CN112949698A - Hyperspectral unmixing method based on non-local low-rank tensor similarity constraint - Google Patents

Hyperspectral unmixing method based on non-local low-rank tensor similarity constraint Download PDF

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CN112949698A
CN112949698A CN202110172750.7A CN202110172750A CN112949698A CN 112949698 A CN112949698 A CN 112949698A CN 202110172750 A CN202110172750 A CN 202110172750A CN 112949698 A CN112949698 A CN 112949698A
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袁媛
董乐
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Abstract

The invention discloses a hyperspectral unmixing method of non-local low-rank tensor similarity constraint, which is used for solving the limitation of a hyperspectral mixed pixel decomposition algorithm to a specific problem, and comprises the steps of firstly expressing a hyperspectral image as a linear combination of an end member and an abundance tensor, then extracting local similarity characteristics in the hyperspectral image, and simulating the local similarity characteristics by adopting a variational deviation regularization term; and finally, extracting image three-dimensional blocks in the hyperspectral image, performing similarity clustering on the image three-dimensional blocks to obtain a fourth-order tensor similarity group, designing a low-rank regular term expression of a fourth-order tensor to constrain all non-local similarities existing in the image, and finally solving to obtain an end member and abundance matrix. The method combines the spectral information and the space of the hyperspectral data, and explores how to fully utilize the spatial structure in the data to assist the non-negative tensor decomposition framework in the unmixing process, so that the method has high unmixing performance.

Description

Hyperspectral unmixing method based on non-local low-rank tensor similarity constraint
Technical Field
The invention belongs to the technical field of image processing, and particularly relates to a hyperspectral unmixing method.
Background
Hyperspectral images (HSI) are entering the field of view of more and more researchers with their superior ability to acquire spectral information and have been successfully applied in many areas. However, due to the objective limitations of imaging conditions, the pixels of hyperspectral images are typically mixed from a variety of different pure substances. Therefore, unmixing is introduced into the image processing task to improve the application efficiency of the hyperspectral image, and the aim is to extract pure substances in the image and determine the proportion distribution of each component on the ground. Colloquially, pure ground objects are called end-members and their fractional fraction in each picture element is called abundance.
From a statistical analysis point of view, unmixing can be seen as a blind source separation problem. non-Negative Matrix Factorization (NMF) is one of the most popular methods for solving this problem, with the goal of finding a set of bases for the data and the expression coefficients of the data under the bases. However, in the NMF framework, since the three-dimensional HSI needs to be converted into a two-dimensional matrix first, this directly destroys the apparent positional relationship of the pixels existing in the original space. In addition, the data is converted into a two-dimensional matrix, so that the corresponding relation between the spectral information and the spatial information is weakened.
To solve the above problem, a method based on non-negative tensor decomposition (NTF) has been used in recent years to directly perform mixed pixel decomposition on three-dimensional hyperspectral data. A hyperspectral image stereoscopic block can be regarded as a third-order tensor without any information loss. Therefore, the unmixing is considered as a non-negative tensor decomposition problem which can more accurately describe the original spatial structure of the data and is a more natural model. Compared with a matrix and tensor, the matrix and tensor are more suitable for describing high-dimensional data, and compared with a nonnegative matrix decomposition framework, the matrix and tensor can improve the performance of a unmixing algorithm to a certain extent.
