Zhen Ye, Shihao Shi, Tao Sun, Lin Bai

Abstract: As a key technique in hyperspectral image pre-processing, dimensionality reduction has received a lot of attention. However, most of the graph-based dimensionality reduction methods only consider a single structure in the data and ignore the interfusion of multiple structures. In this paper, we propose two methods for combining intra-class competition for locally preserved graphs by constructing a new dictionary containing neighbourhood information. These two methods explore local information into the collaborative graph through competing constraints, thus effectively improving the overcrowded distribution of intra-class coefficients in the collaborative graph and enhancing the discriminative power of the algorithm. By classifying four benchmark hyperspectral data, the proposed methods are proved to be superior to several advanced algorithms, even under small-sample-size conditions.

Keywords: intra-class competition; graph construction; hyperspectral image; dimensionality reduction

1 Introduction

Hyperspectral images (HSI) based on remote sensing technology development consist of hundreds of spectral bands [1−5]. Due to the rich spectral information of hyperspectral images, it has significant advantages over conventional images in areas such as target recognition. However,overly rich spectral information can make data processing difficult and is prone to Hughes phenomena that reduce classification accuracy [6]. In addition, the existence of the “dimensional curse”would require a more advanced hardware environment for processing hyperspectral data [7].Therefore, dimensionality reduction techniques have been investigated to solve the problem.

Projection strategy-based methods are viewed as a common strategy for dimensionality reduction, which is achieved by projecting hyperspectral images into a lower dimensional feature space. As a representation of the application of projection techniques, the goal of principal component analysis (PCA) is to find a projection matrix and make it satisfy the maximum variance or minimum reconstruction error [8,9]. Linear discriminant analysis (LDA), as a supervised learning-based projection dimensionality reduction technique, aims at finding a projection matrix that maximizes the generalized Rayleigh quotient [10]. However, both PCA and LDA have limitations. In the feature extraction phase of pattern recognition, PCA is not optimal for classification tasks. When the input data is multimodal, it is difficult for LDA to perform the required dimensionality reduction. As a solution idea, a graph-based approach based on manifold learning is proposed.

Graph theory, employed to describe the relationship between vertices in a graph, is widely used in social network structure analysis, HSI dimensionality reduction, and other areas [11,12].The application of graph theory to dimensionality reduction is achieved by extracting the geometric structure from the HSI and maintaining similarity. The graph-based strategy solves the generalized eigenvalue problem by constructing a graph-rights matrix to obtain a low-dimensional representation of the problem [13−15]. Locality preserving projections (LPP), a classical graph theory approach, maintains the local structure of the data by constructing nearest neighbor graphs[16]. On the basis of LPP, locality preserving Nonnegative matrix factorization (LPNMF) introduces the concept of non-negative matrix factorization (NMF) [17]. NMF, as a matrix decomposition method, achieves nonlinear dimensional approximation reduction based on the assurance that the decomposed data are all non-negative[18,19].

The obvious drawback of graph-based methods based on the Euclidean distance between two points is that they are sensitive to noise [20]. To solve this problem, sparse representation is proposed as a solution, due to the fact that the sparse coefficients, which are weights of the graph power matrix, represent most of the information in the HSI and satisfies the minimize reconstruction errors [21]. Sparse representations can be broadly classified intoℓ0-norm based sparse representations andℓ1-norm based sparse representations, whereℓ1-norm is of interest as an ideal sparse representation constraint [22,23].However, the sparse representation withℓ0-norm constraint is not a convex optimization problem and the optimal solution is more difficult to obtain, so the sparse representation based onℓ1-norm is widely used as a substitute solution.Sparse graph-based discriminant analysis (SGDA) has been applied as a sparse representation method that usesℓ1-norm based optimization to construct sparse graphs [24]. The method further improves the classification accuracy by preserving the sparse connections in the manifold.Since SGDA is not effective in obtaining the global structure of the data, an improved method called sparse and low-rank graph for discriminant analysis of hyperspectral imagery (SLGDA) was proposed, which combined a sparse graph with a low-rank representation to construct a more information-rich graph and maintain the global and local structure of [25,26].

