CN114331976A - Hyperspectral anomaly detection method based on multistage tensor prior constraint - Google Patents
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Abstract
The invention discloses a hyperspectral anomaly detection method based on multilevel tensor prior constraint, which comprises the steps of firstly, segmenting an original hyperspectral image into a background tensor and an anomalous target tensor by tensor decomposition; then, modeling low rank prior of the background tensor and sparse prior of the abnormal target tensor as a truncated nuclear norm TNN regular term and l respectively2,1Norm regularization term, and/or l is created0‑l1The HTV regular term is used for representing the spatial segmentation smoothness prior of the background tensor; and finally, fusing all regular terms together, establishing a new abnormal detection model function, and solving by using an ADMM algorithm to obtain an abnormal target detection result. The invention can improve the abnormality detection precision and reduce the corresponding false alarm rate.
Description
Technical Field
The invention relates to the technical field of hyperspectral data application, in particular to a hyperspectral anomaly detection method based on multilevel tensor prior constraint.
Background
The hyperspectral abnormal target detection technology is a research hotspot in the field of hyperspectral data application, and aims to accurately detect an abnormal target in a hyperspectral image without any target prior information. To achieve this, many anomaly detection methods are proposed, including a Reed-xiaoli (rx) detection method, a detection method based on collaborative representation, a detection method based on low rank sparse characteristics, and the like. The detection method based on the low-rank sparse characteristic can extract the abnormal target sparse characteristic of the global background low-rank characteristic, and receives wide attention and research. However, since only the address characteristic of the background and the sparse characteristic of the target are utilized and the structural feature of the hyperspectral image itself is ignored, the hyperspectral abnormal target detection accuracy needs to be improved.
Disclosure of Invention
The technical problem to be solved by the invention is to provide a hyperspectral anomaly detection method based on multistage tensor prior constraint, which can improve anomaly detection precision and reduce corresponding false alarm rate.
In order to solve the technical problem, the invention provides a hyperspectral anomaly detection method based on multilevel tensor prior constraint, which comprises the following steps of:
Step 2, unfolding a background tensor along a spectrum dimension to apply regularization characterization segmentation smoothing prior;
step 3, unfolding the background tensor along a space dimension to apply regularization to represent a low-rank prior;
step 4, unfolding the abnormal tensor along the spectrum dimension to apply regularization characterization sparse prior;
step 5, constructing a new Lagrangian function by utilizing a unified framework according to the prior characterization in the step 2, the step 3 and the step 4;
step 6, optimizing the function in the step 5 by using an ADMM algorithm;
step 7, calculating an abnormal target tensor S*＝Lspe(S) by
Obtaining an abnormality detection map
Wherein L isspeExpressing the inversion of the matrix expanded along the spectral dimension into a tensor, M denotes a total of M spectral bands, S*(i, j, l) represents the element of the ith row, jth column, and ith band of the anomaly tensor.
Preferably, in step 2, the expanding the background tensor along the spectral dimension to apply the regularization characterization piecewise smoothing prior is specifically: tensor of background
Spread into a two-dimensional matrix along the spectral dimension
Creating l0-l1The mixed total variation regularization term is
Where x and y are two spatial dimensions of the hyperspectral image, D represents a discrete difference operator for the periodic boundary, ξ represents a diagonal matrix with the binary elements 0 and 1, ζ represents the index of the diagonal matrix, DxDiscrete difference operator representing the horizontal direction, DyDiscrete difference operator representing the vertical direction, DdIs a one-dimensional finite difference operator, function
Aiming at strengthening the image edges.
Preferably, in step 3, the background tensor is oriented along the spatial dimensionThe unfolding to apply regularization characterization low rank prior is specifically: give two matrices
And
satisfies PPT＝QQT＝Ir×rCreating a truncated kernel norm regularization term for the background matrix X as
Given, where r represents the number of largest singular values,
denotes the nuclear norm, ω, of XmDenotes the mth maximum singular value of X, Tr (-) denotes the trace of the matrix.
Preferably, in step 4, the unfolding the anomaly tensor along the spectral dimension to apply regularization characterization sparse prior specifically is: tensor of abnormal object
Spread as a matrix S along the spectral dimensionspeCreating SspeL of2,1Norm regularization term of
Where d represents the spectral dimension length and represents the value of S in row i and column j.
Preferably, in step 5, according to the prior characterization of step 2, step 3, and step 4, a new lagrangian function is constructed by using a unified framework, specifically: fusing all spatial-spectral regularization terms to create a new Lagrangian function
Wherein A is1,A2,A3And A4To representAuxiliary variable,. phi.,. B1And B2Representing lagrange multipliers, α, τ being two normal numbers to balance the contributions of the terms, μ and σ representing nonnegative penalty parameters, G ═ DX, E ═ DxXDd、F＝DyXDd，Yspe、XspeRepresenting a hyperspectral matrix and a background matrix, respectively, spread along the spectral dimension.
