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Nonlinear Hierarchical Matrix Factorization-Based Tensor Ring Approximation for Multi-dimensional Image Recovery

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Abstract

Recently, tensor ring (TR) approximation has received increasing attention in multi-dimensional image processing. In TR approximation, the key backbone is the shallow matrix factorizations, which approximate the circular unfolding of the multi-dimensional image. However, the shallow matrix factorization limits the standard TR approximation’s ability to represent images with complex details and textures. To address this limitation, we propose a nonlinear hierarchical matrix factorization-based tensor ring (NHTR) approximation. Specifically, instead of the shallow matrix factorization, we introduce the nonlinear hierarchical matrix factorization in NHTR approximation to approximate circularly \(\lceil \frac{N}{2}\rceil \)-modes unfoldings of an N-th order tensor. Benefiting from the powerful expressive capability of the nonlinear hierarchical matrix factorization, the proposed NHTR approximation can faithfully capture fine details of the clean image compared to classical tensor ring approximation. Empowered with the proposed NHTR, we build a multi-dimensional image recovery model and establish a theoretical error bound between the recovered image and the clean image based on the proposed model. To solve the highly nonlinear and hierarchical optimization problem, we develop an efficient alternating minimization-based algorithm. Experiments on multispectral images and color videos conclusively demonstrate the superior performance of our method over the compared state-of-the-art methods in multi-dimensional image recovery.

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Funding

This research is supported by NSFC (No. 12171072 and 12371456), Sichuan Science and Technology Program (No. 2024NSFJQ0038 and 2023ZYD0007), National Key Research and Development Program of China (No. 2020YFA0714001), Fund of Hubei Key Laboratory of Inland Shipping Technology (No. NHHY2023003)

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, Sichuan, People’s Republic of China

    Wei-Hao Wu, Ting-Zhu Huang, Xi-Le Zhao, Hao Zhang & Zhi-Long Han

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  1. Wei-Hao Wu
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  2. Ting-Zhu Huang
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  3. Xi-Le Zhao
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  4. Hao Zhang
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Correspondence to Ting-Zhu Huang or Xi-Le Zhao.

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Wu, WH., Huang, TZ., Zhao, XL. et al. Nonlinear Hierarchical Matrix Factorization-Based Tensor Ring Approximation for Multi-dimensional Image Recovery. J Sci Comput 101, 31 (2024). https://doi.org/10.1007/s10915-024-02670-7

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