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Applied Mathematics and Computation 216 (2010) 3195–3199

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Applied Mathematics and Computation

journal homepage: www.elsevier.com/locate/amc

Fusion (2D)2PCALDA: A new method for face recognition

Guohong Huang

Department of Information, Guangdong University of Technology, Guangzhou 510006, PR China

a r t i c l e

Keywords: Fusion face image F(2D)2PCA LDA Face recognition Feature extraction

i n f o

a b s t r a c t

This paper proposes an ef?cient face representation and recognition method, which combines the both information between rows and those between columns from two-directional 2DPCA on fusion face image and the optimal discriminative information from column-directional 2DLDA. Experiment results on ORL and Yale face database demonstrate the effectiveness of the proposed method. ? 2010 Elsevier Inc. All rights reserved.

1. Introduction Principal component analysis (PCA) and Linear discriminant analysis (LDA) are two well-known feature extraction and data representation techniques widely used in the areas of pattern recognition for feature extraction and dimension reduction. PCA performs dimensionality reduction by projecting the original data onto the lower dimensional linear subspace spanned by the leading eigenvectors of the data’s covariance matrix. PCA minimizes the reconstruction error in the sense of least square errors, so PCA can ?nd the most representative features. However, PCA is not ideal for general classi?cation tasks, since it ignores the class label information. LDA is a supervised learning algorithm, and its goal is to ?nd a linear transformation that maximizes the between-class scatter and minimizes the within-class scatter of the training set, which preserves the discriminating information. It is generally believed that LDA-based face recognition methods are superior to those based on PCA since LDA deals directly with class discrimination [1]. Recently, two new techniques 2DPCA [2] and 2DLDA [3], derived from the PCA and LDA technique, respectively, were proposed successively, which directly extract features from image matrices and compute eigenvectors of the so-called image covariance matrices and scatter matrices. 2DPCA and 2DLDA are much more ef?cient than their 1D versions, since they require much less time for training and feature extraction, and have obtained promising experimental results in the areas of feature extraction and dimension reduction. However, the works [4] indicate that traditional 2DPCA performing in the horizontal (or vertical) direction is equivalent to performing PCA on the rows (or column) of the image if each row (or column) is viewed as a computational unit, which implies some structure information between row and column direction cannot be uncovered simultaneously. For 2DLDA, it has the same problem with 2DPCA, that is, 2DLDA in the horizontal direction considers the discriminative information on the rows, but neglects partial vertical discriminative information. Fortunately, the idea of 2DPCA and 2DLDA was developed and two-directional 2DPCA ((2D)2PCA) [5] and two-directional 2DLDA ((2D)2LDA) [6] were proposed, which can effectively alleviate this problem. Motivated by (2D)2PCA and (2D)2LDA, Sanguansat et al. proposed 2DPCA plus 2DLDA method [7] and Qi et al. proposed (2D)2PCALDA[8] method. It is observed that previous four methods always consist of two steps: ?rst a feature matrix is obtained by 2DPCA (or 2DLDA)-based technique in the horizontal direction; second the feature matrix is projected onto the space via 2DPCA (or 2DLDA)-based technique in the vertical direction. As we see that the second step in the vertical direction cannot utilize the entire vertical discriminative information available to the image and these methods don’t use both horizontal and vertical information equally but emphasize extracting horizontal discriminant features.

E-mail address: h_guohong@163.com 0096-3003/$ - see front matter ? 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.04.042

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Zhang et al. proposed diagonal principal component analysis (DiaPCA+2DPCA) [9], which seeks the optimal projective vectors from diagonal face images and therefore the correlations between variations of rows and those of columns of images can be kept. Recently, a fusion method [10] was proposed, which uses the row direction 2DPCA and column direction 2DPCA equally to form the feature matrices for recognition. Yu et al. proposed a complex version of 2DLDA [11], which performs 2DLDA vertically and, at the same time, horizontally. Thus both vertical and horizontal discriminant information can be extracted separately. In this paper, inspired by their successes, especially by the works [8,9], we present a similar model. The feature matrix can be obtained by projecting an image onto fusion two-directional 2DPCA and column-directional 2DLDA simultaneously, which can re?ect both information between rows and those between columns, since the rows (columns) in the transformed fusion face images simultaneously integrate the information of rows and columns in original images. Through the entanglement of row and column information, it is expected that some useful structure information for recognition in original images may be found. The rest of this paper is organized as follows: In Section 2, the idea of the proposed method is described; the results of the experiments are presented in Section 3 and conclusions are given in Section 4. 2. The proposed method 2.1. A new fusion two-directional 2DPCA method (F(2D)2PCA) Suppose that there are N training face images, denoted by n ? n matrix Ai 2 Rn?n (i = 1, . . ., N). For each training face image, de?ne the corresponding fusion face image as follows:

