Nonconvex high-order TV and $ \ell_0 $ norm-based method for image restoration

Ben Wang; Zijie Lan; Ben Wang; Zijie Lan

doi:10.3934/era.2025152

Electronic Research Archive

2025, Volume 33, Issue 6: 3431-3449. doi: 10.3934/era.2025152

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Research article

Nonconvex high-order TV and $ \ell_0 $ norm-based method for image restoration

Ben Wang ^{1
,
,},
Zijie Lan ²

1.
School of Mathematics and Statistics, Hunan University of Science and Technology, Xiangtan 411201, China
2.
School of Mathematical Sciences, Guangxi Minzu University, Nanning 530006, China

Received: 10 March 2025 Revised: 15 May 2025 Accepted: 29 May 2025 Published: 03 June 2025

In order to obtain high-quality restoration outcomes, this paper proposed a new nonconvex variational model tailored for reconstructing blurred images corrupted by impulse noise. The newly designed optimization framework integrated nonconvex high-order total variation regularization with $ \ell_0 $ norm data fidelity. This combination aided in overcoming the staircase artifacts and maintaining sharp contours while tackling the poor performance of $ \ell_1 $ norm-based data fidelity in handling high-density impulse noise. To deal with the nonconvex optimization problem, we designed an efficient alternating direction method of multipliers by combining the variable splitting technique with the iteratively reweighted $ \ell_1 $ algorithm. Finally, numerous experiments demonstrated that the suggested scheme achieves competitive performance in terms of objective metrics and visual quality.
- image restoration,
- impulse noise,
- nonconvex function,
- $ \ell_0 $ norm,
- alternating direction method of multipliers
Citation: Ben Wang, Zijie Lan. Nonconvex high-order TV and $ \ell_0 $ norm-based method for image restoration[J]. Electronic Research Archive, 2025, 33(6): 3431-3449. doi: 10.3934/era.2025152

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Abstract

In order to obtain high-quality restoration outcomes, this paper proposed a new nonconvex variational model tailored for reconstructing blurred images corrupted by impulse noise. The newly designed optimization framework integrated nonconvex high-order total variation regularization with $ \ell_0 $ norm data fidelity. This combination aided in overcoming the staircase artifacts and maintaining sharp contours while tackling the poor performance of $ \ell_1 $ norm-based data fidelity in handling high-density impulse noise. To deal with the nonconvex optimization problem, we designed an efficient alternating direction method of multipliers by combining the variable splitting technique with the iteratively reweighted $ \ell_1 $ algorithm. Finally, numerous experiments demonstrated that the suggested scheme achieves competitive performance in terms of objective metrics and visual quality.

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