Multi-level fusion of graph based discriminant analysis for hyperspectral image classification

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1 DOI /s Multi-level fusion of graph based discriminant analysis for hyperspectral image classification Fubiao Feng 1 Qiong Ran 1 Wei Li 1 Received: 28 May 2016 / Revised: 28 October 2016 / Accepted: 18 November 2016 Springer Science+Business Media New York 2016 Abstract Based on the graph-embedding framework, sparse graph-based discriminant analysis (SGDA), collaborative graph-based discriminant analysis (CGDA) and low rankness graph based discriminant analysis (LGDA) have been proposed with different graph construction. However, due to the inherent characteristics of l 1 -norm, l 2 -norm and nuclearnorm, single graph may be not optimal in capturing global and local structure of the data. In this paper, a multi-level fusion strategy is proposed in combining the three graph construction methods: 1) multiple graphs-based discriminant analysis (MGDA) in feature level with adaptive weights; 2) decision level fusion with D-S theory (GDA-DS), followed by a typical support vector machine (SVM) classification. Experimental results on three hyperspectral images datasets demonstrate that results with the fused strategy prevails with better classification performance. Keywords Hyperspectral data Dimensionality reduction Graph embedding Multi-level fusion D-S evidence theory 1 Introduction Hyperspectral imagery consists of hundreds of narrow contiguous wavelength bands which include detailed spectral information about materials [5, 17]. Due to the fact that these bands are often highly correlated, many dimensionality reduction algorithms have been developed to decrease computational complexity and remove redundant features that may deteriorate classification performance [19]. Qiong Ran ranqiong@mail.buct.edu.cn 1 College of Information Science and Technology, Beijing University of Chemical Technology, Beijing, , China

2 The major strategies for dimensionality reduction contain band selection and projectionbased techniques. As for band selection, the aim is to find a small subset of original bands including sufficient information [7, 16, 29]. And for the projection-based techniques, original bands are projected into a lower dimensionality subspace based on some criterion function. The traditional projection-based approaches include principle component analysis (PCA) and Fisher s linear discriminate analysis (LDA). There are a lot of modified versions that have been developed such as kernel PCA (KPCA) [8], noise-adjusted subspace LDA (NA-SLDA) [14], and kernel LDA (KDA) [15]. Some other comparatively excellent techniques include local Fisher discriminative analysis (LFDA) [19], and a kernel version of LFDA (KLFDA) [18]. Some feature extraction methods in [3, 11, 25] also have significant contributions. Besides aforementioned algorithms, there is also another branch called manifold learning for dimensionality reduction. Manifold learning started with ISOMAP [30] and locally linear embedding (LLE) [27]. ISOMAP utilizes the geodesic distance instead of Euclidean distance in multi-dimensionality scaling (MDS) [12] between pixel-pair to reduce the dimensionality of data. LLE preserves the local linear structure of data and performances well. After that, many manifold learning algorithms were proposed, such as Laplacian eigenmaps (LE) [2], locality preserving projection (LPP) [24], neighborhood preserving embedding (NPE) [9]. LE applies the Laplacian matrix to obtain the locally neighbor information. LPP and NPE are non-linearized versions of LE and LLE, respectively. Recently, graph theory is widely applied in dimensionality reduction (DR). A general graph-embedding (GE) framework was proposed in [33]to describe many existing DR techniques. Based on this framework, many graphs have been constructed. Sparse graph-based discriminant analysis (SGDA) was proposed [20, 22] by preserving the sparse connection in a block-structured affinity matrix with class-specific labeled samples. In SGDA, a graph is constructed by an l 1 -norm optimization, which is different from traditional methods (e.g., k-nearest neighbor with Euclidean distance [26]). Furthermore, weights in an l 1 -graph derived via a sparse representation can automatically select more discriminative neighbors in the feature space. However, the l 1 -graph tends to represent each sample individually, which lacks global constraint on its solutions although capable of preserving locally linear structures. In collaborative graph-based discriminant analysis (CGDA) [21], a graph is constructed by an l 2 -norm instead of l 1 -norm optimization in SGDA. With collaborative representation among labeled samples, the solution can be nicely expressed in closed form. But the l 2 -graph is difficult to distinguish the the erroneous datum. In low rankness graph based discriminant analysis (LGDA) [1, 6], low-rank representation (LRR) with nuclearnorm has been proved to be excellent at preserving global data structures using a rank function. Different from the l 1 -graph, similar samples have similar representations under a common base (dictionary) in the low-rank graph. An ideal graph should reveal the true intrinsic complexity including the local neighborhood structure as well as global geometrical structure (e.g., subspaces structure, manifolds, and multiple clusters), especially for high-dimensional data. Thus, in this work, multi-level fusion strategies [4, 10] are designed to effectively fuse multiple graphs considering limitations of the different graphs, that is, multiple graph-based discriminant analysis (MGDA) and graph-based discriminant analysis with D-S evidence theory (GDA-DS). In our first strategy, the l 1 -graph, the l 2 -graph and low-rank graph are combined via a weighted summarization where the weights are used to trade off global and local structure. Then, graph-embedding-based discriminant analysis is employed for dimensionality reduction, which attempts to find an ideal projection capturing essential data structure in an informative graph with sparsity, collaboration and low-rank, solved as a generalized eigenvalue

