Neurocomputing. Laplacian bidirectional PCA for face recognition. Wankou Yang a,n, Changyin Sun a, Lei Zhang b, Karl Ricanek c. Letters.

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1 Neurocomputing 74 (2010) Contents lists available at ScienceDirect Neurocomputing journal homepage: Letters Laplacian bidirectional PCA for face recognition Wankou Yang a,n, Changyin Sun a, Lei Zhang b, Karl Ricanek c a School of Automation, Southeast University, Nanjing , China b Biometrics Research Centre, Dept. of Computing, Hong Kong Polytechnic University, Hong Kong c Face Aging Group, Dept. of Computer Science, UNC Wilmington, USA article info Article history: Received 24 May 2010 Received in revised form 11 August 2010 Accepted 26 August 2010 Communicated by Qi Li Available online 16 October 2010 Keywords: 2DPCA BDPCA Laplacian Face recognition abstract Two-dimensional principal components analysis (2DPCA) needs more coefficients than principal components analysis (PCA) for image representation and hence needs more time for classification. The bidirectional PCA (BDPCA) is proposed to overcome these drawbacks of 2DPCA. Both 2DPCA and BDPCA, however, can work only in Euclidean space. In this paper, we propose Laplacian BDPCA (LBDPCA) to enhance the robustness of BDPCA by extending it to non-euclidean space. Experimental results on representative face databases show that LBDPCA works well and it surpasses BDPCA. & 2010 Elsevier B.V. All rights reserved. 1. Introduction Face recognition (FR) has been an active research area in the computer vision and pattern recognition community for more than two decades [1]. One of the most popular techniques for FR is the so-called subspace learning method, which aims to reveal the distinctive features of high dimensional data in a lower dimensional subspace. Principal component analysis (PCA) is among the most well-known and widely used subspace learning method [2]. In the PCA-based FR techniques, the 2D face image matrices need to be transformed into 1D image vectors. The resulting high dimensional face image vector space makes it difficult to accurately evaluate the covariance matrix because of the relatively small number of face training samples [3]. Furthermore, computing the eigenvectors of a high dimensional covariance matrix is very timeconsuming. To overcome the above so-called small-sample-size (SSS) or dimensionality-disaster problem, Yang et al. [3] proposed the wellknown two-dimensional PCA (2DPCA). Different from PCA, 2DPCA is based on 2D image matrices rather than 1D image vectors. That is, the image matrix is not stretched to a vector for covariance calculation. Instead, an image covariance matrix is constructed directly using the original 2D images, and then the eigenvectors are used for image feature extraction. Compared with the covariance matrix of PCA, the size of the covariance matrix in 2DPCA is significantly reduced. As a result, 2DPCA evaluates the image n Corresponding author. address: wankou_yang@yahoo.com.cn (W. Yang). covariance matrix more accurately and computes the corresponding eigenvectors more efficiently than PCA. 2DPCA also yields higher FR rate than PCA. Xu et al. [4] analyze the theoretical similarities and differences between 2DPCA and PCA. A drawback of 2DPCA is that it needs more coefficients than PCA for image representation. Thus, 2DPCA needs more memory to store its features and costs more time in classification. To overcome these problems, Zuo et al. [5,6] proposed the bidirectional PCA (BDPCA) and Zhang and Zhou [7] proposed the 2-directional 2-dimensional PCA ((2D) 2 PCA).. Their ideas are very similar. BDPCA assumes that the transform kernel of PCA is separable and it is a natural extension to the classical PCA [5,6] and is a generalization of 2DPCA [3]. BDPCA and (2D) 2 PCA can do image feature extraction by reducing the dimension in both column and row directions. We find that when transforming samples into new space, 2DPCA, BDPCA and (2D) 2 PCA do not take into account whether the local structure of samples is preserved. On the other hand, in many classification applications, such as in the application where the nearest neighbor classifier is used, to preserve the local structure information is also important. Inspired by the success of locality preserving projections (LPP) [8], marginal Fisher analysis (MFA) [9], 2DLPP [10] and 2DDSLPP [11]. In this paper, we propose the Laplacian BDPCA (LBDPCA) to enhance the robustness of 2DPCA and BDPCA. Different from 2DPCA and BDPCA, we formulate the scatter of samples on local patches of the images by the weighted summation of distances, which has been successfully used in manifold learning [8,12] in characterizing the underlying clustering structures of samples. The rest of this paper is organized as follows. Section 2 reviews briefly BDPCA. Section 3 presents the proposed method in detail /$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi: /j.neucom

