CN115527117A - Hyperspectral image anomaly detection method based on high-order tensor representation - Google Patents

Hyperspectral image anomaly detection method based on high-order tensor representation Download PDF

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CN115527117A
CN115527117A CN202211208070.7A CN202211208070A CN115527117A CN 115527117 A CN115527117 A CN 115527117A CN 202211208070 A CN202211208070 A CN 202211208070A CN 115527117 A CN115527117 A CN 115527117A
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tensor
hyperspectral image
order tensor
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陈希爱
范慧杰
田建东
贾慧迪
王震
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Shenyang Institute of Automation of CAS
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    • G06—COMPUTING; CALCULATING OR COUNTING
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    • G06V20/00—Scenes; Scene-specific elements
    • G06V20/10—Terrestrial scenes
    • G06V20/194—Terrestrial scenes using hyperspectral data, i.e. more or other wavelengths than RGB
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Abstract

The invention relates to a hyperspectral image anomaly detection method based on high-order Tensor representation, which introduces order-raising operation to perform high-order Tensor representation on data, and embeds proper regularization constraint into hyperspectral priori knowledge while keeping a hyperspectral integral structure by using a Tensor low-Tensor-Train rank approximate framework. Aiming at the space dimension segmented continuous prior of the background tensor, adopting a space spectrum total variation fractional norm regular term to express; aiming at the spectral dimension low-rank prior of the background, expressing by using logarithm and norm regular terms; aiming at the sparse prior of the existing group of the abnormal target tensor, L is adopted 2,1 And expressing a norm regular term. And finally, convex optimization is carried out on a target equation by adopting an alternating direction multiplier method under a tensor frame, the aim of effectively extracting an abnormal target is achieved, and the method is clear in principle, superior in experimental verification effect and strong in robustness. The invention provides theory, model and support for the processing and analysis of the hyperspectral remote sensing image by the tensor expression model.

Description

Hyperspectral image anomaly detection method based on high-order tensor representation
Technical Field
The invention relates to the field of remote sensing image processing, in particular to a hyperspectral image anomaly detection method based on high-order tensor representation.
Background
The hyperspectral image contains abundant spectral information, has the advantage of integrating maps in distinguishing different material characteristics, and the hyperspectral abnormal target detection has the unsupervised characteristic and is widely applied to the fields of military investigation, geological mining, post-disaster rescue and the like. However, in the traditional hyperspectral anomaly detection method, the original hyperspectral image is directly reduced into a two-dimensional matrix from a three-dimensional structure by using low-rank sparse decomposition, so that the space overall structure of the hyperspectral image is damaged, a large amount of false alarms are caused, and an anomaly target cannot be efficiently separated from a background. Therefore, how to effectively separate the abnormal target from the background by using the priori knowledge of the hyperspectral integrated atlas structure in the spatial domain and the spectral domain becomes a key problem while maintaining the hyperspectral integrated atlas structure.
The traditional method is to measure similarity such as distance, density, angle, cluster-based method, etc., and these algorithms perform close to each other in low dimension because the core assumption is that "the representation of the abnormal point is different from the normal point and is a minority of pie". Most similar algorithms face a dimensionality disaster, i.e., common similarity measures (such as euclidean distance) tend to fail on high-dimensional data. To solve this problem, many methods have been proposed including: dimension reduction or feature space selection methods, such as detection and merger on multiple low-dimensional spaces. Restoring the eigensubspace of the data distribution is a key step and data analysis in many applications of machine learning. One of the most common analytical methods is Principal Component Analysis (PCA). Standard PCA is sensitive to outliers: even a single but severe outlier may reduce the effectiveness of the model. Another problem with high dimensional data is scalability. The operation overhead for measuring the similarity is very large, the complexity of most distance measures is high, and the reduction of the complexity by using a data structure for optimization or dynamic planning is also a common exploration direction. The optimal situation is also the control dimension, and it is the root of the problem to find a better data representation.