3 decomposition problem. In our second strategy, D-S evidence theory [34] is used to fuse results of SGDA, CGDA and LGDA after SVM classification. This method is quite effective for adjusting each graph s deviation. The rest of this paper is organized as follows. The multi-level fusion strategies are discussed in Section 2. Classification results on the real hyperspectral datasets are reported in Section 3. Section 4 gives the conclusion. 2 Proposed dimensionality reduction methods The graph-embedding framework employs undirected weighted graphs for the dimensionality reduction task. Consider an N-band hyperspectral dataset with M labeled samples denoted as X = {x i } M i=1 in a RN 1 feature space. An intrinsic graph is denoted as G ={X, W}, and a penalty graph is represented as G p ={X, W p },wherew and W p are M M matrices representing edge weights between vertices. W expresses the similarities between vertices while W p captures similarity relationships are to be supressed by the dimensionality reduction process. Different graphs are constructed with specific definitions of the intrinsic and penalty graph, such as the l 1 -graph, l 2 -graph and low-rank graph. For these graphs, limitations present with the inherent characteristics. To capture global and local structure of the data optimally, we propose multi graph-based discriminant analysis(mgda) and Graph-Based Discriminant Analysis with D-S evidence theory (GDA-DS) to perform multi-level fusion of the graphs. 2.1 Multiple graph-based discriminant analysis (MGDA) In an l 1 -graph, for each pixel x i in the dictionary X, the sparse representation (SR) vector is calculated by solving the l 1 -norm optimization problem [13], arg min wi 1 s.t. Xw i = x i, (1) w i where w i =[w i1,w i2,,w im ] is a vector of size M 1and 1 denotes the l 1 -norm which sums up the absolute values of all entries. An l 2 -graph is constructed with the following objective function, arg min w i w i 2 s.t. Xw i = x i, (2) where w i =[w i1,w i2,,w im ] is a vector of size M 1and 2 denotes the l 2 -norm. As for all the data, (1), (2) can be further expressed as, arg min W F s.t. XW = X, (3) W where when F is 1, W =[w 1, w 2,, w M ] denotes the weight matrix of size M M whose column w i is the sparse representation vector corresponding to x i, diag(w) = 0; when F is 2, it denotes l 2 graph. A low-rank graph is constructed with the following objective function, arg min W W s.t. X = XW, (4) where represents the nuclear norm of a matrix. The equation can be re-formulated as arg min W + λ X XW 2,1, (5) W

