Classification of Multi-Frequency Polarimetric SAR Images Based on Multi-Linear Subspace Learning of Tensor Objects

Transcription

1 Remote Sens. 2015, 7, ; doi: /rs OPEN ACCESS remote sensing ISSN Artice Cassification of Muti-Frequency Poarimetric SAR Images Based on Muti-Linear Subspace Learning of Tensor Objects Chun Liu *, Junjun Yin, Jian Yang and Wei Gao Department of Eectronic Engineering, Tsinghua University, Beijing , China; E-Mais: J.Y.); J.Y.); W.G.) * Author to whom correspondence shoud be addressed; E-Mai: iuchun12@mais.tsinghua.edu.cn; Te.: ; Fax: Academic Editors: Josef Kendorfer and Prasad S. Thenkabai Received: 23 Apri 2015 / Accepted: 13 Juy 2015 / Pubished: 20 Juy 2015 Abstract: One key probem for the cassification of muti-frequency poarimetric SAR images is to extract target features simutaneousy in the aspects of frequency, poarization and spatia texture. This paper proposes a new cassification method for muti-frequency poarimetric SAR data based on tensor representation and muti-inear subspace earning MLS). Firsty, each ce of the SAR images is represented by a third-order tensor in the frequency, poarization and spatia domains, with each order of tensor corresponding to one domain. Then, two main MLS methods, i.e., muti-inear principa component anaysis MPCA) and muti-inear extension of inear discriminant anaysis MLDA), are used to earn the third-order tensors. MPCA is used to anayze the principa component of the tensors. MLDA is appied to improve the discrimination between different and covers. Finay, the ower dimension subtensor features extracted by the MPCA and MLDA agorithms are cassified with a neura network NN) cassifier. The cassification scheme is accessed using muti-band poarimetric SAR images C-, L- and P-band) acquired by the Airborne Synthetic Aperture Radar AIRSAR) sensor of the Jet Propusion Laboratory JPL) over the Fevoand area. Experimenta resuts demonstrate that the proposed method has good cassification performance in comparison with the cassic muti-band Wishart cassifier. The overa cassification accuracy is cose to 99%, even when the number of training sampes is sma. Keywords: cassification; poarimetric SAR; muti-frequency; tensor

2 Remote Sens. 2015, Introduction Land cover cassification is one of the most important appications of poarimetric synthetic aperture radar SAR). The poarimetric SAR image can be cassified more accuratey than the singe-channe SAR image, since poarization aows muti-channe measurements for the observed scene. At the same time, different frequencies are sensitive to different surface scaes. The combined use of muti-frequency data can improve the cassification accuracy, as iustrated by many researchers [1 8]. However, it is not easy to automaticay find effective target features for muti-band image cassification. Some features may be effective at one frequency in distinguishing between targets, but they may fai at another frequency for these targets. Thus, it is crucia to find effective target features for cassification of muti-frequency poarimetric SAR data. In the past two decades, some cassification schemes of muti-band poarimetric SAR data have been proposed. In view of the estabishment of statistica characteristics of the scattering coherent matrix, Lee et a. [1] proposed a generaized maximum ikeihood ML) cassifier, assuming that each frequency is statisticay independent and that the probabiity density function PDF) of the coherent matrix in singe frequency foows the compex Wishart distribution. Assuming that the coherent matrix in two frequencies is statisticay dependent, Fami et a. [2] introduced a entropyh)/anisotropya)/apha-wishart cassification scheme based on the compex Wishart density function of a 6 6 coherent matrix, which is constructed using the singe-ook compex data from the two frequency data. Datasets are cassified with the improved Wishart cassifier after an initia cassification with the H/A/apha cassifier. Frery [3] presented severa statistica distributions, incuding the Wishart distribution, K-distribution and G0-distribution, to mode the statistics of different terrains of different frequencies, and the spatia context information was described with the muti-cass Potts mode. Then, the statistica information and contextua information are fused with the Bayesian framework. Gao et a. [4] put forward a muti-frequency cassification agorithm by modeing the statistics of and with the mixture Gaussian or mixture Wishart PDFs, and then, an ML cassifier is estabished for and covers. From the anaysis of physica scattering characteristics of muti-band poarimetric SAR data, Freeman [5] proposed a hierarchica cassification agorithm using different backscatter parameters based on scattering characteristics of different farmands in different frequencies. Kouskouas [6] estimated the PDF of principa scattering parameters of different terrains in different frequencies with the maximum entropy ME) method and then cassified short vegetation with the Bayesian hierarchica cassifier. From the view of combining scattering features in different frequencies and then performing the cassification with cassic cassifiers, Chen [7] presented a cassification scheme using the dynamic earning neura network NN) cassifier to cassify the combined features from muti-frequency. Lardeux [8] used the support vector machine SVM) cassifier to cassify the high dimensiona combined feature vector. It has been demonstrated on the statistica anaysis of muti-frequency poarimetric data that some scattering components are statisticay correated for singe-frequency data, such as the HH and VV data being strongy correated in homogeneous areas [9]. The scattering characteristics are aso correated for different frequency data; they may be entirey reevant in some terrains, such as the bare soi at the L and C bands. In addition, objects usuay have texture structures, and the scattering of one pixe may be correated with its neighbors. From the view of reducing the poarization, frequency and spatia

