Tensor ring decomposition with data-driven for color image completion

Yingpin Chen; Peiqi Zhuo; Hualin Zhang; Yuan Liao; Zhixiang Chen; Ronghuan Zhang; Yujuan Xu; Yingpin Chen; Peiqi Zhuo; Hualin Zhang; Yuan Liao; Zhixiang Chen; Ronghuan Zhang; Yujuan Xu

doi:10.3934/era.2026023

Electronic Research Archive

2026, Volume 34, Issue 1: 477-508. doi: 10.3934/era.2026023

Previous Article Next Article

Research article Special Issues

Tensor ring decomposition with data-driven for color image completion

1.
School of Physics and Information Engineering, Minnan Normal University, Zhangzhou 363000, China
2.
Key Laboratory of Light Field Manipulation and System Integration Applications in Fujian Province, Minnan Normal University, Zhangzhou 363000, China

Received: 22 October 2025 Revised: 01 December 2025 Accepted: 16 December 2025 Published: 16 January 2026

Tensor ring (TR) decomposition has demonstrated remarkable capability in capturing low-rank tensor structures, achieving significant success in color image completion tasks. However, relying solely on the tensor low-rank prior cannot fully capture the details of a color image. Thus, traditional model-driven TR decomposition models often fail to recover satisfactory local detail. A natural idea is to introduce a deep neural network with end-to-end training to improve the detail quality of image reconstruction. Thus, we proposed a novel hybrid model that integrates model-driven TR decomposition with data-driven deep learning regularization. The proposed framework introduces an energy functional regularization term based on the FFDNET architecture, which enhances the robustness of rank selection while preserving global low-rank and local details. In particular, we assumed that the unfolding matrices of the TR factors exhibited low-rank properties. Thus, nuclear-norm regularization constraints were incorporated on the TR factors to enhance their global low-rank characteristics. Additionally, a deep prior regularization term derived from the FFDNET network was introduced to preserve the local details of the target tensor. We further developed an efficient alternating direction method for the multiplier algorithm to address the associated optimization problem. Extensive experiments on color images have demonstrated that the proposed method outperforms denoising approaches, yielding satisfactory results. The code for implementing the proposed method is available at https://github.com/110500617/TRDD.
- tensor ring decomposition,
- deep learning,
- FFDNET,
- ADMM,
- data-driven
Citation: Yingpin Chen, Peiqi Zhuo, Hualin Zhang, Yuan Liao, Zhixiang Chen, Ronghuan Zhang, Yujuan Xu. Tensor ring decomposition with data-driven for color image completion[J]. Electronic Research Archive, 2026, 34(1): 477-508. doi: 10.3934/era.2026023

Related Papers:

Abstract

Tensor ring (TR) decomposition has demonstrated remarkable capability in capturing low-rank tensor structures, achieving significant success in color image completion tasks. However, relying solely on the tensor low-rank prior cannot fully capture the details of a color image. Thus, traditional model-driven TR decomposition models often fail to recover satisfactory local detail. A natural idea is to introduce a deep neural network with end-to-end training to improve the detail quality of image reconstruction. Thus, we proposed a novel hybrid model that integrates model-driven TR decomposition with data-driven deep learning regularization. The proposed framework introduces an energy functional regularization term based on the FFDNET architecture, which enhances the robustness of rank selection while preserving global low-rank and local details. In particular, we assumed that the unfolding matrices of the TR factors exhibited low-rank properties. Thus, nuclear-norm regularization constraints were incorporated on the TR factors to enhance their global low-rank characteristics. Additionally, a deep prior regularization term derived from the FFDNET network was introduced to preserve the local details of the target tensor. We further developed an efficient alternating direction method for the multiplier algorithm to address the associated optimization problem. Extensive experiments on color images have demonstrated that the proposed method outperforms denoising approaches, yielding satisfactory results. The code for implementing the proposed method is available at https://github.com/110500617/TRDD.