The research introduces the upgrade operation to carry out high-order Tensor expression on the data, and the same hyperspectral integral structure is kept by using a Tensor low-Tensor-Train rank approximate frameworkThen, the appropriate regularization constraints are embedded into the high spectral prior knowledge. If the space dimension of the background tensor is segmented and continuously known a priori, a space spectrum fully-variable fractional norm regular term is adopted for expression; aiming at the spectral dimension low-rank prior of the background, expressing by using logarithm and norm regular terms; aiming at the sparse prior of the existing group of the abnormal target tensor, L is adopted 2,1 And expressing a norm regular term. And finally, convex optimization is carried out on a target equation by adopting an alternating direction multiplier method under a tensor frame, the aim of effectively extracting an abnormal target is achieved, and the method is clear in principle, superior in experimental verification effect and strong in robustness. The research is helpful for clarifying the relationship between various global priors and space local characteristics of the hyperspectral data and the abnormal detection capability, provides theories, models and supports for the tensor expression model in the hyperspectral remote sensing image processing and analysis, and provides a new thought for the hyperspectral image processing aspect in the future.
Disclosure of Invention
Aiming at the technical defects, the invention aims to provide a method for processing the abnormal detection problem of the hyperspectral image by using tensor low TT rank decomposition, space spectrum total variation and group sparseness based on logarithm and norm optimization. In order to fully utilize the high spatial-spectral correlation of the high-multispectral image, an order raising operation is introduced, high-order Tensor expression is carried out on data, and Tensor low-Tensor-Train decomposition is utilized to approximately recover the high-order Tensor data. Aiming at the space dimension subsection continuous prior of the background tensor, adopting a space spectrum total variation norm regular term to express; aiming at the spatial spectrum dimension low-rank prior of the background, expressing by using logarithm and norm regular terms; aiming at the sparse prior of the existing group of the abnormal target tensor, L is adopted 2,1 And expressing a norm regular term. And finally, under a tensor framework, convex optimization is carried out on the target function by adopting an alternative direction multiplier method, so that the aim of effectively extracting the abnormal target is fulfilled.
The technical scheme adopted by the invention for solving the technical problems is as follows:
the hyperspectral image anomaly detection method based on high-order tensor expression comprises the following steps of performing the following steps on the basis of an initial model, and then performing iterative optimization solution repeatedly to obtain a hyperspectral image anomaly detection model based on high-order tensor expression, so as to realize hyperspectral image anomaly detection, and comprises the following steps:
step 1, utilizing a group sparse regular term to mine the difference of the line space structure of each spectrum image and a shared sparse structure, and detecting an abnormal value;
step 2, performing rank approximation on the high-order tensor data by using the logarithm and the optimized high-order tensor weighted low TT rank, and mining the correlation contained in the tensor data to keep the background;
and 3, adopting a space spectrum total variation regularization term for depicting the spectrum space segmentation continuity.
The low-rank spectral image anomaly detection model carries out low-rank reconstruction on the spectral image and achieves hyperspectral image anomaly detection.
The establishment process of the prior initial model is as follows:
l is used to assume that the abnormal values are distributed along the second dimension of the tensor, and the abnormal values have sparseness compared with the data dimension 2,1 Norm, the anomaly detection problem for any higher order tensor is modeled as follows:
Figure BDA0003873556680000031
wherein the content of the first and second substances,
Figure BDA0003873556680000032
is an a priori constrained term of the data;
Figure BDA0003873556680000033
for the observation image to be processed of the input model,
Figure BDA0003873556680000034
for the purpose of background low-rank data,
Figure BDA0003873556680000035
abnormal data output for the model;
introducing a low order prior term
Figure BDA0003873556680000036
Obtaining a prior initial model:
Figure BDA0003873556680000037
the hyperspectral image anomaly detection model based on the high-order tensor expression is as follows:
Figure BDA0003873556680000038
Figure BDA0003873556680000039
wherein λ is t Is a total variation weight factor, λ r Is a low rank weighting factor, beta is a group sparsity weighting factor, mu is a Lagrange factor, alpha n Is a weight factor of the tensor mode matrix,
Figure BDA00038735566800000310
for the spatial spectrum total variation regularization term,
Figure BDA00038735566800000311
to weight the low TT rank terms with logarithms and optimized higher order tensors,
Figure BDA00038735566800000312
are group sparse regularization terms.
In the logarithm and optimized high-order tensor weighted low TT rank item, the high-order tensor is high-order tensor data obtained by conducting an upscaling operation on an image, and the logarithm and the norm are used for conducting tensor rank optimization approximation.