Along with the development of image processing technology, the unmixing performance based on the NTF framework is continuously improved. When tensor Decomposition is initially applied for mixed pel Decomposition, the most basic Tucker Decomposition and CP Decomposition (Canonical polymeric Decomposition) are mostly used. Classical Methods such as "c. chattichritos, e.kofidis, m. morante, and s. theodoris, Blind fmri source un-mixing via high-order regulator complexes, Journal of Neuroscience Methods, 2018" determine the coding coefficient obtained after CPD decomposition as an abundance map, and consider the number of rank-one-quantity after decomposition as the number of end-members. However, CPD-based methods decompose the data into a sum of a finite rank-tensor, which is usually much larger than the number of end-members. In addition, in an actual hyperspectral image, neither the end-members nor the rank of the abundance component is 1. These causes cause difficulty in associating the respective quantities with the end-members and the abundance map, respectively, after the CPD decomposition, and also cause poor unmixing accuracy. In "y.qian, f.xiong, s.zeng, j.zhou, and y.y.tang, Matrix-vector non-guided tensor factorization for blinding non-dispersive of hyperspectral image, IEEE Transactions on Geoscience and remove Sensing, vol.55, No.3, pp.1776-1792,2017", a Matrix-vector based tensor unmixing method is proposed which decomposes the data of the third order tensor into the sum of several component tensors, each component tensor being the outer product of an end-member vector and an abundance Matrix. However, the method ignores the spatial similarity of the hyperspectral data, and the data have smoothness on a local space according to the known prior information. Thus, "f.xiong, y.qian, j.zhou, and y.y.tang, Hyperspectral non-mixed vision total variation regulated non-negative transducer orientation, IEEE Transactions on Geoscience and remove Sensing, vol.57, No.4, pp.2341-2357,2019," based on the previous study, add global bias constraint terms to the NTF framework, further constraining the similarity characteristics of the abundance map in local space. Although the method combines spatial and spectral information to constrain an NTF framework to improve the unmixing performance, the problem that the internal spatial structure of the original hyperspectral data cannot be sufficiently explored still exists, so that an unmixing algorithm based on the NTF method needs to be further explored.
Disclosure of Invention
In order to overcome the defects of the prior art, the invention provides a hyperspectral unmixing method of non-local low-rank tensor similarity constraint, which aims to solve the limitation of a hyperspectral mixed pixel decomposition algorithm to a specific problem, firstly expresses a hyperspectral image as a linear combination of an end member and an abundance tensor, then extracts local similarity characteristics in the hyperspectral image, and simulates the local similarity characteristics by adopting a variation deviation regular term; and finally, extracting image three-dimensional blocks in the hyperspectral image, performing similarity clustering on the image three-dimensional blocks to obtain a fourth-order tensor similarity group, designing a low-rank regular term expression of a fourth-order tensor to constrain all non-local similarities existing in the image, and finally solving to obtain an end member and abundance matrix. The method combines the spectral information and the space of the hyperspectral data, and explores how to fully utilize the spatial structure in the data to assist the non-negative tensor decomposition framework in the unmixing process, so that the method has high unmixing performance.
The technical scheme adopted by the invention for solving the technical problem comprises the following steps:
step 1: inputting an original hyperspectral image to be unmixed
Figure BDA0002939294520000021
According to a linear unmixing model
Figure BDA0002939294520000022
Expressed as an end-member matrix
Figure BDA0002939294520000023
Sum abundance tensor
Figure BDA0002939294520000024
Linear combination of (a):
Figure BDA0002939294520000025
wherein
Figure BDA0002939294520000026
Representing noise in an original hyperspectral image to be unmixed;
the unmixed model objective function based on the nonnegative tensor decomposition is as follows:
Figure BDA0002939294520000031
Figure BDA00029392945200000323
wherein, P represents the number of the end members,
Figure BDA0002939294520000032
tensor of representation abundance
Figure BDA00029392945200000322
