Hyperspectral anomaly detection method for priori coupling driven non-convex tensor representation

Abstract

The invention discloses a hyperspectral anomaly detection method based on prior coupling driving non-convex tensor representation, and belongs to the technical field of hyperspectral image processing. Aiming at the problem that background representation is not ideal due to the fact that an existing tensor representation method depends on a plurality of independent regularization items for background prior modeling and a loose convex substitute is adopted for approximation, a basic tensor representation model is constructed, and meanwhile, a prior-coupled non-convex background regularization item and an abnormal regularization item are constructed; constructing an augmented Lagrangian equation, converting the model into an unconstrained optimization problem, performing model iteration according to the augmented Lagrangian equation, and taking an abnormal tensor when the iteration is completed as an optimal abnormal tensor; and obtaining an anomaly detection result according to the optimal anomaly tensor. According to the method, the global low-rank priori and the local smoothness priori of the background are coupled in a non-convex regularization item, and the background priori is better described, so that the accuracy of anomaly detection is improved, and the false alarm rate is reduced.

Description

Technical Field

[0001] The present invention belongs to the technical field of hyperspectral image processing, and in particular relates to a hyperspectral anomaly detection method using a priori coupling driven non-convex tensor representation. Background Art

[0002] Hyperspectral imagery captures spectral data across hundreds of continuous bands, achieving extremely high spectral resolution, enabling the distinction of different ground substances. This technology has widespread application in both civilian and military fields. Hyperspectral anomaly detection, which aims to isolate anomalous pixels that differ significantly from the surrounding background, is crucial in areas such as camouflage detection, environmental monitoring, and public safety. Unlike traditional hyperspectral target detection, hyperspectral anomaly detection can extract unknown substances without relying on prior knowledge of the target spectrum, offering greater flexibility.

[0003] Existing hyperspectral anomaly detection methods mainly include those based on statistical theory and matrix decomposition. However, these methods often flatten hyperspectral images into two-dimensional matrices, destroying the spatial information in hyperspectral images and resulting in poor performance when processing complex scenes. In recent years, hyperspectral anomaly detection methods based on tensor representation have attracted widespread attention due to their ability to preserve the inherent three-dimensional structure of hyperspectral images. They effectively model the prior structures of background and anomalies through regularization terms, achieving more accurate hyperspectral anomaly detection. However, existing tensor representation methods rely on multiple independent regularization terms to model background priors, and the use of loose convex surrogates for approximation leads to suboptimal background representation. Therefore, it is of great significance to design a background regularization term that uses a single regularization term to represent multiple background priors and avoids the use of loose convex surrogates for approximation. Summary of the Invention

[0004] To address the problem that existing tensor representation methods rely on multiple independent regularization terms to model background priors and use loose convex surrogates for approximation, resulting in suboptimal background representation, this paper provides a hyperspectral anomaly detection method that uses prior-coupled non-convex tensor representations. This method couples the global low-rank prior and the local smoothness prior of the background in a single non-convex regularization term, better characterizing the background prior, thereby improving anomaly detection accuracy and reducing false alarm rates.

[0005] In order to achieve the above object, the present invention adopts the following technical solutions:

[0006] A hyperspectral anomaly detection method based on a priori coupling driven non-convex tensor representation, the method comprising the following steps:

[0007] Step 1: Construct a basic tensor representation model and construct a priori coupled non-convex background regularization term and build Anomaly regularization term;

[0008] The specific operation of step 1 is as follows:

[0009] Step 1.1: Represent the hyperspectral image tensor to be detected as , where , and represent the height, width and number of spectral bands, respectively;

[0010] Construct a basic tensor representation model and convert it into a constrained minimization problem, as follows:

[0011]

[0012] where and represent the regularization term of the encoded background tensor and the regularization term of the encoded anomaly tensor , respectively;

[0013] Step 1.2: Construct a prior-coupled non-convex background regularization term , as follows:

[0014]

[0015] where represents the mode of the tensor, represents the gradient tensor obtained by using the difference operator along the mode of the background tensor , represents the tensor kernel norm of induced by the non-convex Laplacian function, which is defined as:

[0016]

[0017] where the non-convex Laplacian function is defined as , represents the singular value, represents the th front slice matrix after the discrete Fourier transform of the gradient tensor along the third mode;

[0018] The prior-coupled non-convex background regularization term The two parts include a first part of a non-convex Laplacian tensor kernel norm used to depict a global low-rankness prior of the background tensor, and a second part of a gradient tensor capturing local variations of the background tensor in three modes used to depict a local smoothness prior of the background tensor. The global low-rankness prior and the local smoothness prior are coupled by using a single non-convex regularization term, thereby providing a more robust background characterization.