References

[1]	H. Jiao, Q. Liu, K. Liang, L. An, Low-rank quaternion tensor completion by truncated nuclear norm regularization for color video recovery, Neurocomputing, 661 (2026), 131905.https://doi.org/10.1016/j.neucom.2025.131905 doi: 10.1016/j.neucom.2025.131905
[2]	Y. Chen, Z. Peng, M. Li, F. Yu, F. Lin, Seismic signal denoising using total generalized variation with overlapping group sparsity in the accelerated ADMM framework, J. Geophys. Eng. , 16 (2019), 30-51.https://doi.org/10.1093/jge/gxy003 doi: 10.1093/jge/gxy003
[3]	D. N. H. Thanh, L. T. Thanh, N. N. Hien, S. Prasath, Adaptive total variation L1 regularization for salt and pepper image denoising, Optik, 208 (2020), 163677.https://doi.org/10.1016/j.ijleo.2019.163677 doi: 10.1016/j.ijleo.2019.163677
[4]	L. Wang, Y. Chen, F. Lin, Y. Chen, F. Yu, Z. Cai, Impulse noise denoising using total variation with overlapping group sparsity and Lp-pseudo-norm shrinkage, Appl. Sci. , 8 (2018), 2317.https://doi.org/10.3390/app8112317 doi: 10.3390/app8112317
[5]	M. Ding, X. Fu, X. L. Zhao, Bilinear hyperspectral unmixing via tensor decomposition, in 2023 31st European Signal Processing Conference (EUSIPCO), IEEE, (2023), 640-644.https://doi.org/10.23919/EUSIPCO58844.2023.10289962
[6]	M. Ding, X. Fu, X. L. Zhao, Fast and structured block-term tensor decomposition for hyperspectral unmixing, IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens. , 16 (2023), 1691-1709.https://doi.org/10.1109/jstars.2023.3238653 doi: 10.1109/jstars.2023.3238653
[7]	Z. L. Han, T. Z. Huang, X. L. Zhao, H. Zhang, W. H. Wu, Nested fully-connected tensor network decomposition for multi-dimensional visual data recovery, IEEE Trans. Circuits Syst. Video Technol. , 34 (2024), 10092-10106.https://doi.org/10.1109/TCSVT.2024.3401134 doi: 10.1109/TCSVT.2024.3401134
[8]	Y. Luo, X. Zhao, Z. Li, M. K. Ng, D. Meng, Low-rank tensor function representation for multi-dimensional data recovery, IEEE Trans. Pattern Anal. Mach. Intell. , 46 (2023), 3351-3369.https://doi.org/10.1109/TPAMI.2023.3341688 doi: 10.1109/TPAMI.2023.3341688
[9]	S. Liu, J. Leng, X. L. Zhao, H. Zeng, Y. Wang, J. H. Yang, Learnable spatial-spectral transform-based tensor nuclear norm for multi-dimensional visual data recovery, IEEE Trans. Circuits Syst. Video Technol. , 34 (2023), 3633-3646.https://doi.org/10.1109/TCSVT.2023.3316279 doi: 10.1109/TCSVT.2023.3316279
[10]	Z. L. Han, T. Z. Huang, X. L. Zhao, H. Zhang, Y. Y. Liu, Multi-dimensional data recovery via feature-based fully-connected tensor network decomposition, IEEE Trans. Big Data, 10 (2023), 386-399.https://doi.org/10.1109/TBDATA.2023.3342611 doi: 10.1109/TBDATA.2023.3342611
[11]	Y. Chen, Y. Huang, L. Wang, H. Huang, J. Song, C. Yu, et al., Salt and pepper noise removal method based on stationary Framelet transform with non‐convex sparsity regularization, IET Image Process., 16 (2022), 1846-1865.https://doi.org/10.1049/ipr2.12451
[12]	Y. Chen, H. Xie, J. Zhang, H. Yang, H. Wang, L. Wang, Infrared image deblurring with Framelet regularization in the background of salt-and-pepper noise, J. Mod. Opt. , 71 (2024), 1-15.https://doi.org/10.1080/09500340.2024.2383277 doi: 10.1080/09500340.2024.2383277