Bi ? Ai ? AT : i

Based on the fusion face image, de?ne the fusion face covariance matrix as

?1?

G?

N ?T ? ? 1 X? Bi ? B Bi ? B ; N i?1

?2?

P where B ? 1=N i Bi is the mean fusion face image. According to Eq. (2), the projective vectors x1, x2, . . .,xd can be obtained by computing the d eigenvectors corresponding to the d biggest eigenvalues of G. Since the size of G is only n ? n, computing its eigenvectors can be ef?cient. Let Xopt = [x1, x2, . . ., xd] denote the projective matrix, projecting training faces Ais onto Xopt, yielding n ? d feature matrices, which can re?ect both information between rows and those between columns.

F i ? Ai X opt :

2.2. Column direction 2DLDA

?3?

2DLDA is an effective feature extraction and discrimination approach in face recognition. Formally, it can brie?y be formulated as follows: Suppose that fAk gN are the training images, which contain C classes, and the ith class Ci has ni samples P k?1 C i?1 ni ? N . Let z denote an n dimensional unitary column vector. The idea of column 2DLDA is to project image Ak onto z, and yield a projected vector Yk, Yk = zTAk. In fact, column 2DLDA tries to seek optimal discriminating vectors Zopt = [z1, z2, . . ., zd] maximizing the following criterion given by

J?z? ?

tr?SB ? ; tr?Sw ?

?4?

where tr(SB) and tr(Sw), respectively, denote the trace of between-class and within-class scatter matrices of projected samples. The SB and Sw can be computed as following:

SB ? Sw ?

C C T 1 X h iT h i 1X i ni Y ? Y Yi ? Y ? ni zT ?Ai ? A? zT ?Ai ? A? ; N i?1 N i?1 Ni ni C C T iT h i 1 X X i 1 X Xh T i Y k ? Y i ?Y ik ? Y i ? ? z ?Ak ? Ai ? zT ?Aik ? Ai ? ; N i?1 k?1 N i?1 k?1

?5? ?6?

where Ai and A denote the means of the ith class and the whole training set, respectively. Aik is the kth image in the class Ci. So

# C T 1 X i i z ? zT GB z; tr?SB ? ? z ni A ? A A ? A N i?1 " # ni C T X X i i i i T 1 z ? zT Gw z: tr?Sw ? ? z A ? A Ak ? A N i?1 k?1 k

T

"

?7? ?8?

G. Huang / Applied Mathematics and Computation 216 (2010) 3195–3199

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In Eqs. (7), (8), T denotes matrix transpose, GB and Gw, respectively, are image between-class and within-class scatter matrices:

GB ? Gw ?

C T 1 X i ni A ? A Ai ? A ; N i?1 ni C T 1 X X i A ? Ai Aik ? Ai : N i?1 k?1 k

?9? ?10?

Obviously, the optimal discrimination vectors Zopt are the eigenvector corresponding to the dominant eigenvalues of eigenstructure G?1 Gb . It has been proved that the optimal value for the discriminating vectors Zopt is composed of the orthonormal w eigenvectors z1, z2, . . ., zm of G?1 Gb corresponding to the m largest eigenvalues. w 2.3. Feature extraction and classi?cation Suppose that we have obtained the projection matrices Xopt and Zopt, projecting the n ? n image A onto Xopt and Zopt simultaneously to yield a m ? d matrix C

C ? ?Z opt ?T AX opt :

?11?