[0019] Step 1.3: Constructing An anomaly regularization term is constructed by using a tensor norm to depict a group sparsity prior of the anomaly pixels, The anomaly regularization term is formulated as follows:

[0020]

[0021] wherein, represents an anomaly tensor in the coordinate ;

[0022] By using the constructed prior-coupled non-convex background regularization term and the anomaly regularization term, a prior-coupled driven non-convex tensor representation model is constructed, i.e., the following minimization problem:

[0023]

[0024] wherein, is a trade-off parameter used to control the balance between the non-convex background regularization term and the anomaly regularization term.

[0025] Step 2: Constructing an augmented Lagrangian equation to convert the model into an unconstrained optimization problem, and performing model iteration according to the augmented Lagrangian equation, and the anomaly tensor at the iteration completion time is taken as the optimal anomaly tensor;

[0026] The specific operation of the step 2 is:

[0027] Step 2.1: Based on the alternating direction multiplier method, the model is solved, and first, an augmented Lagrangian equation is constructed to convert the model into an unconstrained optimization problem, and the formula is:

[0028]

[0029] wherein, , is a Lagrange multiplier, represents a Frobenius norm;

[0030] Step 2.2: According to the augmented Lagrangian equation, the model is iterated, and the specific steps are as follows:

[0031] Step 2.2.1: Input hyperspectral tensor to be detected , trade-off parameter ;

[0032] Step 2.2.2: Initialize the current iteration number to 0, the maximum iteration number to 100, the penalty parameter to , the maximum value of to , the step size to 1.2, , , , to 0, the error to stop iteration to ;

[0033] Step 2.2.3: Extract the sub-optimization problem involving each tensor from the augmented Lagrangian equation and solve, where:

[0034] 1) The sub-optimization problem involving the background tensor is as follows:

[0035]

[0036] where denotes the background tensor of the th iteration, denotes the anomaly tensor of the th iteration, denotes in the th iteration, denotes the gradient tensor in the th iteration process, denotes the Lagrange multiplier in the th iteration process;

[0037] The solution of the extracted sub-problem is as follows:

[0038]

[0039] where is a tensor with all 1s, denotes the Hadamard product, represents the conjugate transpose, is the difference tensor corresponding to , and denote the multi-dimensional fast Fourier transform and its inverse operation, respectively; is an intermediate variable;

[0040] 2) The sub-problem involving the gradient tensor is given by

[0041]

[0042] Let and compute the tensor singular value decomposition of where is the tensor product, the sub-problem of extraction is solved using an adaptive singular value thresholding method as follows:

[0043]

[0044] where , and denote the left singular tensor, the diagonal tensor, and the right singular tensor , respectively, the tensor transpose of , and denote the discrete Fourier transform and its inverse along the third mode, respectively, satisfies i.e.,

[0045]

[0046] where is the singular value vector;

[0047] 3) The sub-problem involving the anomaly tensor is extracted as follows:

[0048]

[0049] The sub-problem of extraction is solved using a soft thresholding operator, where the th wavelet is given by

[0050]

[0051] where , denotes taking the th frontal slice of a tensor, denotes the Frobenius norm;

[0052] 4) The multiplier in the augmented Lagrangian equation is solved as follows:

[0053] ​

[0054] Step 2.2.4: check the convergence condition as follows:

[0055]

[0056] If the above convergence condition is met, go to the next step, if not, return to step 2.2.3 until the convergence condition is met;

[0057] Step 2.2.5: output the encoding anomaly tensor at the iteration as the optimal anomaly tensor .