[13]	S. Gu, L. Zhang, W. Zuo, X. Feng, Weighted nuclear norm minimization with application to image denoising, in 2014 IEEE Conference on Computer Vision and Pattern Recognition, IEEE, (2014), 2862-2869.https://doi.org/10.1109/CVPR.2014.366
[14]	W. Dong, L. Zhang, G. Shi, X. Li, Nonlocally centralized sparse representation for image restoration, IEEE Trans. Image Process. , 22 (2012), 1620-1630.https://doi.org/10.1109/TIP.2012.2235847 doi: 10.1109/TIP.2012.2235847
[15]	J. Mairal, F. Bach, J. Ponce, G. Sapiro, A. Zisserman, Non-local sparse models for image restoration, in 2009 IEEE 12th International Conference on Computer Vision, IEEE, (2009), 2272-2279.https://doi.org/10.1109/ICCV.2009.5459452
[16]	J. Yang, Y. Chen, Z. Chen, Infrared image deblurring via high-order total variation and Lp-pseudonorm shrinkage, Appl. Sci. , 10 (2020), 2533.https://doi.org/10.3390/app10072533 doi: 10.3390/app10072533
[17]	H. Wang, Y. Cen, Z. He, Z. He, R. Zhao, F. Zhang, Reweighted low-rank matrix analysis with structural smoothness for image denoising, IEEE Trans. Image Process. , 27 (2017), 1777-1792.https://doi.org/10.1109/TIP.2017.2781425 doi: 10.1109/TIP.2017.2781425
[18]	L. Chiantini, G. Ottaviani, On generic identifiability of 3-tensors of small rank, SIAM J. Matrix Anal. Appl. , 33 (2012), 1018-1037.https://doi.org/10.1137/110829180 doi: 10.1137/110829180
[19]	F. L. Hitchcock, The expression of a tensor or a polyadic as a sum of products, J. Math. Phys. , 6 (1927), 164-189.https://doi.org/10.1002/sapm192761164 doi: 10.1002/sapm192761164
[20]	X. Liu, S. Bourennane, C. Fossati, Denoising of hyperspectral images using the PARAFAC Model and statistical performance analysis, IEEE Trans. Geosci. Remote Sens. , 50 (2012), 3717-3724.https://doi.org/10.1109/tgrs.2012.2187063 doi: 10.1109/tgrs.2012.2187063
[21]	N. Renard, S. Bourennane, J. Blanc-Talon, Denoising and dimensionality reduction using multilinear tools for hyperspectral images, IEEE Geosci. Remote Sens. Lett., 5 (2008), 138-142.https://doi.org/10.1109/lgrs.2008.915736
[22]	Y. Chen, W. He, N. Yokoya, T. Z. Huang, Hyperspectral image restoration using weighted group sparsity-regularized low-rank tensor decomposition, IEEE Trans. Cybern. , 50 (2019), 3556-3570.https://doi.org/10.1109/tcyb.2019.2936042 doi: 10.1109/tcyb.2019.2936042
[23]	X. Bai, F. Xu, L. Zhou, Y. Xing, L. Bai, J. Zhou, Nonlocal similarity based nonnegative tucker decomposition for hyperspectral image denoising, IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens. , 11 (2018), 701-712.https://doi.org/10.1109/jstars.2018.2791718 doi: 10.1109/jstars.2018.2791718
[24]	Y. Wang, J. Peng, Q. Zhao, Y. Leung, X. L. Zhao, D. Meng, Hyperspectral image restoration via total variation regularized low-rank tensor decomposition, IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens. , 11 (2017), 1227-1243.https://doi.org/10.1109/jstars.2017.2779539 doi: 10.1109/jstars.2017.2779539
[25]	Y. Chen, T. Z. Huang, X. L. Zhao, Destriping of multispectral remote sensing image using low-rank tensor decomposition, IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens. , 11 (2018), 4950-4967.https://doi.org/10.1109/jstars.2018.2877722 doi: 10.1109/jstars.2018.2877722
