SINGLE-SAMPLE-PER-PERSON-BASED FACE RECOGNITION USING FAST DISCRIMINATIVE MULTI-MANIFOLD ANALYSIS

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1 SINGLE-SAMPLE-PER-PERSON-BASED FACE RECOGNITION USING FAST DISCRIMINATIVE MULTI-MANIFOLD ANALYSIS Hsin-Hung Liu 1, Shih-Chung Hsu 1, and Chung-Lin Huang 1,2 1. Department of Electrical Engineering, National Tsing-Hua University, Hsin-Chu, Taiwan Department of Applied Informatics and Multimedia, Asia University, Taichung, Taiwan Abstract This paper presents a single sample per person (SSPP)-based face recognition method. Based on the Discriminative Multi-manifold Analysis (DMMA), we propose an accelerative face recognition method which consists of three modules. First, for one person one training image sample, we use a modified of K-means method to cluster two groups of people. Second, we divide the face images into nonoverlapping local patches and apply DMMA. Third, we repeat the previous two steps to obtain the binary tree projection matrix of fast DMMA. In the experiments, we test the AR database and FERET database to verify the effectiveness of SSPP-based fast DMMA face recognition process in both accuracy and speed. I. INTRODUCTION Face recognition provides a non-touch way for identity verification. Usually, it requires a lot of training samples which are divided into two categories: MSPP (multiple samples per person) and SSPP (single sample per person). The MSPP can extract more information and also provide the stability. However, in reality, we need to deal with the training set which provides only one photo one person. If we cannot collect enough samples for each person, then MSPP may fail. Some the literatures propose using manifold for training method. The discriminative multimanifold analysis (DMMA) method based on SSPP achieves very high recognition rate. Unfortunately, the speed of recognition slow down very fast if the number of objects increases. We can speed up face recognition through the structure of binary tree without sacrificing the accuracy. Face recognition methods can be divided into two categories [1]: geometric-based methods and appearancebased methods. The former gets the parameters based on the distinctive features of face and relative position of facial sense organs such as eyebrow, eyes, nose, mouth, and chin. The latter focuses on the supervised, semisupervised, and unsupervised training for face recognition. It obtains the intrinsic characteristics of the image, and projects it into the low dimensional feature subspace for recognition which shows good performance and simple operation. These methods include principal component analysis (PCA) [2], Fisher s linear discriminant (FLD) [3], marginal Fisher analysis (MFA) [4] and locality preserving projections (LPPS) [5]. The appearance-based methods require many training samples. However, we often face the situation of insufficient training samples. The intra-personal variation is inaccurate due to the number of training samples is smaller than feature dimensionality. To solve the single sample per person (SSPP) problem, some methods [6] are proposed which can be divided into unsupervised learning, virtual sample generation, generic learning, and image partitioning. The unsupervised technique does not label the training samples, because it neglects the utilization of intra-personal variations. The most representative method is PCA or eigenface [2]. The PCA uses a lowdimensional subspace to maximum the data variance which has become the baseline algorithm of face recognition. There are other PCA methods, such as (PC) A [7] and 2DPCA[8]. The unsupervised learning methods do not suffer the problem of SSPP, but only consider the inter-personal variations and abandon the intra-personal variations. Their performance is poor for different facial expressions variations or lighting variations. The second category methods apply the virtual sample generation method to extract the intra-personal variations information based on the ideas of training by generating each person extra sample inside the database. In [9], SVD method is used to decompose the face image into two complementary parts: a smooth general appearance image and a difference image. This method can improve the problem of SSPP to a certain extent, but it needs to generate the virtual sample that is not an easy job. There exists highly correlated information between the training samples and the generated images and too much redundancy in the discriminative feature subspace. In the third category, an additional generic training set with each person has more than one training sample. It extracts the discriminatory information to directly identify the persons with only one training sample. In [10], a solution is proposed for pose-invariant face recognition. By collecting a generic training set to extract a pose-invariant subspace. Even though this method can alleviate the face recognition of SSPP problem, but this method depends heavily on the generic training set. It is still very difficult for the system to be applied in the real case APSIPA APSIPA 2014

