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A simplified GLRAM algorithm for face recognition
ARTICLE IN PRESS Neurocomputing 72 (2008) 212–217
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Neurocomputing journal homepage: www.elsevier.com/locate/neucom
A simplified GLRAM algorithm for face recognition Chong Lu a, Wanquan Liu b,, Senjian An b a b
Yili Normal College, Yili 835000, PR China Curtin University of Technology, Perth 6102, WA Australia
a r t i c l e in f o
a b s t r a c t
Available online 3 September 2008
In this paper we propose a new face recognition method based on the generalized low rank approximations of matrices (GLRAM). First, we investigate the GLRAM and its associated coupled subspace analysis and then propose a new simplified algorithm, which is named as SGLRAM aiming at deriving the projection matrices for GLRAM. We implement all these algorithms (GLRAM SGLRAM) for face recognition on the ORL and YaleB databases and the experiments show that the SGLRAM can produce comparable high performance compared to the approached of two-dimensional principal component analysis (2DPCA) and GLRAM. However, it will cost much less time than the GLRAM in training and save more space than the 2DPCA in testing. & 2008 Published by Elsevier B.V.
Keywords: Two-dimensional principal component analysis Singular value decomposition Low rank approximation of matrices Face recognition
1. Introduction The problem of dimension reduction has recently received broad attentions in areas such as machine learning, computer vision and information retrieval [1,2,8]. The aim of dimensional reduction is to obtain a lower dimensional compact data representation with few loss of information. There are several important algorithms for dimensional reduction based on vectors selection which may be called vector space model [13,20]. With this model, each datum is treated as a vector and the whole data collection will be a single data matrix, where each column corresponds to a data point. Many algorithms have been proposed based on this model for dimensional reduction in different areas such as face recognition [13], machine learning [5] and information retrieval [4]. A well-known technique based on this vector space model for dimensional reduction is the low rank approximation by using singular value decomposition (SVD). An appealing property of this type of low rank approximation via SVD is that it can achieve the smallest reconstruction error among all approximations with the constrain of same rank in Euclidean distance. However, applications of this technique to high dimensional data, such as images and videos, will quickly meet some critical problems like algorithm complexity, high computational cost, etc. Recently, Yang et al. [18] proposed an algorithm called twodimensional principal component analysis (2DPCA) for face recognition, in which the image covariance (scatter) matrix is directly computed from the image matrix representations. Associated with
Corresponding author.
E-mail addresses: [email protected] (C. Lu), [email protected] (W. Liu), [email protected] (S. An). 0925-2312/$ - see front matter & 2008 Published by Elsevier B.V. doi:10.1016/j.neucom.2007.11.046
this problem, a new type of low rank approximation is proposed recently by Ye et al. [19] based on a matrix space model instead of the vector space model. In addition, Lu et al. [10] presented a necessary condition for the optimality of the GLRAM and Xu et al. [16] proposed an algorithm called coupled subspaces analysis (CSA) based on GLRAM. With those techniques, each datum is treated as a matrix instead of a vector and the whole data collection will be a set of data matrices. In this way, the dimension curse in vector space model disappears. After publications [16,18,19], many applications have been investigated by using this matrix model successfully [15,18]. In this paper, we will study three different methods, the 2DPCA [18], the generalized low rank approximations of matrices (GLRAM) [19] and CSA [16] and then investigate their advantages and shortcomings in face recognition in training and testing stages. The advantage of 2DPCA is that it has unique optimal closed-form solution in training stage. Its main foul is that only one side of image is projected and it will use more time in testing stage, specially for images with large dimensions. The GLRAM and CSA can projected the testing images on both sides in result of efficient testing, however, there is no explicit closed formula for the projection matrices as pointed out in [10] and consequently the complexity is very high in training stage. Based on these observations, we then propose a new algorithm to deriving the projection matrices L and R for the GLRAM directly and call this new algorithm as SGLRAM. Though these obtained projection matrices are not optimal theoretically, they can achieve satisfactory performance in face recognition with the same high performance as 2DPCA and the GLRAM. Then the complexity analysis in training and testing stages are given for the SGLRAM. Finally, we implement all these algorithms for face recognition on the ORL, AMP, Yale and YaleB face databases and compared their performance in recognition accuracy and computational cost.
