Weighted Sparse Cauchy Nonnegative Matrix Factorization Hyperspectral Unmixing Based on Spatial-Spectral Constraints

Shanxue Chen; Zhiyuan Hu

doi:10.3788/LOP213319

Laser & Optoelectronics Progress, Volume. 60, Issue 10, 1028006(2023)

Weighted Sparse Cauchy Nonnegative Matrix Factorization Hyperspectral Unmixing Based on Spatial-Spectral Constraints

Shanxue Chen^1,2 and Zhiyuan Hu^1,3、*

¹Chongqing University of Posts and Telecommunications, School of Communication and Information Engineering, Chongqing, 400065, China

²Engineering Research Center of Mobile Communications of the Ministry of Education, Chongqing, 400065, China

³Chongqing Key Laboratory of Mobile Communications Technology, Chongqing, 400065, China

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Figures & Tables(16)

Fig. 1. Comparison of different loss functions

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Fig. 2. Spectra of five ground objects in simulated data set

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Fig. 3. Average SAD and RMSE on Jasper Ridge data set for different $α$ and $β$

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Fig. 4. Comparison of SSCNMF algorithm for extracting endmembers on Jasper Ridge data set with real endmembers

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Fig. 5. Comparison of abundance maps on Jasper Ridge data set with real abundance maps by different algorithms

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Fig. 6. Comparison of SSCNMF algorithm for extracting endmembers on Urban data set with real endmembers

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Fig. 7. Comparison of abundance maps on Urban data set with real abundance maps by different algorithms

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Table 1. SSCNMF algorithm process

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Table 1. SSCNMF algorithm process

Input：hyperspectral image matrix $R$ ，parameters $δ$ ， $α$ ，and $β$ ；

Initialization：initialize end element matrix $W^{0}$ and abundance matrix $H^{0}$ using VCA-FCLS；

Step1：calculate error matrix $E^{t} = \frac{R - W^{t} H^{t}}{γ_{t}}$ ；

Step2：calculate auxiliary matrix $X_{i j}^{t} = \frac{1}{1 + {(E_{i j}^{t})}^{2}}$ ；

Step3：calculate $e^{t} = \frac{1}{L * N} \sum R_{i j}$ ；

Step4：calculate $γ_{t + 1} = γ_{t} * \sqrt[]{\frac{1}{e^{0} - 1}}$ ；

Step5： $X^{t}$ cancel outliers to $X^{t + 1}$ ；

Step6：applying ASC constraints $R_{f} = [\begin{matrix} R \\ δ 1_{n}^{T} \end{matrix}]$ ， $W_{f} = [\begin{matrix} W \\ δ 1_{n}^{T} \end{matrix}]$ ， $X_{f} = [\begin{matrix} X \\ δ 1_{n}^{T} \end{matrix}]$ ；

Step7：update abundance matrix $H_{k j}^{t + 1}$ according to Eq.（20）；

Step8：update error matrix $E^{t + 1} = \frac{R - W^{t} H^{t + 1}}{γ^{t}}$ ；

Step9：update auxiliary matrix $X_{i j}^{t + 1} = \frac{1}{1 + {(E_{i j}^{t + 1})}^{2}}$ ；

Step10：update element matrix $W_{i k}^{t + 1}$ according to Eq.（22）；

Repeat above steps until stop condition is met；

Output：element matrix $W$ and abundance matrix $H$ .

Table 2. SAD values after adding different levels of Gaussian white noise to each algorithm

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Table 2. SAD values after adding different levels of Gaussian white noise to each algorithm

SNR /dB	MVCNMF	L_1/2-NMF	Cauchy NMF	SSRNMF	SSWNMF	SSCNMF
10	0.2573	0.2674	0.2466	0.2213	0.2160	0.2042
15	0.2041	0.1901	0.1843	0.1594	0.1565	0.1505
20	0.1655	0.1518	0.1511	0.0913	0.0944	0.0889
25	0.1098	0.0942	0.1035	0.0811	0.0735	0.0679
30	0.0917	0.0886	0.0872	0.0641	0.0537	0.0503

Table 3. RMSE values after adding different levels of Gaussian white noise to each algorithm

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Table 3. RMSE values after adding different levels of Gaussian white noise to each algorithm

SNR /dB	MVCNMF	L_1/2-NMF	Cauchy NMF	SSRNMF	SSWNMF	SSCNMF
10	0.4274	0.4118	0.4208	0.3473	0.3358	0.3069
15	0.3781	0.3710	0.3543	0.2873	0.2569	0.2381
20	0.2718	0.2556	0.2371	0.1519	0.1501	0.1477
25	0.1405	0.1289	0.1421	0.1088	0.0912	0.0784
30	0.1175	0.1052	0.1056	0.0733	0.0698	0.0595