2 The fourth category methods apply the image partitioning method by dividing the face image into several local patches and applying the discriminant learning techniques to extract the feature. In [11], Martinez proposes a method by dividing each face image into six elliptical parts and learning a local probabilistic model to recognize the person. Tan et al. [12] extend self-organizing maps(soms) to train each face subspace, but these methods need to partition the face image into several local patches for the training such as eyes, nose, mouth, and eyebrow. So, it cannot be modeled accurately by a simple distribution. To let each patch of same object be in the same manifold, we have to consider each patch corresponds to a point inside the manifold [13~16]. To segment the face region for face alignment, we find out the locations of two eyes to calculate the tilt angle of the line segment connecting two eyes. Based on the line segment and tile angle, we may segment the face region from the picture and then adjust the face region to upright position. Due to the different size of face images, we have to resize the segmented face image into the specific size (60x60) and divide the resize face image into nonoverlapping local patches. We convert the set of local patches into feature vectors representing manifold of a person, and facilitate DMMA training by using manifolds. Finally, we develop the projection matrices in binary tree structure to process the DMMA projection, and use locally linear embedding (LLE) method to recognize the object. Face recognition is through the tree searching to the leaf node that indicates the identity of the input face image II DISCRIMINATIVE MULTI-MANIFOLD ANALYSIS To address the SSPP-based face recognition, the DMMA method[15, 16] partitions the image into several local patches as the feature for training, and converts the face recognition to the manifold-manifold matching problem. The DMMA method takes advantage of the geometrical information of local patches distribute in the space for the objects. In the preprocessing stage, we extract the face region from five personal photos and resize into specific size (60x60). Each face image (60x60) is partitioned into the 36 non-overlapping patches of specific sizes (10x10). Each local patch is represented as a 100 dimension of vector and converted to feature vector through PCA. There are highly overlapped manifolds corresponding to different objects because the intra object patches have large appearance variation. The similarity of the same local patches of different objects is larger than the different local patches of the same object. Figure 1(a) shows that the eyes patches of two different objects are the similar than eyes and nose patches of the same object. Therefore, the local patches of manifold will be high overlapped in the high dimension space. The local patches of the same object may not group together. The similar patches of different object will be close to each other. Our goal is to find manifold projection so that the local patches of same object will be closer and the local patches of different objects will be well-separated after the projection as shown in Figure 1(b). We can obtain more discriminative information and get much larger manifold margin in projection space for face recognition. Object1 Object2 (a) (b) Object 1 Objact 2 Figure 1. The DMMA method. (a) The distribution of local patches of two objects in the original space. (b) the distribution of local patches after the DMMA projection. Given a set of training samples with N individuals,,,, : is the i th person and the size of training image is m n where 1 i N. We divide each image into t pieces of non-overlapping local patches, and each size of local patch is a b where. The training set of N people can be described as,,, and,,,,, where i represented as the i th individual. is the i th manifold which mean the image patch set of the i th person. is the total number of a single object local patches. is the r th local patches of the i th person, and 1. We generate a set of feature projection matrices of N objects,,, where and 1 i N. This projection matrix converts the local patch set of original dimensions d to dimension in a lower dimension feature space. The local patches of same object become closer, whereas the local patches of different objects are well separated to maximize manifold margin. Assume we have to represent the i th manifold with r th local patch in Figure 2(a). We can divide the neighbor of these manifolds into two categories: (1) N intra intramanifold neighbors as shown in Figure 2(b). (2) N inter inter-manifold neighbors as shown in Figure 2(c). Our goal is find he lower dimensional feature space by minimizing the intra-manifold variance and maximizing the inter-manifold distance so that manifold margin between the i th person and others objects simultaneously is maximized as shown in Figure 2(d).