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The rest of this paper is organized as follows. We give a brief overview of the 2DPCA [18], GLRAM [19] and CSA based on the matrix space model. The difference between CSA and GLRAM is that CSA subtracts the mean in objective criterion function and the original GLRAM does not. The experiment shows that the accuracy of face recognition using CSA is a little higher than that of GLRAM. A new algorithm is proposed in Section 3 via investigating 2DPCA, GLRAM and CSA. We then will analyze the computational cost and storage request on the three algorithms, respectively. In Section 4, extensive experiments are conducted on image database ORL, AMP, Yale and YaleB. Conclusions and directions for future work are summarized in Section 5.
2. 2DPCA and GLRAM In this section we will briefly outline the 2DPCA, the GLRAM and the CSA in order to present the contribution of this paper clearly. 2.1. 2DPCA Yang et al. [18] proposed the 2DPCA algorithm for face recognition and we briefly describe this technique here in order to clearly present the proposed algorithm later. Given input image matrices Ai 2 Rmn , i ¼ 1 . . . N, where mn is the number of pixels in the image, and N is the number of the images. The following criterion is adopted for 2DPCA in [18]: JðXÞ ¼ trðSX Þ ¼ trfX > ½EðA EAÞ> ðA EAÞXg
(1)
where SX is the covariance matrix of Ai ði ¼ 1; 2; . . . ; MÞ with the projection matrix X and E is the expectation of a stochastic variable. In fact, the covariance matrix G 2 Rmm with finite N images is given by G¼
N 1X ðA AN ÞðAj AN Þ> N j¼1 j
(2)
P where AN ¼ ð1=NÞ N i¼1 Ai is the mean image matrix of the N training samples. Alternatively the criterion in (1) can be expressed by the following: JðXÞ ¼ trðX > GXÞ
(3)
where X is a unitary column vector. The matrix X that maximizes the criterion is called the optimal projection axis. The optimal projection X opt is a set of unitary vectors that maximize JðXÞ, i.e., the eigenvectors of G corresponding to the large eigenvalues [18]. We usually choose a subset of only d eigenvectors corresponding to largest d eigenvalues to be included in the model, i.e., fX 1 . . . X d g ¼ arg max JðXÞ satisfying X > i X j ¼ 0(iaj; i; j ¼ 1; 2; . . . ; d). With the above mentioned solution, each image can thus be optimally approximated in the least-squares sense up to a predefined reconstruction error. Every input image Ai will project into a point in the d-dimensional subspace spanned by the selected eigenmatrix X opt [18]. In face recognition, an optimal projection matrix X opt will be derived with a given dimension d and training samples. This optimal projection matrix will be used for face recognition in testing stage using the nearest neighbor approach as described in [18]. It should be noted that the testing image is only reduced in one side (column or row) in the testing stage. 2.2. GLRAM The traditional low rank approximation of matrices was formulated as follows: given A 2 RWn , find a matrix B 2 RWn
213
with rank(B)¼ k, such that B ¼ arg
min kA BkF
rankðBÞ¼k
(4)
where the Frobenius norm kMkF of a matrix M ¼ ðM ij Þ is defined as sffiffiffiffiffiffiffiffiffiffiffiffiffiffi X 2ffi kMkF ¼ M ij ij
This problem has a closed-form solution via SVD. However, the computational cost is high when W is large as stated in [19]. In order to overcome the computational complexity in the above rank approximation problem, Ye [19] proposed a matrix space model for low rank approximation of matrices called the GLRAM, which can be stated as follows. Let Ai 2 Rmn , for i ¼ 1; 2; . . . ; N; be the N data points in the training set. The aim is to compute two matrices L 2 Rmr and R 2 Rnc with orthogonal columns and N matrices M i 2 Rrc for i ¼ 1; 2; . . . ; N such that LM i R> approximates Ai well for all i. Mathematically, this can be formulated as an optimization problem min >
N X
L> L¼Ir ;R R¼Ic i¼1
kAi LM i R> k2F
(5)
Until now, no closed-form solution has been found for the proposed GLRAM. In order to find an optimal solution for this GLARM, Ye in [19] has proved that the proposed GLRAM is equivalent to the following optimization problem. max JðL; RÞ ¼ L;R