The weighting group sparse term is used for describing the smoothness of the spectral images of different spectral bands and mining the difference of the line space structures of the image blocks of the spectral bands and the shared sparse structure.
The space spectrum total variation regularization term is used for describing the continuity of spectrum space segmentation and inhibiting noise.
In the repeated iteration optimization solving process, when the error of the two adjacent image recovery result data is within the threshold range, the image is judged to be in accordance with the convergence from the image reconstruction to the current turn, and the iteration is stopped.
The error function is:
Figure BDA00038735566800000313
wherein the content of the first and second substances,
Figure BDA00038735566800000314
in order to input the observation data, it is,
Figure BDA00038735566800000315
for low rank data,. Epsilon.for anomalous data, and tol for a given threshold.
The repeated iteration optimization solution adopts an ADMM algorithm.
The invention has the following beneficial effects and advantages:
1. the method adopts logarithm and an optimized high-order tensor weighted low TT rank to carry out rank approximation on tensor data, and the correlation contained in the tensor data is mined.
2. The method adopts the group sparse regular term to excavate the difference of the line space structure of each spectrum image and the shared sparse structure, and detects abnormal values.
3. The method adopts a space spectrum total variation regularization term to depict the spectrum space segmentation continuity.
4. The method is superior to the existing algorithm in the aspect of hyperspectral anomaly detection, and has better low-rank reconstruction performance and robustness.
Drawings
FIG. 1 is a general framework diagram of the process herein;
FIG. 2 is an image of two scenes in a data set and a reference map thereof;
FIG. 3 is a qualitative detection graph of anomalous target detection for each of the prior art methods and the present method under the scenario of example 1;
fig. 4 is a qualitative detection diagram of anomalous target detection for the prior methods and the present method in the example 2 scenario.
Detailed Description
The present invention will be described in further detail with reference to examples.
An anomaly detection method based on high-order tensor expression is provided, and an efficient algorithm for solving the model by using an alternative-direction multiplication operator is designed. A hyperspectral image anomaly detection method based on high-order tensor expression is a hyperspectral image anomaly detection method realized by logarithm and norm optimized tensor low TT rank decomposition, space spectrum total variation and group sparseness. The hyperspectral image contains abundant spectral information, has the advantage of integrating maps in distinguishing different material characteristics, and the hyperspectral abnormal target detection has the unsupervised characteristic and is widely applied to the fields of military investigation, geological mining, post-disaster rescue and the like. The traditional hyperspectral anomaly detection method cannot efficiently separate an anomalous target from a background. Therefore, how to effectively separate the abnormal target from the background by using the priori knowledge of the hyperspectral integrated map structure in the spatial domain and the spectral domain becomes a key problem while maintaining the hyperspectral integrated map structure. In order to solve the problems, the invention provides an effective hyperspectral image anomaly detection method by using a high-order tensor expression method. The research introduces the step-up operation, the high-order Tensor expression is carried out on the data, and a Tensor low-Tensor-Train rank approximate framework is utilized, so that the proper regularization constraint is embedded into the hyperspectral priori knowledge while the hyperspectral integral structure is kept. Aiming at the space dimension subsection continuous prior of the background tensor, adopting a space spectrum total variation norm regular term to express; aiming at the spectral dimension low-rank prior of the background, expressing by using logarithm and norm regular terms; aiming at the sparse prior of the existing group of the abnormal target tensor, L is adopted 2,1 And expressing a norm regular term. Finally, convex optimization is carried out on a target equation by adopting an alternating direction multiplier method under a tensor framework, the purpose of effectively extracting an abnormal target is achieved, the principle is clear, the experimental verification effect is excellent, and the robustness is strong. The research is helpful to clarify a plurality of global priors, spatial local characteristics and anomalies of the hyperspectral dataThe relation between the detection capabilities provides theories, models and supports for the tensor expression model in the hyperspectral remote sensing image processing and analysis, and provides a new idea for the hyperspectral image processing aspect in the future.
