Subspace-based coupled tensor decomposition for hyperspectral blind fusion

Kunjing Yang; Minru Bai; Renwei Dian; Ting Lu

doi:10.3934/ipi.2024045

Article Contents

2025, Volume 19, Issue 3: 560-591. Doi: 10.3934/ipi.2024045

This issue Previous Article Dynamic MRI reconstruction via weighted nuclear norm and total variation regularization Next Article Sampling strategies in Bayesian inversion: A study of RTO and Langevin methods

Subspace-based coupled tensor decomposition for hyperspectral blind fusion

1.
School of Mathematics, Hunan University, Changsha, Hunan, China
2.
School of Robotics, Hunan University, Changsha, Hunan, China
3.
College of Electrical and Information Engineering, Hunan University, Changsha, Hunan, China

^*Corresponding author: Minru Bai
^*Corresponding author: Minru Bai

Received: December 2023

Revised: August 2024

Early access: November 2024

Published: June 2025

The second author is supported by [Natural Science Foundations of China under Grant number 11971159, 12071399.].

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Abstract

Fusing hyperspectral images (HSIs) and multispectral images (MSIs) is a widely employed technique for obtaining high spatial resolution HSIs (HR-HSIs). Many Tucker decomposition-based methods have been proposed and achieved commendable performance. However, the spatial-spectral correlation of HSIs may not be fully leveraged. To address this issue, we initially divide the HR-HSI into a coefficient tensor and a dictionary by utilizing its spectral correlation. Then, the coefficient tensor is further decomposed into three factor tensors through triple decomposition, which not only fully explores the spatial structure of the image, but also reduces the dimensionality of the problem. Meanwhile, a triple rank dynamic update scheme is proposed for avoiding cumbersome parameter tuning processes. Additionally, the degradation operators can be updated as dictionaries within the framework of triple decomposition, rendering our approach applicable in blind fusion scenarios. Furthermore, nonlocal self-similarity is utilized to cluster similar cubes together, reinforcing the low-rank prior of the data, then we minimize tensor average rank by developing an equivalent surrogate function with a difference of convex (DC) structure. An effective Gauss-Seidel difference of convex algorithm (GS-DCA) is presented to solve the proposed model. Finally, we provide a detailed convergence analysis of each algorithm based on Kurdyka-Lojasiewicz (KL) property. Extensive numerical experiments on hyperspectral datasets verify the effectiveness of our method.

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Mathematics Subject Classification: Primary: 94A08; Secondary: 68U10.

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Figure 1. Low-rank triple decomposition

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Figure 2. (a) The relevance matrix of spectral bands of Pavia University in SNRh = 10dB. (b) The relation between $ r $ and PSNR under various noise levels in Indian Pines

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Figure 3. The reconstructed images and corresponding error maps of TriDNLR and Hysure in fully blind cases. Left: Pavia University (bands 20, 40 and 60). Right: Indian Pines (bands 30, 60 and 90). (a) Hysure. (b) TriDNLR. (c) Ground truth

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Figure 4. The first row shows the reconstructed images of the compared methods for Peppers at 5th, 10th, 15th bands in SNRh = 10dB and SNRm = 15dB, and the second row are corresponding error images of these methods. (a) STEREO. (b) SCOTT. (c)TriD. (d) TriDLR. (e) TriDNLR. (f) Ground truth

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Figure 5. The reconstructed images and error maps of the compared methods. (a) Hysure. (b) CSTF. (c) LTTR. (d) LTMR. (e) IRTenSR. (f) TriDNLR. (G) Ground truth

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Figure 6. The PSNR of different bands of the reconstructed HSIs obtained by compared methods. Left: Pavia University in the case of SNRh = 10dB and SNRm = 15dB. Right: Indian Pines in the case of SNRh = 15dB and SNRm = 20dB

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Table 1. The evaluation metrics of the compared methods (Nonblind)

Method	PSNR	SSIM	ERGAS	SAM	RMSE	TIMES
Pavia University
SCOTT	30.07	0.76	4.57	7.57	8.23	0.70s
TriD	30.30	0.78	3.76	7.11	8.04	0.65s
TriDLR	30.99	0.80	2.01	6.47	7.46	3.58s
TriDNLR	31.86	0.85	1.84	5.58	6.76	5.61s
Peppers
SCOTT	33.91	0.79	2.03	29.67	5.46	2.47s
TriD	35.48	0.85	1.69	25.40	4.72	1.24s
TriDLR	36.09	0.88	1.54	23.26	4.41	5.51s
TriDNLR	36.73	0.91	1.34	22.06	4.17	6.42s

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Table 2. The experimental results of TriDNLR and Hysure (Blind)

Dataset	PSNR	SSIM	ERGAS	SAM	RMSE	PSNR	SSIM	ERGAS	SAM	RMSE
TriDNLR						Hysure
Pavia U	26.81	0.78	3.33	9.37	13.82	25.67	0.61	4.06	14.45	17.67
Indian	25.27	0.73	3.83	6.02	14.62	24.52	0.60	3.99	5.97	16.45
Peppers	31.95	0.86	2.65	26.27	7.70	30.69	0.72	2.55	21.48	8.90

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Table 3. The experimental results on Pavia University in different noise levels (dB) for five compareds method (Semiblind: only the spectral degradation operator is known)