N X
kL> Ai Rk2F
i¼1
¼
N X
Tr½ L> Ai RR> A> i L
(6)
i¼1
where L and R satisfy LL> ¼ Ir ; RR> ¼ Ic : Actually, this is a nonlinear optimization problem which is difficult to solve as pointed out in [10]. In order to simplify the problem, Ye [19] observed the following fact in Theorem 1 and proposed an Algorithm GLRAM stated below instead of solving the above optimization problem directly. Theorem 1. Let L, R and fM i gN i¼1 be the optimal solution to the minimization problem in GLRAM. Then (1) For a given R, L consists of the r eigenvectors of matrix M L ðRÞ ¼
N X
Ai RR> A> i
(7)
i¼1
corresponding to the largest r eigenvalues. (2) For a given L, R consists of the c eigenvectors of the matrix M R ðLÞ ¼
N X
> A> i LL Ai
(8)
i¼1
corresponding to the largest c eigenvalues. Based on the above observations, the following algorithm for solving the GLRAM is proposed in [19], which is called the Algorithm GLRAM. Algorithm GLRAM [19] Input: matrices fAi g, r and c. Output: matrices L, R and fM i g 1. Obtain initial L0 for L and set i ¼ 1. 2. While not convergent
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3. From the matrix M R ðLÞ ¼ eigenvectors
PN
> > j¼1 Aj Li1 Li1 Aj
compute the c
R ffj gcj¼1
of M R corresponding to the largest c
R
R
eigenvalues. 4. Let Ri ¼ ½f1 ; . . . ; fc
PN
> > j¼1 Aj Ri Ri Aj L r eigenvectors ffj gj¼1 corresponding to the L L 6. Let Li ¼ ½f1 ; ; fr
5. From the matrix M L ðRÞ ¼
compute the r largest r eigenvalues.
7. i ¼ i þ 1 8. Endwhile 9. L ¼ Li1 10. R ¼ Ri1 11. Mj ¼ L> Aj R It should be noted here that the convergence of this algorithm cannot be guaranteed theoretically in general [10] and therefore the implementation from steps 2 to 8 may never end. In order to investigate the convergence issue of the proposed algorithm, the following criteria are defined in [19]: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u1 X RMSRE ¼ t kA LM i R> k2F N i¼1 i where RMSRE stands for the root mean square reconstruction error. It is easy to show that each iteration in the Algorithm GLRAM will decrease the RMSRE value and also it is obvious that the sequences of RMSRE is bounded. Mathematically, this will guarantee that the sequences of RMSRE converge. This iterative algorithm GLRAM is proposed to obtain the optimal L, R in [19] and experiments showed the proposed algorithm is very time costive in general.
remove the difficulty for finding the optimal solutions in the GLRAM and can use its advantage on reduced dimensions on both sides for testing. Of course, we expect the performance in face recognition with the approximate solution will not decline significantly compared to the performance with the optimal projection matrices. Such expectation is based on the fact that each side solution is optimal with explicit closed-form solution. Next, we will describe this new algorithm in detail. In order to utilize the idea in the CSA, the samples Ai (i ¼ 1; 2; . . . ; NÞ are assumed to subtract the mean A as M i . In this case, Eqs. (7) and (8) can be rewritten as ML ðRÞ ¼
N X
M i RR> M > i
(9)
i¼1
MR ðLÞ ¼
N X
> M> i LL M i
(10)
i¼1
We intend to obtain the one side optimal solution for the GLRAM. For such purpose, we assumed that R and L are the identity matrices first. With this assumption, we can obtain P PN > > ML ðRÞ ¼ N i¼1 M i M i and M R ðLÞ ¼ i¼1 M i M i . Further, treat M L ðRÞ and MR ðLÞ as covariance matrices in the 2DPCA. Then two couples of eigenmatrices U and V can be obtained by SVD corresponding to the ML ðRÞ and M R ðLÞ, respectively. Now, we have the following Algorithm SGLRAM: Algorithm SGLRAM Input: matrices fAi g, r and c, i ¼ 1; 2; . . . ; N: Output: matrices L, R and fQ i g P 1. Calculate M i ¼ Ai 1=N N i¼1 Ai . 2. Calculate M L ðRÞ and M R ðLÞ as following: N X ML ðRÞ ¼ Mi M> i i¼1
2.3. Coupled subspace analysis Xu et al. [16] proposed the CSA algorithm which is actually based on the GLRAM. The only difference is that in CSA Ai is P replaced by Ai ð1=NÞ N i¼1 Ai . Our preliminary experiments show that CSA performs slightly better than the GLRAM in face recognition due to the average value difference.