3 Remote Sens. 2015, correation and extracting principa component of the data, Lee [10] deveoped a generaized principa component transform method to extract the principa component of muti-frequency poarimetric SAR images, and this method can maximize the signa-to-noise ratio and be taiored to the specke noise characteristics. Trizna [11] presented a projection pursuit PP) cassification method, which projects scattering features of mutiband poarimetric SAR images onto the subspace domain. Azimi-Sadjadi [12] extracted saient features of poarimetric data with principa component anaysis PCA) and then cassified data with a neura network NN) cassifier. Chamundeeswari [13] used PCA to integrate the textura measures and waveet components and cassified the fusion features with the K-means cassifier. Ainsworth [14,15] used two reated noninear dimensionaity reduction methods, i.e., ocay inear embedding LLE) and Isomap, to identify ow dimensiona structures of data; the subspace data are then cassified with the geodesic distance. Reducing the correation of the muti-band data and extracting principa features either with PCA, PP, LLE or Isomap methods, the poarimetric and spatia features of a frequencies wi be represented as a combined high dimensiona feature vector. This wi ead to the oss of the structura information of the origina data, which can be separated into three different domains. In the ast few years, a new principa feature extraction method was proposed based on the toos of mutiinear agebra, by which not ony the structure of the origina data is maintained, but aso effective principa features can be extracted. Many tensor decomposition methods summarized by Koda [16] and ow rank approximations of higher-order tensors, such as Lieven, which is presented in [17], have been appied to many fieds, such as computer vision and signa processing. Using the too of mutiinear agebra, Lu et a. [18] extended PCA to the tensor data and proposed the mutiinear principa component anaysis MPCA) method, such that the principa features of tensor structures can be extracted. Yan et a. [19] extended the inear discriminant anaysis LDA) to the tensor data and proposed the mutiinear extension of inear discriminant anaysis MLDA) to extract features of tensor structures. Because the three-domain or even muti-domain features are simutaneousy earned, the MPCA and MLDA methods are effective and usefu when the features of the data are high dimensiona and muti-domain. When these methods are used in target cassification and recognition [18,20], the performance is better than that of the method of PCA or LDA. To possess the structure information and extract features more effectivey, this paper proposes a new cassification method for muti-frequency poarimetric SAR data based on tensor representation and mutiinear subspace earning. MPCA and MLDA are combined to earn the muti-band poarimetric SAR data, which are represented by tensors. Firsty, each ce of the data is represented by a third-order tensor according to the frequency, poarization and spatia domains. Then, the MPCA agorithm is used to anayze the principa components of tensor data in a three domains, and MLDA is appied to improve the discrimination between different casses. Finay, the ower dimension subtensor feature is unfoded to a vector and then is cassified by an NN cassifier. The rest of this paper is organized as foows. In Section 2, tensor agebra is summarized, and the MPCA and MLDA agorithms are described briefy. In Section 3, the proposed scheme is presented in detai. The cassification resuts with muti-frequency poarimetric SAR data wi be shown and anayzed in Section 4. A comparison wi be aso provided. Section 5 concudes the study.