Compared with traditional techniques, although the method based on sparse graphs withℓ1-norm has made progress in HSI dimensionality reduction, it suffers from problems such as the relationship between global data is not considered and high requirements on the number of training samples. In addition, estimating similar matrices usingℓ1-norm minimization requires high computational cost [27].Therefore, to improve the computational efficiency and reduce the sparsity of similar matrix coefficients, collaborative graph-based discriminant analysis (CGDA) has been designed to useℓ2-norm minimization instead ofℓ1-norm minimization [28,29].Since CGDA emphasizes the joint action of all elements, the collaborative representation is significantly stronger than the sparse representation in maintaining the global geometric structure of the data. This further validates the view of some scholars that it is more important to explore the global structural information than to maintain the local structure in the data.However, while CGDA performs well in maintaining global structure, it performs unsatisfactorily in maintaining local manifold structure,which is a drawback of collaborative representation. To enhance the intra-class representation,block collaborative graph-based discriminant analysis (BCGDA) introduces the block concept into CGDA, i.e., constructing collaborative graphs in subgraphs and saving them as diagonal block matrices [30,31]. Since BCGDA uses elements of the same kind as the target element to construct the collaboration graph, its effective weighting of the intra-class coefficients improves the performance of the collaboration representation in the intra-class. Different from BCGDA, an unsupervised approach of collaboration-competition preserving graph embedding (CCPGE) combined the concepts of collaboration and competition to construct a “collaboration-competition” diagram,taking advantage of the difference in focus between the two concepts to effectively learn the global geometry and local manifold structure [32].

Due to the limitations of collaborative representation, the coefficients of similar matrix constructed by CGDA and BCGDA may be too dense in local regions. Based on this, two supervised dimensionality reduction approaches are proposed, named as competition-introduced collaborative graph-based discriminant analysis (CCGDA) and competition-introduced block collaborative graph-based discriminant analysis (CBCGDA). In order to combine the local information in the data with the global geometric information of the collaborative representation, we design a new dictionary to hold the local information adding it as a constraint to the construction of the collaborative graph. Since the role of this constraint is to enhance the influence of the neighboring elements of the target element in the process of constructing the graph, this constraint is named as a competition constraint. The main contributions of this work are as follows:

1) Two improved local preserving graphs for dimensionality reduction of HSI are designed,which optimize the objective function using a closed solution due to collaborative representation. By adding intra-lass competitive constraint,the proposed graphs can effectively exploit the competition relationship of intra-class samples and preserve their local manifold features.

2) Corresponding to the proposed graphs, CCGDA and C-BCGDA have their own characteristics for supervised dimensionality reduction of HSI. C-CGDA is based upon inter-class collaborative representation, which introduces intra-class competitive constraint to exploit the relationship of inter-class and intra-class samples. C-BCGDA changes the situation that collaborative representation treats all elements equally through intra-class competition constraints, and uses local information to make the weight coefficient distribution more reasonable.

3) The proposed approaches make full use of the adjacent labeled samples, which is equivalent to expand the labeled samples and then solve the small-sample-size (SSS) problem and improve the discriminant capability in classification task.

The remainder of this paper is distributed as follows. Section 2 is divided into two major parts to discuss the dimensionality reduction methods:A review of related works, mainly including CGDA and BCGDA algorithms and graph subspace embedding, is presented; The derivation of formulas for the proposed methods C-CGDA and CBCGDA and feasibility analysis are presented. In Section 3, we present an extensive body of experimental evaluations, following with Section 4 wherein we make several concluding remarks.