2 488 W. Yang et al. / Neurocomputing 74 (2010) In Section 4, experiments on face image databases are performed to demonstrate the effectiveness of the proposed method. Conclusions are made in Section Bidirectional PCA (BDPCA) BDPCA is a straightforward image projection technique where a k col k row feature matrix Y of an m n image X (k col rm, k row rn) can be obtained by Y ¼ W T col X W row ð1þ where W col is the column projector and W row is the row projector. Let fx 1,X 2,..., gbe a training set of N images. By representing the ith image matrix X i as an m-set of 1 n row vectors, the row total scatter matrix S row t is defined by S row t ¼ 1 P N Nm ðx i XÞ T ðx i XÞ, where X is the mean matrix of all training images. We choose the row eigenvectors corresponding to the first k row largest eigenvalues of S row t to construct the row projector W row ¼½w row 1,w row 2,..., w row Š. krow By treating an image matrix X i as an n-set of m 1 column vectors, the column total scatter matrix S col t is defined by S col t ¼ 1 Nn P N ðx i XÞðX i XÞ T. We then choose the column eigenvectors corresponding to the first k col largest eigenvalues of S col t to construct the column projector W col ¼½w col 1,wcol 2,..., wcol k col Š. Note that BDPCA is a generalization of 2DPCA, while 2DPCA can be regarded as a special case of BDPCA with W col ¼I m, where I m is an m m identity matrix [4]. 3. Laplacian bidirectional PCA 3.1. Fundamentals For non-gaussian or manifold data, we usually deal with them from local patches because non-gaussian data can be viewed locally Gaussian and a curved manifold can be viewed locally Euclidean [13,14]. The weighted summation of distance can characterize the underlying clustering structures of samples and it has been successfully used in manifold learning [8]. In this section, we employ the weighted summation of distance to enhance the robustness of 2DPCA and BDPCA. We first formulate the Laplacian scatter matrices using similarity weighting, and then propose the Laplacian 2DPCA (L2DPCA) the Laplacian BDPCA (LBDPCA) for face feature extraction. The variables used in this paper are listed in Table 1. Let us consider a set of N sample images X 1,..., taken from a (m n) dimensional image space. The similarity between two Table 1 List of notation. Notation Description samples is defined as S ij ¼ expð 99x i x j 99 2 =tþ where x i is the vector of image matrix X i. Obviously, for any x i, x j and parameter t, 0rS ij r1 always holds. Furthermore, the similarity function is a strictly monotonically decreasing function with respect to the distance between two samples X i, X j. The image scatter matrix G t of 2DPCA, the row total scatter matrix S row t and the row total scatter matrix S row t of BDPCA can be reformulated as follows: G t ¼ 1 N S row t ¼ 1 Nm S col t ¼ 1 Nn ðx i XÞ T ðx i XÞp XN j ¼ 1 ðx i XÞ T ðx i XÞp XN ðx i XÞðX i XÞ T p XN j ¼ 1 ðx i X j Þ T ðx i X j Þ j ¼ 1 ðx i X j Þ T ðx i X j Þ ðx i X j ÞðX i X j Þ T By using the similarity weight, we can get the Laplacian image scatter matrix of L2DPCA as G ¼ XN j ¼ 1 S ij ðx i X j Þ T ðx i X j ÞpX ð2þ ð3þ ð4þ ð5þ S ij Xi T X i S ij Xi T X j ¼ X T ðl I m ÞX where X T ¼½X1 T,..., XT NŠ and L¼D S is the Laplacian matrix. Here D is a diagonal matrix with D ii ¼ P j S ij being the column (or row) sum of S. The weight S ij incurs a heavy penalty when samples X i and X j are far apart from each other. We can choose the eigenvectors corresponding to the first k largest eigenvalues of G to construct the projector W L of L2DPCA: W L ¼½w L1,w L2,..., w Lk Š, and then use the transformation Y¼X W L to extract the L2DPCA feature matrix Y of image matrix X. The Laplacian row total scatter matrix S row can be defined as S row ¼ X S ij ðx i X j Þ T ðx i X j Þp X S ij Xi T X i S ij Xi T X j ¼ X T ðl r I m ÞX ð7þ where L r ¼D S is the Laplacian