[26]	W. Gong, Z. Huang, L. Yan, Enhanced low-rank and sparse tucker decomposition for image completion, in 2024 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), IEEE, (2024), 2425-2429.https://doi.org/10.1109/ICASSP48485.2024.10448445
[27]	M. E. Kilmer, K. Braman, N. Hao, R. C. Hoover, Third-order tensors as operators on matrices: A theoretical and computational framework with applications in imaging, SIAM J. Matrix Anal. Appl. , 34 (2013), 148-172.https://doi.org/10.1137/110837711 doi: 10.1137/110837711
[28]	M. E. Kilmer, C. D. Martin, Factorization strategies for third-order tensors, Linear Algebra Appl. , 435 (2011), 641-658.https://doi.org/10.1016/j.laa.2010.09.020 doi: 10.1016/j.laa.2010.09.020
[29]	Y. B. Zheng, T. Z. Huang, X. L. Zhao, T. X. Jiang, T. Y. Ji, Mixed noise removal in hyperspectral image via low-fibered-rank regularization, IEEE Trans. Geosci. Remote Sens. , 58 (2019), 734-749.https://doi.org/10.1109/TGRS.2019.2940534 doi: 10.1109/TGRS.2019.2940534
[30]	D. Goldfarb, Z. Qin, Robust low-rank tensor recovery: Models and algorithms, SIAM J. Matrix Anal. Appl. , 35 (2014), 225-253.https://doi.org/10.1137/130905010 doi: 10.1137/130905010
[31]	Y. Hu, Z. Tang, T. X. Jiang, X. L. Zhao, G. Liu, Degradation accordant plug-and-play for low-rank tensor recovery, Pattern Recognit. , 172 (2026), 112612.https://doi.org/10.1016/j.patcog.2025.112612 doi: 10.1016/j.patcog.2025.112612
[32]	Y. B. Zheng, X. L. Zhao, H. C. Li, C. Li, T. Z. Huang, Q. Zhao, Tensor network decomposition for data recovery: Recent advancements and future prospects, Neural Networks, 191 (2025), 107808.https://doi.org/10.1016/j.neunet.2025.107808 doi: 10.1016/j.neunet.2025.107808
[33]	J. A. Bengua, H. N. Phien, H. D. Tuan, M. N. Do, Efficient tensor completion for color image and video recovery: Low-rank tensor train, IEEE Trans. Image Process., 26 (2017), 2466-2479.https://doi.org/10.1109/TIP.2017.2672439 doi: 10.1109/TIP.2017.2672439
[34]	Q. Zhao, G. Zhou, S. Xie, L. Zhang, A. Cichocki, Tensor ring decomposition, preprint, arXiv: 160605535.
[35]	J. Miao, K. I. Kou, H. Cai, L. Liu, Quaternion tensor left ring decomposition and application for color image inpainting, J. Sci. Comput. , 101 (2024), 1.https://doi.org/10.1007/s10915-024-02624-z doi: 10.1007/s10915-024-02624-z
[36]	P. L. Wu, X. L. Zhao, M. Ding, Y. B. Zheng, L. B. Cui, T. Z. Huang, Tensor ring decomposition-based model with interpretable gradient factors regularization for tensor completion, Knowledge-Based Syst. , 259 (2023), 110094.https://doi.org/10.1016/j.knosys.2022.110094 doi: 10.1016/j.knosys.2022.110094
[37]	Y. B. Zheng, T. Z. Huang, X. L. Zhao, Q. Zhao, T. X. Jiang, Fully-connected tensor network decomposition and its application to higher-order tensor completion, in Proceedings of the AAAI Conference on Artificial Intelligence, AAAI Press, (2021), 11071-11078.https://doi.org/10.1609/aaai.v35i12.17321
[38]	M. Ding, T. Z. Huang, X. L. Zhao, T. H. Ma, Tensor completion via nonconvex tensor ring rank minimization with guaranteed convergence, Signal Process. , 194 (2022), 108425.https://doi.org/10.1016/j.sigpro.2021.108425 doi: 10.1016/j.sigpro.2021.108425
