Tensor Collaborative Graph Discriminant Analysis Method for Feature Extraction of Remote Sensing Images

Abstract

Provided is a method for feature extraction of a remote sensing image based on tensor collaborative graph discriminant analysis, including: taking each of pixels as a center for intercepting a three-dimensional tensor data block; dividing experimental data into a training set and a test set in proportion; computing a Euclidean distance between a current training pixel and each class of training data; configuring a L2 norm collaborative representation model with a weight constraint; acquiring a projection matrix of each dimension of each of the three-dimensional tensor data block; and utilizing a low-dimensional projection matrix to obtain a training set and a test set, expanding the training set and the test set into a form of column vectors according to a feature dimension, inputting extracted low-dimensional features into a support vector machine classifier for classification, to determine a class of the test set, and evaluating, by a classification effect, performance of feature extraction.

Claims

1. A method for feature extraction of a remote sensing images based on tensor collaborative graph discriminant analysis, comprising: setting a size of a square sliding window, taking a first pixel of input original hyperspectral data as a starting point, and taking each of pixels as a center for intercepting a three-dimensional tensor data block; dividing experimental data into a training set and a test set in proportion according to three-dimensional tensor data blocks, and expanding each of the three-dimensional tensor data blocks into a column vector according to a spectral dimension; computing a Euclidean distance between a current training pixel and each class of training data, to construct a diagonal weight constraint matrix; configuring a L2 norm collaborative representation model with a weight constraint, to compute a representation coefficient of the current training pixel under each class of training data, to construct a graph weight matrix and a tensor locality preserving projection model; acquiring a projection matrix of each dimension of each of the three-dimensional tensor data blocks according to the tensor locality preserving projection model; and utilizing a low-dimensional projection matrix to obtain a training set and a test set which are represented by three-dimensional low dimensions, expanding the training set and the test set into a form of column vectors according to a feature dimension, inputting extracted low-dimensional features into a support vector machine classifier for classification, to determine a class of the test set, and evaluating, by a classification effect, performance of feature extraction.

2. The method for feature extraction of the remote sensing images based on the tensor collaborative graph discriminant analysis as claimed in claim 1, wherein the original hyperspectral data H ∈ R.sup.A×B×D is cut into third-order tensor blocks according to the size of the square sliding window, A and B respectively represents two spatial dimensions of the original hyperspectral data, D represents a spectral dimension of the original hyperspectral data, and R represents a real number space.

3. The method for feature extraction of the remote sensing images based on the tensor collaborative graph discriminant analysis as claimed in claim 1, wherein the size of the square sliding window is configured as w×w, one third-order tensor data block is represented as K ∈ R.sup.w×w×D, the training set obtained by division in proportion consists of N samples comprising C classes, and is represented as X=[x.sub.1, x.sub.2, . . . , x.sub.N] ∈ R.sup.w×w×w×D×N, and an l-th class of samples is represented as X.sup.l=[x.sub.1.sup.l, x.sub.2.sup.l, . . . , x.sub.N.sub.l.sup.l] ∈ R.sup.w×w×D×N.sup.l, l=1, 2, . . . , C, $N=\underset{l=1}{\overset{C}{.Math.}}{N}_l,$ x.sub.i represents an i-th data block in the training set, 1≤i≤N, N.sub.l represents the number of an l-th class of training samples, and x.sub.i.sup.l represents an i-th data block in the l-th class of training samples.

4. The method for feature extraction of the remote sensing images based on the tensor collaborative graph discriminant analysis as claimed in claim 3, wherein the test set obtained by division in proportion consists of M samples, and is represented as Y=[y.sub.1, y.sub.2, . . . , y.sub.M] ∈ R.sup.w×w×D×M, y.sub.j represents a j-th test data block, 1≤j≤M.

5. The method for feature extraction of the remote sensing images based on the tensor collaborative graph discriminant analysis as claimed in claim 1, wherein in construction of the diagonal weight constraint matrix, data blocks in the training set obtained by division in proportion are divided into C data sub-sets according to classes, an l-th data sub-set is X.sup.l, and has N.sub.l samples in total, an i-th sample x.sub.i.sup.l in the l-th data sub-set X.sup.l is expanded into a form of a vector x.sub.i.sup.l according to a modulus 3 and has a Euclidean distance Γ.sub.ij.sup.l=∥x.sub.i.sup.l−x.sub.j.sup.l∥.sub.2 from a j-th sample in the l-th data sub-set, and (N.sub.l−1) Euclidean distances are obtained, 1≤j≤N.sub.l, j≠i, ∥.Math.∥.sub.2 represents an L2 norm.