matrix. Similarly we choose the row eigenvectors corresponding to the first k row largest eigenvalues of S row to construct the row projector W Lrow ¼½w row L1,wrow L2,..., wrow Š. Lkrow The Laplacian column total scatter matrix S col can be defined as LS col t ¼ X S ij ðx i X j ÞðX i X j Þ T p X S ij ðx i Xi T X ixj T Þ¼XðL c I n ÞX T ð8þ where L c ¼D S is the Laplacian matrix. We then choose the column eigenvectors corresponding to the first k col largest eigenvalues of S col to construct the column projector W Lcol ¼½w col Finally, we use the transformation Y ¼ W T Lcol X W Lrow L1,wcol L2 ð6þ,..., wcol Lk col Š. ð9þ X i X G t G S row S col W L W Lrow W Lcol S ij L L r L c The ith image matrix,,2,..., N The set of the image matrices The total scatter matrix of 2DPCA The Laplacian image scatter matrix of L2DPCA The Laplacian row total scatter matrix The Laplacian column total scatter matrix The projector of L2DPCA The row projector of LBDPCA The column projector of LBDPCA The similarity between two samples The Laplacian matrix of L2DPCA The row Laplacian matrix of LBDPCA The column Laplacian matrix of LBDPCA to extract the LBDPCA feature matrix Y of image matrix X. Note that L2DPCA can be regarded as a special case of LBDPCA with W Lcol ¼I m, where I m denotes an m m identity matrix. It is easy to see that LBDPCA degrades to BDPCA if t approaches positive infinity in the similarity function Eq. (2). So, LBDPCA is a general version of BDPCA and inherits its strengths LBDPCA algorithm The proposed LBDPCA algorithm is summarized as follows: Step 1: Construct the similar matrix S using Eq. (2);

3 W. Yang et al. / Neurocomputing 74 (2010) Fig. 1. Images of one person in FERET. Fig. 2. The recognition rates of 2DPCA versus the dimension of features. Fig. 4. The recognition rates of the proposed LBDPCA versus the parameter t. Fig. 3. The recognition rates of BDPCA and the proposed LBDPCA versus the dimension of features. Step 2: Calculate the Laplacian row total scatter matrix S row and the Laplacian column total scatter matrix S col using Eqs. (7) and (8). Then the projection axes W Lrow and W Lcol are calculated. Step 3: Extract the sample features using Eq. (9) for classification Connections with MFA and LPCA MFA is developed using the graph embedding framework [9]. MFA combines locality and class label information to represent the intra-class compactness and inter-class separability. MFA takes advantage of the partial structural information of classes and neighborhoods of samples; however, it is difficult to decide the number of nearest neighbors of each sample and the number of shortest pairs from different classes in MFA. In addition, MFA is based on face image vectors and overlooks the image structural information. Laplacian PCA (LPCA) [15] is the extension of PCA to a more general form by locally optimizing the weighted scatter. The proposed LBDPCA is the extension of BDPCA to a more general form by locally optimizing the weighted scatter. In LPCA, the reductive coding length as a new dissimilarity is introduced to model the local geometrical structure. It is based on the global alignment of local scatter, which is complex to be implemented. In LPCA, the sorting algorithm is implemented for each sample to calculate the coding length and calculate the local scatter, which causes the computational complexity to grow with the increasing of the sample size. In the proposed method, we use the weighted distance sum of any two data points to preserve the local geometrical structure, which is widely used in manifold learning and easier to be implemented. In LPCA, the image matrix is first transformed into vectors and the image structural information is ignored. In our proposed method, we directly work on image matrix to preserving the image structural information. 4. Experimental results The features of 2DPCA, BDPCA and LBDPCA are matrices. The distance of two feature matrices can be calculated using either a vector-based or matrix-based method [5]. In a vector-based method, a feature matrix is first mapped to a vector and then a vector-based distance measure is used. In a matrix-based method, the distance between two feature matrices can be directly computed. In the experiments, we use the vector-based method to calculate the distance of two feature matrices. Two face image databases, namely, the FERET database and the ORL database are used to compare the proposed approach with the following algorithms: PCA (Eigenface) [2], LDA (Fisherface) [16], 2DPCA