[0058] Step 3: according to the optimal anomaly tensor, the anomaly detection result is obtained;

[0059] The specific operation of the step 3 is:

[0060] According to the optimal anomaly tensor , the anomaly detection result is obtained, and the formula is as follows:

[0061]

[0062] Wherein, Indicates the entry of the tensor In the coordinate .

[0063] Compared with the prior art, the present application has the following advantages:

[0064] The present application uses a single background regularization term to describe the global low rank prior and local smoothness prior of the background, eliminates the selection of the weight parameter between multiple background regularization terms, and uses a tighter non-convex Laplace function to improve the approximation accuracy, thereby improving the accuracy of the background representation and having certain user friendliness.

[0065] The present application uses the tensor Norm to describe the group sparsity of the abnormal pixel, and improves the accuracy of the abnormal representation.

[0066] The present application uses the model solution based on the alternating direction multiplier method to ensure the closed-form solution of all problems, thereby improving the stability of the anomaly detection.

[0067] The detection step of the present application is clear in meaning, and the obtained anomaly detection result has higher credibility. BRIEF DESCRIPTION OF DRAWINGS

[0068] Figure 1 The flowchart of the present application;

[0069] Figure 2 From left to right, the pseudo-color image of the hyperspectral image to be detected, the anomaly reference image, and the anomaly detection result obtained by using the present application are shown.​ DETAILED DESCRIPTION

[0070] For a further understanding of the application, reference will be made to the following detailed description. It is emphasized that the application is not limited to the specific examples listed herein, and that the examples are presented for the purpose of deepening the overall understanding of the disclosure.

[0071] A hyperspectral anomaly detection method of a priori coupled driving non-convex tensor representation, the method comprising the following steps:

[0072] Step 1: constructing a basic tensor representation model, while constructing a priori coupled non-convex background regularization term and constructing an anomaly regularization term;

[0073] The specific operation of the step 1 is as follows:

[0074] Step 1.1: representing the hyperspectral image to be detected as , wherein , and represent the height, width and spectral band number respectively;

[0075] Constructing a basic tensor representation model, which is converted into a constrained minimization problem, the formula is as follows:

[0076]

[0077] wherein, and represent the regularization term of the encoding background tensor and the regularization term of the encoding anomaly tensor respectively;

[0078] Step 1.2: constructing a priori coupled non-convex background regularization term , the formula is as follows:

[0079]

[0080] wherein, represents the mode of the tensor, represents the gradient tensor obtained by using the difference operator along the mode of the background tensor , represents the tensor kernel norm of induced by the non-convex Laplacian function, which is defined as:

[0081] ​

[0082] Among them, the non-convex Laplace function Defined as , Indicates the singular values, Represents the gradient tensor The third mode after discrete Fourier transform A front slice matrix;

[0083] Prior coupled non-convex background regularization term It consists of two parts, the first part of which is the non-convex Laplace tensor nuclear norm, which is used to characterize the global low-rank prior of the background tensor, and the second part is the gradient tensor, which captures the local changes of the background tensor on the three modes and is used to characterize the local smoothness prior of the background tensor. A single non-convex regularizer is used to couple the global low-rank prior with the local smoothness prior, thus providing a more robust background representation.

[0084] Step 1.3: Build Abnormal regularization term, using tensors The norm characterizes the group sparsity prior of abnormal pixels, The formula for the anomaly regularization term is as follows:

[0085]

[0086] in, Represents an abnormal tensor In coordinates Items under;

[0087] Utilize the constructed prior coupled non-convex background regularization term and The abnormal regularization term constructs a non-convex tensor representation model driven by prior coupling, which is the following minimization problem:

[0088]

[0089] in, is a trade-off parameter that controls the balance between the non-convex background regularization term and the anomaly regularization term.

[0090] Step 2: Construct the augmented Lagrangian equation and convert the model into an unconstrained optimization problem. According to the augmented Lagrangian equation, perform model iteration and use the anomaly tensor at the end of the iteration as the optimal anomaly tensor.