3 (a) (b) and from the same object, then their low-dimensional representations will be similar. These inter-manifold neighbors and intra-manifold neighbors are used to maximize manifold margins to separate different manifolds where k 1 =8 and k 2 =3. It is difficult to obtain the feature projection matrices of a group which contain N people simultaneously. We can solve this problem by iterative maximization process. First, we have to initialize,,,, then solve sequentially. So we rewrite (1) as (c) Figure 2. The DMMA method: (a) The points with the same color denote the local patches from the same object and those with different colors denote the local patches from different objects. (b) Three intra-manifold neighbors (, and ) for a local patch. (c) Eight intermanifold neighbors (,,,,,, and ) for a local patch. (d) After DMMA, the manifold margin between the i th person and others objects is maximized. The optimize formula of DMMA can be shown as:,,., (,,, ) (1) (,,, ) (,,, ) where and represents the p th k 1 -nearest intermanifold neighbors and the q th k 2 -nearest intra-manifold neighbors of respectively. and are the weight coefficients based on the similarity of with and with. It applies KNN (K-nearest neighbors) to obtain the k 1 inter-manifold neighbors and k 2 intra-manifold neighbors as, 0, h, 0, h ( ) ( ) where ( ) and ( ) represent k 1 -intermanifold neighbors of and k 2 -intra-manifold neighbors of. in (1) ensures that if and are close from the different objects, then their lowdimensional representations are separated as far as possible. in (1) ensures that if and are close (d) (2) (3) ( )( ( ) ) ( ( ) ) (4) where, and, During the training of, the calculations of and do not affect the optimization training of. So, we can ignore these two parts. ( ) can be rewritten as ( )( ) (5). where ( ) can be rewritten as ( )( ) (6). where We can obtain the by solving the equation ( ) where,,,, is a set of eigenvectors corresponding to the eigenvalues 1,2,3,,. The eigenvalues are sorted as, and,,,, is the projection matrix. The selection of feature dimension is different than the other manifold learning algorithms. Generally, other manifold learning algorithms select the optimal feature dimension. In our training method, we obtain the feature dimension by analyzing the eigenvalues because it takes much time to find the set of optimal dimension parameters. is not positive semi-definite matrix. According to the order of the eigenvalues, we select the eigenvectors which are greater than to maximize the equation (4). It is because when the samples are projected to one specific eigenvector corresponding to an eigenvalue with the following formula ( ) (7)

4 when λ 0, based on the direction of, the intermanifold distance is larger than the intra-manifold distance and it is correctly classified. We obtain the W sequentially using DMMA algorithm. Figure 3 shows the 4 th object (in yellow patches) in DMMA training. We use the same and different colors to represent intra-manifold and inter-manifold, and then we show the result of lowdimensional feature space distribution from the projection matrix. In Figure 3, we can see the distances of different manifolds are well separated so that it can achieve more effective identification. Figure 3. The 4 th object (in yellow patches) in DMMA training. In [16], they select 200 different objects from FERET database for DMMA training. They fix the other two parameters and test the remaining ones to obtain the optimal result. Similar to [16], we fix the patches size into 10 10, with inter-parameter =15 and intra-parameter 5, respectively. III FACE RECOGNITION USING FAST DMMA Here, we introduce fast DMMA for SSPP-based face recognition. It consists of (1) segmenting the face region from the input image and divide single sample per person into non-overlapping local patches as our preprocessing and feature extraction, (2) applying the DMMA to process the training of manifold projection matrix, (3) proposing an accelerated technology using binary tree and (4) developing recognition method. We solve the speed problem of original DMMA due to excessive training objects for SSPP-based face recognition. 