2 Dimension Reduction Methods

2.1 Related Work

2.1.1 Graph-Embedding Subspace Learning

Set aNdimensional data withMlabeled samples asX=[x1,x2,···,xM]. An intrinsic graph is denoted asG=(X,W) and a penalty graph is definedGP=(X,WP), whereWis the similarity matrix andWPis the matrix of edge weight of sizeM×M[22]. The graph-based approach to dimensionality reduction builds on graph embedding, which aims to find a low-dimensional representation of a high-dimensional graph, i.e., to find a projection matrixPof sizeN×R(R≪N)whereRis the dimensionality after dimensionality reduction that generates a low-dimensional subspace by

Thus, based on the preservation of certain geometric and statistical properties of the image,the objective function can be expressed as

which can be solved as a generalized eigenvalue problem

whereΛis a diagonal matrix ofRminimal nonzero eigenvalues with eigenvalues corresponding toReigenvectors in the projection matrixP.

Since the choice of intrinsic graphGplays a decisive role in taking the effect of graph-based dimensionality reduction algorithms, it is very important to choose the appropriate similar matrix construction graph [31].

whereXidoes not containxiitself andwiis a collaborative representation vector of size(M−1)×1 corresponding toxi. Thus, the objective function of CGDA can be expressed as

whereλis the regularization parameter that acts as a balance between the residual part and the regularization term and makes the least-squares solution more stable. The solution to (5) is

wherewl,jis the collaborative representation vector corresponding toxl,jof sizeN×1,λis the regularization parameter of the stable least squares solution. The collaborative representation vector can be calculated as

Due to the strong correlation between the bands of HSI, it may not be accurate to use original data to construct the similarity matrix [33].Since the non-zero coefficients in the weight matrix may not come from intra-class samples but from inter-class samples, this limits the classification ability of collaborative expressions. Therefore, constructing the weight matrix using intraclass samples or eliminating the influence of coefficients between inter-class data can avoid this problem and effectively-enhance the differentiation between classes, where the weight matrix is designed in block-diagonal form

First pitch, strike one. Harry didn t swing, but he kept his eyes open. I pumped my fist and gave it a little swing, encouraging him to swing. He smiled, got into his awkward stance and waited. He swung, eyes open, but missed. Strike two. That was the first real swing Harry had ever taken. Who cared if we won the game? I considered Harry a winner already.

2.2 Local Preserving Graphs Using Intra-Class Competitive Representation

Since CGDA uses inter-class data to jointly represent the target element, it results in the coefficients of the weight matrix being equally divided among all the elements at the time of construction, making the coefficients of elements of the same kind as the target element too small to give an inaccurate representation of the target element. Therefore, BCGDA uses intra-class elements to construct the weight matrix, so that the weight coefficients are evenly distributed among the intra-class elements, which effectively improves the inter-class differentiation. However,BCGDA still suffers from the effect that the collaborative representation treats all elements equally (i.e., the collaborative representation only considers the global structure of the data and ignores the local information contained between the target element and its neighboring elements), making the differentiation between intra-class coefficients in the weight matrix insignificant. Thus, we investigate competitive and collaborative representation for HSI and add an intra-class competitive constraint into graph of CGDA and BCGDA, respectively.

2.2.1 C-CGDA

In [32], an unsupervised collaboration-competition graph is constructed to exploit global geometric structures as well as local manifold structures. However, it simply combines competitive representation with collaborative representation by inter-class samples neglecting the relationship between intra-class and inter-class samples.Therefore, a new dictionary is designed to store the nearest neighbors of the target element in the same kind of samples, thus ensuring that the local information is obtained from around the target element. As shown in Fig.1, the C-CGDA construction diagram is divided into two main parts, the first part is to use collaborative representation to represent all elements together to represent the target element, while the second part of the competition constraint is to use neighboring elements to represent the target element. The common effect of these two parts is to change the collaborative representation to treat all elements equally and to elevate the status of neighboring elements in the construction of the weight matrix, thus combining local information with global structural information.

