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

Abstract

The application relates to a hyperspectral image anomaly detection method based on high-order tensor representation, introduces a promotion operation, performs high-order tensor expression on data, utilizes a tensor low Tensor-Train rank approximation framework, embeds suitable regularization constraints into hyperspectral prior knowledge while retaining the overall structure of the hyperspectral image, adopts a spatial spectral total variation norm regularization term expression for the spatial dimension segmentation continuous prior of the background tensor, adopts a logarithmic sum norm regularization term expression for the spectral dimension low rank prior of the background, and adopts an L 2,1 norm regularization term expression for the group sparse prior of the anomaly target tensor. Finally, in the tensor framework, the target equation is subjected to a convex optimization through an alternating direction multiplier method, the anomaly target is effectively extracted, the principle is clear, the experimental verification effect is superior, and the robustness is strong. The application provides theory, a model and support for tensor expression model processing and analysis of hyperspectral remote sensing images.

Description

TECHNICAL FIELD

[0001] The present application 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

[0002] Hyperspectral image contains rich spectral information, and has the advantage of "graph-spectrum integration" in distinguishing the characteristics of different substances. Hyperspectral anomaly target detection is widely used in military reconnaissance, geological mining and post-disaster rescue due to its unsupervised characteristics. However, the traditional hyperspectral anomaly detection method directly reduces the original hyperspectral image from a three-dimensional structure to a two-dimensional matrix by using low-rank sparse decomposition, which destroys the spatial overall structure of the hyperspectral image and causes a large number of false alarms, and cannot efficiently separate the abnormal targets from the background. Therefore, how to preserve the hyperspectral "graph-spectrum integration" structure while utilizing the spatial domain and spectral domain prior knowledge to effectively separate the abnormal targets from the background has become a key problem.

[0003] The traditional method is to measure similarity such as distance, density, angle, cluster-based method, etc. These algorithms are actually close in low dimension, because the core assumption is that the representation of abnormal points is different from that of normal points, and is a minority. However, most similar algorithms will face the curse of dimensionality, that is, common similarity measures (such as Euclidean distance) often fail on high-dimensional data. In order to solve this problem, many methods have been proposed, including dimensionality reduction or feature space selection methods, such as detecting in multiple low-dimensional spaces and then merging. The intrinsic subspace that restores the data distribution is a key step in many applications of machine learning and data analysis. The most commonly used analysis method is principal component analysis (PCA). Standard PCA is sensitive to outliers: even a single but severe outlier can reduce the effectiveness of the model. Another problem brought by high-dimensional data is scalability. The operation cost of measuring similarity is large, and most distance metrics have high complexity. Using data structures for optimization or dynamic programming to reduce complexity is also a common exploration direction. The most ideal case is to control the dimensionality and find a better data representation, which is the root of the problem.

[0004] The present application introduces a high-order tensor representation, and uses a tensor low-Tensor-Train rank approximation framework to preserve the overall structure of the hyperspectral image while embedding appropriate regularization constraints into the hyperspectral prior knowledge. For example, the spatial dimension segmentation continuous prior of the background tensor is expressed by using a spatial-spectral total variation norm regularization term; the low-rank prior of the background in the spectral dimension is expressed by using a logarithmic sum norm regularization term; and the group sparse prior of the abnormal target tensor is expressed by using a L 2,1The norm regularization term is used for expression. Finally, in the tensor framework, the objective function is optimized by using the alternating direction multiplier method to achieve the purpose of effectively extracting abnormal targets. SUMMARY

[0005] In view of the above technical deficiencies, the present application aims to provide a method for processing the anomaly detection problem of hyperspectral images based on tensor low-TT rank decomposition, spatial-spectral total variation and group sparsity based on logarithmic sum and norm optimization. In order to make full use of the high spatial-spectral correlation of high-multiple spectral images, the ascending operation is introduced, the data is expressed as high-order tensor, and the high-order tensor data is approximately restored by using tensor low-Tensor-Train decomposition. For the spatial dimension segmentation continuous prior of the background tensor, the spatial-spectral total variation norm regularization term is used for expression; for the low rank prior of the background in spatial-spectral dimension, the logarithmic sum norm regularization term is used for expression; for the group sparsity prior of the abnormal target tensor, the L 2,1 The norm regularization term is used for expression. Finally, in the tensor framework, the objective function is optimized by using the alternating direction multiplier method to achieve the purpose of effectively extracting abnormal targets.