4 where 2,1 represents the l 2,1 norm [35] and λ is a regularization parameter. The symmetric low-rank graph can be represented as, W lgda = ( W + W T )/2. In the first strategy of multi-level fusion of graph, multiple graphs, i.e., W sgda, W cgda and W lgda, are fused via a weighted summarization, W mgda = ω 1 W sgda + ω 2 W cgda + ω 3 W lgda s.t. ω 1 + ω 2 + ω 3 = 1,ω 1 0,ω 2 0,ω 3 0, (6) where ω 1, ω 2 and ω 3 are parameters to balance these three terms. Through the combination, the resulting graph can simultaneously reflect global and local structure of the data. Note that when ω 2 = 0andω 3 = 0, the graph reduces to an l 1 -graph, and when ω 1 = 0and ω 3 = 0, the graph reduces to an l 2 graph, and when ω 1 = 0andω 2 = 0, the graph reduces to a low rank graph. Figure 1 shows the model of fusion with multiple graphs. 2.2 Graph-embedding subspace learning and analysis on MGDA A graph-embedding subspace learning framework [32, 33]seeks to find an N K projection matrix P (with K N) which results in a low-dimensional subspace Y = P T X,where desired statistical or geometrical characteristics are preserved. The objective function can be described as, P = argmin P T x i P T x j 2 Wi,j P T XL p X T P i =j = argmin tr(p T XLX T P), (7) P T XL p X T P where L is the Laplacian matrix of graph G, L = D W, for the fusion strategy, W = W mgda, D is a diagonal matrix with the i th diagonal element being D ii = M j=1 W i,j.ifa penalty graph used, L p may be the Laplacian matrix of the penalty graph G p or a simple scale normalization constraint [33]. The optimal projection matrix P can be obtained as, P = arg min P P T XLX T P P T XL p X T P, (8) Fig. 1 The flowchart of fusion with multiple graphs

5 which can be further transferred into a generalized eigenvalue decomposition problem, XLX T P = XL p X T P, (9) where is a diagonal eigenvalue matrix. For an N K projection matrix P, it is constructed by the K eigenvectors corresponding to the K smallest nonzero eigenvalues. By constructing the optimized graph W mgda, the projected vectors are expected to be better centralized. In this work, MGDA is proposed to reduce dimensionality of the hyperspectral image. However, spectral information can be easily affected by many factors, such as differences in illumination conditions, geometric features of material surfaces, atmospheric affects [28]. So it is necessary to discuss the statistical distributions of objects in hyperspectral data. We expect that MGDA can outperform other graphs. The aforementioned graph matrices (i.e. SGDA, CGDA, LGDA and the proposed MGDA) using 3-class synthetic data (about 400 samples per class) are illustrated in Fig. 2. As the result shows, SGDA represents the data sparsely in Fig. 2a; CGDA can obtain good results via collaborative representation in Fig. 2b; and LGDA is more robust to noise in Fig. 2c. Whereas MGDA fuses advantages of the above three graphs in a graph in Fig. 2d. Figure 3 illustrates class distributions and classification results along the first two dimensions. Figure 3a is the original data(3-class synthetic data), where class 1 is represented by Fig. 2 Visualization of different graph weights using 3-class synthetic data

6 Fig. 3 Two-dimensional 3-class synthetic data classified (with accuracy) by different graph-based methods red plus,class2isblue square, and class 3 is black circle. Some points from class 2 and 3 are hard to be distinguished. And class 1 is composed of two parts, one of which is overlapped with class 2, marked with a black circle. It is obvious that MGDA is better than

7 others in dealing with the inseparable data. Distinguishing of points from class 2 and 3 with ambiguous edges and the class 1 and 2 with overlapping areas are both dealt with perfectly. To some extend, MGDA adequately captures different advantages and performs optimally. 2.3 Graph-based discriminant analysis with D-S evidence theory (GDA-DS) In this subsection, D-S evidence does a decision level fusion to decide the label of pixel, by fusing the results of SGDA, CGDA and LGDA after SVM classification [23]. D-S evidence theory is widely applied in information fusion. As a reasoning method under uncertainty, the D-S evidence theory has two main characters: 1) can deal with distributions compared to Bayesian probability theory; 2) with direct expression uncertain and don t know Frame of discernment If denotes the set of C corresponding to C labels, let ={ 1, 2,, C } be a frame of discernment. It is composed of C mutually exhaustive and exclusive hypotheses. The power set of is the set including all the 2 C possible subsets of, represented as P( ): P( ) ={, 1, 2,, C, { 1, 2 }, { 1, 3 },, } where denotes the null set. The { C } subsets containing only one element are called singletons Basic Probability Assignment (BPA) function In the aforementioned frame, a BPA function m is defined by m : 2 [0, 1], whichis also called mass function. And this function conforms to the following conditions: { m(a) = 1 A (10) m( ) = 0 where m(a) denotes the proportion of all terms and available evidence for each term. If subset A contains only one element, A is called unit focal element; and if subset A contains multiple elements, it is called multiple focal element Belief and plausibility functions In a frame and given BPA m, a belief function Bel is defined as: Bel(A) = B A m(b) (11) and a plausibility function Pl is defined as: Pl(A) = 1 Bel(A) = B A = m(b) (12) where A and B A. Pl(A) is also called plausible function or upper limit function. Bel(A) denotes the degree that proposition A is true; Pl(A) denotes the degree that proposition A is not true. So the probable range of proposition A can be expressed as [Bel(A),Pl(A)], i.e. uncertainly. And the internelspan Pl(A) Bel(A) represents the ignorance in proposition A. the relationship of Bel(A) and Pl(A)is shown in Fig. 4.