Fig.1 Schematic diagram of the proposed C-CGDA with intra-class competition for hyperspectral image analysis

The method introduces intra-class competitive constraint to inter-class collaborative representation exploiting the relationship of inter-class and intra-class samples to significantly improve discriminant capability. For C-CGDA, which can be viewed as a combination of intra-class competition constraints and CGDA, then the objective function can be expressed as

In this work, the weight matrixWis a diagonal block matrix constructed by (10) and (11),then the generalized eigenvalue problem is solved by (4).

Although BCGDA improves the shortcomings of CGDA in terms of too small intra-class coefficients, it is limited by the characteristics of collaborative representation, which ignores the local information around the target elements. Therefore, in this paper, C-BCGDA is proposed to improve the distribution of intra-class coefficients.As shown in Fig.2, the C-BCGDA is also divided into two major parts, similar to the C-CGDA, and the presence of the competition constraint makes will raise the weights of neighboring elements, thus using local information to improve the distribution of intra-class coefficients.Therefore, the minimization problem is expressed as

Fig.2 Schematic diagram of the proposed C-BCGDA with intra-class competition for hyperspectral image analysis

2.3 Analysis on C-CGDA and C-BCGDA

From the above, it is clear that the two improvement methods both use the competition constraint to improve the intra-class coefficient distribution. Therefore, four methods (CGDA,BCGDA, C-CGDA and C-BCGDA) are used to construct weight matrices for each of the three types of synthetic data and analyze the differences of the four weight matrices.

Fig.3 shows the similarity matrixWconstructed by learning from the four methods, i.e.,CGDA, C-CGDA, BCGDA and C-BCGDA.Since the collaborative representation uses all data equally to represent the target elements, the weighting coefficient will be divided among all elements. However, the number of intra-class elements is small in proportion to all elements, and the weight coefficients are mostly distributed to elements of other classes, so the intra-class coefficients of the CGDA-constructed weight matrix have small values and poor differentiation between coefficients, as shown in Fig.3(a).

Fig.3 Comparison of graph W using CGDA, BCGDA, C-CGDA and C-BCGDA on three class synthetic dataset: (a) CGDA; (b)BCGDA; (c) C-CGDA; (d) C-BCGDA

To boost the representation coefficients of the intra-class elements, BCGDA uses the intraclass elements as the training set to construct the weight matrix. As shown in Fig.3(b), the value of the coefficient of the weight matrix constructed by BCGDA is significantly higher than that of CGDA, but the value of the coefficient still suffers from low differentiation, which is because the collaborative representation does not take into account the local information of the data.

As a practical matter, elements close to the target element (neighboring elements generally lift higher similarity to the target element) need to be assigned higher weights, while the weights of more distant elements should become smaller.To enhance the values of neighboring elements in the weight matrix, we introduce intra-class competition constraints into CGDA and BCGDA, respectively. As shown in Fig.3(c), compared with the weight matrix constructed by CGDA, the values of the coefficients in the weight matrix constructed by C-CGDA are significantly higher after introducing local information, and the distinction between the coefficients is higher. Also in Fig.3(d), compared to BCGDA, the C-BCGDA constructs a weight matrix with greater differentiation among the coefficients and a sparser distribution.

In addition, the coefficient distribution of the weight matrix constructed by C-CGDA is sparser in Fig.3(c) compared with Fig.3(d).This is because C-CGDA uses inter-class data as the training set, which will inevitably assign some of the coefficients to elements of other classes, thus making the intra-class coefficient distribution more sparse. From the above, the existence of intra-class competition constraint improves the drawback that the collaboration graph only considers the global structure, and the integration of local and global information makes the construction of the weight matrix more reasonable.