Table 4. SAD values after adding salt and pepper noise of different densities to each algorithm
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Table 4. SAD values after adding salt and pepper noise of different densities to each algorithm
D MVCNMF L_1/2-NMF Cauchy NMF SSRNMF SSWNMF SSCNMF
0.1 0.1565 0.1497 0.0831 0.0990 0.1074 0.0579
0.2 0.1996 0.1755 0.1463 0.1529 0.1406 0.0892
0.3 0.2136 0.2438 0.1986 0.1820 0.1885 0.1002
0.4 0.3853 0.3313 0.2965 0.2999 0.2734 0.1282

Table 5. RMSE values after adding salt and pepper noise of different densities to each algorithm
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Table 5. RMSE values after adding salt and pepper noise of different densities to each algorithm
D MVCNMF L_1/2-NMF Cauchy NMF SSRNMF SSWNMF SSCNMF
0.1 0.1281 0.1209 0.1093 0.0891 0.0857 0.0567
0.2 0.1687 0.1524 0.1678 0.1356 0.1426 0.0722
0.3 0.2349 0.2388 0.2116 0.1982 0.2031 0.1274
0.4 0.2968 0.2784 0.2849 0.2523 0.2382 0.1519

Table 6. SAD values of different algorithms on Jasper Ridge data set

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Table 6. SAD values of different algorithms on Jasper Ridge data set

Category	MVCNMF	L_1/2-NMF	Cauchy NMF	SSRNMF	SSWNMF	SSCNMF
Tree	0.1474	0.1316	0.0913	0.1270	0.1112	0.1078
Water	0.1168	0.1239	0.1279	0.0792	0.0875	0.0558
Soil	0.0889	0.1179	0.1028	0.1457	0.0845	0.1203
Road	0.1358	0.1146	0.1256	0.1231	0.0981	0.0886
Mean	0.1224	0.1220	0.1119	0.1187	0.0953	0.0931

Table 7. RMSE values of different algorithms on Jasper Ridge data set
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Table 7. RMSE values of different algorithms on Jasper Ridge data set
Category MVCNMF L_1/2-NMF Cauchy NMF SSRNMF SSWNMF SSCNMF
Mean 0.2223 0.2011 0.2022 0.1871 0.1431 0.1367

Table 8. SAD values of different algorithms on Urban data set

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Table 8. SAD values of different algorithms on Urban data set

Category	MVCNMF	L_1/2-NMF	Cauchy NMF	SSRNMF	SSWNMF	SSCNMF
Asphalt	0.2445	0.3580	0.2267	0.2772	0.2310	0.1806
Glass	0.3683	0.3324	0.2863	0.2118	0.2027	0.2015
Tree	0.2874	0.1935	0.3473	0.1347	0.1965	0.2204
Roof	0.1904	0.1471	0.2481	0.1923	0.1618	0.1647
Mean	0.2727	0.2578	0.2771	0.2040	0.1980	0.1918

Table 9. RMSE values of different algorithms on Urban data set
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Table 9. RMSE values of different algorithms on Urban data set
Category MVCNMF L_1/2-NMF Cauchy NMF SSRNMF SSWNMF SSCNMF
Mean 0.3629 0.3408 0.3426 0.3288 0.2623 0.2594

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Shanxue Chen, Zhiyuan Hu. Weighted Sparse Cauchy Nonnegative Matrix Factorization Hyperspectral Unmixing Based on Spatial-Spectral Constraints[J]. Laser & Optoelectronics Progress, 2023, 60(10): 1028006

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Paper Information

Category: Remote Sensing and Sensors

Received: Dec. 23, 2021

Accepted: Feb. 25, 2022

Published Online: Apr. 24, 2023

The Author Email: Zhiyuan Hu (308776453@qq.com)

DOI:10.3788/LOP213319

Topics

laser devices and laser physics

Lasers and Laser Optics

Laser physics

laser manufacturing

Instrumentation, Measurement and Metrology

Table 1. SSCNMF algorithm process

Table 1. SSCNMF algorithm process

Table 2. SAD values after adding different levels of Gaussian white noise to each algorithm

Table 2. SAD values after adding different levels of Gaussian white noise to each algorithm

Table 3. RMSE values after adding different levels of Gaussian white noise to each algorithm

Table 3. RMSE values after adding different levels of Gaussian white noise to each algorithm

Table 4. SAD values after adding salt and pepper noise of different densities to each algorithm

Table 4. SAD values after adding salt and pepper noise of different densities to each algorithm

Table 5. RMSE values after adding salt and pepper noise of different densities to each algorithm

Table 5. RMSE values after adding salt and pepper noise of different densities to each algorithm

Table 6. SAD values of different algorithms on Jasper Ridge data set

Table 6. SAD values of different algorithms on Jasper Ridge data set

Table 7. RMSE values of different algorithms on Jasper Ridge data set

Table 7. RMSE values of different algorithms on Jasper Ridge data set

Table 8. SAD values of different algorithms on Urban data set

Table 8. SAD values of different algorithms on Urban data set

Table 9. RMSE values of different algorithms on Urban data set

Table 9. RMSE values of different algorithms on Urban data set