8 Fig. 4 Confidence interval schematic Combination rule of evidence Suppose m 1,m 2 are two mass functions. according to Dempsters orthogonal rule [31], we have 0,C = m(c) = (m 1 m 2 )(C) = m 1 (A)m 2 (B) (13) A B=C,C = where K is classic friction coefficient, and represented as: K = m 1 (A)m 2 (B)(0 K 1) (14) A B= K measures the degree of the conflict between m 1 and m 2. The denominator 1 K in (13) is a normalization factor. K = 0 denotes there is no conflict between m 1 and m 2, whereas K = 1 denotes complete contradiction between m 1 and m 2. If there are more than two mass functions, such as m 1,m 2,,m n, the combination rule is as bellow: m 1 (A 1 )m 2 (A 2 ) m n (A n ) A 1 A 2 A n =A (m 1 m 2 m n )(A) = (15) 1 K where K is represented as: K = m 1 (A 1 )m 2 (A 2 ) m n (A n ) (16) A 1 A 2 A n = In our second strategy, we use SVM classification to obtain the probability of different labels with different graphs. After SVM classification, P 1n is obtained for SGDA; P 2n is obtained for CGDA; and P 3n is obtained for LGDA, where n = 1, 2,,C, C is the number of class labels. And P 1n,P 2N P 3n are mass functions like m 1,m 2,m 3. The system structure is shown in Fig K 3 Experiments and analysis In this section, the proposed multiple graph-based discriminant analysis (MGDA) and graph-based discriminant analysis with D-S evidence theory (GDA-DS) are validated with three popular hyperspectral datasets. The typical SVM is employed as the classifier.

9 Fig. 5 System structure of DS based on SVM classifier 3.1 Hyperspectral data The first data employed was acquired using National Aeronautics and Space Administration s (NASA) Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) sensor and was collected over northwest Indiana s Indian Pine test site in June The image represents Fig. 6 Parameter tuning of λ for SGDA, CGDA and LGDA

10 Table 1 Classification accuracy (%) versus the value of w 1 and w 2 for MGDA in Indian Pine dataset with 10 % training samples per class a classification scenario with pixels and 220 bands in 0.4- to 2.45-μm region of visible and infrared spectrum with a spatial resolution of 20 m. The scenario contains two-thirds agriculture, and one-third forest. In this work, a total of 202 bands is used after removal of water-absorption bands, and twelve classes are used in this study.we moved four classes out because the number of each class is less than 100. There are 10 % training samples per class (randomly selected) and a total of 9,155 testing samples. The second data was also collected by the AVIRIS sensor, capturing an area over Salinas Valley, California, with a spatial resolution of 3.7 m. The image comprises pixels with 204 bands after 20 water absorption bands are removed. It mainly contains vegetables, bare soils, and vineyard fields. There are 16 different classes, and 10 % training samples per class and a total of 48,714 testing samples. The third data was acquired by the AVIRIS instrument over the Kennedy Space Center (KSC), Florida, on March 23, AVIRIS acquires data in 224 bands of 10 nm width with center wavelengths from nm. The KSC data, acquired from an altitude of Table 2 Classification accuracy (%) versus the value of w 1 and w 2 for MGDA in Salinas dataset with 10 % training samples per class