[0006] The technical scheme adopted by the present application to solve its technical problems is:

[0007] The hyperspectral image anomaly detection method based on high-order tensor representation is based on the prior initial model and the following steps are performed, and then iteratively optimized to obtain a hyperspectral image anomaly detection model based on high-order tensor representation, and the hyperspectral image anomaly detection is realized, comprising:

[0008] Step 1, using group sparse regularization term, for mining the spatial structure difference of each spectral image row and the shared sparse structure, for detecting abnormal values;

[0009] Step 2, using the high-order tensor weighted low-TT rank based on logarithmic sum optimization to approximate the rank of high-order tensor data, for mining the correlation contained in the tensor data, and maintaining the background;

[0010] Step 3, using spatial-spectral total variation regularization term, for describing the spectral spatial segmentation continuity.

[0011] The model performs low-rank reconstruction on the spectral image to realize the anomaly detection of the hyperspectral image.

[0012] The establishment process of the prior initial model is as follows:

[0013] Assuming that the outliers are distributed along the second dimension of the tensor and the outliers are sparse compared to the data scale, the L 2,1 The outlier detection problem of any high-order tensor is modeled as follows:

[0014]

[0015] where, is the prior constraint term of the data; is the input model of the observation image to be processed, is the background low-rank data, is the output model of the outlier data;

[0016] The low-order prior term is introduced The prior initial model is obtained:

[0017]

[0018] The hyperspectral image outlier detection model based on high-order tensor representation is:

[0019]

[0020]

[0021] where, λ t is the total variation weight factor, λ r is the low-rank weight factor, β is the group sparse weight factor, μ is the Lagrange factor, α n is the tensor mode matrix weight factor, is the spatial-spectral total variation regularization term, is the high-order tensor weighted low TT rank term using log-sum optimization, is the group sparse regularization term.

[0022] In the high-order tensor weighted low TT rank term using log-sum optimization, the high-order tensor is the high-order tensor data obtained by the order operation on the image, and the tensor rank optimization is approximated by using the log-sum norm.

[0023] The weighted group sparse term is a description of the smoothness of the spectral image of different spectral bands, and the spatial structure difference and shared sparse structure of each spectral image block are mined.

[0024] The spatial-spectral total variation regularization term describes the spectral spatial segment continuity and suppresses noise.

[0025] In the repeated iterative optimization solving process, when the error of the image restoration result data of the adjacent two times is within the threshold range, it is judged that the image reconstruction to the current round meets the convergence, and the iteration is stopped.

[0026] The error function is:

[0027]

[0028] wherein, is the input observation data, is the low rank data, epsilon is the abnormal data, and tol is a given threshold value.

[0029] The repeated iterative optimization solution is an ADMM algorithm.

[0030] The present application has the following beneficial effects and advantages:

[0031] 1. The method of the present application uses logarithmic sum optimization of high-order tensor weighted low TT rank to approximate the rank of tensor data, and excavates the correlation contained in the tensor data.

[0032] 2. The method of the present application uses group sparse regularization terms to excavate the differences in spatial structure of each spectral image and the shared sparse structure, and detects abnormal values.

[0033] 3. The method of the present application uses spatial spectral total variation regularization terms to depict the spectral spatial segment continuity.

[0034] 4. The method of the present application is superior to existing algorithms in terms of hyperspectral anomaly detection, and has better low rank reconstruction performance and robustness. BRIEF DESCRIPTION OF DRAWINGS

[0035] Figure 1 is the overall framework diagram of the method in this paper;

[0036] Figure 2 is the image of two scenes in the data set and its reference map;

[0037] Figure 3 is the qualitative detection map of the abnormal target detection of each existing method and the method of the present application under the scene of example 1;

[0038] Figure 4 is the qualitative detection map of the abnormal target detection of each existing method and the method of the present application under the scene of example 2. DETAILED DESCRIPTION

[0039] The present application will be further described in detail below in combination with examples.