4 Remote Sens. 2015, Mutiinear Subspace Learning of Tensor Objects 2.1. Basic Tensor Agebra The tensor agebra has been discussed by [16,17]. A vector in a given basis is expressed as a one-dimensiona array; a matrix is represented by a two-dimensiona array; and a tensor with respect to a basis is represented by a muti-dimensiona array. An n-th-order tensor is denoted as X = [x i1...i n ] K I 1 I 2... I n, where n is the order of the tensor, which means that there are n path arrays. An eement of X is denoted by x i1...i n, and I k is the dimension of the k-th path array. The inner product of tensors A, B K I 1 I 2... I N is A, B = i 1... i n a i1...i n b i1...i n. The Frobenius norm of a tensor A is A F = A, A, and the corresponding distance of tensors A, B is A B F. The n-mode product of tensor A K I 1 I 2... I N and matrix U K Jn In are defined as A n U) i1,...,i n 1,j,i n+1,...,i N = In a i1 i 2...i N u jin. The n-mode unfoding of tensor A K I 1 I 2... I N is i n=1 a matrix A n) K In I n, where I n = k n k) I. A n) is tensor unfoding in which the I n path index is kept constant, and the others paths are embedded into the I n I n matrix sequentiay. The n-mode product of tensor B = A n U is equivaent to the n-mode unfoding of tensor B n) = U A n). Just as the singuar vaue decomposition SVD) of a matrix, any tensor has a simiar higher order decomposition, named the higher order singuar vaue decomposition HOSVD) or Tucker decomposition. An I 1 I 2... I n -tensor A can be expressed by an n-mode product, as foows. A = Σ 1 U 1) 2 U 2)... N U N) 1) where Σ is the core tensor of the decomposition. Σ = A 1 U 1)T 2 U 2)T... N U N)T is aso an I 1 I 2... I n -tensor. U i) is an I n I n orthogona matrix. T denotes the matrix transpose operator. According to the equivaent equations of the n-mode product and the n-mode unfoding, Equation 1) can be represented as a Kronecker product of matrices by unfoding tensors A and Σ as foows. A n) = U n) Σ n) U n+1) U N) U 1) U n 1)) T 2) 2.2. MPCA Simiar to PCA for one-dimensiona vector data, the MPCA of tensor data can be described as foows [18]. Supposing that {X m, m = 1, 2,..., M} is a set of M tensor sampes in K I 1 I 2... I N, the objective of MPCA is to search a set of projection matrices {U n) K In Jn, J n I n, n = 1,..., N}, such that the energy of the new projected tensors is maximum, where { } U n) are semi-orthogona matrices, which project the origina tensor space I 1 I 2... I N into a tensor subspace J 1 J 2... J N, and the projected tensors are {Y m K J 1 J 2... J N, m = 1, 2,..., M}, in which Y m = X m 1 U 1)T 2 U 2)T... N U N)T. The energy of new tensors can be denoted as W y = M Ym Ȳ m 2, and the objective function is: m=1 F {U n), n = 1,..., N} = arg max U 1),U 2),...,U N) W y 3)

5 Remote Sens. 2015, Simiar to the aternating iterative of HOSVD agorithm, Equation 3) can aso be soved by an aternating iterative agorithm. For the given projection matrices Û 1),..., Û n 1), Û n+1),..., Û N), supposing the tensor in W y is unfoding in the -mode, then Û ) is: Û ) = arg max U )T U ) =I = arg max U )T U ) =I M Ym Ȳ m 2 F ) m=1 )) 4) trace U )T Ψ U ) where trace.) is the trace of matrix Ψ = M ) Xm X m ) Φ T Φ X ) T m X m and ) Û+1) ) m=1 = T...Û Φ N)T Û 1)T...Û 1)T. The quantity Equation 4) to be maximized can be recognized as a Rayeigh quotient probem, and coumns of U ) are the corresponding eigenvectors of the argest J eigenvaues of matrix Ψ. If the eigenvaues of Ψ are denoted as λ t) 1, λ t) 2,..., λ t) n in the t-th iteration, J t) is the minimum number when the energy ratio of output tensors to input tensors exceeds a threshod ρ, i.e., J t) = min k { λ 1 t) + λ 2 t) λ k t) λ 1 t) + λ 2 t) λ n t) ρ where ρ is usuay set to The aternating iterative agorithm of MPCA can be described as beow. Input eements: tensor sampes { X m K I 1 I 2... I N, m = 1,..., M }, the threshod ρ and the maximum number of iterations Γ. Output eements: the projected matrices U ), = 1, 2,..., N and the output tensors { Ym K J 1 J 2... J N, m = 1,..., M }. Step 1: Cacuate the mean of the tensor set X m = 1 M M m=1 X m, and initiaize U )0) N =1 = I I I, where I I I is the identity matrix. Step 2: In the t-th iteration, cacuate Ψ t) with the -mode unfoding of tensor in W y, Ψ t) = M m=1 Xm X m )) Φt 1) where Φ t) Û+1) ) = t 1)T...Û N)t 1)T Û 1)t 1)T...Û 1)t 1)T. T } 5) Φ t 1) X m X m ) T ) 6) Step 3: Cacuate the eigenvaue decomposition Ψ t) = UΛU T. Set U )t) I J t) consists of the eigenvectors corresponding to the argest J t) eigenvaues using the criterion Equation 5). Step 4: If the termination condition U )t) U )t 1) < I J t) ε, = 1,..., N or the number of iterations t = Γ is satisfied, then go to Step 5; otherwise, return to Step 2. Step 5: Output the projections U )Γ), = 1, 2,..., N and output tensors {Y m, m = 1,..., M} 2.3. MLDA Simiar to the LDA of one-dimensiona vector data, the MLDA of tensor data can be described as foows [19,20]. Supposing that {X m, m = 1, 2,..., M} is a set of M tensor sampes in K I 1 I 2... I N,