4 490 W. Yang et al. / Neurocomputing 74 (2010) [3], BDPCA [5], 2DLPP [10], LPCA [15] and MFA [9]. The experiments are implemented on AMD Athlon(tm) 64 Processor Lenovo computer with 512M RAM and programmed in the MATLAB language (Version 7.01) Experiments on the FERET database The FERET face image database is a result of the FERET program, which was sponsored by the US Department of Defense through the DARPA Program [17,18]. It has become a standard database for testing and evaluating state-of-the-art face recognition algorithms. The proposed algorithm is tested on a subset of the FERET database. This subset includes 1400 images of 200 individuals (each individual has seven images). It involves variations in facial expression, illumination and pose. In our experiment, the facial portion of each original image was automatically cropped based on the location of eyes and the cropped image was resized to pixels. Some example images of one person are shown in Fig Selection of the projection axis In the experiments, we used the first 4 images per class for training and the remaining images for testing. First, 2DPCA method is used for feature extraction. The number of selected eigenvectors (projection vectors) varies from 1 to 20. Let k denote the projection vector number, then the dimension of corresponding projected feature vector is 40 k. Finally, a nearest neighbor classifier with cosine distance is employed to classify in the projected feature space. The recognition rates versus k are shown in Fig. 2. We can see that 2DPCA achieves the top recognition rate when k¼6. Next, BDPCA and the proposed LBDPCA are used for feature extraction. Let k¼6, and we let m denote the projection vector number for the feature extraction along column direction, which Table 2 Recognition rate on the FERET database Table 3 Average recognition rate on the FERET database. varies from 2 to 40 (2:2:40). A nearest neighbor classifier with cosine distance is employed for classification. We can see that BDPCA achieves the top recognition rate when m equals to 22 and LBDPCA achieves the top recognition rate when m equals to 30. The recognition rates of LBDPCA versus dimension of features are shown in Fig. 3. The parameter t is set as t¼50. Fig. 4 shows the recognition rates of LBDPCA versus the parameter t. It can be been seen that LBDPCA has a stable performance with the variant of parameter t Comparison of the performance In the experiments, we used the first 4 images per class for training and the remaining images for testing. PCA, LDA, 2DPCA, BDPCA, 2DLPP, MFA, LPCA, and the proposed LBDPCA are involved in the comparison. Note that LDA involves a PCA phase. In this phase, we kept nearly 98% the image energy to select the number of principal components and there is m¼250. The nearest neighbor classifier with Cosine distance was employed for classification. The maximal recognition rate of each method is listed in Table 2. We then randomly selected four images for training and the rest images for testing. The average recognition rates and standard deviation are shown in Table 3. From Tables 2 and 3 we can see that the proposed LBDPCA has the top recognition rate. BDPCA can be used for image feature extraction by reducing the dimensionality in both column and row directions. So BDPCA and the proposed LBDPCA consider the structural information embedded in the original images. Moreover, LBDPCA considers the distribution information of the original images using the sample similarity weights, and hence it leads to better results than BDPCA. To further evaluate the performance, we select first l (l¼3, 4, 5) images per class for training and the remaining images for testing. The experimental results are shown in Table 4. We can see that LBDPCA has the best results. Table 5 Recognition rate on the ORL database. l¼ l¼ l¼ Mean Std Table 6 Recognition rate on the ORL database. l¼5 l¼6 l¼7 Table 4 Recognition rate on the FERET database with different training sample number. l¼ l¼ l¼ PCA (0.0119) (0.0093) (0.0136) LDA (0.0140) (0.0106) (0.0131) 2DPCA (0.0113) (0.0111) (0.0100) BDPCA (0.0123) (0.0126) (0.0168) 2DLPP (0.0130) (0.0092) (0.0154) LPCA (0.0097) (0.0158) (0.0133) MFA (0.0111) (0.0128) (0.0125) LBDPCA (0.0120) (0.0100) (0.0148) Fig. 5. Ten images of one person in ORL.

