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Nonlinear Transform Induced Tensor Nuclear Norm for Tensor Completion

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Abstract

The linear transform-based tensor nuclear norm (TNN) methods have recently obtained promising results for tensor completion. The main idea of these methods is exploiting the low-rank structure of frontal slices of the targeted tensor under the linear transform along the third mode. However, the low-rankness of frontal slices is not significant under the linear transforms family. To better pursue the low-rank approximation, we propose a nonlinear transform-based TNN (NTTNN). More concretely, the proposed nonlinear transform is a composite transform consisting of the linear semi-orthogonal transform along the third mode and the element-wise nonlinear transform on frontal slices of the tensor under the linear semi-orthogonal transform. The two transforms in the composite transform are indispensable and complementary to fully exploit the underlying low-rankness. Based on the suggested low-rankness metric, i.e., NTTNN, we propose a low-rank tensor completion model. To tackle the resulting nonlinear and nonconvex optimization model, we elaborately design a proximal alternating minimization algorithm and establish the theoretical convergence guarantee. Extensive experimental results on hyperspectral images, multispectral images, and videos show that our method outperforms the linear transform-based state-of-the-art LRTC methods in terms of PSNR, SSIM, SAM values and visual quality.

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Data Availability

All datasets are publicly available.

Notes

  1. https://engineering.purdue.edu/~biehl/MultiSpec/hyperspectral.html.

  2. http://www.ehu.eus/ccwintco/index.php/Hyperspectral_Remote_Sensing_Scenes#anomaly_detection.

  3. https://www.cs.columbia.edu/CAVE/databases/multispectral/.

  4. http://trace.eas.asu.edu/yuv/.

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Funding

This research is supported by NSFC (Nos. 61876203, 12171072, 12001432, 12171189, and 11801206), the Applied Basic Research Project of Sichuan Province (No. 2021YJ0107), the Key Project of Applied Basic Research in Sichuan Province (No. 2020YJ0216), and National Key Research and Development Program of China (No. 2020YFA0714001).

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, China

    Ben-Zheng Li, Xi-Le Zhao & Ting-Zhu Huang

  2. School of Mathematics and Statistics, Northwestern Polytechnical University, Xian, Shanxi, China

    Teng-Yu Ji

  3. School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, Hubei, China

    Xiong-Jun Zhang

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  1. Ben-Zheng Li
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  2. Xi-Le Zhao
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Correspondence to Xi-Le Zhao.

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This research is supported by NSFC (Nos. 61876203, 12171072, 12001432, 12171189, and 11801206), the Applied Basic Research Project of Sichuan Province (No. 2021YJ0107), the Key Project of Applied Basic Research in Sichuan Province (No. 2020YJ0216), and National Key Research and Development Program of China (No. 2020YFA0714001)

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Li, BZ., Zhao, XL., Ji, TY. et al. Nonlinear Transform Induced Tensor Nuclear Norm for Tensor Completion. J Sci Comput 92, 83 (2022). https://doi.org/10.1007/s10915-022-01937-1

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