6 Remote Sens. 2015, the objective of MLDA is to search a set of projection matrices {U n) K In Jn, J n I n, n = 1,..., N}, such that the inter-cass distance is maximum and the intra-cass variance is minimum, where { U n)} are semi-orthogona matrices, which project the origina tensor space I 1 I 2... I N into a tensor subspace J 1 J 2... J N, and the projected tensors are {Y m K J 1 J 2... J N, m = 1, 2,..., M} in which Y m = X m 1 U 1)T 2 U 2)T... N U N)T. The objective can be given as foows, U ) N =1 = arg max U 1),...,U N) C c=1 n c X c 1 U 1)T... N U N)T X 1 U 1)T... N U N)T 2 M i=1 X i 1 U 1)T... N U N)T X ci 1 U 1)T... N U N)T 2 7) where C is the tota cass number, n c is the sampe number in cass c, X c is the average tensor of the cass c and X is the tota average tensor of a sampes. The numerator of Equation 7) is inter-cass variances measured by the sum of the distances between X c and X, and the denominator is intra-cass variances measured by the sum of the distances between each tensor X i and its center tensor X ci in cass c i. Simiar to the aternating iterative agorithm of MPCA, the optimization probem Equation 7) can aso be soved by an aternating iterative agorithm. For given matrices Û 1),..., Û n 1), Û n+1),..., Û N), unfoding the tensors in objective function with the -mode, then: Û ) = arg max = arg max C c=1 nc X c X) 1 Û 1)T... N Û N)T 2 M i=1 X i X ci ) 1 Û 1)T... N Û N)T 2 U )T U ) =I ) trace U )T S B) U ) ) trace U )T S W) U ) U )T U ) =I 8) C c=1 n c X c X ) ) ΦT Φ X c X ) )T where S B) = S W) = M i=1 Xi X ci )) ΦT Φ X ) i X ci ) T is the inter-cass variance, is the intra-cass variance when the tensor is unfoded in -mode and Φ has been defined in Equation 4). The quantity Equation 8) is aso a Rayeigh quotient probem, and the projected matrix U ) consists of the eigenvectors corresponding to the most significant J eigenvaues of the matrix S W) 1 S B). If the eigenvaues of S 1 W) S B) are λ t) 1, λ t) 2,..., λ t) n at the t-th iteration, then J t) can be chosen using criterion Equation 5). The aternating iterative agorithm of MLDA is as foows. Input eements: tensor sampes { X m K I 1 I 2... I N, m = 1,..., M }, the threshod ρ, the maximum number of iterations Γ and the cass number C. Output eements: the projected matrices U ), = 1, 2,..., N and the output tensors { Ym K J 1 J 2... J N, m = 1,..., M }. Step 1: Cacuate the tota average tensor X m = 1 M M m=1 X m. The average tensor of the cass c is X c = 1 Nc N c i=1 X i, and initiaize U ) 0) N =1 = I I I. Step 2: In the t-th iteration, cacuate S t) B) and S t) W) with the -mode unfoding of the tensor, as foows. where Φ t) = S B) t) = S W) t) = C c=1 n c M i=1 X c X ) ) Φt 1)T Xi X ci )) Φt 1)T Φ t 1) X c X ) ) T ) Φ t 1) X i X ci )) Û+1) t 1)T...Û N)t 1)T Û 1)t 1)T...Û 1)t 1)T ). T ) 9)

7 Remote Sens. 2015, Step 3: Cacuate the eigenvaue decomposition consists of the eigenvectors corresponding to the argest J t) the criterion constraint Equation 5). Step 4: If the termination condition S t)) W) 1SB) t) = UΛU T. Set U )t) I J t) eigenvaues of U )t) U )t 1) < I J t) S W) t)) 1SB) t) under ε, = 1,..., N or the number of iterations t = Γ is satisfied, then go to Step 5; otherwise, return to Step 2. Step 5: Output the projections U )Γ), = 1, 2,..., N, and output the tensors {Y m, m = 1,..., M}. 3. The Proposed Agorithm for Muti-Frequency Poarimetric SAR Data Since the property of a ce in muti-frequency poarimetric SAR can be described in spatia, poarization and frequency domains, each ce of the image can be represented by a muti-order tensor. MPCA can be used to extract the principa component of the tensor data and meanwhie to possess the structure of the data in three different directions. This means that the principa component extraction can preserve the main characteristics of mutiband data in three directions simutaneousy. For the principa component of severa kinds of targets, MLDA is usefu to improve the discrimination between different casses. This means that we can aso improve the discrimination in three directions for the mutiband data. After extracting the important subtensor features. We can cassify data by measuring subtensor features of different casses with the Frobenius distance of tensors. However, the subtensor of one cass is of high-variabiity if the mutipicative specke of data is strong. The distance measure is unstabe for cassification, which wi resut in poor cassification. Therefore, we cassify the subtensors with an NN cassifier since the PDF of the subtensors is difficut to derive. The proposed agorithm is described as foows Tensor Representation of Muti-Frequency Poarimetric SAR Data In the case of a singe-frequency and singe-ook poarimetric SAR, each resoution ce is expressed by a 2 2 compex matrix named the Sincair matrix, which contains scattering coefficients of a target in a pair of orthogona poarization bases. [ ] S = S HH S V H where H, V denotes the orthogona poarization bases and the eement S qp is the backscatter coefficient when the transmitting poarization is p and the receiving poarization is q. In a monostatic case, the reciprocity theorem hods, and the Sincair matrix is symmetric, i.e.,s HV = S V H. Then, the matrix can be reduced to a three-dimensiona scattering vector k. In the compex Paui basis set, the vector becomes: k = 1 [ ] T S pp + S qq S pp S qq 2S pq 11) 2 For the muti-ook case, the coherency matrix is often used, as foows. T = 1 L T 11 T 12 T 13 k i k H i = T L 21 T 22 T 23 12) i=1 T 31 T 32 T 33 S HV S V V 10)