3 Experimental Results and Analysis

In this section, experiments are carried out on MATLAB on a 3 GHz machine with 16 GB of RAM. In order to analyze the experimental results efficiently and conveniently, we perform classification experiments on a support vector machine (SVM), which is a generalized linear classifier that classifies data in a supervised manner and has powerful capabilities to classify both linear and non-linear data [34]. For further exploring the effect of parameters of the proposed methods, the following work utilizes a cross-validation strategy for parameter optimization and employs the overall classification accuracy (OA),Kappa coefficient (KA) and average accuracy(AA) to evaluate the performance of each method.

3.1 Experiment Data

Four real datasets are used, including the Indian Pines dataset, the University of Pavia dataset,the Center of Pavia dataset and the Houston dataset.

1)Indian Pines: The Indian Pines dataset is a typical HSI dataset of size 145×145 collected with an AVIRIS sensor at the Indian Pines test site in northwestern Indiana. The dataset consists of 224 bands with a resolution of 20 m.After removing noise bands, the data contains 220 available bands. In the experiment, we extracted 150 samples in each class as a training set. The detailed information is indicated in Tab. 1.Class Class name Train Test

Tab. 1 Class labels and train-test distribution of samples for the Indian Pines dataset

2) University of Pavia: The University of Pavia dataset is composed of part of the data taken by the Reflection Optical Spectral Imager(ROSI) in Pavia. The image size is 610×340,contains 115 bands, and the resolution is 1.3 m.After removing noise bands, the data contains 103 available bands. In the experiments, we extract 200 samples from each class as a training set. The detailed information is provided in Tab .2.

3) Center of Pavia: The Center of Pavia dataset is taken by the Reflective Optics System Imaging Spectrometer (ROSIS) at the Pavia City Centre. The image size is 1 096×715 and the resolution is 1.3 m. After removing noise bands,the dataset contains 102 available bands. In the experiment, we extracted 100 samples from each class as a training set. Specific information is listed in Tab. 3.

Tab. 2 Class labels and train-test distribution of samples for the University of Pavia dataset

Tab. 3 Class labels and train-test distribution of samples for the Center of Pavia dataset

4) Houston: The Houston dataset is composed of a dataset of size 349×1905 collected in the Houston area using the ITRES CASI-1 500 sensor. The dataset consists of 144 bands in the hyperspectral range of 380 nm to 1 050 nm with a resolution of 2.5 m. In our experiments, we extract 100 samples from each class as the training set. The specific information is shown in Tab. 4.

Tab. 4 Class labels and train-test distribution of samples for the Houston dataset

3.2 Parameter Tuning

In order to obtain stable classification performance, parameter optimization is required, which is divided into regularized parameter optimization and dimensionality reducingRoptimization.First, the number of nearest neighbors of an elementkis set to 5 in the Indian Pines dataset, 5 in the University of Pavia dataset, 7 in the Center of Pavia dataset and 7 in the Houston dataset, respectively, based on a priori knowledge.Then, for the proposed C-BCGDA and C-CGDA,the range chosen for parameterλis set to 0.05 to 1 (0.05 to 0.4 for C-CGDA), the range for parameterβis set to 0.1 to 2.1 (0.05 to 0.4 for C-CGDA), and the range for reduced dimensionRis set from 2 to 40.

Fig.4 illustrates the overall accuracy (OA)performance of C-BCGDA for different parameters under the four datasets. In Fig.4(a), when the parameterλis unchanged, the classification performance of C-BCGDA increases with increasing values of the parameterβ. But when the parameterβis fixed, the classification performance of C-BCGDA decreases with increasing values of the parameterλ. Therefore,λis set to 0.05 and is set to 2.1. In Fig .4 (b), it is obvious that when the parameterβandλare greater than 1 and 0.3, respectively, the distribution of C-BCGDA classification accuracy tends to be stable.Therefore, the parameters are selected in a stable distribution area to maximize the classification accuracy, that is, setβto 2.1 and setλto 0.5. In Fig.4(c), when the parameterλdoes not change, the classification performance of C-BCGDA increases with the increase of the parameterβ, and when the parameterβdoes not change,the OA first becomes larger and then becomes smaller asλincreases. Therefore, whenλis set to 0.35 andβis set to 2.1, the proposed C-BCGDA performs the best. In Fig.4 (d), similar to the parameter settings for the Houston data set,parameter is set to 0.3 and parameter is set to 2.1.