[0040] An anomaly detection method based on high-order tensor representation is proposed, and an efficient algorithm using alternating direction multiplier operator to solve the model is designed. The anomaly detection method based on high-order tensor representation of hyperspectral image is a kind of hyperspectral image anomaly detection method realized by using log-sum norm optimization, spatial-spectral total variation and group sparsity of tensor low TT rank decomposition. Hyperspectral image contains rich spectral information, and has the advantage of 'graph-spectrum integration' in distinguishing the characteristics of different substances. Hyperspectral anomaly target detection is widely used in military reconnaissance, geological mining and post-disaster rescue due to its unsupervised characteristics. The traditional hyperspectral anomaly detection method cannot efficiently separate the anomaly target from the background. Therefore, how to effectively separate the anomaly target from the background while preserving the hyperspectral 'graph-spectrum integration' structure and utilizing the prior knowledge in the spatial domain and the spectral domain has become a key problem. In order to solve the above problem, the application uses high-order tensor representation method, and proposes an effective hyperspectral image anomaly detection method. The study introduces the upgrading operation, performs high-order tensor representation on the data, uses the tensor low Tensor-Train rank approximation framework, embeds the appropriate regularization constraint into the hyperspectral prior knowledge while preserving the overall structure of the hyperspectral image. For the spatial dimension segmentation continuous prior of the background tensor, the spatial-spectral total variation norm regularization term is adopted; for the spectral dimension low rank prior of the background, the log-sum norm regularization term is adopted; for the group sparse prior of the anomaly target tensor, the L 2,1 norm regularization term is adopted. Finally, the alternating direction multiplier method is used for convex optimization of the target equation in the tensor framework, so as to effectively extract the anomaly target. The principle is clear, the experimental verification effect is superior, and the robustness is strong. The study will help to clarify the relationship between the multiple global priors, spatial local features and anomaly detection ability of hyperspectral data, provide theory, model and support for the tensor expression model in hyperspectral remote sensing image processing and analysis, and provide a new idea for future hyperspectral image processing.

[0041] A. Problem modeling

[0042] It is assumed that the anomaly value is distributed along the second dimension of the tensor (other dimensions are universal), that is, it represents the possible anomaly value, and in addition, the anomaly value has the sparse characteristic compared with the data scale, so the L 2,1 norm can be used to measure. For the anomaly detection problem of any high-order tensor, the following model can be established,

[0043]

[0044] Among them, is the prior constraint term of the data, is the observation image to be processed input into the model, is the background low rank data, is the anomaly data output by the model.

[0045] According to the low-rank property of data for visual application tasks, a low-rank prior term is introduced We have

[0046]

[0047] The advantage of tensor low Tensor-train (TT) rank measure in high-order tensor information mining is due to the balanced mechanism of each module matrix in TT decomposition. For a given tensor Its TT rank is defined as

[0048]

[0049] Introducing the tensor TT rank into formula (2), we have

[0050]

[0051] where is the tensor lifting operation. Higher-order tensor representation provides some important advantages. In the case of higher-order tensors, tensor TT decomposition also becomes more effective due to the effective use of local data structure. Here, we use the Generalization Ket Augmentation (GKA) mechanism to perform lifting operation on three-order spectral data. Taking the initialization block size as 2x2 as an example, the lifting operation of three-order image is defined as follows

[0052]

[0053] where represents the pixel value at position j, and u j are orthogonal bases.

[0054] Since formula (4) is a combination of TT rank, the solution of the rank is an NP-hard problem. We use the log-sum norm to solve the non-convex optimization of the rank. Given a tensor The log-sum norm under the tensor TT decomposition can be expressed as

[0055]

[0056] Proposition 1: For a matrix A, its Log-Sum (LS) norm is defined as

[0057] ‖A‖ Ls =∑ i log(σ i (A)+∈) (7)

[0058] where σ iis the i-th singular value of matrix A, ∈ is a small positive scalar designed to avoid zero. This problem is equivalent to

[0059]

[0060] where α is the low-rank weight factor and C is the input data.