8 Remote Sens. 2015, where H denotes the conjugate transpose; and L is the number of ooks. There is aso a spatia structure for each cass of targets; the scattering matrix of each pixe is correated with its neighborhood pixes; one centra ce can be represented by an N N spatia ces around it. Thus, each ce of singe-frequency poarimetric SAR data can be represented by a 3 3 N N tensor. If the number of frequency bands is M, then each ce can be represented by a fifth-order compex tensor C M 3 3 N N. Since the N N spatia ces are correated with each other, we spread out the N N spatia ce matrix to an N 2 -dimensiona vector. Some eements of the T matrix are aso correated, especiay the degrees of freedom being ony five in the case of singe-ook data; thus, we spread out the 3 3 scattering matrix with a vector as T 11, T 22, T 33, ReT 12 ), ReT 13 ), ReT 23 ), ImT 12 ), ImT 13 ), ImT 23 )), where Re ) is the rea part of a compex eement and Im ) is its imaginary part. In this way, each ce of poarimetric SAR data in M bands can be represented by a third-order rea tensor R 9 M N Principa Tensor Component Extracting Based on MPCA and MLDA After the muti-frequency data have been represented by third-order tensors, the tota dimension of the data is arge. The dimension of each ce is 9 M N 2. Even if M and N are sma, such as M = N = 3, the dimension is 243. If we spread out the tensor with a vector and then process it with a PCA or cassify it with a cassifier directy, the computation is huge. Since the principa component of the data is extracted based on the tensor in the MPCA method, the tensor is processed in three directions simutaneousy. In this way, the dimension is sma in each direction, and thus, the computation wi be reduced greaty. In addition, if we process the data with vector representation, the structure information of the origina data wi be ost. Hence, we extract the principa subtensor component with the MPCA agorithm. After the principa component has been extracted, the MLDA agorithm is appied to improve the discrimination between different casses, and the dimension of the subtensor is reduced at the same time. The computation time of MLDA is aso ess than that of the LDA method, and the structure of the data in three different directions is sti possessed. The specific agorithm fow chart is shown in Figure 1. First, each ce of a of the muti-band data is represented by a third-order tensor R 9 M N 2 ; then, the training sampes are processed with the MPCA agorithm, and the projection matrices and the projected tensors of training sampes are outputted. After the subtensors are trained with the MLDA agorithm, we can get the fina projection matrices and the fina projected tensors of training sampes. Finay, the principa components of the testing data are extracted with the projection matrices that have been trained. Let U ) N =1 be the output projection matrices with the MPCA training agorithm, V ) N =1 be output projection matrices with the MLDA training agorithm and the testing tensors be X i, i = 1,..., M), and then, the extracting subtensors are Z i, where: Z i = X i 1 U 1) V 1)) T 2 U 2) V 2)) T... N U N) V N)) T 13)

9 Remote Sens. 2015, Figure 1. Agorithm fow chart of the subtensor extracting with the proposed agorithm Cassification with Neura Network After the subtensors have been extracted by the MPCA and MLDA agorithms, we can cassify data using the Frobenius distance between testing subtensors and training subtensors. Since the measure of subtensors is unstabe for the mutipicative specke of the origina data, the subtensor is spread out with a vector and cassified with a neura network cassifier. Let the training sampe be X i, i = 1,..., M, the extracted subtensor be Z i, the corresponding vector be z i, the cass abes be C i, C i {1,..., N c }, the output of z i traversing the NN be y i and the corresponding desired output vectors be c i. Then, the training mode of NN is: ) ŵ = arg min y i w) c i ) 2 14) w i where e i = y i w) c i is the error of the i-th sampe and w is the weight of the neura network. After the NN is trained and the weight of the neura network w is obtained, the output cass abe of the test sampe X k is: ĉ k = arg max y k,j w)) 15) j where y k,j w) is the vaue of the output node j for test sampe X k. 4. Experimenta Resuts and Anaysis Muti-frequency poarimetric SAR images acquired by AIRSAR with the P, L and C bands, over Fevoand in the Netherands, are used to test the performance of the proposed cassification agorithm. Paui pseudo-coor images of the three bands are presented in Figure 2a c, and the ground-truth data are shown in Figure 2d, which is drawn according to the paper of Hoekman [21]. There are 14 kinds of crops on this site, incuding potato, fruit, oats, and so on. Figure 2e is the types of crops and the corresponding coors in the ground-truth. To demonstrate the effectiveness of the proposed method, three experiments are designed. In Experiment 1, we want to show the cassification performance and resuts when the training sampes are enough and the dimension of the subtensors is reasonabe by setting high threshods in both the MPCA and MLDA steps. The cassification performance is compared with those of the cassic Wishart cassifier and the NN cassifier for the tensor data, which are unfoded into a vector without dimension