Noise level	Dataset	Evaluation	STEREO	SCOTT	TriD-Blind	TriDLR	TriDNLR
SNRh=2	Pavia U	PSNR$ \uparrow $	23.52	23.74	24.28	24.79	26.78
SNRm=5		SSIM$ \uparrow $	0.47	0.48	0.48	0.49	0.64
		ERGAS $ \downarrow $	4.79	2.33	4.39	4.18	3.28
		SAM $ \downarrow $	13.43	13.88	11.36	11.97	8.14
		RMSE $ \downarrow $	17.53	17.14	16.22	15.27	12.09
	Peppers	PSNR$ \uparrow $	30.10	27.74	30.78	31.36	32.79
		SSIM$ \uparrow $	0.74	0.73	0.74	0.76	0.81
		ERGAS $ \downarrow $	3.46	4.65	2.85	2.72	2.27
		SAM $ \downarrow $	30.17	29.23	28.77	28.82	27.58
		RMSE $ \downarrow $	8.48	10.83	7.67	7.61	6.20
SNRh=10	Pavia U	PSNR$ \uparrow $	27.14	29.00	30.05	30.75	31.55
SNRm=15		SSIM$ \uparrow $	0.69	0.76	0.77	0.80	0.84
		ERGAS $ \downarrow $	3.17	1.28	2.27	2.10	1.90
		SAM $ \downarrow $	9.76	8.73	7.22	6.20	5.41
		RMSE $ \downarrow $	11.56	9.23	8.39	7.65	6.97
	Peppers	PSNR$ \uparrow $	33.88	33.04	34.31	34.69	35.44
		SSIM$ \uparrow $	0.85	0.78	0.85	0.88	0.90
		ERGAS $ \downarrow $	2.12	2.20	1.89	1.83	1.65
		SAM $ \downarrow $	21.46	27.58	25.19	23.13	22.87
		RMSE $ \downarrow $	5.41	5.46	5.29	5.27	4.72
SNRh=15	Pavia U	PSNR $ \uparrow $	28.92	32.52	32.95	33.31	34.01
SNRm=20		SSIM $ \uparrow $	0.77	0.85	0.86	0.88	0.90
		ERGAS $ \downarrow $	2.59	0.87	1.64	1.56	1.44
		SAM $ \downarrow $	8.42	6.29	5.46	4.88	4.49
		RMSE $ \downarrow $	9.39	6.26	5.98	5.73	5.34
	Peppers	PSNR$ \uparrow $	35.08	35.92	35.67	36.73	37.36
		SSIM$ \uparrow $	0.88	0.86	0.87	0.92	0.93
		ERGAS $ \downarrow $	1.73	1.72	1.60	1.39	1.32
		SAM $ \downarrow $	20.75	25.32	25.06	21.60	22.38
		RMSE $ \downarrow $	4.67	4.43	4.63	4.27	3.98

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Table 4. Evaluation metrics of different methods on three HSI datasets (Nonblind)

Noise level	Dataset	Evaluation	Hysure	CSTF	LTTR	LTMR	IRTenSR	TriDNLR
SNRh=2	Pavia U	PSNR $ \uparrow $	25.12	25.35	20.73	25.42	24.57	26.88
SNRm=5		SSIM $ \uparrow $	0.63	0.58	0.44	0.54	0.59	0.65
		ERGAS $ \downarrow $	4.06	3.92	6.82	3.97	4.36	3.26
		SAM $ \downarrow $	9.52	10.04	23.79	11.64	9.51	8.06
		RMSE $ \downarrow $	14.40	14.14	23.54	13.85	15.22	11.92
	Indian	PSNR $ \uparrow $	22.84	23.13	14.31	22.18	22.99	24.03
		SSIM $ \uparrow $	0.57	0.52	0.13	0.50	0.52	0.59
		ERGAS $ \downarrow $	5.08	4.85	13.39	5.50	5.14	4.42
		SAM $ \downarrow $	8.32	7.74	23.68	9.49	7.97	7.03
		RMSE $ \downarrow $	19.13	18.37	49.21	20.40	18.97	16.87
SNRh=10	Pavia U	PSNR $ \uparrow $	28.43	30.14	27.78	30.65	29.97	31.86
SNRm=15		SSIM $ \uparrow $	0.80	0.78	0.77	0.80	0.81	0.85
		ERGAS $ \downarrow $	2.79	2.12	3.07	2.11	2.34	1.84
		SAM $ \downarrow $	6.59	6.74	10.12	7.09	6.82	5.52
		RMSE $ \downarrow $	9.81	8.61	10.82	7.33	8.20	6.76
	Indian	PSNR $ \uparrow $	26.01	25.85	24.95	26.26	25.04	27.39
		SSIM $ \uparrow $	0.70	0.71	0.66	0.72	0.66	0.75
		ERGAS $ \downarrow $	3.46	3.81	4.02	3.40	4.07	3.08
		SAM $ \downarrow $	5.78	5.91	6.83	6.03	6.41	5.36
		RMSE $ \downarrow $	13.57	14.44	15.10	12.92	14.98	11.88
SNRh=15	Pavia U	PSNR $ \uparrow $	32.14	33.06	30.01	33.42	33.54	34.69
SNRm=20		SSIM $ \uparrow $	0.89	0.87	0.78	0.87	0.91	0.92
		ERGAS $ \downarrow $	1.83	1.58	4.79	1.65	1.59	1.34
		SAM $ \downarrow $	5.65	5.56	9.40	5.50	4.66	4.40
		RMSE $ \downarrow $	6.56	5.71	8.08	5.47	5.60	4.86
	Indian	PSNR $ \uparrow $	28.15	27.64	27.09	28.02	27.89	29.27
		SSIM $ \uparrow $	0.79	0.76	0.75	0.80	0.78	0.82
		ERGAS $ \downarrow $	2.75	3.00	3.15	2.76	2.99	2.59
		SAM $ \downarrow $	4.99	5.34	5.39	5.10	4.58	4.78
		RMSE $ \downarrow $	10.78	11.16	11.97	10.96	11.28	10.10

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