3. The proposed algorithm: SGLRAM We have presented the algorithms of the 2DPCA, the GLRAM and the CSA, and found that the GLRAM and CSA are actually the same with double sides in matrix optimization and the 2DPCA is one side matrix optimization. Consequently, the advantage of the 2DPCA is that it has a closed-form solution, which results in less expensive training time, while the advantage of the GLRAM is that the testing image will have lower dimension on both sides of the projected image and lead to less testing time. However, the GLRAM has no explicit closed-form solution, which needs an iterative algorithm to derive its optimal solution and this is usually difficult or costive due to two reasons: one is the convergence issue has not been solved theoretically [10] and even the GLRAM converges, the other problem is that its convergence speed is unknown. Similarly, in testing stage, the 2DPCA will have a high dimension in one side and will use a long time for testing. Based on these observations, we propose a new algorithm called the SGLRAM. The idea is very explicit. We will use the idea in 2DPCA algorithm to obtain one side optimization, respectively, for both left and right sides and then we use these two direct solutions as approximating solutions for the GLRAM. This will
MR ðLÞ ¼
N X
M> i Mi
i¼1
3. Calculate the left eigenmatrix U for M L ðRÞ and the right eigenmatrix V for M R ðLÞ. These two matrices can be obtained directly by doing SVD on the M L ðRÞ and M R ðLÞ, respectively. 4. Using the eigenmatrices U ¼ L and V ¼ R to obtain Q i ¼ LAi R. From the above algorithm we can use eigenmatrices U, V obtained directly in the Algorithm SGLRAM instead of L and R which were derived iteratively in the GLRAM (CSA). We expect the corresponding proposed face recognition algorithm will produce as same good performance as the GLRAM and the 2DPCA. Obviously, the Algorithm SGLRAM will save time in training compared to the GLRAM and save time in testing compared to the 2DPCA as demonstrated in theoretical analysis in next subsection and experimental results in next section. 3.1. Analysis of computational complexity In this subsection we will analyze the algorithm complexity for the 2DPCA, the GLRAM (CSA) and the SGLRAM. Here we assume that the image size of Ai is m n with nXm. The number of training images is N and the reduced dimension for both sides of GLRAM is d for simplicity. Also we assume that the iteration number of convergence in the GLRAM is r. In training stage, the complexity of the 2DPCA can be calculated easily to be Oðn3 þ NÞ due to the fact that the complexity of one SVD is of Oðn3 Þ [7]. Further the complexity of the SGLRAM is Oðn3 þ m3 þ NÞ and the complexity of the GLRAM (CSA) is of Oðrn3 þ rm3 þ NÞ. One can see
ARTICLE IN PRESS C. Lu et al. / Neurocomputing 72 (2008) 212–217
0.8
YaleB
d ¼ 10
d ¼ 15
d ¼ 20
d ¼ 25
2DPCA SGLRAM
159.069 119.531
172.128 112.402
206.497 134.219
240.737 136.972
ORL
d¼5
d ¼ 10
d ¼ 15
d ¼ 20
d ¼ 25
2DPCA SGLRAM
7.871 5.198
25.447 11.757
34.65 16.434
47.66 22.462
18.467 8.773
Table 2 The RMSRE value ðE þ 003Þ
Recognition accuracy
Table 1 Computational cost (seconds)
0.75
0.7 CSA SGLRAM 2DPCA GLRAM 0.65 0
GLRAM SGLRAM
215
d ¼ 10
d ¼ 15
d ¼ 20
d ¼ 25
d ¼ 30
14.2 14.311
11.963 12.066
10.48 10.57
9.366 9.4697
8.4769 8.5273
that the computational cost is much higher for the GLRAM when r is large. In the testing stage, when we project the training images into a new space the required storage of the 2DPCA is Nnd and the other 2 two is Nd . Due to the fact d5n in usual, the computational cost by the 2DPCA is much higher than the SGLRAM. In order to illustrate this, the experiments on the two image databases of YaleB and ORL are done and the results are shown in Table 1. The experiment includes projecting 320 training images into the feature spaces and recognizing 560 testing images in the YaleB database, and projecting 160 training images into the feature spaces and recognizing 240 testing images in the ORL database, respectively. We have done all the experiments in this paper on a stand alone PC machine (CPU:PIV 1.3 GHz, RAM:256M) with Matlab. One can see that it is usually much faster and more efficient to use the SGLRAM algorithm in the recognition process when the dimension d is large. The recognition performance in face recognition will be demonstrated in the