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Poisson Tensor Completion via Nonconvex Regularization and Nonlocal Self-Similarity for Multi-dimensional Image Recovery

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Abstract

The problem of Poisson tensor completion aims to recover a tensor from partial observations in the presence of Poisson noise. Existing approaches utilized the transformed tensor nuclear norm to explore the low-rankness of a tensor, which is the \(\ell _1\) norm of singular values vectors of all frontal slices of a tensor in the transformed domain. Nevertheless, the \(\ell _1\) norm is suboptimal due to its biased estimate. In this paper, we propose a nonconvex model based on transformed tensor nuclear norm for Poisson tensor completion. In order to explore the global low-rankness of the underlying tensor, a family of nonconvex functions are employed onto the singular values of all frontal slices of a tensor in the transformed domain. Furthermore, the nonlocal self-similarity is incorporated into the nonconvex model to describe the similar structures and characterize the intrinsic details of multi-dimensional images. A proximal alternating minimization algorithm is developed to solve the resulting models, whose convergence is established under very mild conditions. Extensive numerical examples on hyperspectral images, video images, and fluorescence microscope images demonstrate that the proposed approach outperforms several state-of-the-art methods.

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Data Availability

Enquiries about data availability should be directed to the authors.

Notes

  1. The global low-rankness means to explore the low-rankness of the underlying tensor directly.

  2. https://www.cs.rochester.edu/u/jliu/publications.html

  3. https://xu-yangyang.github.io/TMac/

  4. https://github.com/Shiran-Yuan/SPTC

  5. https://xj-zhang.github.io/math/publications.html

  6. https://personalpages.manchester.ac.uk/staff/d.h.foster//Hyperspectral_images_of_natural_scenes_04.html

  7. https://media.xiph.org/video/derf/

  8. http://www.cellimagelibrary.org/images/35532

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Acknowledgements

The authors would like to thank the anonymous referees for their helpful comments and suggestions, which have improved this paper.

Funding

The research of D. Qiu was supported in part by the National Natural Science Foundation of China under Grant No. 12201473 and the Science Foundation of Wuhan Institute of Technology under Grant No. K202256. The research of B. Li was supported in part by the National Natural Science Foundation of China under Grant No. 62377019 and Self-determined Research Funds of CCNU from the Colleges’ Basic Research under Grant No. CCNU24JC004. The research of X. Zhang was supported in part by the National Natural Science Foundation of China under Grant No. 12171189, Hubei Provincial Natural Science Foundation of China under Grant No. JCZRYB202501474, and Fundamental Research Funds for the Central Universities under Grant No. CCNU24ai002.

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Authors and Affiliations

  1. School of Mathematics and Physics, Wuhan Institute of Technology, Wuhan, 430205, China

    Duo Qiu

  2. School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China

    Sijia Xia & Bei Yang

  3. School of Mathematics and Statistics, and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, China

    Bo Li & Xiongjun Zhang

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D. Qiu: The research of this author was supported in part by the National Natural Science Foundation of China under Grant No. 12201473 and the Science Foundation of Wuhan Institute of Technology under Grant No. K202256.

The research of B. Li was supported in part by the National Natural Science Foundation of China under Grant No. 62377019 and Fundamental Research Funds for the Central Universities under Grant No. CCNU24JC004. The research of X. Zhang was supported in part by the National Natural Science Foundation of China under Grant No. 12171189 and Fundamental Research Funds for the Central Universities under Grant No. CCNU24AI002.

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Qiu, D., Xia, S., Yang, B. et al. Poisson Tensor Completion via Nonconvex Regularization and Nonlocal Self-Similarity for Multi-dimensional Image Recovery. J Sci Comput 102, 76 (2025). https://doi.org/10.1007/s10915-025-02801-8

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