Preferably, in step 6, the optimization of the function in step 5 by using the ADMM algorithm specifically includes the following steps:
step 6.1, fix other variables, pass E ═ Sα/2[DxXDd+A1]Updating variable E, where symbol operator
The final update is solved by applying the operator Sε[x]Sgn (x) max (| x | -epsilon, 0);
step 6.2, fix other variables, pass F ═ Sα/2[DyXDd+A2]Updating a variable F;
step 6.3, fixing other variables by G ═ G' + (I- ξ) (DX + a)3) Updating a variable G, wherein I represents an identity matrix, G' ═ ξ (DX + A)3) At the k-th iteration, G'(k)Are arranged in descending order;
step 6.4,
Is written as
Fixing other variables, updating variable A by the above equation3；
Step 6.5, fixing other variables, selecting r maximum singular values by using a Singular Value Decomposition (SVD) method, and solving
Updating P and Q; wherein Σ represents a singular value matrix and SVDs represents a singular value decomposition function;
step 6.6, fixing other variables, update
The sub-optimization problem is modeled in a vector form as
Solving using least squares to update variables
Wherein
Expressed as the Kronecker product; wherein s, a1、a2、g、a3Respectively, are vector forms corresponding to the matrix.
Step 6.7, fix other variables by
Updating variables
And
wherein
And
respectively, the expansion of the tensor along the spatial and spectral dimensions, formulated as X1＝U1(X),X2＝U2(X) and X3＝U3(X)；
Step 6.8, fix other variables, and solve
Updating variable Sspe；
Step 6.9, fix other variablesBy passing
Updating the lagrange multipliers Φ, A separatelynAnd Bn。
The invention has the beneficial effects that: (1) the invention creates a novel hyperspectral anomaly detection model which comprises tensor low-rank prior, tensor sparse prior and global space segmentation smooth prior and can fully utilize the space-spectrum structure information of a hyperspectral image; (2) according to the hyperspectral anomaly detection method, the novel hyperspectral anomaly detection model is solved by using the ADMM algorithm, the hyperspectral anomaly target detection precision is improved, and the false alarm rate is reduced under the same condition.
Drawings
FIG. 1 is a schematic flow chart of the method of the present invention.
FIG. 2(a) is a pseudo-camouflage chart of the San Diego dataset used in the detection method MSSR of the present invention.
FIG. 2(b) is a ground truth diagram of the San Diego dataset used in the detection method MSSR of the present invention.
FIG. 3 is a ROC plot of the San Diego dataset used in the detection method MSSR of the present invention compared to other methods.
FIG. 4 is a graph of the results of the testing of the San Diego dataset used in the testing method MSSR of the present invention in comparison to other methods.
Detailed Description
As shown in fig. 1, a hyperspectral anomaly detection method based on multilevel tensor prior constraint includes the following steps:
Step 2, tensor of background
Spread into a two-dimensional matrix along the spectral dimension
Creating l0-l1The mixed total variation regularization term is
Where x and y are two spatial dimensions of the hyperspectral image, D represents a discrete difference operator for the periodic boundary, ξ represents a diagonal matrix with the binary elements 0 and 1, ζ represents the index of the diagonal matrix, DxDiscrete difference operator representing the horizontal direction, DyDiscrete difference operator representing the vertical direction, DdIs a one-dimensional finite difference operator, function
Aiming at strengthening the image edge;
step 3, two matrixes are given
And
satisfies PPT＝QQT＝Ir×rCreating a truncated kernel norm regularization term for the background matrix X as
Given, where r represents the number of largest singular values,
denotes the nuclear norm, ω, of XmThe mth maximum singular value of X is represented, Tr (-) represents the trace of the matrix;
step 4, tensor of abnormal target
Spread as a matrix S along the spectral dimensionspeCreating SspeL of2,1Norm regularization term of
Wherein d represents the length of the spectral dimension and represents the value of S in the ith row and the jth column;
step 5, fusing all space-spectrum dimensional regular terms and creating a new Lagrangian function
Wherein A is1,A2,A3And A4Representing auxiliary variables, phi, B1And B2Representing lagrange multipliers, α, τ being two normal numbers to balance the contributions of the terms, μ and σ representing nonnegative penalty parameters, G ═ DX, E ═ DxXDd、F＝DyXDd，Yspe、XspeRepresenting a hyperspectral matrix and a background matrix, respectively, spread along the spectral dimension.