[0091] The specific operations of step 2 are:

[0092] Step 2.1: Model solving based on alternating direction multiplier method, first construct the augmented Lagrange equation, convert the model into an unconstrained optimization problem, the formula is:

[0093]

[0094] wherein, , is the Lagrange multiplier, denotes the Frobenius norm;

[0095] Step 2.2: According to the augmented Lagrange equation, the model is iterated, the specific steps are as follows:

[0096] Step 2.2.1: input the hyperspectral tensor to be detected , trade-off parameter ;

[0097] Step 2.2.2: initialize the current iteration number is 0, the maximum iteration number is 100, the penalty parameter is initialized to , the maximum value of is initialized to , the step size is initialized to 1.2, , , , is initialized to 0, and the error of stopping iteration is initialized to ; ;

[0098] Step 2.2.3: extract the sub-optimization problem involving each tensor from the augmented Lagrange equation and solve it, wherein:

[0099] 1) The sub-optimization problem involving the background tensor is as follows:

[0100]

[0101] wherein, denotes the background tensor of the th iteration, denotes the anomaly tensor of the th iteration, denotes in the th iteration, denotes the gradient tensor in the th iteration process, denotes the Lagrange multipliers in the sub-iteration process ;

[0102] The sub-problem of extraction is solved as follows:

[0103]

[0104] where is a tensor of all ones, denotes the Hadamard product, represents the conjugate transpose, is the corresponding difference tensor, and denote the multi-dimensional fast Fourier transform and its inverse operation, respectively; is an intermediate variable;

[0105] 2) The sub-optimization problem involving the gradient tensor is as follows:

[0106]

[0107] Let and compute the tensor singular value decomposition of where is the tensor product, the sub-optimization problem of extraction is solved using an adaptive singular value thresholding method as follows:

[0108]

[0109] where , , and denote the left singular tensor, the diagonal tensor, and the tensor transpose of the right singular tensor , and denote the discrete Fourier transform and the inverse discrete Fourier transform along the third mode, respectively, satisfy i.e.:

[0110]

[0111] where is the singular value vector;

[0112] 3) The sub-optimization problem involving the anomaly tensor is extracted as follows:

[0113]

[0114] The sub-optimization problem of extraction is solved using a soft thresholding operator, where the first band is as follows:​

[0115]

[0116] wherein, , denotes the th frontal slice of the tensor, denotes the Frobenius norm;

[0117] 4) solve the multiplier in the augmented Lagrangian equation:

[0118]

[0119] Step 2.2.4: check the convergence condition as follows:

[0120]

[0121] If the above convergence condition is met, go to the next step, if not, return to step 2.2.3 until the convergence condition is met;

[0122] Step 2.2.5: output the encoding anomaly tensor at the iteration as the optimal anomaly tensor .

[0123] Step 3: according to the optimal anomaly tensor, the anomaly detection result is obtained, as shown in Figure 2 .

[0124] The specific operation of the step 3 is:

[0125] According to the optimal anomaly tensor , the anomaly detection result is obtained, and the formula is as follows:

[0126]

[0127] wherein, denotes the entry of the tensor under the coordinate .

[0128] The contents not described in detail in the specification of the present application belong to the prior art known to those skilled in the art. Although the above describes the specific embodiments of the present application in order to facilitate those skilled in the art to understand the present application, it should be clear that the present application is not limited to the scope of the specific embodiments, and for those skilled in the art, it is obvious that various changes are within the spirit and scope of the present application defined and limited by the appended claims, and all the inventions utilizing the concept of the present application are within the scope of protection.

Claims

1. A hyperspectral anomaly detection method based on prior coupling driven non-convex tensor representation, characterized in that: The method comprises the following steps: Step 1: Construct a basic tensor representation model and construct a priori coupled non-convex background regularization term and build Anomaly regularization term; Step 2: Construct the augmented Lagrangian equation and convert the model into an unconstrained optimization problem. According to the augmented Lagrangian equation, perform model iteration and use the anomaly tensor at the end of the iteration as the optimal anomaly tensor. Step 3: Obtain anomaly detection results based on the optimal anomaly tensor.