11 Table 3 Classification accuracy (%) versus the value of w 1 and w 2 for MGDA in KSC dataset with 40 % training samples per class approximately 20 km, have a spatial resolution of 18 m. After removing water absorption and low SNR bands, 176 bands were used for the analysis. For classification purposes, 13 classes representing the various land cover types that occur in this environment were defined for the site. There are 40 % training samples per class and a total of 3,127 testing samples. 3.2 Parameter tuning We report experiments demonstrating the sensitivity of SGDA, CGDA and LGDA over a wide range of regularization parameters (i.e., λ in (5)), and dimensionality of the projected subspace. Table 4 Classification results with SVM classifier on Indian Pine data set # Class Train Test SGDA CGDA LGDA MGDA GDA-DS 1 Corn-notill Corn-mintill Corn Grass-pasture Grass-trees Hay-windrowed Soybean-notill Soybean-mintill Soybean-clean Wheat Woods Build-Grass-Trees-Drives OA AA KC

12 Table 5 Classification results with SVM classifier on Salinas data set # Class Train Test SGDA CGDA LGDA MGDA GDA-DS 1 Brocoli-green-weeds Brocoli-green-weeds Fallow Fallow-rough-plow Fallow-smooth Stubble Celery Grapes-untrained Soil-vinyard-develop Corn-senesced-green-weeds Lettuce-romaine-4wk Lettuce-romaine-5wk Lettuce-romaine-6wk Lettuce-romaine-7wk Vinyard-untrained Vinyard-vertical-trellis OA AA KC Figure 6 illustrates the classification accuracy of the SGDA, CGDA and LGDA as a function of λ with optimal reduced dimensionality. The parameter is chosen from the region of {0.001, 0.01, 0.1, 1, 10, 100, 1000}. Through cross-validation in the experiments, the Table 6 Classification results with SVM classifier on KSC data set Class Train Test SGDA CGDA LGDA MGDA GDA-DS OA AA KC

13 Fig. 7 Thematic maps resulting from classification for the Indian Pines dataset with 12 classes optimal λ of SGDA, CGDA and LGDA are set to 1000, 0.1 and 10 for the Indian Pines data; As for Salinas data, λ are set to 100, 1 and 10; and in KSC data, λ are set to 10, 1 and 1. The parameters for MGDA where ω 1 for SGDA, ω 2 for CGDA and ω 3 for LGDA (note that ω 3 = 1-ω 1 - ω 2 ) are shown in the Table 1 for Indian Pine data, Table 2 for Salinas data and Table 3 for KSC data. Noted that the remaining dimensionality is 41, 23, 39, respectively. In these tables, we can see that the same graph performs differently with different datasets. And MGDA can exploit the advantage of different graphs and lead to a good result. The feature fusion shows its advantages in assigning weights to all the graphs adaptively. The optimum weights are ω 1 = 0.1, ω 2 = 0.1 and ω 3 = 0.8 for Indian Pine data; ω 1 = 0.1, Fig. 8 Thematic maps resulting from classification for the Salinas dataset with 16 classes

14 Fig. 9 Thematic maps resulting from classification for the KSC dataset with 13 classes ω 2 = 0.5 and ω 3 = 0.4 for Salinas data; ω 1 = 0.1, ω 2 = 0.8 and ω 3 = 0.1 for KSC data. The precisions are improved greatly with indian pines dataset with an enhancement of 2.3 % to 5.3 %. While for the Salinas and KSC dataset, although the original precisions exceed 90 %, an improvement of 0.5 % to 3 % is still reported. 3.3 Classification performance To further proof the proposed methods, the overall classification accuracy (OA), the average classification accuracy (AA) and the Kappa coefficient (KC) ara utilized to evaluate the Fig. 10 Classification accuracy versus reduced dimensionality for methods using the Indian Pines dataset

15 Fig. 11 Classification accuracy versus reduced dimensionality for methods using the Salinas dataset results in Table 4, 5 and 6. The OAs, AAs and KCs of the proposed methods are better than SGDA, CGDA and LGDA. Noted with Indian pine dataset, the KCs of MGDA and GDA-DS exceed 0.8, while those of SGDA, CGDA, LGDA are inferior than 0.8. Besides, MGDA in regions of Corn-notill, Corn-mintill in Table 4, the one of Vinyard-untrained in Table 5, and the one of 4th region in Table 6 performance better than the other single graph methods. The classification map results are shown in Figs. 7, 8 and 9. Figures 10, 11 and 12 illustrate the relation between classification accuracy and the reduced dimensions for the three experimental data. The performance of other traditional Fig. 12 Classification accuracy versus reduced dimensionality for methods using the KSC dataset