3.1 FAST DMMA The original DMMA considers an object as a manifold and treats the face recognition as solving the manifoldmanifold matching problem. The speed of recognition of DMMA method [15, 16] will becomes pretty slow if the numbers of objects becomes very large. Here, we consider developing the binary tree DMMA structure to solve the problem. We consider the manifold-manifold matching problem as binary tree structure as shown in Figure 4. For fast DMMA, we use binary tree structure which only needs to compare 2log (1) time to find the leaf node for face recognition. Figure 4. Binary tree structure of fast DMMA. A binary tree data structure is composed of split nodes and leaf nodes. For each split node, we have a manifoldmanifold matching problem. By solving the matching problem to decide whether the test data move into next left node or next right node. Training binary tree of fast DMMA encounters two major split problems. The first problem is how to randomly split two sets for DMMA training. It may cause the similar objects in different groups which may easily cause the recognition errors during identification stage. Therefore, we simply split two set of similar objects into two manifolds for DMMA training. The second problem is that after the simply classification, we have some ambiguous objects similar to two sets which are treated as the median samples. To simply classify the similar objects, we use k-means clustering method (with K=2) to cluster input face images,,.., from the split node. By using the objective function, we divide into two similar sets, and find the means as,. To find the median sample points, we calculate the average center point of these two mean of and. Based on, we find the nearest neighbors and produce a new set as as shown in Figure 5(a). The set which inside the green dashed line is also the median samples set which may cause identification errors. We use classify the median samples set simultaneously to and using k-means clustering method. Then, we produce two new sets and. Figure 5(b) shows the left node objects subset and right node objects subset as and (8)

5 (a) n( ). If > ω K-means clustering Compute and. Produce, } Generate, }. Get left split node of =,go to Step 2. Get right split node of =, go to Step 2. else =, n( ) Produce manifolds, }. Generate,,.., }. STEP 3 (Output): Output binary tree structure of fast DMMA. (b) Figure 5. Split method. (a)find the average center point and find out neighbors produce a new set as. (b) Split to new subsets and. With and, we combine these two sets corresponding to two manifolds and. In DMMA training, given and, we obtain projection matrix and. Then, we produce split node manifold set, and projection matrices set, for the split node to determine the split direction during recognition stage. The left and right split manifolds are defined as :,,1 :,,1 Based on the DMMA projection matrix, we repeat the split method to find the corresponding manifold using eq. (9) until the number of split node objects, n, is insufficient. Then, we determine this node as leaf node. If the leaf node has few objects, it cannot provide acceleration advantage. The accuracy of original DMMA is higher than fast DMMA. We use the original DMMA training method in leaf node and consider each object as individual manifold to produce leaf node manifold set, and train the individual projection matrix to produce leaf node projection matrices set,,,. The recognition stage is based on the projections provided by leaf node for the corresponding object. We summarize the fast DMMA algorithm as follows. Fast DMMA Algorithm Input: Train sets,.. and,.., the decide leaf node parameters ω. Output: Manifolds and projection matrices of nodes STEP 1 ( Initialization ): Set =. STEP 2 (Split ): (9) 3.2 RECOGNITION For an unknown testing sample T, we partition the sample into t non-overlapping local patches and produce the manifold,,,. Then, we calculate the distance between manifolds using the following formula = arg min (, ), (10) where i=left/right for split node and i=1, N leaf for leaf node. d(, ) is manifold distance between and. Then, we assign a label c to for the split node. To calculate manifold-manifold distance of and, we apply the DMMA projection on the two local patches as = = [,,, ] and = = [,,, ] and generate the low-dimensional representations of the two manifolds respectively. The manifold distance can be defined as (, ) = (, ( )) (11) where ( ) denotes k-nearest neighbors of in. The (, ( )) can be easily used to solve the constrained optimization problem similar to locally linear embedding (LLE) method [14] as, = min G (12) where is reconstruction coefficient of the neighbor G ( ) to. IV. EXPERIMENTAL RESULTS To evaluate binary tree structure of fast DMMA, we choose the public face images database for the experiments. We will show experimental results and compare the results with other research methods. 4.1 DataBase