2. The hyperspectral anomaly detection method based on a priori coupling-driven non-convex tensor representation according to claim 1 is characterized in that: The specific operations of step 1 are: Step 1.1: Represent the hyperspectral image tensor to be detected as ,in , and Represent the height, width and number of spectral bands respectively; Construct a basic tensor representation model and transform it into a constrained minimization problem. The formula is: in, and Represent the encoded background tensor The regularization term and encoding anomaly tensor of Regularization term of ; Step 1.2: Construct a priori coupled non-convex background regularization term , the formula is as follows: ,in, Represents the tensor model, Represents the background tensor along No. Mode using difference operator The obtained gradient tensor, express The tensor nuclear norm induced by the non-convex Laplace function is defined as: , where the non-convex Laplace function Defined as , Indicates the singular values, Represents the gradient tensor The third mode after discrete Fourier transform A front slice matrix; Step 1.3: Build Abnormal regularization term, using tensors The norm characterizes the group sparsity prior of abnormal pixels, The formula for the anomaly regularization term is as follows: ,in, Represents an abnormal tensor In coordinates Items under; Utilize the constructed prior coupled non-convex background regularization term and The abnormal regularization term constructs a non-convex tensor representation model driven by prior coupling, which is the following minimization problem: ,in, is a trade-off parameter that controls the balance between the non-convex background regularization term and the anomaly regularization term.

3. The hyperspectral anomaly detection method based on a priori coupling-driven non-convex tensor representation according to claim 2 is characterized in that: The specific operations of step 2 are: Step 2.1: Solve the model based on the alternating direction multiplier method. First, construct the augmented Lagrangian equation and convert the model into an unconstrained optimization problem. The formula is: ,in, , is the Lagrange multiplier, represents the Frobenius norm; Step 2.2: According to the augmented Lagrange equation, perform model iteration. The specific steps are as follows: Step 2.2.1: Input the hyperspectral tensor to be detected , trade-off parameters ; Step 2.2.2: Initialize the current number of iterations 0, the maximum number of iterations is 100, the penalty parameter Initialized to , The maximum value Initialized to , step length Initialized to 1.2, , , , Initialized to 0, the error at which the iteration stops for ; Step 2.2.3: Extract the sub-optimization problems involved in each tensor from the augmented Lagrangian equation and solve them, where: 1) Involving background tensors The sub-optimization problem is as follows: ,in, Indicates the The background tensor of the iteration, Indicates the The exception tensor of the iteration, Indicates the In the iteration , Indicates the The gradient tensor during the iteration , Indicates the Lagrange multipliers in the iterative process ; The extracted sub-problems are solved as follows: ,in, is a tensor of all 1s, represents the Hadamard product, stands for conjugate transpose, yes The corresponding difference tensor, and Represent multidimensional fast Fourier transform and its inverse operation respectively; is an intermediate variable; 2) Involving gradient tensors The sub-optimization problem is as follows: ,make , and calculate Tensor singular value decomposition of ,in is the tensor product, and the adaptive singular value threshold method is used to solve the extracted sub-optimization problem as follows: ,in, , ,and Represents the left singular tensor, diagonal tensor, and right singular tensor respectively The tensor transpose of and denote the discrete Fourier transform and inverse discrete Fourier transform along the third mode, respectively. satisfy ,Right now: ,in, is the singular value vector; 3) Involving abnormal tensors The sub-optimization problem is extracted as follows: , using the soft threshold operator to solve the sub-optimization problem of extraction, where The bands are as follows: ,in, , Indicates taking the first A front slice, represents the Frobenius norm; 4) Solve the multipliers in the augmented Lagrange equation: ； Step 2.2.4: Check the convergence conditions as follows: If the above convergence conditions are met, proceed to the next step. If not, return to step 2.2.3 until the convergence conditions are met. Step 2.2.5: Output the encoded exception tensor at the end of the iteration as the optimal exception tensor .

4. The hyperspectral anomaly detection method based on a priori coupling-driven non-convex tensor representation according to claim 3 is characterized in that: The specific operations of step 3 are: According to the optimal abnormal tensor , the anomaly detection result is obtained, the formula is as follows: ,in, Representing a tensor In coordinates The following entry.

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