16 Fig. 13 Classification performance of methods with different numbers of training-sample sizes using the experimental datasets classifiers, such as SGDA, CGDA and LGDA, is also included. It is apparent that the performance of MGDA and GDA-DS is always better than SGDA, CGDA and LGDA. While comparing the two methods MGDA and GDA-DS, MGDA performs better for Salinas and KSC dataset, especially with larger reduced dimension. We note that even for an extremely low dimensionality, the performance of MGDA and GDA-DS can be superior to others, which further suggests that the proposed strategy is able to find a transformation that can better centralize the information in the first dimensions. In practical situation, the number of training samples available is insufficient to estimate models for each class. Figure 13 shows the classification performance with different number of training samples in datasets. For Indian Pine data, the ratio to the whole available training set is set to [ ]; And for Salinas data, the ratio is set to [ ]; as for KSC data, the ratio is set to [ ]. It is apparent that the classification performance of MGDA and GDA-DS are consistently better than a single graph at various sample sizes, which implies that the multi-level fusion methods are more robust compared to the single graph-based methods.

17 4 Conclusions Based on the characteristics of l 1 -graph, l 2 and low-rank graph, a strategy with multi-level fusion of graph was proposed effectively. In the first strategy, multiple graphs-based discriminative analysis was designed to exploit the global and local structure simultaneously, while in the second method, graph-based discriminative analysis with D-S evidence theory was used to fuse classification results from separate pipeline. Compared with existing graph-embedding discriminant analysis methods, the proposed MGDA and GDA-DS can significantly reduce the dimensionality while preserving the rich statistical structure of the data. Experimental results on real hyperspectral images verified that the proposed methods consistently outperformed the traditional SGDA, CGDA and LGDA even with a small number of reduced dimensionality. Acknowledgments This work was supported by the National Natural Science Foundation of China under Grants No. NSFC , , and partly by the Fundamental Research Funds for the Central Universities under Grants No. BUCTRC201401, BUCTRC201615, YS1404, XK1521, ZY1504. References 1. Bao B, Liu G, Xu C, Yan S (2012) Inductive robust principal component analysis. IEEE Trans Image Process 21(8): Belkin M, Niyogi P (2003) Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput 15(6): Benediktsson JA, Palmason JA, Sveinsson JR (2005) Classification of hyperspectral data from urban areas based on extended morphological profiles. IEEE Trans Geosci Remote Sens 43(3): Bo C, Lu H, Wang D (2016) Hyperspectral image classification via jcr and svm models with decision fusion. IEEE Geosci Remote Sens Lett 13(2): Bo C, Lu H, Wang D (2016) Robust joint nearest subspace for hyperspectral image classification. Remote Sens Lett 7(10): Candès EJ, Li X, Ma Y, Wright J (2011) Robust principal component analysis? J ACM 3:58 7. Du Q, Yang H (2008) Similarity-based unsupervised band selection for hyperspectral image analysis. IEEE Geosci Remote Sens Lett 5(4): Fauvel M, Chanussot J, Benediktsson JA (2009) Kernel principal component analysis for the classifcation of hyperspectral remote sensing data over urban areas. EURASIP J Appl Signal Process 2009(1): He X, Cai D, Yan S, Zhang H-J (2005) Neighborhood preserving embedding. In: Tenth IEEE International Conference on Computer Vision (ICCV 05) Volume 1, vol. 2. IEEE, pp Kang X, Li S, Benediktsson JA (2014) Spectral-spatial hyperspectral image classification with edgepreserving filtering. IEEE Trans Geosci Remote Sens 52(5): Kang X, Li S, Fang L, Benediktsson JA (2015) Intrinsic image decomposition for feature extraction of hyperspectral images. IEEE Trans Geosci Remote Sens 53(4): Kruskal JB (1964) Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika 29(1): Li W, Du Q, zhang B (2015) Combined sparse and collaborative representation for hyperspectral target detection. Pattern Recogn 48: Li W, Prasad S, Fowler JE (2013) Noise-adjusted subspace discriminant analysis for hyperspectral imagery classification. IEEE Geosci Remote Sens Lett 10(6): Li W, Prasad S, Fowler JE (2014) Decision fusion in kernel-induced spaces for hyperspectral image classification. IEEE Trans Geosci Remote Sens 52(6): Li W, Prasad S, Fowler JE (2014) Hyperspectral image classification using Gaussian mixture model and Markov random field. IEEE Geosci Remote Sens Lett 11(1): Li W, Chen C, Su H, Du Q (2015) Local binary patterns and extreme learning machine for hyperspectral imagery classification. IEEE Trans Geosci Remote Sens 53(7):