10 Remote Sens. 2015, reducing. In Experiment 2, we want to compare the cassification performance of the tensor earning method with the vector earning method, and we aso want to compare the cassification performances of different tensor earning methods, incuding the tensor earning ony using MPCA and ony using MLDA, as we as the proposed method. In Experiment 3, we want to show the robustness of the proposed method and the effectiveness of the extracted subtensor features, and then, the cassification performance is compared when the subtensor is extracted with different threshods in the MPCA and MLDA steps. a) b) c) d) e) Figure 2. The pseudo-coor images and ground-truth: a c) Paui pseudo-coor images of C, L and P bands; d) the ground-truth [21]; e) the types of crops and the corresponding coors Experiment 1 Using the same training sampes, the cassification performance of the proposed method is compared with the NN cassification agorithm with vectors and the Wishart agorithm proposed by Lee [1], in which three bands data are assumed to be statisticay independent. The training sampes are seected randomy from the ground-truth, and 500 sampes are seected for each cass. The testing sampes are a pixes in the ground-truth. The spatia window size is 3 3 in tensor representation. The threshod

11 Remote Sens. 2015, ρ is set to 0.99 in MPCA, and the threshod is set to in MLDA for the proposed agorithm. The cassification resuts are shown in Figure 3. Figure 3a is the cassification resut of the Wishart agorithm; Figure 3b is the cassification resut of the Wishart agorithm after the data have been fitered with a 3 3 window; Figure 3c is the resut of the NN cassifier in which the tensor data are unfoded into a vector without dimension reducing; and Figure 3d is the resut of the proposed agorithm. The dimension of the origina tensor is After being processed with MPCA and MLDA, the dimension of the subtensor is Tabe 1 ists the cassification resut, incuding the correct rates, the tota accuracies and the kappa coefficients of the confusion matrices. a) b) c) d) Figure 3. The cassification resuts: a) the Wishart agorithm; b) the Wishart agorithm with fitered data; c) the NN cassifier with origina data unfoded into a vector; d) the proposed agorithm. From Tabe 1, it can be observed that the cassification performance of the proposed agorithm is good. The correct rates of different crops are a higher than 95%, except corn; the overa accuracy is as high as 98.9%; and the kappa coefficient measuring the overa cassification effect of the proposed agorithm is 97.9%. The performance of the proposed method is improved for a crops compared with the Wishart cassifier and the NN cassifier with the origina data without dimension reducing, and the accuracies of some crops are improved obviousy, such as beet, barey and onions. Athough the overa accuracies of the proposed method and that of the Wishart cassifier with the fitered data are amost equa, the

12 Remote Sens. 2015, accuracy rates of the casses of onion and grass are far better than those of the Wishart cassifier. It can aso be observed that the cassification resut is better than other agorithms from Figure 3. The burr of the proposed method in homogeneous areas is ess than that of the Wishart agorithm. Athough the cassification accuracies are neary equa, the cassification accuracies are better than the NN cassifier without dimension reducing and the Wishart cassifier with the fitered data in the areas that are not marked in the ground-truth, especiay in the grass area from Figure 3. Tabe 1. The cassification resuts of the Wishart agorithm, the NN cassifier with origina data and the proposed agorithm OA is the overa accuracy) 1 14 are the orders corresponding to the crop types shown in Figure 2e): 1, potato; 2, fruit; 3, oats; 4, beet; 5, barey; 6, onions; 7, wheat; 8, beans; 9, peas; 10, maize; 11, fax; 12, rapeseed; 13, grass; 14, ucerne). 1 Tria%) Accuracy OA Kappa Wishart Wishart With fitered data NN cassifier With origina data Proposed method Experiment 2 Cassification performances of the NN cassifier with the two vector earning methods and three tensor earning methods are compared. The two vector earning methods are the PCA and PCA + LDA agorithms; the three tensor earning methods are the MPCA, MLDA and the proposed MPCA + MLDA agorithms. The window size is the same as that used in Experiment 1. To keep the cassification performance reativey good, the average dimensions of subtensors of those five agorithms about the same and the vaues reativey ow, the threshod of the PCA step is set to 0.9 in the PCA method, to 0.95 in the PCA+LDA method, the threshod of MPCA ony to 0.9, the threshod of MLDA ony to 0.99, to 0.97 in the MPCA step and to 0.99 in the MLDA step for the proposed method. The experiment is repeated 100 times. In each tria, the training sampes are seected randomy, and the sampe number for each crop is 300; the testing sampes are a pixes in ground-truth. The average dimensions of the subvectors and subtensors of those agorithms are about the same after being processed, which is 23 for PCA, 32 for MPCA, 21 for MLDA and 21 for the proposed method. The dimension of the PCA + LDA method is determined by the number of casses. Tabe 2 ists the average cassification resut of the different agorithms. According to the resuts shown in Tabe 2, firsty, we can observe that the cassification performances of the tensor-based earning methods are better than those of the vector-based earning methods. Secondy, for the three tensor-based earning methods, MPCA performs worse than the MLDA and the