6 We use two different face recognition databases: AR database [19] and FERET database [20]. These two databases have lots of sample images which are taken in different conditions. These databases are often used for SSPP face recognition research and cited by many papers. We use these two databases and compare the results with other methods. Fb Fa Fb Fa A. AR database Obset A Obset B Obset C Obset D Figure 6. Three objects in four different image subsets of the AR database. B. FERET database FERET database contains over 1,000 objects with 14,051 images. Similar to [7, 9, 12, 21], we use 200 different people with 400 gray-level frontal face images. The size of each photo is and use two images for each object (Fa and Fb) with different races, genders, ages, expressions, illuminations, scales and etc. Fa images are used for training, and Fb images are used for testing. Figure 7 shows that there are four objects in Fa and Fb images. Then, we compare the experimental result with other face recognition methods. We do the comparison between our fast DMMA method and original DMMA method by using 1000 different people from the FERET database. Figure 7. Four objects in Fa and Fb images of FERET. We use 100 different people for weight analysis of local patch. We choose 100 images from set ba (images of frontal face) to be our training samples; 100 images from set bj (images of different facial expressions) and use the images in set be (images of pose angle with 15o) as our testing samples. 4.2 EXPERIMENTAL RESULTS AND ANALYSIS In this section, we analyze the parameter setting and compare the recognition accuracy between our method and other methods. Finally, we show the performance in terms of recognition speed with different number of training samples. A. PARAMETER ANALYSIS We explore the parameter settings that may influence the acceleration speed and recognition accuracy. The parameters are the number of median sample Q R and the leaf node terminative condition. The analysis of Q R is shown in Figure 8 and Table 2. We use 200 different people from the FERET database as split node to cluster the sample image set. Then, we try different ratios of the median samples number and the split node samples number to analyze the recognition accuracy of single split node. We observe that the recognition accuracy become higher when the Q R value is larger, but it also generates a deeper binary tree indicating more complicate tree searching. To find the trade-off between the recognition accuracy and tree searching, we set the ratio as 30%, so that we have Q R = #T {%(Q}:~ 0.3) as the number of median samples in the experiments recognition rate(%) AR database contains over 4,000 color face images (with size ). These color images include different facial expressions, lighting conditions, and occlusions (scarves and sunglasses) frontal views of faces image. There are no restrictions for the participants such as wear, make-up, hair style, and etc. Each participant has a set of color images which taken in two sessions (separated by two weeks). In the experiment, we use 100 different people images with different facial expressions and time. We have four different image subsets which contain 400 images (each subset contains 100 images). Table 1 shows the detail information of each subset. Figure 6 shows there are three objects in four different subsets (Subset A to D) images. Subset A, B, and C are collected with neutral, simile, and anger expressions. Subset D are other collections with neutral, expression. In the experiment, we selected the images of Subset A for training and the remaining 3 subsets for testing. 0% 10% 20% 30% 40% 50% Nmed size (different ratios of split node samples # ) Figure 8. Parameter analysis of ª«size on FERET. Table 1. Different ª«size of tree depth. ª«size Depth 0% Κ۷1Ǒ 10% 7۷17 20% ȑ ۷19 30% 9۷ҟẇ 40% 1ҟ۷ҟ7 50% 1 ۷ҟ9