Matrix C is called the feature matrix, which is used for face recognition. Thus, feature matrix contains both the descriptive information of the image extracted by Fusion Two-directional 2DPCA and the discriminant information of image extracted by column-direction 2DLDA. During training, each training image Ak(k = 1, . . ., N) is projected onto both Xopt and Zopt simultaneously to obtain the respective feature matrix Fk(k = 1, . . ., N). During recognition, let A be a given image for recognition, we ?rst use Eq. (11) to get the feature matrix F, then a Euclidean distance based nearest neighbor classi?er is used for classi?cation. Here the distance between F and Fk can be de?ned as following

dF;F k

v?????????????????????????????????????? u m d uX X i;j ?t ?fk ? f i;j ?2 :

i?1 j?1

?12?

3. Experimental results In this section, we experimentally evaluate our proposed method with (2D)2PCA, (2D)2LDA, (2D)2PCALDA and DiaPCA+2DPCA on two well-known face databases: ORL (http://www.uk.research.att.com/facedatabase.html) and Yale (http://cvc.yale.edu/projects/yalefaces/yalefaces.html). The main objective of this research is to improve the accuracy of face recognition, so in the experiments, the face images are preprocessed by the histogram equalization technique and the geometrical normalization technique suggested by Brunelli and Poggio[12] to reduce the in?uence of background, illumination and the hair. The data set was split into two parts: one part was taken for training; the other part would be used for testing.

100 90

the recognition accuracy(%)

80 70 60 50 40 30 20

(2D)2PCA (2D)2LDA (2D)2PCALDA DiaPCA+2DPCA the proposed method

0

2

4

6

8

10

12

14

16

18

20

dimension of feature vectors

Fig. 1. Recognition performance of different approaches with varying dimension of feature matrices on ORL face database.

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G. Huang / Applied Mathematics and Computation 216 (2010) 3195–3199

The recognition is performed based on a nearest neighbor classi?er in feature space. In order to evaluate the ef?ciency of the algorithm and remove the in?uence caused by the choice of the training set and test set, we repeated each experiment 15 times and all the result data given in this paper is an average of them. 3.1. Results on ORL database The ORL database is used to test the performance of the face recognition algorithms under condition of minor variation of scaling and rotation, which consists of 112 ? 92 sized 400 frontal faces taken at different times. There are 10 images of 40 individuals with variation in pose, illumination, facial expression (open/closed eyes, smiling/not smiling) and facial details (glasses or no glasses). In this experiment, all the face images are preprocessed and cropped into size of 80 ? 80 from original frames based on the location of the two eyes. In order to disclose the relationship between the accuracy and dimension of feature vectors, classi?cation experiments are performed under a series of different dimensions. Five images of each class are selected randomly for training and the rest of the images are used for test. Fig. 1 gives the comparisons of ?ve methods on top recognition accuracy corresponding to the dimension of feature matrices. It can be found from Fig. 1 that the highest classi?cation accuracy of the proposed method is 95.5%, which is comparable with other methods in terms of recognition accuracy. We also compare the performance of the proposed method with other methods for varying number of training samples. p images are randomly selected from each class to construct the training data set, the remaining images being used as the test images. To ensure suf?cient training, a value of at least 2 is used for p. Table 1 shows the top recognition accuracy. It can be found that the proposed method has better recognition accuracy than the other methods. 3.2. Results on Yale database The Yale database is used to examine the performance of the algorithms under the condition of varied facial expression and lighting con?guration, which contains 165 images of 15 individuals (each person has 11 different images) under various facial expressions and lighting conditions. All images are gray with 256 levels and size of 320 ? 243 pixels. In this experiment, they were preprocessed and normalized to the size of 100 ? 100. All experiments were performed with 5 training

Table 1 Comparison of different approaches in terms of top recognition accuracy (%) on ORL database. Methods Number of training samples per class 2 (2D)2PCA (2D)2LDA (2D)2PCALDA DiaPCA + 2DPCA The proposed method 78.50 79.41 80.56 80.62 80.74 4 90.27 91.63 91.68 90.89 92.30 6 92.32 94.57 95.73 92.95 95.98 8 95.95 96.15 97.40 96.30 98.32

100 90

the recognition accuracy(%)

80 70 60 50 40 30 20 10

(2D)2PCA (2D)2LDA (2D)2PCALDA DiaPCA+2DPCA the proposed method

0

2

4

6

8

10

12

14

16

18

20

dimension of feature vectors

Fig. 2. Recognition performance of different approaches with varying dimension of feature vectors on Yale face database.