next section. Here we will evaluate the mathematical comparisons with the proposed criteria in [19]. Recall that vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N X 1u RMSRE ¼ t kA LM i R> k2F N i¼1 i is used in [19] and here we compute this value using the optimal solutions obtained via the SGLRAM and the GLRAM by applying the images in YaleB database as variables. In this experiment, N ¼ 320 is the number of training images, Mi 2 Rrc for i ¼ 1; 2; . . . ; N are obtained matrices through these two algorithms such that LM i R> can approximate Ai well for all i. The experimental results are shown in Table 2. The results in Table 2 implied that the SGLRAM algorithm can produce nearly as same good function values as the GLRAM algorithm and this indicates that the SGLRAM can produce much satisfactory sub-optimal solutions for the GLRAM on the YaleB database. We expect the face recognition performance has the same results, which will be demonstrated in next section.
4. Experimental evaluations In this section we will carry out several experiments to demonstrate the performance of the SGLRAM algorithm. For the chosen databases, we first use two types of input images. One is
5
10 15 Number of feature d
20
25
Fig. 1. Face recognition performance on the ORL database.
the ORL face database, which contains 400 images of 40 individuals (each person has 10 different images) under various facial expressions, wearing or not wearing glasses, different lighting conditions with each images being cropped and resized to 112 92 pixels in this experiment. The other database is the YaleB database [6], which contains 650 face images of 10 people, including frontal views of face with different facial expressions, lighting conditions. With these face databases, we will conduct two group experiments using the 2DPCA, CSA and the SGLRAM, respectively. The first one is on the ORL image face database, where the first four images of each person are used as training and the other six images are used as testing. The second experiment is on the YaleB image face database, where the first eight images of each person are used for training and the other 56 images are used for testing. The results are shown in Figs. 1 and 2, respectively. From Fig. 1 we can see that the performance of the SGLRAM is between the 2DPCA and the GLRAM before they achieve the best performance in the ORL database, i.e., with lower dimension d, the performance for the 2DPCA is better than that for the SGLRAM and GLRAM due to the fact that it includes more effective information with lower dimension d since only the dimension in one side of the image matrix is reduced. In this case effective information kept in the other side of the image matrix plays an important role. With the increasing dimension d, the performance of the SGLRAM and the GLRAM are getting better since the effective information is being enriched. Finally, when d is large enough, all their performances are nearly the same. In Fig. 2 we can see the performances are closely the same when dimension increases up to 30 in the YaleB database. The large gap with d ¼ 40 may be due to the fact that the face images in the YaleB database have different backgrounds. From these experiments, we can see that the proposed SGLRAM algorithm can save much more time in training stage compared to the GLRAM algorithm and run much faster in testing stage compared to the 2DPCA. Also their performances are nearly all the same when the reduced dimension is large enough. In order to show the performance on other databases, we also did extensive experiments on the AMP and the Yale databases. The AMP database contains 975 images with 13 individuals (each person has 75 different images) under various facial expressions and lighting conditions with 64 64 pixels. We use the first five images of each person as training and the others are used for testing. The performance is shown in Fig. 3. On the other hand, the Yale database has 15 individuals with each having 11 images under various facial expressions and
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0.72 0.85 Recognition accuracy
Recognition accuracy
0.7 0.68 0.66 0.64
CSA SGLRAM 2DPCA GLRAM
0.62
0.8 0.75 CSA SGLRAM 2DPCA GLRAM
0.7 0.65 0.6
0.6 0
5
10
15
20
25
30
35
2
40
2.5
3
Number of feature d
5
5.5
6
Fig. 4. Face recognition performance on the Yale database.