7 Assume we use the proposed split method to split a group of objects into two overlapped sub-groups evenly. If 0.3, then each sub-group has about ( 1.3)/2 objects (based on the result of analysis). To find the trade-off between the fast DMMA and original DMMA we need to determine the so-called the minimal group. Once the number of group of objects is less than the number of objects of the minimal group, it will not be split further into two subgroups. We continuously divide a group of objects into two overlapped sub-groups with N med =0.3 N split which is called the overlapped binary tree. The overlapped binary tree with depth h may contain totally about (2/1.3) objects. For the group of objects, the number of sequential operations of conventional DMMA is (2/1.3), whereas, the number of binary tree searching operations of fast DMMA is 2h. If 2h > (2/1.3), then applying fast DMMA has no advantage. Therefore, once h<5.64 (or (2/1.3) <11), the group of objects becomes the minimal group, and the original DMMA is applied to these objects. Once 10, fast DMMA has no advantage compared with original DMMA. Therefore, we set ω=10, once the number of training samples in the leaf node is less than or equal to ω, the advantage of using the fast DMMA no longer exists, and we may simply apply the original DMMA method for the training samples in the leaf node. B. COMPARISON To compare our methods with other methods, we select parameters =10 and = 0.3 respectively for experiment analysis. The other DMMA method parameter settings are the same as original DMMA method. The size of each patch is with the parameters =15, =5, and =4 respectively. Table 3 shows the comparison of our method with [16] by using 100 different people from AR database and 200 different people from FERET database. The performance of our method is slightly worse than original DMMA using oneby-one comparison which is supposed to be the optimal. However, the recognition accuracy of ours is still higher than most other methods. Table 2. Rank-1 recognition accuracy (in %) of different methods on different subsets of the AR and FERET. Methods AR FERET B C D [2] () [7] () [21] [8] (2) [22] [12] [5] [9] [23] [24] [25] [16] OURS C. ACCELERATION ANALYSIS Although our recognition accuracy is slightly worse than original DMMA, but the execution speed has a lot of improvement. We select different number of people from the FERET database and compare the processing speed of fast DMMA method with original DMMA method as shown in Table 4. Table 3. Comparison of our method and the original DMMA method [16] using FERET database. # of Training Objects DMMA OURS 10 22~34 24~48 26~50 26~62 Time Ratio # of Training Objects 1 2.9~ ~ 8.3 6~ ~ DMMA OURS 28~68 30~72 30~82 30~80 30~86 Time ratio 7.4~ ~ ~ ~ ~ 33.3 Figure 9 shows that the recognition time of original DMMA increases as the number of training objects increases. Our proposed method has very little influence for increasing number of training objects. Figure 10 shows the comparison of time ratio improvement which is defined as the ratio of the recognition time of the original DMMA and fast DMMA. We can observe the significant improvement when the number of training objects increases # of comparisons (times) DMMA Worst case of our method Best case of our method Number Of Train Subjects(persons) Figure 9. Number of comparisons by fast DMMA method and original DMMA based on FERET database.

8 Improve (time ratio) 50x 40x 30x 20x 10x Worst case of our method Best case of our method x Number Of Train Subjects (persons) Figure 10. Performance Compare of fast DMMA method with original DMMA method based on FERET database. D. WEIGHT ANALYSIS OF LOCAL PATCH The patch of facial features are not easily affected by variation of background, such as hair style and pose angle. We differentiate these patches into two types and, where indicates the facial organs patch (with red boundary), and indicates the edge patch (with blue boundary). P in denotes the patches which may be affected by the background whereas P out denotes the patches which are not affected. M i is the i th person manifold which can be divided into Pin two sets: M i ={xir } for r = 2, 3, 4, 5, 8, 9, 10, 11, 15, 16, Pout 20, 21, 22, 23, 26, 27, 28, 29, and M i ={xir } for r = 1, 6, 7, 12, 13, 14, 17, 18, 19, 24, 25, 30, 31, 32, 33, 34, 35, 36 which are are easily affected by variation of background. First, we generate M Pin Pin Pout ={M i 1 i N} and M = Pout {M i 1 i N}. Second, we differentiate Pin and P out patches Third we find α=p in or P out, and search ( ) and ( ) as k 1 -inter-manifold and k 2 - intra-manifold neighbors in M α, respectively Figure 11. and of manifold patches. For face recognition, we modify equation (12) as. (, ) = (, ( )) (13) where W α =W Pin for y Tj P in and W α =W Pout for y Tj P out. G k (y Tj ) denotes the k-nearest neighbors of y Tj (if y Tj α, α=p in or P out ) in patch of. Figure 12 shows the different ratio of weight coefficients (W Pin and W Pout ).The recognition accuracy