Fig.4 Parameter tuning of λ and β for the proposed C-BCGDA on four experimental datasets: (a) Indian Pines; (b) University of Pavia; (c) Center of Pavia; (d) Houston

Fig.5 shows the OA performance of C-CGDA for different parameters under the four datasets. Based on the observation, the distributions of Fig.5 (a) and Fig.5(c) are approximately the same. Both OA decreases with increasing parameterλand increases with increasing parameterβ. Therefore, the parameterλis set to 0.05 and the parameterβis set to 0.4 in the Indian Pines datasets and Center of Pavia datasets. In Fig.5(b), since OA first increases and then decreases with increasing parameterλ, it decreases with increasing parameterβ. Hence, parameterλis set to 0.2, and parameterβis set to 0.05. In Fig.5(d), OA is more sensitive to the change of parameterβwhile it is not sensitive to parameterλ.The parameterλis set to 0.05 andβis set to 0.4.

Fig.5 Parameter tuning of λ and β for the proposed C-CGDA on four experimental datasets: (a) Indian Pines; (b) University of Pavia; (c) Center of Pavia; (d) Houston

Fig.6 shows the OA performance of seven methods in different reduced dimensions, including SGDA, LGDA, SLGDA, CGDA, BCGDA,the proposed C-CGDA and C-BCGDA. The dimensionality reduction numberRis an important parameter in the dimensionality reduction strategy, which determines the degree of data compression after dimensionality reduction and the degree of preservation of the original information. Therefore, under the condition that the classification accuracy is stable, selecting the appropriate parameterRwill help the original data to be compressed as much as possible. According to Fig.6, for C-BCGDA, the parameterRis set to 11 for the Indian Pines dataset, 14 for the University of Pavia dataset, 14 for the Center of Pavia dataset and 20 for the Houston dataset.We further evaluate the parameterRselection for the other six methods. For C-CGDA, the parameterRis set to 14, 17, 14 and 20 for above four datasets, respectively. For BCGDA, the parameter is set to 11, 14,11 and 17, respectively.For CGDA, the parameter is set to 11, 14, 11,and 17, respectively. For LGDA, the parameter is set to 11, 14, 11 and 17, respectively. For SLGDA, the parameter is set to 11, 14, 11 and 20, respectively. For SGDA, the parameter is set to 20,14, 11 and 20, respectively.

Fig.6 OA (%) versus reduced dimensionality R of extracted features for methods using the four experimental datasets: (a) Indian Pines; (b) University of Pavia; (c) Center of Pavia; (d) Houston

3.3 Classification Performance

The performance of the chosen seven methods are reported for the four datasets according to overall accuracy (OA), Kappa coefficients (KA)and average accuracy (AA). In addition to the four collaborative-based methods that have been analyzed, SGDA, LGDA and SLGDA are chosen as the comparison methods for the experiments in this paper. In [22], SGDA uses the method of constructing sparse graphs to maintain the sparse relationships in the original data, thus improving the discriminative power. In [25], LGDA effectively maintains the global structure of the data by constructing low-rank representation graphs. In addition, SLGDA improves the ability of the constructed graph in maintaining the global geometric structure by introducing the low-rank representation into SGDA, thus improving the discriminative power of the algorithm.

