Spectral weighted sparse unmixing of hyperspectral images based on framelet transform

Chenguang XU; Hongyu XU; Chunyan YU; Chengzhi DENG

doi:10.37188/OPE.20233109.1404

Journals >Optics and Precision Engineering >Volume 31 >Issue 9 >Page 1404 > Article

Optics and Precision Engineering
Vol. 31, Issue 9, 1404 (2023)

Spectral weighted sparse unmixing of hyperspectral images based on framelet transform

Chenguang XU, Hongyu XU, Chunyan YU, and Chengzhi DENG^*

Author Affiliations

Jiangxi Province Key Laboratory of Water Information Cooperative Sensing and Intelligent Processing，Nanchang Institute of Technology， Nanchang330099， China

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DOI: 10.37188/OPE.20233109.1404 Cite this Article

Chenguang XU, Hongyu XU, Chunyan YU, Chengzhi DENG. Spectral weighted sparse unmixing of hyperspectral images based on framelet transform[J]. Optics and Precision Engineering, 2023, 31(9): 1404 Copy Citation Text

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$True fractional abundances of endmembers in DC1$

Fig. 1. True fractional abundances of endmembers in DC1

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$True fractional abundances of endmembers in DC2$

Fig. 2. True fractional abundances of endmembers in DC2

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Fig. 3. Estimated abundance and difference maps of DC1 after unmixing by five methods under 40 dB

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Fig. 4. Estimated abundance and difference maps of DC2 after unmixing by five methods under 40 dB

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Fig. 5. True and best estimated abundance of DC1 and DC2 after unmixing by five methods under SNR=20 dB

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Fig. 6. Map of different minerals in Cuprite mining district in Nevada

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Fig. 7. Abundance map of cuprite data after unmixing with five methods

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1. Input： spectrum library $S$ and observed data $Y$ .

2. Initialize： $k = 0$ ， $t = 0$ ， $α > 0$ ， $λ > 0$ ， $μ > 0$ ， $U^{0} = {(S^{T} S + 2 I)}^{- 1} (S^{T} Y)$ ， $V_{1}^{0} = S U^{0}$ ， $V_{2}^{0} = W U^{0}$ ， $V_{3}^{0} = U^{0}$ ， $D_{1}^{0} = 0$ ， $D_{2}^{0} = 0$ ， $D_{3}^{0} = 0$ 。

3. Repeat until the stop condition is reached

（1） Update $Κ$ with formula （15），

（2） Repeat until the stop condition is reached

（a） Update variables $U_{}^{t + 1},$ $V_{1}^{t + 1},$ $V_{2}^{t + 1},$ $V_{3}^{t + 1}$ with formulas （19）-（22） respectively.

（b） Update Lagrangian operators：

$D_{1}^{t + 1} = D_{1}^{t} - (S U^{t + 1} - V_{1}^{t + 1}) .$

$D_{2}^{t + 1} = D_{2}^{t} - (W U^{t + 1} - V_{2}^{t + 1}) .$

$D_{3}^{t + 1} = D_{3}^{t} - (U^{t + 1} - V_{3}^{t + 1}) .$

（c） Update inner iteration times： $t = t + 1 .$

（d） variable replacement：

$U_{}^{k + 1} = U_{}^{t + 1}$ ， $D_{}^{k + 1} = D_{}^{t + 1}$ .

（3） Update outer iteration times： $k = k + 1 .$

4. Output： $X$ .

Table 1. Pseudo-code of SFSU

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Algorithms	SNR=20				SNR=30				SNR=40
Algorithms	$S R E / d B$	$p_{s}$	$λ$	$α$	$S R E / d B$	$p_{s}$	$λ$	$α$	$S R E / d B$	$p_{s}$	$λ$	$α$
SUnSAL	2.844 5	0.178 5	0.5	-	6.117 1	0.808 5	0.05	-	11.149 8	0.999 6	0.01	-
CLSUnSAL	5.911	0.927 5	20	-	7.515 0	0.967 8	1.5	-	14.853 7	1	0.5	-
RWCLSUnSAL	0.965 8	0.089 1	0.007	-	5.104 9	0.581 9	0.001	-	8.901 9	0.981 7	0.000 2	-
FSU	8.530 6	0.972 4	0.05	0.005	11.103 4	0.993 1	0.01	0.05	18.273 1	0.999 1	0.02	0.1
SFSU	11.018 6	0.988 6	0.05	0.001	20.871 7	1	0.05	0.01	26.979 0	1	0.07	0.08

Table 2. Results of DC1 after unmixing under different SNRs

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Algorithms	SNR=20				SNR=30				SNR=40
Algorithms	$S R E / d B$	$p_{s}$	$λ$	$α$	$S R E / d B$	$p_{s}$	$λ$	$α$	$S R E / d B$	$p_{s}$	$λ$	$α$
SUnSAL	4.216 7	0.559 7	0.2	-	8.475 8	0.798 3	0.02	-	14.271 2	0.990 2	0.005	-
CLSUnSAL	3.414 8	0.468 6	2.2	-	6.619 8	0.735 8	0.33	-	11.446 8	0.971 5	0.02	-
RWCLSUnSAL	3.683 2	0.533 5	0.03	-	8.185 3	0.812 7	0.01	-	14.868 4	0.973 90	0.000 1	-
FSU	7.447 3	0.744 1	0.05	0.005	12.627 5	0.967 4	0.01	0.05	16.341 2	0.999 6	0.001	0.1
SFSU	8.369 5	0.842 5	0.2	0.006	16.044 6	0.963 2	0.007	0.01	29.523 7	1	0.007	0.1

Table 3. Results of DC2 after unmixing under different SNRs

Chenguang XU, Hongyu XU, Chunyan YU, Chengzhi DENG. Spectral weighted sparse unmixing of hyperspectral images based on framelet transform[J]. Optics and Precision Engineering, 2023, 31(9): 1404

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