Fig. 2. Face recognition performance on the YaleB database.
Table 3 The RMSRE values for AMP
1.05
Recognition accuracy
3.5 4 4.5 Number of feature d
GLRAM SGLRAM
1
CSA SGLRAM 2DPCA GLRAM
GLRAM SGLRAM
0.9 2.5
3
3.5
4
4.5
5
5.5
d¼3
d¼4
d¼5
d¼6
3.7295 3.7533
3.5184 3.5167
3.2822 3.2851
3.0698 3.0909
2.9278 2.9484
Table 4 The RMSRE value for the Yale dataset
0.95
2
d¼2
d¼2
d¼3
d¼4
d¼5
d¼6
9.0254 9.0483
8.2 8.3169
7.6847 7.697
7.2255 7.2879
6.9225 6.9561
6
Number of feature d Fig. 3. Face recognition performance on the AMP database.
lighting conditions with 231 195 pixels. The first five images for each person are used for training and the others are used for testing. The performance is shown in Fig. 4. Correspondingly, the RMSRE values for the GLRAM and the SGLRAM for the AMP database are listed in Table 3 and the same result for Yale database is listed in Table 4. Next, we compare the computational time in testing stage with the 2DPCA on these two databases and they are listed in Table 5. Based on these experiments, we can see that the performances for these four algorithms are nearly the same when the reduced dimension d is chosen properly. In general, when d is smaller, the 2DPCA performs better than all other three approaches since it only used one side dimension reduction and contains more effective information on the other side. With d increasing, the performances of all these algorithms are nearly the same as evidenced from Figs. 1 to 4. However, the computational cost in training stage is large for the GLRAM and CSA algorithms and the computing cost is large in testing stage for the 2DPCA as evidenced from the listed tables. In this case, the proposed SGLRAM is the best choice since it can reduce the computational complexity while keeping the same high performance. However, we observed when the dimension d is large enough, the performances of all the four algorithms on the ORL database are deteriorating due to the fact that the extra information brought by the large d is actually becoming noise. The result on the YaleB
Table 5 Computational cost (seconds) Yale
d¼2
d¼3
d¼4
d¼6
2DPCA SGLRAM
2.444 2.003
2.764 2.063
3.435 2.233
3.906 2.534
AMP
d¼2
d¼3
d¼4
d¼6
2DPCA SGLRAM
6.419 5.658
7.741 6.319
8.993 6.92
11.557 8.282
database is interesting with large d, the GLRAM algorithm performs worse than the proposed algorithm SGLRAM, which deserves further investigation in the future.
5. Conclusions In this paper we investigated the 2DPCA, GLRAM and CSA and then derived a new face recognition algorithm called the SGLRAM. We have demonstrated that this new algorithm inherited the advantages of the 2DPCA and the GLRAM while can avoid their corresponding shortcomings in high computational costs, respectively. Experiments have shown that the SGLRAM algorithm needs much less time in training stage compared to the GLRAM or CSA
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and much more efficient in testing compared to the 2DPCA and can still produce high performance as the 2DPCA and the GLRAM. The results in this paper showed that the sub-optimal solution obtained via the SGLRAM can approximate the optimal solution satisfactorily. In the future, we will investigate the incremental learning algorithms for the SGLRAM along the line of incremental learning algorithm for the 2DPCA in [11] and study its performance. The incremental investigation is hard or impossible for the GLRAM due to the complexity in iteration steps in training while it will be feasible for the SGLRAM based on the simplified procedure. One more possible research is to simplify the procedures in the coupled kernel-based subspace learning [17]. Actually, many existing kernel approaches based on subspace analysis for face recognition [9,3,14] can be explored further. Furthermore, the appropriate dimension d for the GLRAM, the SGLRAM and the 2DPCA needs to be investigated for different databases in the future. This problem is very important for the design of automated face recognition systems in selecting optimal model parameters as demonstrated in eigenface approach [12].