13 Remote Sens. 2015, proposed method when the dimensions of the subtensors are about the same, especiay in Types 4 and 11. The performance of the proposed method is sighty better than the MLDA agorithm. Thirdy, the accuracies of the proposed method are improved compared with the Wishart cassifier with the fitered data for a crops, except Type 10. Tabe 2. The cassification performance of the PCA ony, LDA ony, muti-inear principa component anaysis MPCA) ony, muti-inear extension of inear discriminant anaysis MLDA) ony, the proposed method and Wishart agorithm. 100 Trias%) Accuracy OA Kappa PCA PCA+LDA MPCA MLDA Wishart Proposed method Experiment 3 In the third experiment, the cassification performance is compared when the subtensor is extracted with different threshods in the MPCA and MLDA steps. The threshod is varying from 0.9 to 0.99 for MPCA and varying from 0.92 to for MLDA. The training and testing sampes are seected the same as in Experiment 2, i.e., 100 trias are carried out for each threshod. The variation of cassification accuracies of a 14 crops aong with the dimensiona change of the subtensors are shown in Figure accuracy%) cass1 cass2 cass3 cass4 cass5 cass6 cass7 cass8 cass9 cass10 cass11 cass12 cass13 cass dimension OA Kappa dimension a) b) Figure 4. The accuracy of the proposed agorithm with the dimensiona change of the subtensors; a) correct rates for each crop; b) overa accuracy and kappa coefficient.

14 Remote Sens. 2015, With the increase of the dimension of subtensors, the accuracies of a crops are improved graduay. The ascent rates are fast when the dimension is ess than 20 and tend to be saturated after the dimension exceeds 20, except Crop 10. The variation tendency of the overa accuracy and the kappa coefficient are the same, aso increasing rapidy firsty and then growing sowy. It can be observed that the cassification accuracy for each crop and the overa accuracy wi exceed 90% after the dimension of subtensors is higher than 20, which shows the robustness of the proposed agorithm and the efficiency of the extracted subtensor features. 5. Concusions Taking into consideration the characteristics of frequency, poarization and spatia domain simutaneousy for the muti-frequency poarimetric synthetic aperture radar SAR) data, we have presented a muti-frequency poarimetric SAR cassification scheme based on tensor representation and mutiinear subspace earning MSL). Compared with the cassic method in which the data or features are represented and earned by vector, the proposed method represents the ce of muti-band data with a third-order tensor in the directions of spatia, frequency and poarization. The principe features are extracted more precisey, and the discrimination abiity of the features between casses is improved when using the muti-inear principa component anaysis MPCA) method and the muti-inear extension of inear discriminant anaysis MLDA) method. The proposed agorithm was evauated with muti-frequency poarimetric SAR data of Airborne Synthetic Aperture Radar AIRSAR). The cassification performances of the three tensor earning agorithms, two vector earning agorithms and the cassic Wishart cassifier were compared. The accuracies of the proposed agorithm with the dimensiona change of subtensors were aso tested. Experimenta resuts have demonstrated that the cassification accuracy of the proposed agorithm is better than other agorithms; the performance is promising, even when the dimension of the extracted subtensors is ow. In addition, the extracted subtensors are three-dimensiona, and the overa structure of the origina data in the directions of spatia, frequency and poarization is we kept. Acknowedgments The work was in part supported by the Nationa Natura Science Foundation of China NSFC) No ), in part supported by the key project of the NSFC No ), in part supported by the major research pan of the NSFC No ), in part supported by the Aeronautica Science Foundation of China No ) and in part supported by the Research Foundation of Tsinghua University. The authors woud aso ike to thank the reviewers for their hepfu comments. Author Contributions The genera idea for the cassification method for muti-frequency poarimetric SAR images using muti-inear subspace earning was proposed by Chun Liu. Junjun Yin and Jian Yang revised the paper and put forward many usefu modification suggestions. The experiments were carried out by Chun Liu and Wei Gao.