[39]	W. Wang, V. Aggarwal, S. Aeron, Efficient low rank tensor ring completion, in 2017 IEEE International Conference on Computer Vision (ICCV), IEEE, (2017), 5698-5706.https://doi.org/10.1109/ICCV.2017.607
[40]	L. Yuan, J. Cao, X. Zhao, Q. Wu, Q. Zhao, Higher-dimension tensor completion via Low-rank tensor ring decomposition, in 2018 Asia-Pacific Signal and Information Processing Association Annual Summit and Conference (APSIPA ASC), IEEE, (2018), 1071-1076.https://doi.org/10.23919/APSIPA.2018.8659708
[41]	Y. Chen, W. He, N. Yokoya, T. Z. Huang, X. L. Zhao, Nonlocal tensor-ring decomposition for hyperspectral image denoising, IEEE Trans. Geosci. Remote Sens. , 58 (2019), 1348-1362.https://doi.org/10.1109/tgrs.2019.2946050 doi: 10.1109/tgrs.2019.2946050
[42]	D. Li, D. Chu, X. Guan, W. He, H. Shen, Adaptive regularized low-rank tensor decomposition for hyperspectral image denoising, IEEE Trans. Geosci. Remote Sens. , 62 (2024), 1-17.https://doi.org/10.1109/TGRS.2024.3385536 doi: 10.1109/TGRS.2024.3385536
[43]	X. L. Zhao, W. H. Xu, T. X. Jiang, Y. Wang, M. K. Ng, Deep plug-and-play prior for low-rank tensor completion, Neurocomputing, 400 (2020), 137-149.https://doi.org/10.1016/j.neucom.2020.03.018 doi: 10.1016/j.neucom.2020.03.018
[44]	J. Zhang, Z. Li, L. Wang, Y. Chen, Salt and pepper denoising method for colour images based on tensor low‐rank prior and implicit regularization, IET Image Process. , 17 (2023), 886-900.https://doi.org/10.1049/ipr2.12680 doi: 10.1049/ipr2.12680
[45]	K. Zhang, W. Zuo, L. Zhang, FFDNet: Toward a fast and flexible solution for CNN-based image denoising, IEEE Trans. Image Process., 27 (2018), 4608-4622.https://doi.org/10.1109/TIP.2018.2839891 doi: 10.1109/TIP.2018.2839891
[46]	B. Stephen, P. Neal, C. Eric, P. Borja, E. Jonathan, Distributed optimization and statistical learning via the alternating direction method of multipliers, Found. Trends Mach. Learn. , 3 (2011), 1-122.
[47]	Y. Chen, Y. Liao, Y. He, X. He, Q. Yu, T. Chen, et al., Non-local similar block matching and hybrid low-rank tensor network for colour image inpainting, Digital Signal Process., 162 (2025), 105169.https://doi.org/10.1016/j.dsp.2025.105169
[48]	T. G. Kolda, B. W. Bader, Tensor decompositions and applications, SIAM Rev. , 51 (2009), 455-500.https://doi.org/10.1137/07070111x doi: 10.1137/07070111x
[49]	S. Holtz, T. Rohwedder, R. Schneider, The alternating linear scheme for tensor optimization in the tensor train format, SIAM J. Sci. Comput., 34 (2012), A683-A713.https://doi.org/10.1137/100818893
[50]	Y. Xu, R. Hao, W. Yin, Z. Su, Parallel matrix factorization for low-rank tensor completion, Inverse Probl. Imaging, 9 (2015), 601-624.https://doi.org/10.3934/ipi.2015.9.601 doi: 10.3934/ipi.2015.9.601
[51]	Z. Zhang, S. Aeron, Exact tensor completion using t-SVD, IEEE Trans. Signal Process. , 65 (2016), 1511-1526.https://doi.org/10.1109/TSP.2016.2639466 doi: 10.1109/TSP.2016.2639466
[52]	L. Yuan, C. Li, D. Mandic, J. Cao, Q. Zhao, Tensor ring decomposition with rank minimization on latent space: An efficient approach for tensor completion, in Proceedings of the AAAI Conference on Artificial Intelligence, AAAI Press, (2019), 9151-9158.https://doi.org/10.1609/aaai.v33i01.33019151

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)