for origin DMMA and weighted DMMA are 71 and 75 respectively. P out patch is easily affected due to variation of the facing angles. Increasing the weight W Pout will decrease the recognition accuracy. On the contrary, P in is much more stable, increasing weight W Pin may improve the recognition accuracy. recognition rate(%) Weight of patch(wp in :Wp out ) : : : : : :0.4 Figure12. Weight analysis of local patch for FERET. V. CONCLUSIONS This paper has proposed a fast DMMA by using binary tree structure. The experiments prove that our fast DMMA method can accelerate the process of DMMA under circumstances of very limited accuracy decrement. However, the recognition accuracy of fast DMMA decreases if the pose angle of face has lots of variation. We need to design an effective manifold-manifold distance method to improve the recognition performance. REFERENCES [1] W. Zhao, R. Chellappa, P. Phillips, and A. Rosenfeld, Face recognition: A literature survey, ACM Computing Surveys, vol. 35, no. 4, pp , [2] M. Turk et al., Eigenfaces for Recognition, J. Cognitive Neuroscience, vol. 3, no. 1, pp , [3] P.N. Belhumeur et al., Eigenfaces vs. Fisherfaces: Recognition Using Class Specific Linear Projection, IEEE Trans. on PAMI, vol. 19, no. 7, pp , [4] S. Yan et al. Graph Embedding and Extensions: A General Framework for Dimensionality Reduction, IEEE Trans. on PAMI, vol. 29, no. 1, [5] X. He et al., Face Recognition Using Laplacianfaces, IEEE Trans. on PAMI, vol. 27, no. 3, pp , [6] X. Tan et al., Face recognition from a single image per person: A survey, Pattern Recognition, vol. 39, pp , [7] J. Wu and Z. Zhou, Face Recognition with One Training Image per Person, Pattern Recognition Letters, vol. 23, no. 14, pp , [8] J. Yang et al., Two-Dimensional PCA: A New Approach to Appearance-Based Face Representation and Recognition, IEEE Trans. on PAMI, vol. 26, no. 1, pp , [9] D. Zhang et al., A New Face Recognition Method Based on SVD Perturbation for Single Example Image per Person, Applied Math. and Computation, vol. 163, no. 2, pp , [10] T. Kim and J. Kittler, Locally linear discriminant analysis for multimodally distributed classes for face recognition with a single model image, IEEE Trans. on PAMI, vol. 27, no. 3, pp , [11] A. Martı nez, Recognizing Imprecisely Localized, Partially Occluded,and Expression Variant Faces

9 from a Single Sample perclass, IEEE Trans. on PAMI, vol. 24, no. 6, pp , [12] X. Tan et al., Recognizing Partially Occluded, Expression Variant Faces from Single Training Image per Person with SOM and Soft K-NN Ensemble, IEEE Trans. on NN,, vol.16, no.4, [13] J. B. Tenenbaum, et al., A Global Geometric Framework for Nonlinear Dimensionality Reduction, Science, vol. 290, no. 5500, pp , [14] S. Roweis and L. Saul, Nonlinear Dimensionality Reduction by Locally Linear Embedding, Science, vol. 290, no. 5500, pp , [15] J. Lu, et al., Discriminative multi-manifold analysis for face recognition from a single training sample per person, ICCV, pp , [16] J. Lu, et al., Discriminative Multi-manifold Analysis for Face Recognition from a Single Training Sample per Person, IEEE Trans. on PAMI, pp. 39 ~51, 2013 [17] B. Fei et al. Binary Tree of SVM: A New Fast Multi-class Training and Classification Algorithm, IEEE Trans. on NN, vol. 17, no. 3, [18] G. D. Guo et al., Face recognition by support vector machines, Proc. Int. Conf. Automatic Face and Gesture Recognition, pp , [19] A.M. Martinez and R. Benavente, The AR Face Database, technical report, CVC, [20] P. Phillips, H. Moon, S. Rizvi, and P. Rauss, The FERET Evaluation Methodology for Face- Recognition Algorithms, IEEE Trans. on PAMI, vol. 22, no. 10,pp , [21] S. Chen et al, Enhanced (PC) A for Face Recognition with One Training Image per Person, Pattern Recognition Letters, vol. 25, no. 10, pp , [22] D. Zhang et al. (2D) PCA: Two-Directional Two- Dimensional PCA for Efficient Face Representation and Recognition, Neurocomputing, vol. 69, nos. 1-3, pp , [23] R. Gottumukkal et al., An Improved Face Recognition Technique Based on Modular PCA Approach, Pattern Recognition Letters, vol. 25, no. 4, pp , [24] S. Chen, J. Liu, and Z. Zhou, Making FLDA Applicable to Face Recognition with One Sample Per Person, Pattern Recognition, vol. 37, no. 7, pp , [25] W. Deng, J. Hu, J. Guo, W. Cai, and D. Feng, Robust, Accurate and Efficient Face Recognition from a Single Training Image: A Uniform Pursuit Approach, Pattern Recognition, vol. 43, no. 5, pp , 2010.

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