[0061] The local minimum of which can be expressed as

[0062] A = Udiag(d1, d2, … d n ) V (9)

[0063] where U is the left orthogonal matrix and V is the right orthogonal matrix.

[0064] where d i = D α,∈ (σ i )

[0065]

[0066] where c1 = |x| - ∈ and

[0067] The solution of formula (4) is transformed into

[0068]

[0069] is equivalent to

[0070]

[0071] where λ r is the low-rank weight factor, β is the group sparsity weight factor, and μ is the Lagrange factor,

[0072] To better utilize the piecewise continuity in the spatial and spectral domains of the spectral image, we introduce a spatial-spectral total variation (SSTV) term. This prior can be modeled as

[0073]

[0074] where W is the width of the hyperspectral image, H is the height of the hyperspectral image, S is the number of spectral bands of the hyperspectral image, i is the coordinate in the width direction of the image, j is the coordinate in the height direction of the image, and k is the coordinate in the spectral direction of the image.

[0075] Thus, we obtain the following anomaly detection model with high-order low-TT rank tensor representation

[0076]

[0077] λt is the total variation weight factor, λ r is the low-rank weight factor, β is the group sparsity weight factor, μ is the Lagrange factor, α n is the tensor mode matrix weight factor, is the spatial-spectral total variation regularization term, is the high-order tensor weighted low-TT rank term optimized by log-sum, is the group sparsity regularization term.

[0078] B. Algorithm solution

[0079] In order to solve the coupling problem in the variable solving process, auxiliary variables are introduced for the corresponding variables

[0080]

[0081]

[0082] wherein, is the tensor mode matrix.

[0083] The augmented Lagrange function of the above model is as follows

[0084]

[0085] Update

[0086]

[0087] Update

[0088]

[0089] wherein, the auxiliary tensor The solution of formula (18) is as follows

[0090]

[0091] Wherein, represents the orthogonal projection operation, which is defined by the following formula

[0092]

[0093] Linear operation is defined as

[0094]

[0095] Wherein And (w, h, s) satisfy

[0096]

[0097] for 1≤i≤N w -1,1≤j≤N H -1,1≤k≤N S -1,

[0098] |w i,j,k |≤1,for 1≤i≤N W -1,

[0099] |h i,j,k |≤1,for 1≤j≤N H -1,

[0100] |s i,j,k |≤1,for 1≤k≤N S -1. (22)

[0101] The adjoint matrix operation of l is defined as

[0102]

[0103] w i,j,k =z i,j,k -z i+1,j,k ,for 1≤i≤N W -1,

[0104] h i,j,k =z i,j,k -z i,j+1,k ,for 1≤j≤N H -1,

[0105] s i,j,k =z i,j,k -z i,j,k+1 ,for 1≤k≤N S -1. (23)

[0106] Proof:

[0107]

[0108]

[0109]

[0110] Therefore, the isotropic SSTV norm introduced above can be rewritten as:

[0111]

[0112] where

[0113]

[0114] Then, we get

[0115]

[0116] According to the above problem, equation (19) can be restated as

[0117]

[0118] The objective function is convex in and concave in (p, q, r). It allows us to switch the order of minimization and maximization.

[0119]

[0120] Can be written as

[0121]

[0122] Update

[0123]

[0124] For

[0125]

[0126] Update

[0127]

[0128] Update the Lagrange multiplier

[0129]

[0130]

[0131] The proposed method was tested on the public datasets San Diego and Airport1, Figure 2 The images of the two scenes in the dataset and their reference maps are shown in FIGS. 1, Figure 3 and 2. The proposed method (f) was compared with representative anomaly detection methods (a)-(e), and the results are shown in Table 1, Figure 4 and FIGS. 3 and 4. The AUC index value, which measures the area enclosed by the ROC curve and the coordinate axis, is given in Table 1. The closer to 1, the higher the detection authenticity. Figure 3 and FIGS. 3 and 4. The AUC index value, which measures the area enclosed by the ROC curve and the coordinate axis, is given in Table 1. The closer to 1, the higher the detection authenticity. Figure 4 The qualitative detection results of the abnormal target detection under different scenes are shown in FIGS. 3 and 4, respectively. From the results, it can be seen that our method achieves better detection effect, verifying the effectiveness of the proposed method.