5 W. Yang et al. / Neurocomputing 74 (2010) Experiments on the ORL database Comparison of the recognition performance The ORL ( database contains 40 persons, each having 10 different images. The images of the same person are taken at different times, under slightly varying lighting conditions and with various facial expressions. Some people were with or without glasses in data collection. The heads in images are slightly titled or rotated. The images in the database are manually cropped and rescaled to Fig. 5 shows ten images of one person in ORL. In the experiment, we used the first l (l¼5, 6, 7) images per class for training and the remaining images for testing. In PCA and the PCA stage of LDA, we kept nearly 98% image energy to select the number of principal components. In the proposed LBDPCA, the parameter t is set as t¼50. In LPCA, the number of the nearest neighbors is set as l 1, and the final dimension is set as big as PCA. In MFA, the number of the nearest neighbors in the intra-class compactness graph is set as l 1 and the number of the nearest neighbors in the inter-class separability graph is set as 20 l. Finally a nearest neighbor classifier with cosine distance is employed. The final recognition rates are given in Table 5, from which we can find that the proposed method has the top recognition rate in most cases. In the second experiment, l¼5, 6, 7 images per class were randomly chosen for training, while the remaining images were used for testing. We run the system 10 times. The average recognition rates and standard deviation are shown in Table 6. We further evaluated the performance of 2DPCA, BDPCA, L2DPCA and LBDPCA with different dimensions of features. The variations of recognition rates versus dimension are shown in Figs. 6, 7 and 8. For 2DPCA and L2DPCA, we only showed the performance in the (dn112)-dimensional subspace with d¼2, 4, 6, 8, 10, 12, 14, 16. For BDPCA and LBDPCA, we only showed the performance in the (dnd)-dimensional subspace with d¼6, 8, 10, 12, 14, 16. From Figs. 6 8 and Table 6,we can find that our proposed method has a stable and better performance. The average training time and testing time is shown in Table Comparison of the reconstruction performance In the last experiment, we compare the reconstruction capability of PCA, 2DPCA, BDPCA and the proposed LBDPCA. We chose the first 6 images of per class as the training set. For PCA, we set the number of PCs as 100. For 2DPCA, we set the number of row eigenvectors as 10. For BDPCA and the LBDPCA, we set the number of row eigenvectors as 10, and the number of column eigenvectors as 20. The feature dimension of PCA, 2DPCA, BDPCA and LBDPCA are 100, , and 20 10, respectively. Fig. 9 shows two original images and the reconstructed images by PCA, 2DPCA, BDPCA and LBDPCA. The first row is an image from Fig. 6. Results when l¼5. Fig. 8. Results when l¼7. Table 7 Average training time and testing time on the ORL database(s). l¼5 l¼6 l¼7 Training testing Training testing Training testing PCA LDA DPCA BDPCA LPCA Fig. 7. Results when l¼6. LBDPCA

6 492 W. Yang et al. / Neurocomputing 74 (2010) Fig. 9. The reconstruction capability of PCA, 2DPCA, BDPCA and LBDPCA. The 1st column shows the original images; the 2nd to 5th columns show the reconstructed images by PCA, 2DPCA, BDPCA and LBDPCA, respectively. The image in the first row is from the training set, while the image in the second is from the testing set. the training set. Satisfactorily reconstructed images can be obtained using all of these methods. The second row shows an image from the testing set. We see that the quality of the reconstructed image by PCA deteriorates greatly, while 2DPCA, BDPCA and LBDPCA can still obtain satisfying image reconstruction quality. 5. Conclusions Using the similarity of samples, we extended 2DPCA and BDPCA to non-euclidean space and proposed the Laplacian BDPCA (LBDPCA) method to improve the robustness of 2DPCA and BDPCA. First, the Laplacian row total scatter matrix and the Laplacian column total scatter matrix were defined; then the projectors were obtained by calculating the eigenvectors of the Laplacian row and column total scatter matrices; finally the image matrix was projected onto the projectors to extract the LBDPCA feature. LBDPCA considers the structural information of the image samples and the distribution information embedded in the original images. It can be seen that BDPCA is a special case of LBDPCA. Experimental results on the FERET and ORL face databases showed that LBDPCA works well and has better performance than BDPCA and other unsupervised subspace methods. Acknowledgements This project is supported by NSF of China (Grant nos: , , , ), the Hong Kong RGC General Research Grant (PolyU 5351/08E), China Postdoctoral Science Foundation ( ) and the Fundamental Research Funds for the Central Universities (2010B10014). [5] W. 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Yang, An approach for directly extracting features from matrix data and its application in face recognition, Neurocomputing 71 (10-12) (2008) Wankou Yang received the B.S., M.S. and Ph.D. degrees in the School of Computer Science and Technology, Nanjing University of Science and Technology (NUST), PR China, respectively, in 2002, 2004 and Now he is a Postdoctoral Fellow in the school of automation, Southeast University, PR China. His research interests include pattern recognition, computer vision, digital image processing.