The matrix obtained after expansion along the third dimension, I1,I2,I3Respectively representing original hyperspectral images to be unmixed
Figure BDA0002939294520000033
Length, width and height dimensions;
step 2: maintaining local smoothness of the image by adopting a variation deviation regular term;
calculating variation deviation of an original hyperspectral image to be unmixed in the horizontal and vertical directions:
Figure BDA0002939294520000034
Figure BDA0002939294520000035
wherein the content of the first and second substances,
Figure BDA0002939294520000036
and
Figure BDA0002939294520000037
respectively representing original hyperspectral images to be unmixed
Figure BDA0002939294520000038
At spatial position (i)1,i2Variation deviation in the horizontal and vertical directions at p);
mixing L with1The regularization term is applied to equation (2), resulting in:
Figure BDA0002939294520000039
wherein the content of the first and second substances,
Figure BDA00029392945200000310
a variation deviation regular term is represented,
Figure BDA00029392945200000311
is composed of
Figure BDA00029392945200000312
And
Figure BDA00029392945200000313
synthesizing;
biasing variational variables by a regularization term
Figure BDA00029392945200000314
The addition to equation (1) enables the abundance tensor to maintain local smoothness during the unmixing process, the objective function becomes:
Figure BDA00029392945200000315
Figure BDA00029392945200000316
wherein λ isTVA weight coefficient representing a variation deviation regular term;
and step 3: the original hyperspectral image to be unmixed
Figure BDA00029392945200000317
The method comprises sliding and dividing the two dimensions of length and width by using a window with the step length of q and the size of p multiplied by p, keeping the height direction unchanged, and dividing the obtained object
Figure BDA00029392945200000318
Decomposition into p × p × I3A dimensional block of a size;
then, clustering the image blocks by using a K-means + + algorithm to obtain K fourth-order tensor similarity groups, wherein the K-th similarity group is expressed as:
Figure BDA00029392945200000319
wherein N iskRepresenting the number of image blocks in each similarity group of the fourth order tensor,
Figure BDA00029392945200000320
a set representing similarity blocks included in the kth group;
defining non-local low tensor rank regularization terms
Figure BDA00029392945200000321
The simulated abundance tensor is low-rank, and is shown in a formula (4):
Figure BDA0002939294520000041
wherein ls (a) is defined as:
Figure BDA0002939294520000042
A(i)represents the ith slice, σ, obtained along the third dimension by a three-dimensional tensor formed by expanding the 4 th order tensor in the fourth dimensioni(A) The ith characteristic value of A is shown, and epsilon is a non-negative adjusting parameter;
rank regularization term of low tensor
Figure BDA0002939294520000043
Adding the obtained product to the formula (3) to obtain a final objective function:
Figure BDA0002939294520000044
Figure BDA0002939294520000045
wherein λ isNLWeight coefficients representing a non-local low tensor rank regularization term,
Figure BDA0002939294520000046
an ith band representing a kth similarity block;
and 4, step 4: and (3) carrying out iterative solution on the objective function formula (5) by using an ADMM algorithm, wherein the objective function is deformed into:
Figure BDA0002939294520000047
Figure BDA0002939294520000048
Figure BDA0002939294520000049
Figure BDA00029392945200000410
Figure BDA00029392945200000411
Figure BDA00029392945200000412
Figure BDA00029392945200000413
wherein
Figure BDA00029392945200000414
Solving the values of the variables as follows:
Figure BDA00029392945200000415
Figure BDA00029392945200000416
Figure BDA00029392945200000417
Figure BDA00029392945200000418
Figure BDA00029392945200000419
Figure BDA00029392945200000420
where μ denotes the Lagrangian multiplier,
Figure BDA0002939294520000051
and 5: is obtained by the step 4
Figure BDA0002939294520000052
Final end member matrix M and abundance tensor after unmixing
Figure BDA0002939294520000053
Figure BDA0002939294520000054
Figure BDA0002939294520000055
Step 6: evaluating the performance of the unmixing method;
and (3) quantitatively evaluating the end member matrix M obtained in the step (5) by taking the spectrum angle SAD as an unmixing precision evaluation index, wherein the evaluation index is expressed as:
Figure BDA0002939294520000056
wherein
Figure BDA0002939294520000057
Representing a true end-member matrix; the value of the spectral angle SAD is positively correlated with the performance of the unmixing method.