To avoid any bias induced by random sampling, the classification results of all methods are averaged over 20 trials. Tabs. 5-8 provide the classification performance of aforementioned methods for the Indian Pines dataset, the Pavia of University dataset, the Center of Pavia dataset and the Houston dataset, respectively. Since the classification results of C-CGDA and C-BCGDA were similar, C-BCGDA is used here as a representative for the result analysis. In Tab. 5,the performance of C-BCGDA is obviously superior to other algorithms. The OA of C-BCGDA is 2.2% higher than SLGDA, 16.5% higher than SGDA, 2.6% higher than LGDA, 6.9% higher than CGDA and 1.2% higher than BCGDA. In Tab. 6, the advantage of C-BCGDA is more pronounced. The OA of C-BCGDA yields 1.1% higher than BCGDA, 2.5% higher than CGDA, 2.5%higher than LGDA, 2.1% higher than SLGDA and 8.7% higher than SGDA. The results in Tabs. 7-8 also indicate that C-BCGDA has better performance in the Center of Pavia dataset and the Houston dataset compared to other methods.

Tab. 5 Classification accuracy (%) and time cost for the Indian Pines dataset

Tab. 6 Classification accuracy (%) and time cost for the University of Pavia dataset

Tab. 7 Classification accuracy (%) and time cost for the Center of Pavia dataset

Tab. 8 Classification accuracy (%) and time cost for the Houston dataset

Tab. 9 Classification accuracy (%) versus different numbers of training samples per class for the Indian Pines dataset

Tab. 10 Classification accuracy (%) versus different numbers of training samples per class for the University of Pavia dataset

Tab. 11 Classification accuracy (%) versus different numbers of training samples per class for the Center of Pavia dataset

Tab. 12 Classification accuracy (%) versus different numbers of training samples per class for the Houston dataset

To quantify the computational efficiency of the proposed approach to other DR strategies,training times (in seconds) are provided in the end rows of Tabs. 5-8 for the four datasets. From Tabs.5-8, it can be seen that the required computational time for the proposed C-BCGDA calculation is approximate to BCGDA, while far better than the other five methods. The reason is that C-BCGDA and BCGDA use intra-class data to construct similar matrices instead of using global data to calculate collaborative representations, which saves the time for calculating collaborative representations of inter-class samples. In addition, both CGDA and C-CGDA are time consuming. First, CGDA is calculated using inter-class samples when constructing the weight matrix, and the amount of samples is several times higher than BCGDA which uses intra-class samples, thus leading to a significant increase in computational cost. On top of that, since local competitive constraint is added, C-CGDA requires more computational resources than CGDA, but it is still within the acceptable range.

In general, the number of tagged training samples is not sufficient to better predict the model for each class, so it is meaningful to detect how the dimension reduction algorithm performs at different numbers of training samples.To avoid randomness in the experimental results,both the regularization parameters (λandβ)and the dimensionality-reducingRare fixed, and the values of OA averaged over the 20 trails. As can be seen from Tabs. 9-12, we set the number of training samples per category for the Indian Pines and the University of Pavia datasets from 125 to 250, the range for the Pavia of Center dataset from 70 to 120 and the range for the Houston dataset from 50 to 100. The experimental results show that the proposed C-CGDA and C-BCGDA perform better than the other five methods at different sample sizes in all four datasets (even under small-sample conditions).

As shown in Figs. 1-2, the multi-structural information of C-CGDA and C-BCGDA is obtained by the joint action of collaborative representation and competing constraints, while CGDA and BCGDA only use the collaborative representation representing the global information in the upper part of the figure, i.e., only a single structural information of the data is considered.In the experimental session, it is clear to understand that graph-based methods (C-CGDA and C-BCGDA) that fuse multi-structural information have better classification results than singlestructural methods (CGDA, BCGDA, etc.). This is because methods that consider only a single structure in the data often have obvious drawbacks (such as CGDA takes into account the global structural information but ignores the relationships between local data, making the differentiation of intra-class coefficients insignificant),while methods that utilize multi-structured information in the data can complement the strengths of each structure to achieve the goal of further mining the information in the original data. For example, SLGDA combines sparse representation and low-rank representation, taking into account both global and local information in the original data, and thus it outperforms LGDA and SGDA, which utilize a single structural information, in most cases. In addition, our experiments in a small sample conditions further illustrate the advantages of the proposed method in preserving the information implied by the original data.