Step 6, optimizing the function in the step 5 by using an ADMM algorithm; the method specifically comprises the following steps:
step 6.1, fix other variables, pass E ═ Sα/2[DxXDd+A1]Updating variable E, where symbol operator
The final update is solved by applying the operator Sε[x]Sgn (x) max (| x | -epsilon, 0);
step 6.2, fix other variables, pass F ═ Sα/2[DyXDd+A2]Updating a variable F;
step 6.3, fixing other variables, through G ═ G'+(I-ξ)(DX+A3) Updating a variable G, wherein I represents an identity matrix, G' ═ ξ (DX + A)3) At the k-th iteration, G'(k)Are arranged in descending order;
step 6.4,
Is written as
Fixing other variables, updating variable A by the above equation3；
Step 6.5, fixing other variables, selecting r maximum singular values by using a Singular Value Decomposition (SVD) method, and solving
Updating P and Q; where Σ denotes a singular value matrix and SVDs denotes a singular value decomposition function.
Step 6.6, fixing other variables, update
The sub-optimization problem is modeled in a vector form as
Solving using least squares to update variables
Wherein
Expressed as the Kronecker product; wherein s, a1、a2、g、a3Respectively, are vector forms corresponding to the matrix.
Step 6.7, fix other variables by
Updating variables
And
wherein
And
respectively, the expansion of the tensor along the spatial and spectral dimensions, formulated as X1＝U1(X),X2＝U2(X) and X3＝U3(X)；
Step 6.8, fix other variables, and solve
Updating variable Sspe；
Step 6.9, fix other variables by
Updating the lagrange multipliers Φ, A separatelynAnd Bn；
Step 7, calculating an abnormal target tensor S*＝Lspe(S) by
Obtaining an abnormality detection map
. Wherein L isspeExpressing the inversion of the matrix expanded along the spectral dimension into a tensor, M denotes a total of M spectral bands, S*(i, j, l) represents the element of the ith row, jth column, and ith band of the anomaly tensor.
In order to better embody the advantages of the tensor decomposition-based multispectral spatial representation (MSSR) of the present invention, the detection method of the present invention is compared with several existing advanced detection algorithms in the following description with reference to a specific example.
Example (b):
the comparison method is as follows: and (4) carrying out abnormal target detection on the real hyperspectral image San Diego, and comparing the detection precision which can be achieved by each method. The detection accuracy is measured by using the Receiver Operating Characteristic (ROC) and the area under the curve (AUC), wherein the ROC curve represents the corresponding relation between the false alarm rate and the detection rate of each threshold segmentation result, the AUC value is obtained by calculating the area under the ROC curve of the anomaly detector, and under the same false alarm rate, the higher the AUC value is, the higher the detection performance is. The hyperspectral image used contained a number of spatial pixels of 100 x 100, the background mainly contained asphalt, road, roof and shadows, and three aircraft occupying 58 pixels of the image were considered anomalous targets. The pseudo camouflage image of the hyperspectral image and the ground truth image used are shown in fig. 2(a) and 2(b), and the detailed information thereof is shown in table 1.
TABLE 1 Hyperspectral image parameter Table
Table 2 shows AUC comparison of each method abnormality detection result, fig. 3 shows ROC curve comparison of each method abnormality detection result, and fig. 4 shows tag map comparison of each method abnormality detection result. From the results, the method provided by the invention has better detection performance compared with other methods.
TABLE 2 AUC comparison table of abnormality detection results of each method
In conclusion, the invention provides a novel hyperspectral anomaly detection method, namely MSSR. It skillfully combines the prior attributes (low rank, sparsity and segmentation smoothness) with the decomposition of the hyperspectral data tensor. Different regularization methods are used for different dimensions of the tensor to embed the priors. Wherein the dimension along the background spectrumIs represented by a truncated kernel norm, and the piecewise smoothing of the background spatial dimension is represented by l0-l1Mixed total variation regularization expression, sparse prior abnormal component is expressed by l2,1Norm regularization representation. In addition, tensor decomposition representation can effectively extract global structural features, thereby better separating the anomaly from the background. And (4) fusing all regularization constraints into a convex optimization function, and performing iterative optimization by using an ADMM algorithm. Finally, when the iterations converge, a detection map is obtained. In experiments, the performance of our proposed MSSR proved to be robust and superior to several advanced anomaly detection methods.
