper noise through implementation of the proximity algorithm. As we varied the fractional order $ \alpha $ from 0.8 to 1.9 in increments of 0.1, we observed that different images correspond to different optimal values of α.

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