A. Problem modeling
L is used as the abnormal value is distributed along the second dimension of the tensor (the other dimensions are general), namely the abnormal value represents possible abnormal value, and the abnormal value has the sparse characteristic compared with the data dimension 2,1 The norm. The anomaly detection problem for any higher order tensor can be modeled as follows,
Figure BDA0003873556680000051
wherein the content of the first and second substances,
Figure BDA0003873556680000052
is a prior constraint term for the data,
Figure BDA0003873556680000053
for the observation image to be processed of the input model,
Figure BDA0003873556680000054
for the purpose of background low-rank data,
Figure BDA0003873556680000055
anomaly data output for the model.
Introducing a low-order prior term according to the low-order characteristic of data for the visual application task
Figure BDA0003873556680000056
To obtain
Figure BDA0003873556680000057
Tensor low-Tensor-train (TT) rank measurement has the advantages in high-order Tensor information mining, and benefits from a balance mechanism of each mode matrix in TT decomposition. For oneA given tensor
Figure BDA0003873556680000058
Its TT rank is defined as
Figure BDA0003873556680000059
The tensor TT rank is introduced into the formula (2), and we have
Figure BDA00038735566800000510
Wherein the content of the first and second substances,
Figure BDA00038735566800000511
is a tensor upscaling operation. Higher order tensor expressions offer some important advantages. Tensor TT decomposition also becomes more efficient due to the efficient use of the local structure of the data, with higher tensor orders. Here, we use the Generalization scaling mechanism, generalized temperature Augmentation (GKA), to scale up the third-order spectral data. Taking the initialization block size of 2 × 2 as an example, the upscaling operation of the third-order image is defined as follows
Figure BDA00038735566800000512
Wherein
Figure BDA00038735566800000513
Representing the pixel value at the ordinary segment j,
Figure BDA00038735566800000514
and u j Are all orthogonal bases.
Since the solution of formula (4) to the TT rank of the combined property is an NP-hard problem, we use the logarithm and norm to perform a non-convex optimization solution of the rank. Given tensor
Figure BDA00038735566800000515
The logarithm and norm under the TT decomposition of the tensor can be expressed as
Figure BDA0003873556680000061
Proposition 1: for matrix A, its Log-Sum (LS) norm is defined as
‖A‖ Ls ＝∑ i log(σ i (A)+∈) (7)
Wherein sigma i Is the ith singular value of the matrix a, ∈ is designed to avoid a small positive scalar of zero. The problem is equivalent to
Figure BDA0003873556680000062
Where α is a low rank weighting factor and C is the input data.
Its local minimum can be expressed as
A＝Udiag(d 1 ,d 2 ,…d n )V (9)
Wherein U is a left orthogonal matrix and V is a right orthogonal matrix.
Wherein d is i ＝D α,∈ (σ i )
Figure BDA0003873556680000063
Wherein c is 1 = x | - ∈ and
Figure BDA0003873556680000064
the solution of equation (4) is converted into
Figure BDA0003873556680000065
Is equivalent to
Figure BDA0003873556680000066
Wherein λ is r Is a low rank weighting factor, beta is a group sparsity weighting factor, mu is a Lagrange factor,
to better exploit the segmentation continuity in the Spectral image Spatial and Spectral domain, we introduce the Spatial-Spectral Total Variation (SSTV), a priori modelable as
Figure BDA0003873556680000071
Wherein, W is the width of the hyperspectral image, H is the height of the hyperspectral image, S is the number of the hyperspectral image spectral segments, i is the coordinate in the image width direction, j is the coordinate in the image height direction, and k is the coordinate in the image spectral segment direction.
So far, we obtain an anomaly detection model represented by the following high-order low TT rank tensor
Figure BDA0003873556680000072
λ t Is a total variation weight factor, λ r Is a low rank weighting factor, beta is a group sparse weighting factor, mu is a Lagrange factor, alpha n Is a weight factor of the tensor mode matrix,
Figure BDA0003873556680000073
for the spatial spectrum total variation regularization term,
Figure BDA0003873556680000074
to weight the low TT rank terms with logarithms and optimized higher order tensors,
Figure BDA0003873556680000075
are group sparse regularization terms.
B. Algorithm solution
In order to solve the coupling problem in the process of variable solution, auxiliary variables are introduced into corresponding variables
Figure BDA0003873556680000076
Figure BDA0003873556680000077
Wherein the content of the first and second substances,
Figure BDA0003873556680000078
is a tensor mode matrix.