In addition, there is a phenomenon that CCGDA performs similarly to C-BCGDA. The reason can be analyzed as follows. The biggest difference between C-CGDA and C-BCGDA is that one uses inter-class data as training samples,while the other uses intra-class data as training samples. However, the local competition constraint aims to raise the weight factor of neighboring elements in the construction graph, that is, to reduce the role of non-neighboring elements (either inter-class or intra-class elements).For C-CGDA and C-BCGDA, the neighboring elements of the target element within the class are similar. Therefore, the competition constraint will substantially reduce the difference between C-CGDA and C-BCGDA due to different training samples, and thus have similar classification results.

Figs. 7-10 perform the classification maps for the datasets corresponding to the methods as listed in Tabs. 5-8, respectively. According to the three figures, when compared with other five methods (i.e., SGDA, LGDA, SLGDA, CGDA and BCGDA), the proposed C-CGDA and CBCGDA provide obvious improvement of performance as well as smoother classification maps with fewer erroneous outliers. For example, Cornmin till and Soybean-no in Fig.7 for the Indian Pines dataset, Asphalt, Gravel and Bricks in Fig.8 for the Pavia of University dataset, Asphalt and Bitumen in Fig.9 for the Center of Pavia dataset, Residential and Parking Lot1 in Fig.10 for the Houston dataset.

Fig.7 Thematic maps resulting from classification for the Indian Pines dataset: (a) Ground truth; (b) SGDA:71.95%; (c)LGDA:84.98%; (d) SLGDA:85.48%; (e) CGDA:80.65%; (f) BCGDA:86.33%; (g) C-CGDA:87.67%; (h) C-BCGDA:87.67%

Fig.8 Thematic maps resulting from classification for the University of Pavia dataset: (a) Ground truth; (b) SGDA:81.47%; (c) LGDA:87.67%; (d) SLGDA:88.00%; (e) CGDA87.65%; (f) BCGDA:88.96%; (g) C-CGDA:90.60%; (h) C-BCGDA:90.17%

Fig.9 Thematic maps resulting from classification for the Center of Pavia dataset: (a) Ground truth; (b) SGDA:93.17%; (c)LGDA:96.78%; (d) SLGDA:96.90%; (e) CGDA:96.02%; (f) BCGDA:97.27%; (g) C-CGDA:97.72%; (h) C-BCGDA:97.68%

Fig.10 Thematic maps resulting from classification for the Houston dataset: (a) Ground truth; (b) SGDA:79.19%; (c) LGDA:86.79%;(d) SLGDA:88.20%; (e) CGDA:83.09%; (f) BCGDA:87.57%; (g) C-CGDA:89.55%; (h) C-BCGDA:89.52%

4 Conclusion

In this paper, two effective supervisory methods for HSI dimensionality reduction, competition-introduced collaborative graph-based discriminant analysis (C-CGDA) and competition-introduced block collaborative graph-based discriminant analysis (C-BCGDA), were proposed. The C-BCGDA has been designed to take into account the collaborative representation and competition of the intra-class data, further preserving the potential structure between the intra-class data while maintaining the global geometry and providing additional classification information. For C-CGDA, its graph is an inter-class collaborative representation combining with intra-class competitive constraint, thus alleviating the problem of overly dense distribution of intra-class coefficients in weight matrix. According to the experimental results evaluated using four benchmark hyperspectral datasets, the proposed C-CGDA and C-BCGDA have reflected better performance than other methods (e.g., SGDA, LGDA,SLGDA, CGDA and BCGDA), in terms of classification accuracy (even under small-sample-size conditions), Kappa coefficient and average accuracy. In summary, the proposed methods are able to preserve the intrinsic structure of HSI data and improve the discriminant classification performance.