The invention has the following beneficial effects:
the method treats the hyperspectral mixed pixel decomposition task as a high-order data tensor decomposition problem, can completely keep the spectrum and spatial information of original data in the unmixing process, explores the spatial structure of the data by using a low-rank tensor regular term and a global variation regular term to further constrain an NTF (normalized root class function) frame, and can effectively improve the unmixing performance and the robustness.
Drawings
FIG. 1 is a flow chart of the method of the present invention.
Fig. 2 is a schematic diagram of Jasper ridge data in the embodiment of the present invention.
Fig. 3 is a diagram illustrating Samon data according to an embodiment of the present invention.
FIG. 4 is a graph of abundance of end members after unmixing Jasper ridge data in an embodiment of the present invention.
FIG. 5 is a graph of end-member spectra after unmixing Jasper ridge data in an embodiment of the present invention.
FIG. 6 is a graph of abundance of each end-member after unmixing the Samon data in an example of the invention.
FIG. 7 is a graph of the spectra of the end members of the Samon data after unmixing in an embodiment of the present invention.
Detailed Description
The invention is further illustrated with reference to the following figures and examples.
As shown in fig. 1, a robust multi-temporal multispectral image change detection method includes the following steps:
step 1: inputting an original hyperspectral image to be unmixed
Figure BDA0002939294520000061
According to a linear unmixing model
Figure BDA0002939294520000062
Expressed as an end-member matrix
Figure BDA0002939294520000063
Sum abundance tensor
Figure BDA0002939294520000064
Linear combination of (a):
Figure BDA0002939294520000065
wherein
Figure BDA0002939294520000066
Representing noise in an original hyperspectral image to be unmixed;
the unmixed model objective function based on the nonnegative tensor decomposition is as follows:
Figure BDA0002939294520000067
Figure BDA0002939294520000068
wherein, P represents the number of the end members,
Figure BDA0002939294520000069
tensor of representation abundance
Figure BDA00029392945200000610
The matrix obtained after expansion along the third dimension, I1,I2,I3Respectively representing the original hyperspectrum to be unmixedImage of a person
Figure BDA00029392945200000611
Length, width and height dimensions;
step 2: maintaining local smoothness of the image by adopting a variation deviation regular term;
calculating variation deviation of an original hyperspectral image to be unmixed in the horizontal and vertical directions:
Figure BDA00029392945200000612
Figure BDA00029392945200000613
wherein the content of the first and second substances,
Figure BDA00029392945200000614
and
Figure BDA00029392945200000615
respectively representing original hyperspectral images to be unmixed
Figure BDA00029392945200000616
At spatial position (i)1,i2Variation deviation in the horizontal and vertical directions at p);
mixing L with1The regularization term is applied to equation (2), resulting in:
Figure BDA00029392945200000617
wherein the content of the first and second substances,
Figure BDA00029392945200000618
a variation deviation regular term is represented,
Figure BDA00029392945200000619
is composed of
Figure BDA00029392945200000620
And
Figure BDA00029392945200000621
synthesizing;
biasing variational variables by a regularization term
Figure BDA00029392945200000622
The addition to equation (1) enables the abundance tensor to maintain local smoothness during the unmixing process, the objective function becomes:
Figure BDA00029392945200000623
Figure BDA00029392945200000624
wherein λ isTVA weight coefficient representing a variation deviation regular term;
and step 3: the original hyperspectral image to be unmixed
Figure BDA0002939294520000071
The method comprises sliding and dividing the two dimensions of length and width by using a window with the step length of q and the size of p multiplied by p, keeping the height direction unchanged, and dividing the obtained object
Figure BDA0002939294520000072
Decomposition into p × p × I3A dimensional block of a size;