15 Remote Sens. 2015, Conficts of Interest The authors decare no confict of interest. References 1. Lee, J.S.; Grunes, M.R.; Kwok, R. Cassification of muti-ook poarimetric SAR imagery based on compex Wishart distribution. Int. J. Remote Sens. 1994, 15, Fami, L.F.; Pottier, E.; Lee, J.S. Unsupervised cassification of mutifrequency and fuy poarimetric SAR images based on the H/A/Apha-Wishart cassifier. IEEE Trans. Geosci. Remote Sens. 2001, 39, Frery, A.C.; Correia, A.H.; Da Freitas, C. Cassifying muti-frequency fuy poarimetric imagery with mutipe sources of statistica evidence and contextua information. IEEE Trans. Geosci. Remote Sens. 2007, 45, Gao, W.; Yang, J.; Ma, W. Land cover cassification for poarimetric SAR images based on mixture modes. Remote Sens. 2014, 6, Freeman, A.; Wiasenor, J.; Kein, J.D. On the use of muti-frequency and poarimetric radar backscatter features for cassification of agricutura crops. Int. J. Remote Sens. 1994, 15, Kouskouas, Y.; Uaby, F.T.; Pierce, L.E. The Bayesian hierarchica cassifier BHC) and its appication to short vegetation using muti-frequency poarimetric SAR. IEEE Trans. Geosci. Remote Sens. 2004, 42, Chen, K.S.; Huang, W.P. Cassification of mutifrequency poarimtric SAR imagery using a dynamic earning neura network. IEEE Trans. Geosci. Remote Sens. 1996, 34, Lardeux, C.; Frison, P.L.; Tison, C.; Souyris, J.C.; Sto, B.; Fruneau, B.; Rudant, J.P. Support vector machine for muti-frequency SAR poarimetric data cassification. IEEE Trans. Geosci. Remote Sens. 2009, 47, Lee, J.S.; Grunes, M.R.; Mango, S.A. Specke reduction in mutipoarization, muti-frequency SAR imagery. IEEE Trans. Geosci. Remote Sens. 1991, 29, Lee, J.S.; Hoppe, K. Principa components transformation of muti-frequency poarimetric SAR imagery. IEEE Trans. Geosci. Remote Sens. 1992, 30, Trizna, D.B.; Bachmann, C.; Setten, M.; Aan, N.; Toporkov, J.; Harris, R. Projection pursuit cassification of mutiband poarimetric SAR and images. IEEE Trans. Geosci. Remote Sens. 2001, 39, Azimi-Sadjadi, M.R.; Ghaoum, S.; Zoughi, R. Terrain cassification in SAR images using principa components anaysis and neura networks. IEEE Trans. Geosci. Remote Sens. 1993, 31, Chamundeeswari, V.V.; Singh, D.; Singh, K. An anaysis of texture measures in PCA-based unsupervised cassification of SAR images. IEEE Trans. Geosci. Remote Sens. 2009, 6, Ainsworth, T.L.; Lee, J.S. Optima poarimetric decomposition variabes-non-inear dimensionaity reduction. IEEE Inter. Geosci. Remote Sens. 2001, 2,

16 Remote Sens. 2015, Ainsworth, T.L.; Lee, J.S. Poarimetric SAR image cassification-expoiting optima variabes derived from mutipe-image datasets. In Proceedings of the IEEE Internationa Geoscience and Remote Sensing Symposium, Anchorage, AK, USA, September 2004; Voume Koda, T.G.; Bader, B.W. Tensor decompositions and appications. SIAM Rev. 2009, 51, De Lathauwer, L.; de Moor, B.; Vandewae, J. On the best rank-1 and rank-r 1, r 2,..., rn) approximation of higher-order tensors. SIAM J. Matrix Ana. App. 2000, 21, Lu, H.; Pataniotis, K.N.; Venetsanopouos, A.N. MPCA: Mutiinear principa component anaysis of tensor objects. IEEE Trans. Neura Netw. 2008, 19, Yan, S.; Xu, D.; Yang, Q.; Zhang, L.; Tang, X.; Zhang, H.J. Discriminant anaysis with tensor representation. In Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, San Diego, CA, USA, June 2005; Voume 1, pp Tao, D.; Li, X.; Wu, X.; Maybank, S.J. Genera tensor discriminant anaysis and gabor features for gait recognition. IEEE Trans. PAMI 2007, 29, Hoekman, D.H.; Vissers, M.A. A new poarimetric cassification approach evauated for agricutura crops. IEEE Trans. Geosci. Remote Sens. 2003, 41, c 2015 by the authors; icensee MDPI, Base, Switzerand. This artice is an open access artice distributed under the terms and conditions of the Creative Commons Attribution icense