The augmented Lagrangian function of the above model is shown below
Figure BDA0003873556680000081
Updating
Figure BDA0003873556680000082
Figure BDA0003873556680000083
Updating
Figure BDA0003873556680000084
Figure BDA0003873556680000085
Wherein the auxiliary tensor
Figure BDA0003873556680000086
The solution of formula (18) is as follows
Figure BDA0003873556680000087
Where # represents an orthogonal projection operation, defined by the following equation
Figure BDA0003873556680000088
Linear operation
Figure BDA0003873556680000089
Is defined as
Figure BDA00038735566800000810
Wherein
Figure BDA00038735566800000811
And (w, h, s) satisfy
Figure BDA00038735566800000812
for 1≤i≤N w -1,1≤j≤N H -1,1≤k≤N S -1,
|w i,j,k |≤1,for 1≤i≤N W -1,
|h i,j,k |≤1,for 1≤j≤N H -1,
|s i,j,k |≤1,for 1≤k≤N S -1.
(22)
The adjoint operation of l is defined as
Figure BDA0003873556680000091
w i,j,k ＝z i,j,k -z i+1,j,k ,for 1≤i≤N W -1,
h i,j,k ＝z i,j,k -z i,j+1,k ,for 1≤j≤N H -1,
s i,j,k ＝z i,j,k -z i,j,k+1 ,for 1≤k≤N S -1.
(23)
And (3) proving that:
Figure BDA0003873556680000092
Figure BDA0003873556680000093
Figure BDA0003873556680000094
thus, the isotropic SSTV norm introduced above can be rewritten as:
Figure BDA0003873556680000095
wherein
Figure BDA0003873556680000101
Then, we obtain
Figure BDA0003873556680000102
In light of the above problem, equation (19) can be reformulated as
Figure BDA0003873556680000103
The objective function is
Figure BDA0003873556680000104
Is convex and concave in (p, q, r).It allows us to switch the order of minimization and maximization.
Figure BDA0003873556680000105
Can be written as
Figure BDA0003873556680000106
Updating
Figure BDA0003873556680000107
Figure BDA0003873556680000111
For the
Figure BDA0003873556680000112
Figure BDA0003873556680000113
Updating
Figure BDA0003873556680000114
Figure BDA0003873556680000115
Updating lagrange multipliers
Figure BDA0003873556680000116
Figure BDA0003873556680000117
Figure BDA0003873556680000118
The method carries out algorithm tests on the public data sets San Diego and Airport1, and images of two scenes in the data sets and reference images thereof are shown in FIG. 2. We compared the proposed method (f) with representative abnormality detection methods (a) to (e), and the results are shown in table 1, fig. 3, and fig. 4. The AUC index value is given in the table 1 and used for measuring the area enclosed by the coordinate axis under the ROC curve, and the closer to 1 represents the higher the detection authenticity. The qualitative detection graph results of the abnormal target detection under different scenes are shown in fig. 3 and fig. 4, respectively. From the results, the method achieves better detection effect and verifies the effectiveness of the method.
TABLE 1
Figure BDA0003873556680000121
While the foregoing is directed to the preferred embodiment of the present invention, it will be understood by those skilled in the art that various changes and modifications may be made without departing from the spirit and scope of the invention as defined in the appended claims.

Claims (10)

1. The hyperspectral image anomaly detection method based on high-order tensor representation is characterized by comprising the following steps of performing iterative optimization solution on the basis of an initial model, obtaining a hyperspectral image anomaly detection model based on high-order tensor representation, and realizing hyperspectral image anomaly detection, wherein the hyperspectral image anomaly detection method comprises the following steps of:
step 1, utilizing a group sparse regular term to mine the difference of the line space structure of each spectrum image and a shared sparse structure, and detecting an abnormal value;
step 2, performing rank approximation on the high-order tensor data by using the logarithm and the optimized high-order tensor weighted low TT rank, and mining the correlation contained in the tensor data to keep the background;
and 3, adopting a space spectrum total variation regularization term for depicting the spectrum space segmentation continuity.
2. The method for detecting the abnormality of the hyperspectral image based on the high-order tensor representation according to claim 1, wherein the model carries out low-rank reconstruction on the spectral image to realize the detection of the abnormality of the hyperspectral image.