6. The method for feature extraction of the remote sensing images based on the tensor collaborative graph discriminant analysis as claimed in claim 1, wherein in response to the Euclidean distance Γ.sub.ij.sup.l being computed without containing the Euclidean distance between x.sub.i.sup.l and x.sub.i.sup.l, and the (N.sub.l=1) Euclidean distances are taken as diagonal elements of a symmetric matrix, to construct an l-th class of diagonal weight constraint matrix Γ.sup.l′ ∈ R.sup.(N.sup.l.sup.−1)×(N.sup.l.sup.−1) as follows: ${\Gamma }^{l^{\prime }}=\left[\begin{array}{cccc}{\Gamma }_{i1}^l& 0& .Math.& 0\\ 0& {\Gamma }_{i2}^l& .Math.& 0\\ .Math.& .Math.& \ddots & .Math.\\ 0& 0& .Math.& {\Gamma }_{{\mathrm{iN}}_l}^l\end{array}\right].$

7. The method for feature extraction of the remote sensing images based on the tensor collaborative graph discriminant analysis as claimed in claim 1, wherein in a construction process of the L2 norm collaborative representation model with weight constraint, an L2 norm is used for achieving sparsity constraint of a representation coefficient of the training sample x.sub.i.sup.l and reducing complexity of the model, improving representation capability of the representation coefficient by the diagonal weight constraint matrix, an within-class representation method is used, and a training sample x.sub.i.sup.l uses the same l-th class samples for representation learning, and the L2 norm collaborative representation model with weight constraint is constructed as follows:
α.sub.i.sup.l=arg min∥x.sub.i.sup.l−X.sup.l′α.sub.i.sup.l∥.sub.2.sup.2+λ∥Γ.sup.l′α.sub.i.sup.l∥.sub.2.sup.2, wherein arg min represents a minimum value of an objective function, X.sup.l′=[x.sub.1.sup.l, . . . , x.sub.i−1.sup.l, x.sub.i+1.sup.l, . . . , x.sub.N.sub.l.sup.l] ∈ R.sup.Dw.sup.2.sup.×(N.sup.l.sup.−1) represents a dictionary, in which elements include (N.sub.l−1) samples except for x.sub.i.sup.l and a dimension of the sample is Dw.sup.2, ∥.Math.∥.sub.2.sup.2 represents a square of the L2 norm of the matrix, α.sub.i.sup.l represents the representation coefficient in response to x.sub.i.sup.l taking X.sup.l′ as the dictionary, and λ represents a regularization parameter.

8. The method for feature extraction of the remote sensing images based on the tensor collaborative graph discriminant analysis as claimed in claim 1, wherein the L2 norm collaborative representation model is based on a L2 norm, and an optimal solution α.sub.i.sup.l=(X.sup.l′TX.sup.l′+λ.sup.2Γ.sup.l′TΓ.sup.l′).sup.−1X.sup.l′Tx.sub.i.sup.l of the representation coefficient α.sub.i.sup.l is obtained by means of derivation, wherein T represents a transpose of the matrix, and (.Math.).sup.−1 represents an inverse of the matrix.