then, clustering the image blocks by using a K-means + + algorithm to obtain K fourth-order tensor similarity groups, wherein the K-th similarity group is expressed as:
Figure BDA0002939294520000073
wherein N iskRepresenting the number of image blocks in each similarity group of the fourth order tensor,
Figure BDA0002939294520000074
a set representing similarity blocks included in the kth group;
defining non-local low tensor rank regularization terms
Figure BDA0002939294520000075
The simulated abundance tensor is low-rank, and is shown in a formula (4):
Figure BDA0002939294520000076
wherein ls (a) is defined as:
Figure BDA0002939294520000077
A(i)represents the ith slice, σ, obtained along the third dimension by a three-dimensional tensor formed by expanding the 4 th order tensor in the fourth dimensioni(A) The ith characteristic value of A is shown, and epsilon is a non-negative adjusting parameter;
rank regularization term of low tensor
Figure BDA0002939294520000078
Adding the obtained product to the formula (3) to obtain a final objective function:
Figure BDA0002939294520000079
Figure BDA00029392945200000710
wherein λ isNLWeight coefficients representing a non-local low tensor rank regularization term,
Figure BDA00029392945200000711
an ith band representing a kth similarity block;
and 4, step 4: and (3) carrying out iterative solution on the objective function formula (5) by using an ADMM algorithm, wherein the objective function is deformed into:
Figure BDA00029392945200000712
Figure BDA00029392945200000713
Figure BDA00029392945200000714
Figure BDA00029392945200000715
Figure BDA00029392945200000716
Figure BDA00029392945200000717
Figure BDA00029392945200000718
wherein
Figure BDA0002939294520000081
Solving the values of the variables as follows:
Figure BDA0002939294520000082
Figure BDA0002939294520000083
Figure BDA0002939294520000084
Figure BDA0002939294520000085
Figure BDA0002939294520000086
Figure BDA0002939294520000087
where μ denotes the Lagrangian multiplier,
Figure BDA0002939294520000088
and 5: is obtained by the step 4
Figure BDA0002939294520000089
Final end member matrix M and abundance tensor after unmixing
Figure BDA00029392945200000810
Figure BDA00029392945200000811
Figure BDA00029392945200000812
Step 6: evaluating the performance of the unmixing method;
and (3) quantitatively evaluating the end member matrix M obtained in the step (5) by taking the spectrum angle SAD as an unmixing precision evaluation index, wherein the evaluation index is expressed as:
Figure BDA00029392945200000813
wherein
Figure BDA00029392945200000814
Representing a true end-member matrix; the value of the spectral angle SAD is positively correlated with the performance of the unmixing method.
The specific embodiment is as follows:
1. simulation conditions
The invention is in the central processing unit
Figure BDA0002939294520000091
And (3) simulating by using MATLAB software on an i 5-34703.2 GHz CPU and a memory 4G, WINDOWS 7 operating system. The data used in the simulation experiments are public database Jasper ridge data and Samon data as shown in fig. 2 and 3, respectively. The image size of Jasper ridge data is 100 pixels × 100 pixels, and includes 188 spectral bands at the 4 spectral end members. The image size of the Samon data is 95 pixels × 95 pixels, and includes a total of 156 spectral bands of 3 spectral end members.
2. Emulated content
And carrying out a simulation experiment of hyperspectral mixed pixel decomposition on the three simulation databases according to the following steps.
a) Firstly, on Jasper ridge data, obtaining an end member matrix and an abundance tensor according to the steps 1, 2, 3, 4 and 5 by adopting the method, and obtaining the unmixing precision of the method by utilizing the step 6;
secondly, existing algorithms SGSNMF, ULTRA-V, SULRSR-TV, NMF-QMV and MV-NTF are adopted on Jasper ridge data to perform unmixing respectively and obtain SAD results.
Among them, SGSNMF is proposed in the documents "X.Wang, Y.Zhong, L.Zhang, and Y.Xu", "Spatial group regulated non-reactive matrix catalysis for hyperspectral irradiation", "IEEE Transactions on Geoscience and remove Sensing, vol.55, No.11, pp.6287-6304,2017".