3. The hyperspectral image abnormality detection method based on the higher-order tensor representation according to claim 1 is characterized in that the establishment process of the prior initial model is as follows:
l is used to assume that the outliers are distributed along the second dimension of the tensor, and are sparse compared to the data scale 2,1 Norm, the anomaly detection problem for any higher order tensor is modeled as follows:
Figure FDA0003873556670000011
wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003873556670000012
is an a priori constrained term of the data;
Figure FDA0003873556670000013
for the observation image to be processed of the input model,
Figure FDA0003873556670000014
for the purpose of background low-rank data,
Figure FDA0003873556670000015
abnormal data output for the model;
introducing a low order prior term
Figure FDA0003873556670000016
Obtaining a prior initial model:
Figure FDA0003873556670000017
4. the method for detecting the abnormality of the hyperspectral image based on the higher order tensor representation according to claim 1, wherein the hyperspectral image based on the higher order tensor representation abnormality detection model is as follows:
Figure FDA0003873556670000018
Figure FDA0003873556670000019
wherein λ is t Is a total variation weight factor, λ r Is a low rank weighting factor, beta is a group sparsity weighting factor, mu is a Lagrange factor, alpha n Is a weight factor of the tensor mode matrix,
Figure FDA00038735566700000110
for the spatial spectrum total variation regularization term,
Figure FDA00038735566700000111
to weight the low TT rank terms with logarithms and optimized higher order tensors,
Figure FDA0003873556670000021
are group sparse regularization terms.
5. The hyperspectral image abnormality detection method based on high-order tensor representation according to claim 4 is characterized in that in the log-sum optimized high-order tensor weighted low TT rank term, the high-order tensor is high-order tensor data obtained by performing an up-scaling operation on an image, and the tensor rank optimization approximation is performed by using a log and a norm.
6. The method for detecting the abnormality of the hyperspectral image based on the higher-order tensor representation according to claim 4, wherein the weighted set sparse terms are descriptions of smoothness of spectral images in different spectral bands, and are used for mining differences of line space structures of image blocks in the spectral bands and shared sparse structures.
7. The hyperspectral image anomaly detection method based on high-order tensor representation according to claim 4 is characterized in that the spatial spectrum total variation regularization term describes the continuity of spectral space segments and suppresses noise.
8. The hyperspectral image anomaly detection method based on high-order tensor representation according to claim 1 is characterized in that in the repeated iterative optimization solution process, when the error of the image recovery result data of two adjacent times is within a threshold range, the image is judged to be in accordance with convergence when being reconstructed to the current turn, and the iteration is stopped.
9. The hyperspectral image abnormality detection method based on the higher-order tensor representation according to claim 8 is characterized in that the error function is:
Figure FDA0003873556670000022
wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003873556670000023
in order to input the observation data, it is,
Figure FDA0003873556670000024
for low rank data,. Epsilon.is anomalous data, and tol is a given threshold.
10. The hyperspectral image abnormality detection method based on high-order tensor representation according to claim 1 is characterized in that the iterative optimization solution is an ADMM algorithm.
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Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN116245779A (en) * 2023-05-11 2023-06-09 四川工程职业技术学院 Image fusion method and device, storage medium and electronic equipment
CN117333681A (en) * 2023-09-26 2024-01-02 中国科学院国家空间科学中心 Hyperspectral abnormal target detection method and system based on weighted sparsity
CN117333681B (en) * 2023-09-26 2024-05-14 中国科学院国家空间科学中心 Hyperspectral abnormal target detection method and system based on weighted sparsity

Cited By (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN116245779A (en) * 2023-05-11 2023-06-09 四川工程职业技术学院 Image fusion method and device, storage medium and electronic equipment
CN116245779B (en) * 2023-05-11 2023-08-22 四川工程职业技术学院 Image fusion method and device, storage medium and electronic equipment
CN117333681A (en) * 2023-09-26 2024-01-02 中国科学院国家空间科学中心 Hyperspectral abnormal target detection method and system based on weighted sparsity
CN117333681B (en) * 2023-09-26 2024-05-14 中国科学院国家空间科学中心 Hyperspectral abnormal target detection method and system based on weighted sparsity

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