9. The method for feature extraction of the remote sensing images based on the tensor collaborative graph discriminant analysis as claimed in claim 1, wherein during solving the projection matrix, the tensor locality preserving projection method is used for solving projection of three dimensions in the corresponding tensor data block, which is shown in the following expressions: $\min \underset{i,j}{.Math.}{.Math.{\stackrel{\_}{X}}_{1,\left(n\right)}{\times }_n{U}_n-{\stackrel{\_}{X}}_{j,\left(n\right)}{\times }_n{U}_n.Math.}^2{W}_{i,j}$ $\min \mathrm{Tr}\left(U_n\left(\underset{\mathrm{ij}}{.Math.}\left({\stackrel{︵}{X}}_i^n-{\stackrel{︵}{X}}_j^n\right){\left({\stackrel{︵}{X}}_i^n-{\stackrel{︵}{X}}_j^n\right)}^T{W}_{\mathrm{ij}}\right)U_n^T\right)$ $s.t.\mathrm{Tr}\left(U_n\left(\underset{\mathrm{ij}}{.Math.}{\stackrel{︵}{X}}_i^n{\stackrel{︵}{X}}_i^{\mathrm{nT}}{C}_{\mathrm{ii}}\right)U_n^T\right)=1$ wherein min represents a minimum value of an objective function, Σ represents summation operation, X.sub.i,(n) represents operation of an i-th data block according to a n-mode, ×.sub.n represents multiplication of the n-mode, U.sub.n represents the n-mode projection matrix, W.sub.i,j represents an element of the graph weight matrix with a row number being i and a column number being j, Tr(.Math.) represents a trace of the matrix, and {circumflex over (X)}.sub.i.sup.n represents expansion of the n-th modulus of the i-th data block.

10. The method for feature extraction of the remote sensing images based on the tensor collaborative graph discriminant analysis as claimed in claim 1, wherein during computation of the low-dimensional features of the training set and the test set, the low-dimensional features {circumflex over (X)}=X×.sub.1U.sub.1×.sub.2U.sub.2×.sub.3U.sub.3 and Ŷ=Y×.sub.1U.sub.1×.sub.2U.sub.2×.sub.3U.sub.3 of the training set and the test set which are represented by the three-dimensional low-dimensions are computed according to projection matrices U.sub.1, U.sub.2 and U.sub.3 on three dimensions, wherein {circumflex over (X)} and Ŷ respectively represents the low-dimensional features of the training set X and the test set Y which are represented by the three-dimensional low dimensions.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0017] FIG. 1 is a schematic diagram of tensor collaborative graph discriminant analysis based feature extraction method for remote sensing images according to an embodiment of the present disclosure;

[0018] FIG. 2 is a flow chart of tensor collaborative graph discriminant analysis method for image feature extraction according to an embodiment of the present disclosure;

[0019] FIG. 3 is a schematic diagram of expansion of a module three of a three-order tensor according to an embodiment of the present disclosure;

[0020] In order to make the objectives, technical solutions and advantages of the present disclosure clearer, the present disclosure will be further described in detail below in combination with particular embodiments with reference to the accompanying drawings.

DETAILED DESCRIPTION

[0021] With reference to FIGS. 1-3, an embodiment of the present disclosure includes: firstly, set a size of a square sliding window, take a first pixel of hyperspectral data as a starting point, and take each of pixels as a center for intercepting a three-dimensional tensor data block; divide experimental data into a training set and a test set in proportion according to the obtained tensor data blocks, and expand each of the data blocks into a column vector according to a spectral dimension; compute a Euclidean distance between a current training pixel and each class of training data, to construct a diagonal weight constraint matrix; then configure an L2 norm collaborative representation model with a constraint, to compute a representation coefficient of the current training pixel under each class of training data, so as to construct a graph weight matrix and a tensor locality preserving projection model; obtain a projection matrix of each dimension of the corresponding tensor data block by means of the tensor locality preserving projection model; and finally, utilize a low-dimensional projection matrix to obtain a training set and a test set which are represented by three-dimensional low dimensions, expand the training set and the test set into a form of column vectors according to a feature dimensionality, input extracted low-dimensional features into a support vector machine classifier for classification, to determine a class of the test set to obtain a determination result, and evaluate the performance of feature extraction by a classification effect of determination results.

[0022] With reference to FIG. 2, the embodiment of the present disclosure specifically includes:

[0023] At step 1, in an optional embodiment, the input original hyperspectral data H ∈ R.sup.A×B×D is divided into third-order tensor blocks according to the size of the square sliding window, and the tensor data blocks are divided into a training set and a test set in a certain proportion, where A and B represent two spatial dimensions of the hyperspectral data respectively, D represents a spectral dimension of the hyperspectral data, and R represents a real number space.