18 18. Li W, Prasad S, Fowler JE, Bruce LM (2011) Locality-preserving discriminant analysis in kernelinduced feature spaces for hyperspectral image classification. IEEE Geosci Remote Sens Lett 8(5): Li W, Prasad S, Fowler JE, Bruce LM (2012) Locality-preserving dimensionality reduction and classification for hyperspectral image analysis. IEEE Trans Geosci Remote Sens 50(4): Ly N, Du Q, Fowler JE (2014) Collaborative graph-based discriminant analysis for hyperspectral imagery. IEEE J Selected Topics Appl Earth Observations Remote Sens 7(6): Ly NH, Du Q, Fowler JE (2014) Collaborative graph-based discriminant analysis for hyperspectral imagery. IEEE J Selected Topics Appl Earth Observations Remote Sens 7(6): Ly N, Du Q, Fowler JE (2014) Sparse Graph-based discriminant analysis for hyperspectral imagery. IEEE Trans Geosci Remote Sens 52(7): Melgani F, Bruzzone L (2004) Classification of hyperspectral remote sensing images with support vector machines. IEEE Trans Geosci Remote Sens 42(8): Niyogi X (2004) Locality preserving projections. In: Neural information processing systems, vol. 16. MIT, p Plaza A, Martínez P, Plaza J, Pérez R (2005) Dimensionality reduction and classification of hyperspectral image data using sequences of extended morphological transformations. IEEE Trans Geosci Remote Sens 3: Rohban MH, Rabiee HR (2012) Supervised neighborhood graph construction for semi-supervised classification. Pattern Recogn 45(4): Roweis ST, Saul LK (2000) Nonlinear dimensionality reduction by locally linear embedding. Science 290(5500): Shaw G, Manolakis D (2002) Signal processing for hyperspectral image exploitation. IEEE Signal Process Mag 19: Su H, Yang H, Du Q, Sheng Y (2011) Semisupervised band clustering for dimensionality reduction of hyperspectral imagery. IEEE Geosci Remote Sens Lett 8(6): Tenenbaum JB, De Silva V, Langford JC (2000) A global geometric framework for nonlinear dimensionality reduction. Science 290(5500): Vapnik V, Vapnik V (1998) Statistical learning theory, vol 1. Wiley, New York 32. Wright J, Ma Y, Mairal J, Sapiro G, Huang T, Yan S (2010) Sparse representation for computer vision and pattern recognition. Proc IEEE 98(6): Yan S, Xu D, ZHang B, Zhang H, Yang Q, Lin S (2007) Graph embedding and extensions: A general framework for dimensionality reduction. IEEE Trans Pattern Anal Mach Intell 29(1): Zeng D, Xu J, Xu G (2008) Data fusion for traffic incident detector using ds evidence theory with probabilistic svms. J Comput 3(10): Zhuang L, Gao H, Lin Z, Ma Y, Zhang X, Yu N (2012) Non-negative low rank and sparse graph for semisupervised learning. In: Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern recognition, Providence, Rhode Island, pp Fubiao Feng is currently pursuing the M. S. degree in Beijing University of Chemical Technology, Beijing, China. His supervisor is Dr. Wei Li and co-supervisor is Dr. Qiong Ran.

19 Qiong Ran received her Ph.D. degrees from the Institute of Remote Sensing Applications, Chinese Academy of Sciences (CAS), Beijing, China, in She has published over 10 papers in China and abroad. She is currently with the College of Information Science and Technology at Beijing University of Chemical Technology, Beijing, China. Her research interests include image acquisition, image processing, hyperspectral image analysis and applications. Wei Li received his Ph.D. degree in electrical and computer engineering from Mississippi State University, Starkville, in Subsequently, he spent one year as a postdoctoral researcher at the University of California, Davis. He is currently with the College of Information Science and Technology at Beijing University of Chemical Technology, Beijing, China. His research interests include statistical pattern recognition, hyperspectral image analysis, and data compression.