[0132] Table 1

[0133]

[0134] The above describes the preferred embodiments of the present application. It should be noted that, for those skilled in the art, several improvements and refinements can be made without departing from the principles of the present application, and these improvements and refinements should be considered as falling within the scope of the present application.

Claims

1. A hyperspectral image anomaly detection method based on high-order tensor representation, characterized in that, On the basis of the prior initial model, the following steps are performed, and then iterative optimization is repeatedly solved to obtain a hyperspectral image anomaly detection model based on high-order tensor representation, and hyperspectral image anomaly detection is realized; the method comprises: The process of establishing the prior initial model is as follows: Assuming that the outliers are distributed along the second dimension of the tensor and that the outliers are sparse compared to the scale of the data, we measure the outliers using the L 2,1 norm, the outlier detection problem for any high-order tensor can be modeled as follows: ； wherein, is a prior constraint term of data; is an input model of an observation image to be processed, is background low-rank data, is abnormal data of a model output; Introducing low-order prior terms to obtain the prior initial model: ； The prior initial model is improved according to steps 1-3, and then iterative optimization is repeatedly solved; Step 1: a group sparse regularization term is used to mine the spatial structure difference and shared sparsity structure of each spectral image block, and to detect outliers; Step 2: a logarithmic sum optimized high-order tensor weighted low TT rank is used to perform rank approximation on high-order tensor data, to mine the correlation contained in the tensor data, and to maintain the background; Step 3: a spatial-spectral total variation regularization term is used to describe the spectral spatial segment continuity; A hyperspectral image anomaly detection model based on high-order tensor representation for realizing hyperspectral image anomaly detection is obtained: ； ； wherein, is a total variation weight factor, is a low-rank weight factor, is a group sparsity weight factor, is a Lagrangian factor, is a tensor mode matrix weight factor, is a spatial-spectral total variation regularization term, is a high-order tensor weighted low-TT rank term optimized with log-sum, is a group sparsity regularization term.

2. The high dimensional tensor representation based hyperspectral image anomaly detection method according to claim 1, characterized in that, The model performs low-rank reconstruction on spectral images to realize hyperspectral image anomaly detection. 3.The high dimensional tensor representation based hyperspectral image anomaly detection method of claim 1, wherein, In the logarithmic sum optimized high-order tensor weighted low TT rank term, the high-order tensor is high-order tensor data obtained by ascending the order of the image, and the tensor rank optimization approximation is performed by using the logarithmic sum norm.

4. The high dimensional tensor representation based hyperspectral image anomaly detection method of claim 1, wherein, The group sparse regularization term is a description of the smoothness of different spectral images, and is used to mine the spatial structure difference and shared sparsity structure of each spectral image block.

5. The high dimensional tensor representation based hyperspectral image anomaly detection method of claim 1, wherein, The spatial-spectral total variation regularization term describes the spectral spatial segment continuity and suppresses noise. 6.The high dimensional tensor representation based hyperspectral image anomaly detection method of claim 1, wherein, In the process of iterative optimization, when the error of the image restoration result data of two adjacent times is within a threshold range, it is judged that the image reconstruction to the current round meets the convergence, and the iteration is stopped.

7. The high dimensional tensor representation based hyperspectral image anomaly detection method according to claim 6, characterized in that, The error is obtained by the following error function: ； wherein, is the input observation data, is the low rank data, is the anomaly data, is a given threshold value. 8.The high dimensional tensor representation based hyperspectral image anomaly detection method of claim 1, wherein, The iterative optimization is solved by using the ADMM algorithm.

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