[0024] The size of the square sliding window is configured as w×w, the third-order tensor data block obtained by cutting may be represented as K ∈ R.sup.w×w×D, the training set obtained by division in proportion consists of N samples including C classes, and is represented as X=[x.sub.1, x.sub.2, . . . , x.sub.N] ∈ R.sup.w×w×D×N, and an l-th class of samples is represented as X.sup.l=[x.sub.1.sup.l, x.sub.2.sup.l, . . . , x.sub.N.sub.1.sup.l] ∈ R.sup.w×w×D×N.sup.l, l=1, 2, . . . , C,

[00001]$N=\underset{l=1}{\overset{C}{.Math.}}{N}_l,$

x.sub.i representing an i-th data block in the training set, 1≤i≤N, N.sub.l representing the number of an l-th class of training samples, and x.sub.i.sup.l representing an i-th data block in the l-th class of training samples.

[0025] The test set consists of M samples, and is represented as Y=[y.sub.1, y.sub.2, . . . , y.sub.M] ∈ R.sup.w×w×D×M, y.sub.j representing a j-th test data block, 1≤j≤M.

[0026] With reference to FIG. 3, at step 2, in construction of the diagonal weight constraint matrix, the data blocks in the training set obtained by division in proportion are divided into C data sub-sets according to classes, an l-th data sub-set is X.sup.l, and has N.sub.l samples in total, an i-th sample x.sub.i.sup.l the l-th data sub-set X.sup.l is expanded into a form of a vector x.sub.i.sup.l according to a module three and has a Euclidean distance Γ.sub.ij.sup.l=∥x.sub.i.sup.l−x.sub.j.sup.l∥.sub.2 from a j-th sample in the l-th data sub-set, and (N.sub.l−1) Euclidean distances are finally obtained, 1≤j≤N.sub.l, j≠i, ∥.Math.∥.sub.2 representing as an L2 norm. The embodiment of the present disclosure uses an within-class representation method, and therefore, in response to the Euclidean distance Γ.sub.ij.sup.l being computed, the Euclidean distance between x.sub.i.sup.l and itself is not included. The (N.sub.l−1) Euclidean distances are taken as diagonal elements of a symmetric matrix, to construct an l-th class of diagonal weight constraint matrix Γ.sup.l′ ∈ R.sup.(N.sup.l.sup.−1)×(N.sup.l.sup.−1) as follow formula:

[00002]${\Gamma }^{l^{\prime }}=\left[\begin{array}{cccc}{\Gamma }_{i1}^l& 0& .Math.& 0\\ 0& {\Gamma }_{i2}^l& .Math.& 0\\ .Math.& .Math.& \ddots & .Math.\\ 0& 0& .Math.& {\Gamma }_{{\mathrm{iN}}_l}^l\end{array}\right]$

[0027] At step 3, in construction of the collaborative representation model with a weight constraint, a L2 norm is used to achieve sparsity constraint of a representation coefficient of the training sample x.sub.i.sup.l, and to reduce complexity of the model, and moreover, representation capability of the representation coefficient is improved by the weight constraint matrix. The embodiment of the present disclosure uses the within-class representation method. That us, a training sample x.sub.i.sup.l only uses the same l-th class samples for representation learning, and the collaborative representation model with a weight constraint is constructed as follow formula:

α.sub.i.sup.l=arg min∥x.sub.i.sup.l−X.sup.l′α.sub.i.sup.l∥.sub.2.sup.2+λ∥Γ.sup.l′α.sub.i.sup.l∥.sub.2.sup.2,

[0028] where arg min represents a minimum value of an objective function, X.sup.l′=[x.sub.1.sup.l, . . . , x.sub.i−1.sup.l, x.sub.i+1.sup.l, . . . , x.sub.N.sub.i.sup.l] ∈ R.sup.Dw.sup.2.sup.×(N.sup.l.sup.−1) represents a dictionary, in which elements include (N.sub.l−1) samples except for x.sub.i.sup.l and a dimension of the sample is Dw.sup.2, ∥.Math.∥.sub.2.sup.2 represents a square of the L2 norm of the matrix, α.sub.i.sup.l represents the representation coefficient in response to x.sub.i.sup.l taking X.sup.l′ as the dictionary, and λ represents a regularization parameter.

