A non-convex tensor RPCA model with TL1 penalty for image restoration

Xiao Guo; Chuanpei Xu; Zhibin Zhu; Benxin Zhang; Xiao Guo; Chuanpei Xu; Zhibin Zhu; Benxin Zhang

doi:10.3934/jimo.2026108

Journal of Industrial and Management Optimization

2026, Volume 22, Issue 6: 2942-2962. doi: 10.3934/jimo.2026108

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Research article

A non-convex tensor RPCA model with TL1 penalty for image restoration

1.
School of Electronic Engineering and Automation, Guilin University of Electronic Technology, Guilin, 541004 China
2.
School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, 541004, China

Received: 04 February 2026 Revised: 06 May 2026 Accepted: 15 May 2026 Published: 26 May 2026
65K05, 90C30

For high-dimensional image restoration, tensor robust principal component analysis (TRPCA) is a highly effective method. However, since it uniformly penalizes all singular values, it tends to over-penalize larger singular values, thereby failing to adequately preserve the primary structural information of the image. In contrast, the non-convex transformed $ L_1 $ penalty (TL1) function demonstrates a stronger ability to protect larger singular values and reduce estimation bias, offering a more accurate approximation of the rank function. Inspired by this, we propose a TRPCA with TL1 penalty model (TL1-TRPCA), which non-convexly substitutes the low-rank function with the TL1, and apply the alternating direction method of multipliers (ADMM) to solve the model. Extensive experiments on color image, hyperspectral image, and gray video datasets show that the proposed method achieves superior performance compared to several state-of-the-art approaches. Our code is available at https://github.com/zbx913/Tensor-RPCA-with-TL1-Penalty.
- TRPCA,
- image restoration,
- TL1 penalty,
- ADMM
Citation: Xiao Guo, Chuanpei Xu, Zhibin Zhu, Benxin Zhang. A non-convex tensor RPCA model with TL1 penalty for image restoration[J]. Journal of Industrial and Management Optimization, 2026, 22(6): 2942-2962. doi: 10.3934/jimo.2026108

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Abstract

For high-dimensional image restoration, tensor robust principal component analysis (TRPCA) is a highly effective method. However, since it uniformly penalizes all singular values, it tends to over-penalize larger singular values, thereby failing to adequately preserve the primary structural information of the image. In contrast, the non-convex transformed $ L_1 $ penalty (TL1) function demonstrates a stronger ability to protect larger singular values and reduce estimation bias, offering a more accurate approximation of the rank function. Inspired by this, we propose a TRPCA with TL1 penalty model (TL1-TRPCA), which non-convexly substitutes the low-rank function with the TL1, and apply the alternating direction method of multipliers (ADMM) to solve the model. Extensive experiments on color image, hyperspectral image, and gray video datasets show that the proposed method achieves superior performance compared to several state-of-the-art approaches. Our code is available at https://github.com/zbx913/Tensor-RPCA-with-TL1-Penalty.

References

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