ULTRA-V is set forth in the documents "T.Imbiriba, R.A.Borsio, and J.C.M.Bermudez," Low-rank transducer modeling for hyperspectral approximation for spectral variability, "IEEE Transactions on Geoscience and Remote Sensing, vol.58, No.3, pp.1833-1842,2020".
SULRSR-TV is proposed in the literature "H.Li, R.Feng, L.Wang, Y.Zhong, and L.Zhang," Superpixel-based weighed low-rank and total variation mapping for hyperspectral Remote Sensing image, "IEEE Transactions on diagnostics and record Sensing, pp.1-19,2020.
NMF-QMV is proposed in the literature "L.Zhuang, C. -H.Lin, M.A.Figueiredo, and J.M.Bioucas-Dias", "Regulation parameter selection in minor volume hyperspectral subtraction", IEEE Transactions on Geoscience and Remote Sensing, vol.57, No.12, pp.9858-9877,2019 ".
MV-NTF is proposed in the literature "Qian, F.Xiong, S.Zeng, J.Zhou, and Y.Y.Tang," Matrix-vector non-inductive transducer factor for cladding non-dispersive imaging, "IEEE Transactions on Geoscience and Remote Sensing, vol.55, No.3, pp.1776-1792,2017".
The SAD calculation results are shown in table 1 and fig. 5.
TABLE 1 SAD results on Jasper edge data
SGSNMF ULTRA-V SULRSR-TV NMF-QMV MV-NTF NLTR
Tree 0.2069 0.1774 0.1783 0.2649 0.2512 0.2613
Water 0.2063 0.4246 0.2329 0.4267 0.2479 0.0793
Soil 0.1317 0.1499 0.1302 0.1699 0.1703 0.1614
Road 0.0555 0.2998 0.5109 0.0474 0.3889 0.0437
Mean 0.1626 0.2846 0.3017 0.4245 0.2760 0.1364
b) The same experimental procedure as a) was used to perform the experiments on Samon data and the SAD results are shown in table 2, fig. 6:
TABLE 2 SAD results on Samon data
Figure BDA0002939294520000101
As can be seen from the SAD results in tables 1 and 2 and the visualization results in fig. 4, 5, 6 and 7, compared with other existing methods, the present invention has significantly improved unmixing accuracy, because the present invention fully utilizes the information of the original hyperspectral data, explores the internal nonlocal similar characteristics of the data, maintains the local smoothness of the data, and obtains better hyperspectral unmixing accuracy for establishing a better unmixing model, further verifying the advancement of the present invention.

Claims (1)

1. A hyperspectral unmixing method based on non-local low-rank tensor similarity constraint is characterized by comprising the following steps:
step 1: inputting an original hyperspectral image to be unmixed
Figure FDA0002939294510000011
According to a linear unmixing model
Figure FDA0002939294510000012
Expressed as an end-member matrix
Figure FDA0002939294510000013
Sum abundance tensor
Figure FDA0002939294510000014
Linear combination of (a):
Figure FDA0002939294510000015
wherein
Figure FDA0002939294510000016
Representing noise in an original hyperspectral image to be unmixed;
the unmixed model objective function based on the nonnegative tensor decomposition is as follows:
Figure FDA0002939294510000017
Figure FDA0002939294510000018
wherein, P represents the number of the end members,
Figure FDA0002939294510000019
tensor of representation abundance
Figure FDA00029392945100000110
The matrix obtained after expansion along the third dimension, I1,I2,I3Respectively representing original hyperspectral images to be unmixed
Figure FDA00029392945100000111
Length, width and height dimensions;
step 2: maintaining local smoothness of the image by adopting a variation deviation regular term;
calculating variation deviation of an original hyperspectral image to be unmixed in the horizontal and vertical directions:
Figure FDA00029392945100000112
Figure FDA00029392945100000113
wherein the content of the first and second substances,
Figure FDA00029392945100000114
and
Figure FDA00029392945100000115
respectively representing original hyperspectral images to be unmixed
Figure FDA00029392945100000116
At spatial position (i)1,i2Variation deviation in the horizontal and vertical directions at p);