[0029] At step 4, the collaborative representation model with a weight constraint is solved. The collaborative representation model is based on the L2 norm, and an optimal solution α.sub.i.sup.l=(X.sup.l′TX.sup.l′+λ.sup.2Γ.sup.l′TΓ.sup.l′).sup.−1X.sup.l′Tx.sub.i.sup.l of the representation coefficient α.sub.i.sup.l may be obtained by means of derivation, where T represents a transpose of the matrix, and (.Math.).sup.−1 represents an inverse of the matrix.

[0030] At step 5, in construction of the graph weight matrix, according to the representation coefficient α.sub.i.sup.l=[α.sub.i,1.sup.l, α.sub.i,2.sup.l, . . . , α.sub.i,N.sub.l.sub.−1.sup.l], a graph weight coefficient of the l-th class is obtained, which is represented as follow formula:

[00003]${\left(W_l\right)}_{i,j}=\{\begin{array}{cc}0,& i=j\\ {\alpha }_{i,j}^l& i>j\\ {\alpha }_{i,j-1}^l& i<j\end{array}$

[0031] finally, the graph weight matrix constructed by the training samples is as follow formula:

[00004]$W=\left[\begin{array}{cccc}{W}_1& 0& .Math.& 0\\ 0& W_2& .Math.& 0\\ .Math.& .Math.& \ddots & .Math.\\ 0& 0& .Math.& W_C\end{array}\right],$

[0032] where W.sub.i represents an i-th class of intra-class weight matrix, i=1,2, . . . , C, and C represents the total number of classes in hyperspectral data.

[0033] At step 6, during solving the projection matrix, the embodiment uses the tensor locality preserving projection algorithm to solve projection of three dimensions in the hyperspectral data block, which is shown in the following formulas:

[00005]$\min \underset{i,j}{.Math.}{.Math.{\stackrel{\_}{X}}_{1,\left(n\right)}{\times }_n{U}_n-{\stackrel{\_}{X}}_{j,\left(n\right)}{\times }_n{U}_n.Math.}^2{W}_{i,j}$$\min \mathrm{Tr}\left(U_n\left(\underset{\mathrm{ij}}{.Math.}\left({\stackrel{︵}{X}}_i^n-{\stackrel{︵}{X}}_j^n\right){\left({\stackrel{︵}{X}}_i^n-{\stackrel{︵}{X}}_j^n\right)}^T{W}_{\mathrm{ij}}\right)U_n^T\right)$$s.t.\mathrm{Tr}\left(U_n\left(\underset{\mathrm{ij}}{.Math.}{\stackrel{︵}{X}}_i^n{\stackrel{︵}{X}}_i^{\mathrm{nT}}{C}_{\mathrm{ii}}\right)U_n^T\right)=1$

[0034] where min represents a minimum value of an objective function, Σ represents summation operation, X.sub.i(n) represents operation of an i-th data block according to a n-mode, ×.sub.n represents multiplication of the n-mode, U.sub.n represents the n-mode projection matrix, W.sub.i,j represents an element of the graph weight matrix having a row number being i and a column number being j, Tr(.Math.) represents a trace of the matrix, and {circumflex over (X)}.sub.i.sup.n represents expansion of the n-th modulus of the i-th data block.

[0035] At step 7, during computation of the low-dimensional features of the training set and the test set, the low-dimensional features {circumflex over (X)}=X×.sub.1U.sub.1×.sub.2U.sub.2×.sub.3U.sub.3 and Ŷ=Y×.sub.1U.sub.1×.sub.2U.sub.2×.sub.3U.sub.3 of the training set and the test set are computed according to projection matrices U.sub.1, U.sub.2 and U.sub.3 on three dimensions obtained in step 6,

[0036] where {circumflex over (X)} and Ŷ represents the low-dimensional features of the training set X and the test set Y.

[0037] At step 8, the support vector machine classifier is used to compute classes of samples of the test set after feature extraction, the low-dimensional features {circumflex over (X)} of the training set are used to train the support vector machine classifier, and then, the low-dimensional features Ŷ of the test set are classified, so as at least to test performance of a feature extraction method according to accuracy of classification of the classes of the samples of the test set.

[0038] The objective, the technical solution and the beneficial effects of the present disclosure are further described in detail by means of the above mentioned embodiments, and it should be understood that what is mentioned above is only the particular embodiment of the present disclosure and is not intended to limit the present disclosure. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present disclosure are intended fall within the scope of protection of the present disclosure.