G. Huang / Applied Mathematics and Computation 216 (2010) 3195–3199 Table 2 Comparison of different methods in terms of top recognition accuracy on Yale database. Methods Number of training samples per class 2 (2D)2PCA (2D)2LDA (2D)2PCALDA DiaPCA + 2DPCA The proposed method 68.46 64.81 73.43 69.02 74.03 4 89.27 86.63 92.68 90.39 92.70 6 92.32 90.51 94.73 92.95 94.88 8

3199

95.38 95.15 98.89 95.43 98.89

images and 6 test images per person for a total of 75 training images and 90 test images. There was no overlap between the training and test sets. Fig. 2 gives the comparisons of ?ve methods on top recognition accuracy with the varying dimension of feature matrices. It can be found from Fig. 2 that the highest classi?cation accuracy of the proposed method is 96.7%, which has better recognition accuracy than other methods. Table 2 shows the top recognition accuracy obtained by different methods for varying number of training samples. It can be easily ascertained from Table 2 that the proposed method obtains same or even better recognition accuracy compared with other methods. The recognition rate achieves 98.89%when the number of training samples is more than 8. 4. Conclusion In this paper, an ef?cient face representation and recognition method is proposed. The major advantage of the proposed is to combine the both information between rows and those between columns from fusion two-directional 2DPCA on fusion face image and the optimal discriminative information from column-directional 2DLDA. Experimental results on ORL and Yale database shows that the proposed method obtains same or even better recognition accuracy than (2D)2PCA, (2D)2LDA, (2D)2PCALDA and DiaPCA+2DLDA. References

[1] P.N. Belhumeur, J.P. Hespanha, D.J. Kriegman, Eigenfaces versus ?sherfaces: recognition using class speci?c linear projection, IEEE Trans. Pattern Anal. Mach. Intell. 19 (7) (1997) 711–720. [2] J. Yang, D. Zhang, A.F. Frangi, J.Y. Yang, Two dimensional PCA: a new approach to appearance-based face representation and recognition, IEEE Trans. Pattern Anal. Mach. Intell. 26 (1) (2004) 131–137. [3] L. Ming, B. Yuan, 2D-LDA: a statistical linear discriminant analysis for image matrix, Pattern Recognit. Lett. 26 (5) (2005) 527–532. [4] L. Wang, X. Wang, J. Feng, On image matrix based feature extraction algorithms, IEEE Trans. Systems Man Cybern.—Part B. 36 (1) (2006) 194–197. [5] D. Zhang, Z.H. Zhou, (2D)2PCA: two-directional two-dimensional PCA for ef?cient face representation, Neurocomputing 69 (1–3) (2005) 224–231. [6] S. Noushath, G. Hemantha Kumar, P. Shivakumar, (2D)2LDA: an ef?cient approach for face recognition, Pattern Recognit. 39 (7) (2006) 1396–1400. [7] P. Sanguansat, W. Asdornwised, S. Jitapunkul, S. Marukatat, Two-dimensional linear discriminant analysis of principle component vectors for face recognition, in: IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), vol. 2, 2006, pp. 345–348. [8] Y. Qi, J. Zhang, (2D)2PCALDA: an ef?cient approach for face recognition, Appl. Math. Comput. 213 (1) (2009) 1–7. [9] D.Q. Zhang, Z.H. Zhou, Diagonal principal component analysis for face recognition, Pattern Recognit. 39 (1) (2006) 140–142. [10] Y.G. Kim, Y.J. Song, U.D. Chang, D.W. Kim, T.S. Yun, J.H. Ahn, Face recognition using a fusion method based on bidirectional 2DPCA, Appl. Math. Comput. 205 (2) (2008) 601–607. [11] W. Yu, Z. Wang, W. Chen, A new framework to combine vertical and horizontal information for face recognition, Neurocomputing 72 (2009) 1084– 1091. [12] R. Brunelli, T. Poggio, Face recognition: features versus templates, IEEE Trans. Pattern Anal. Mach. Intell. 15 (10) (1993) 1042–1052.

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