Claims (6)
1. A hyperspectral anomaly detection method based on multilevel tensor prior constraint is characterized by comprising the following steps:
step 1, inputting an original hyperspectral image
Decomposing the data into a background tensor and an abnormal tensor, and initializing the background tensor
The abnormal target tensor S is 0;
step 2, unfolding a background tensor along a spectrum dimension to apply regularization characterization segmentation smoothing prior;
step 3, unfolding the background tensor along a space dimension to apply regularization to represent a low-rank prior;
step 4, unfolding the abnormal tensor along the spectrum dimension to apply regularization characterization sparse prior;
step 5, constructing a new Lagrangian function by utilizing a unified framework according to the prior characterization in the step 2, the step 3 and the step 4;
step 6, optimizing the function in the step 5 by using an ADMM algorithm;
step 7, calculating an abnormal target tensor S*＝Lspe(S) by
Obtaining an abnormality detection map
Wherein L isspeExpressing the inversion of the matrix expanded along the spectral dimension into a tensor, M denotes a total of M spectral bands, S*(i, j, l) represents the element of the ith row, jth column, and ith band of the anomaly tensor.
2. The hyperspectral anomaly detection method based on multilevel tensor prior constraint according to claim 1, wherein in the step 2, the unfolding the background tensor along the spectrum dimension to apply regularization characterization piecewise smoothing prior specifically comprises: tensor of background
Spread into a two-dimensional matrix along the spectral dimension
Creating l0-l1The mixed total variation regularization term is
Where x and y are two spatial dimensions of the hyperspectral image, D represents a discrete difference operator for the periodic boundary, ξ represents a diagonal matrix with the binary elements 0 and 1, ζ represents the index of the diagonal matrix, DxDiscrete difference operator representing the horizontal direction, DyDiscrete difference operator representing the vertical direction, DdIs a one-dimensional finite difference operator, function
Aiming at strengthening the image edges.
3. The hyperspectral anomaly detection method based on multilevel tensor apriori constraint according to claim 1, wherein in step 3, the expansion of the background tensor along the spatial dimension to apply regularization to characterize the low rank prior is specifically as follows: give two matrices
And
satisfies PPT＝QQT＝Ir×rCreating a truncated kernel norm regularization term for the background matrix X as
Given, where r represents the number of largest singular values,
denotes the nuclear norm, ω, of XmDenotes the mth maximum singular value of X, Tr (-) denotes the trace of the matrix.
4. The hyperspectral anomaly detection method based on multilevel tensor prior constraint according to claim 1, wherein in step 4, unfolding the anomaly tensor along the spectral dimension to apply regularization characterization sparse prior specifically is: unfolding the anomaly target tensor S into a matrix S along the spectral dimensionspeCreating SspeL of2,1Norm regularization term of
Where d represents the spectral dimension length and represents the value of S in row i and column j.
5. The hyperspectral anomaly detection method based on multilevel tensor prior constraint according to claim 1 is characterized in that in step 5, according to prior characterization of step 2, step 3 and step 4, a unified framework is used for constructing a new Lagrangian function, specifically: fusing all spatial-spectral regularization terms to create a new Lagrangian function
Wherein A is1,A2,A3And A4Representing auxiliary variables, phi, B1And B2Representing lagrange multipliers, α, τ being two normal numbers to balance the contributions of the terms, μ and σ representing nonnegative penalty parameters, G ═ DX, E ═ DxXDd、F＝DyXDd，Yspe、XspeRepresenting a hyperspectral matrix and a background matrix, respectively, spread along the spectral dimension.
6. The hyperspectral anomaly detection method based on multilevel tensor apriori constraint according to claim 1, wherein in step 6, the optimization of the function in step 5 by using an ADMM algorithm specifically comprises the following steps:
step 6.1, fix other variables, pass E ═ Sα/2[DxXDd+A1]Updating variable E, where symbol operator
The final update is solved by applying the operator Sε[x]Sgn (x) max (| x | -epsilon, 0);
step 6.2, fix other variables, pass F ═ Sα/2[DyXDd+A2]Updating a variable F;
step 6.3, fixing other variables by G ═ G' + (I- ξ) (DX + a)3) Updating a variable G, wherein I represents an identity matrix, G' ═ ξ (DX + A)3) At the k-th iteration, G'(k)Are arranged in descending order;
step 6.4,
Is written as
Fixing other variables, updating variable A by the above equation3；
Step 6.5, fixing other variables, selecting r maximum singular values by using a Singular Value Decomposition (SVD) method, and solving
Updating P and Q; wherein Σ represents a singular value matrix and SVDs represents a singular value decomposition function;
step 6.6, fixing other variables, and modeling the sub-optimization problem of updating chi in a vector mode
Solving using least squares, updating the variable χ, wherein
Expressed as the Kronecker product; wherein s, a1、a2、g、a3Respectively in the form of vectors corresponding to the matrix;
step 6.7, fix other variables by
Updating variables
And
wherein
And
respectively, the expansion of the tensor along the spatial and spectral dimensions, formulated as X1＝U1(X),X2＝U2(X) and X3＝U3(X)；
Step 6.8, fix other variables, and solve
Updating variable Sspe；
Step 6.9, fix other variables by
Updating the lagrange multipliers Φ, A separatelynAnd Bn。
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