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Operations Research Transactions >

2022 , Vol. 26 >Issue 3: 31 - 43

DOI: https://doi.org/10.15960/j.cnki.issn.1007-6093.2022.03.003

A tensor completion method based on tensor train decomposition and its application in image restoration

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  • 1. School of Sciences, Hangzhou DianZi University, Hangzhou 310018, Zhejiang, China

Received date: 2022-01-18

  Online published: 2022-09-07

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Abstract

Low-rank tensor completion is widely used in data recovery, and the tensor completion model based on tensor train (TT) decomposition works well in color image, video and internet data recovery. This paper proposes a tensor completion model based on the third-order tensor TT decomposition. In this model, the sparse regularization and the spatio-temporal regularization are introduced to characterize the sparsity of the kernel tensor and the inherent block similarity of the data, respectively. According to the structural characteristics of the problem, some auxiliary variables are introduced to convert the original model into a separable form equivalently, and the method of combining proximal alternating minimization (PAM) and alternating direction multiplier method (ADMM) is used to solve the model. Numerical experiments show that the introduction of two regular terms is beneficial to improve the stability and practical effect of data recovery, and the proposed method is superior to other methods. When the sampling rate is low or the image is structurally missing, the presented method is more effective.

Key words： tensor completion; tensor train (TT) decomposition; proximal alternating minimization; alternating direction method of multipliers; image restoration

Cite this article

Wenhui XIE, Chen LING, Chenjian PAN . A tensor completion method based on tensor train decomposition and its application in image restoration[J]. Operations Research Transactions, 2022 , 26(3) : 31 -43 . DOI: 10.15960/j.cnki.issn.1007-6093.2022.03.003

References

1 Kolda T G , Bader B W . Tensor decompositions and applications[J]. SIAM Review, 2009, 51 (3): 455- 500.
2 Kilmer M , Braman K , Hao N , et al. Third-order tensors as operators on matrices: a theoretical and computational framework with applications in imaging[J]. SIAM Journal on Matrix Analysis and Applications, 2013, 34 (1): 148- 172.
3 Oseledets I . Tensor-train decomposition[J]. SIAM Journal of Scientific Computing, 2011, 33 (5): 2295- 2317.
4 Liu J , Musialski P , Wonka P , et al. Tensor completion for estimating missing values in visual data[J]. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2013, 35 (1): 208- 220.
5 Shang K , Li Y F , Huang Z H . Iterative $p$-shrinkage thresholding algorithm for low Tucker rank tensor recovery[J]. Information Sciences, 2019, 482, 374- 391.
6 Zhang Z M, Ely G, Aeron S, et al. Novel methods for multilinear data completion and de-noising based on tensor-SVD[C]//IEEE Conference on Computer Vision and Pattern Recognition, 2014: 3842-3849. Doi: 10.1109/CVPR.2014.485.
7 Zhang Z M , Aeron S C . Exact tensor completion using $t$-SVD[J]. IEEE Transactions on Signal Processing, 2017, 65 (6): 1511- 1526.
8 Lu C Y , Feng J S , Chen Y D , et al. Tensor robust principal component analysis: Exact recovery of corrupted low-rank tensors via convex optimization[J]. IEEE Conference on Computer Vision and Pattern Recognition, 2016, 5249- 5257.
9 Qiu D , Bai M R , Ng M K , et al. Robust low-rank tensor completion via transformed tensor nuclear norm with total variation regularization[J]. Neurocomputing, 2021, 435, 197- 215.
10 Acar E , Dunlavy D M , Kolda T G , et al. Scalable tensor factorizations for incomplete data[J]. Chemometrics and Intelligent Laboratory Systems, 2011, 106 (1): 41- 56.
11 Zhao Q B , Zhang L Q , Cichocki A . Bayesian CP factorization of incomplete tensors with automatic rank determination[J]. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2015, 37 (9): 1751- 1763.
12 Zeng C , Jiang T X , Ng M K . An approximation method of CP rank for third-order tensor completion[J]. Numerische Mathematik, 2021, 147, 727- 757.
13 Mu C, Huang B, Wright J, et al. Square deal: Lower bounds and improved relaxations for tensor recovery[C]. International Conference on Machine Learning, 2014, 2: 73-81.
14 Bengua J A , Phien H N , Tuan H D , et al. Efficient tensor completion for color image and video recovery: Low-rank tensor train[J]. IEEE Transactions on Image Processing, 2017, 26 (5): 2466- 2479.
15 Ko C Y , Batselier K , Yu W J , et al. Fast and accurate tensor completion with total variation regularized tensor trains[J]. IEEE Transactions on Image Processing, 2020, 29, 6918- 6931.
16 Du S Q , Shi Y Q , Hu W J , et al. Robust tensor factorization for color image and grayscale video recovery[J]. IEEE Access, 2020, 8, 174410- 174423.
17 Zhang Z Y , Ling C , He H J , et al. A tensor train approach for internet traffic data completion[J]. Annals of Operations Research, 2021,
18 Attouch H , Bolte J , Svaiter B F . Convergence of descent methods for semialgebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods[J]. Mathematical Programming, 2013, 137 (1/2): 91- 129.
19 Ma T H , Lou Y F , Huang T Z . Truncated $\ell_{1-2}$ models for sparse recovery and rank minimization[J]. SIAM Journal on Imaging Sciences, 2017, 10 (3): 1346- 1380.
20 Hale E T , Yin W T , Zhang Y . Fixed-point continuation for $\ell_{1}$-minimization: methodology and convergence[J]. SIAM Journal on Optimization, 2008, 19 (3): 1107- 1130.
21 Eckstein J , Bertsekas D P . On the douglas-rachford splitting method and the proximal point algorithm for maximal monotone operators[J]. Mathematical Programming, 1992, 55 (1): 293- 318.
22 Chen Y L , Hsu C T , Liao H Y M . Simultaneous tensor decomposition and completion using factor priors[J]. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2014, 36 (3): 577- 591.
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