一个基于张量火车分解的张量填充方法及在图像恢复中的应用

doi:10.15960/j.cnki.issn.1007-6093.2022.03.003

摘要/Abstract

摘要：

低秩张量填充在数据恢复中有广泛应用, 基于张量火车(TT) 分解的张量填充模型在彩色图像和视频以及互联网数据恢复中应用效果良好。本文提出一个基于三阶张量TT分解的填充模型。在模型中, 引入稀疏正则项与时空正则项, 分别刻画核张量的稀疏性和数据固有的块相似性。根据问题的结构特点, 引入辅助变量将原模型等价转化成可分离形式, 并采用临近交替极小化(PAM) 与交替方向乘子法(ADMM) 相结合的方法求解模型。数值实验表明, 两正则项的引入有利于提高数据恢复的稳定性和实际效果, 所提出方法优于其他方法。在采样率较低或图像出现结构性缺失时, 其方法效果较为显著。

关键词: 张量填充, 张量火车分解, 临近交替极小化, 交替方向乘子法, 图像恢复

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

中图分类号:

O221.2

谢文蕙, 凌晨, 潘晨健. 一个基于张量火车分解的张量填充方法及在图像恢复中的应用[J]. 运筹学学报, 2022, 26(3): 31-43.

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.

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方法	SR/%
方法	3	5	7	10	15	20	25	30
TTFM	13.92	14.72	14.86	15.12	14.55	12.83	11.42	10.46
TTC-TV	3.44	4.25	4.95	6.17	8.11	10.12	12.15	14.35
HaLRTC	4.09	4.39	4.39	4.38	4.17	3.81	3.64	3.53
STDC	37.48	44.99	48.65	59.09	58.51	57.59	56.45	54.68
TNN	6.30	6.95	7.08	7.34	7.56	7.07	7.39	7.45
IpST	7.57	8.55	8.60	8.89	9.11	8.48	8.48	7.56
BCPF	137.87	130.90	121.80	137.08	138.16	141.59	146.84	157.33

方法	(5)	(6)	(7)	(8)
方法	PSNR/RSE/Time/s/Iter	PSNR/RSE/Time/s/Iter	PSNR/RSE/Time/s/Iter	PSNR/RSE/Time/s/Iter
TTFM	$ \bf{27.86}/\bf{0.13} $/8.89/42	$ \bf{35.18}/\bf{0.05} $/7.96/37	$ \bf{23.77}/\bf{0.16} $/7.95/38	$ \bf{26.28}/\bf{0.08} $/8.75/38
TTC-TV	23.16/0.22/13.27/3	31.26/0.08/43.08/3	20.85/0.22/10.46/3	24.72/0.09/35.61/3
HaLRTC	11.36/0.86/1.18/25	32.36/0.07/2.81/50	21.61/0.21/2.57/46	13.70/0.33/2.28/41
STDC	25.85/0.16/44.38/80	30.74/0.09/38.32/80	22.25/0.19/50.14/80	23.10/0.11/38.41/80
TNN	11.31/0.87/1.40/21	32.23/0.07/6.68/76	21.08/0.22/5.50/71	11.72/0.41/1.51/21
IpST	11.31/0.87/1.62/20	33.02/0.07/3.30/37	22.63/0.18/3.99/45	11.72/0.41/1.73/20
BCPF	11.30/0.87/195.41/13	28.01/0.13/177.45/12	20.66/0.23/82.79/12	11.61/0.42/156.17/11

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