mixing L with1The regularization term is applied to equation (2), resulting in:
Figure FDA00029392945100000117
wherein the content of the first and second substances,
Figure FDA00029392945100000118
a variation deviation regular term is represented,
Figure FDA00029392945100000119
is composed of
Figure FDA00029392945100000120
And
Figure FDA00029392945100000121
synthesizing;
biasing variational variables by a regularization term
Figure FDA00029392945100000122
The addition to equation (1) enables the abundance tensor to maintain local smoothness during the unmixing process, the objective function becomes:
Figure FDA00029392945100000123
Figure FDA00029392945100000124
wherein λ isTVA weight coefficient representing a variation deviation regular term;
and step 3: the original hyperspectral image to be unmixed
Figure FDA00029392945100000125
The method comprises sliding and dividing the two dimensions of length and width by using a window with the step length of q and the size of p multiplied by p, keeping the height direction unchanged, and dividing the obtained object
Figure FDA0002939294510000021
Decomposition into p × p × I3A dimensional block of a size;
then, clustering the image blocks by using a K-means + + algorithm to obtain K fourth-order tensor similarity groups, wherein the K-th similarity group is expressed as:
Figure FDA0002939294510000022
wherein N iskRepresenting the number of image blocks in each similarity group of the fourth order tensor,
Figure FDA0002939294510000023
a set representing similarity blocks included in the kth group;
defining non-local low tensor rank regularization terms
Figure FDA0002939294510000024
The simulated abundance tensor is low-rank, and is shown in a formula (4):
Figure FDA0002939294510000025
wherein ls (a) is defined as:
Figure FDA0002939294510000026
A(i)represents the ith slice, σ, obtained along the third dimension by a three-dimensional tensor formed by expanding the 4 th order tensor in the fourth dimensioni(A) The ith characteristic value of A is shown, and epsilon is a non-negative adjusting parameter;
rank regularization term of low tensor
Figure FDA0002939294510000027
Adding the obtained product to the formula (3) to obtain a final objective function:
Figure FDA0002939294510000028
Figure FDA0002939294510000029
wherein λ isNLWeight coefficients representing a non-local low tensor rank regularization term,
Figure FDA00029392945100000210
an ith band representing a kth similarity block;
and 4, step 4: and (3) carrying out iterative solution on the objective function formula (5) by using an ADMM algorithm, wherein the objective function is deformed into:
Figure FDA00029392945100000211
Figure FDA00029392945100000212
Figure FDA00029392945100000213
Figure FDA00029392945100000214
Figure FDA00029392945100000215
Figure FDA00029392945100000216
Figure FDA00029392945100000217
wherein
Figure FDA00029392945100000218
Solving the values of the variables as follows:
Figure FDA0002939294510000031
Figure FDA0002939294510000032
Figure FDA0002939294510000033
Figure FDA0002939294510000034
Figure FDA00029392945100000313
Figure FDA0002939294510000035
where μ denotes the Lagrangian multiplier,
Figure FDA0002939294510000036
and 5: is obtained by the step 4
Figure FDA0002939294510000037
Final end member matrix M and abundance tensor after unmixing
Figure FDA0002939294510000038
Figure FDA0002939294510000039
Figure FDA00029392945100000310
Step 6: evaluating the performance of the unmixing method;
and (3) quantitatively evaluating the end member matrix M obtained in the step (5) by taking the spectrum angle SAD as an unmixing precision evaluation index, wherein the evaluation index is expressed as:
Figure FDA00029392945100000311
wherein
Figure FDA00029392945100000312
Representing a true end-member matrix; the value of the spectral angle SAD is positively correlated with the performance of the unmixing method.
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