7 W. Yang et al. / Neurocomputing 74 (2010) Changyin Sun is a professor in School of Automation at Southeast University, China. He received the M.S. and Ph.D. degrees in Electrical Engineering from Southeast University, Nanjing, China, respectively, in 2001 and His research interests include Intelligent Control, Neural Networks, SVM, Pattern Recognition, Optimal Theory, etc. He has received the First Prize of Nature Science of Ministry of Education, China. He has published more than 40 papers. He is an associate editor of IEEE Transactions on Neural Networks, Neural Processing Letters and International Journal of Swarm Intelligence Research, Recent Patents on Computer Science. He is an IEEE Member. Lei Zhang received the B.S. degree in 1995 from Shenyang Institute of Aeronautical Engineering, Shenyang, PR China, the M.S. and Ph.D degrees in Electrical and Engineering from Northwestern Polytechnical University, Xi an, P.R. China, respectively, in 1998 and From 2001 to 2002, he was a research associate in the Dept. of Computing, The Hong Kong Polytechnic University. From January 2003 to January 2006 he worked as a Postdoctoral Fellow in the Dept. of Electrical and Computer Engineering, McMaster University, Canada. Since January 2006, he has been an Assistant Professor in the Dept. of Computing, The Hong Kong Polytechnic University. His research interests include Image and Video Processing, Biometrics, Pattern Recognition, Multisensor Data Fusion and Optimal Estimation Theory, etc. Dr. Zhang is a member of IEEE and an Associate Editor of IEEE Trans. on SMC-C. Karl Ricanek Jr. is an Associate Professor at University of North Carolina Wilmington in the Computer Science Department. Prof. Ricanek is the founder and director of the Face Aging Group Research Lab ( Group.com) at UNCW where he has been the primary project lead on more than $5 Million USD in Department of Defense and intelligence funded research since Prof. Ricanek is the Director for the Institute for Interdisciplinary Studies in Identity Sciences (I2SIS) that was awarded in August He is one of four chosen researchers to form the Office of the Director of National Intelligence (ODNI) Center of Academic Excellence in Identity Sciences (CASIS). His research is mainly focused in developing algorithms for modeling age-progression for the mitigation of severe performance degradation of face recognition technology due aging. He has also been involved in research for robust age-estimation from facial images and for gender and race classification from facial images. He has extensive research background in pattern recognition, artificial intelligence, and machine and computer vision. He has authored or co-authored over 40 referred articles in biometrics and pattern recognition and 3 book chapters since 2003 when he returned to academia. He is program committee member on several Biometric and related conferences such as IEEE Biometric: Theory, Applications, and Systems (IEEE BTAS), IEEE Transactions on Pattern and Machine Intelligence (IEEE PAMI), SPIE Biometrics Conference (SPIE BC), International Joint Conference on Neural Networks (IJCNN) and many others. He is invited editor for IEEE Computer Magazine on Identity Sciences and chair for IEEE Automatic Face and Gesture Special Session on Craniofacial Aging and Age-estimation 2011.