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[15] L. Wang, X. Wang, X. Zhang, J. Feng, Equivalence of two dimensional pca to line-based pca, Pattern Recognition Lett. (2005) 57–60. [16] D. Xu, S. Yan, L. Zhang, Z. Liu, H.J. Zhang, Coupled subspaces analysis, in: Proceedings of CVPR’2005, 2005, pp. 204–208. [17] S. Yan, D. Xu, L. Zhang, B. Zhang, H. Zhang, Coupled kernel based subspace learning, in: Proceedings of Computer Vision and Pattern Recognition (CVPR), 2005, pp. 645–650. [18] J. Yang, D. Zhang, A. Frangi, J. Yang, Two-dimensional pca: a new approach to appearance-based face representation and recognition, IEEE Trans. Pattern Anal. Mach. Intell. 5 (1) (2004) 131–137. [19] J. Ye, Generalized low rank approximations of matrices, in: Proceedings of the Twenty-first International Conference on Machine Learning, 2004, pp. 887–894. [20] W. Zhao, R. Chellappa, P. Phillips, Face recognition: a literature survey, ACM Comput. Surv. 35 (2003) 399–458. Chong Lu received the BS degree in Applied Mathematics from YiLi Normal College in 1990 and obtained the certificate of MS in Computer Application in 2002. He worked as Visiting Scholar in Curtin University of Technology from November 2005 to November 2006. His research interests include image processing, computational learning theory and pattern recognition. He became Associate Professor at YiLi Normal College in 2002.
References [1] S. An, W.Q. Liu, S. Venkatesh, R. Tjahyadi, A fast dimension reduction algorithm for kernel based classification, in: Proceedings of International Conference on Machine Learning and Cybernetics, 2005, pp. 3369–3376. [2] M.S. Bartlett, J.R. Movellan, T.J. Sejnowski, Face recognition by independent component analysis, IEEE Trans. Neural Networks 13 (6) (2003) 1450–1464. [3] P. Belhumeur, J. Hespanha, D. Kriegman, Eigenfaces vs fisherfaces: recognition using class specific linear projection, IEEE Trans. Pattern Anal. Mach. Intell. 19 (1997) 711–720. [4] M. Berry, S. Dumais, G. Obrie, Using linear algebra for intelligent information retrieval, SIAM Rev. 37 (1995) 573–595. [5] V. Castelli, A. Thomasian, C.-S. Li, Csvd: clustering and singular value decomposition for approximate similarity searches in high dimensional space, IEEE Trans. Knowl. Data Eng. 15 (3) (2003) 671–685. [6] hftp://plucky.cs.yale.edu/CVC/pub/images/yalefaceB/Tarsetsi. Yale database. [7] hhttp://www.library.cornell.edu/nr/bookcpdf.htmli. Svd. 2006. [8] Y. Hua, W.Q. Liu, Generalized Karhunen–Loeve transform, IEEE Signal Process. Lett. 5 (6) (1998) 141–142. [9] Q. Liu, R. Huang, H. Lu, S. Ma, Face recognition using kernel based Fisher discriminant analysis, in: Proceedings of the Fifth International Conference on Automatic Face and Gesture Recognition, 2002, pp. 205–211. [10] C. Lu, W.Q. Liu, S. An, Revisit to the problem of generalized low rank approximation of matrices, in: Proceedings of International Conference on Intelligent Computing, 2006, pp. 450–460. [11] C. Lu, W.Q. Liu, S. An, S. Venkatesh, Face recognition via incremental 2dpca, in: First International Workshop on Multimodal Information Retrieval at IJCAI 2007, 2007, pp. 1–6. [12] W.Q.L. Ronny Tjahyadi, S. Venkatesh, Automatic parameters selection for eigenfaces, Pac. J. Optim. (2006) 277–288. [13] M. Turk, A. Pentland, Eigenfaces for recognition, J. Cognitive Neurosci. 13 (1) (1991) 71–86. [14] M. Vasilescu, D. Terzopoulos, Multilinear subspace analysis for image ensembles, in: Proceedings of Computer Vision and Pattern Recognition (CVPR), vol. 2, 2003, pp. 93–99.
Wanquan Liu received the BSc degree in Applied Mathematics from Qufu Normal University, PR China, in 1985, the MSc degree in Control Theory and Operation Research from Chinese Academy of Science in 1988, and the PhD degree in Electrical Engineering from Shanghai Jiaotong University, in 1993. He once hold the ARC Fellowship and JSPS Fellowship and attracted research funds from different resources. He is currently a Senior Lecturer in the Department of Computing at Curtin University of Technology. His research interests include large-scale pattern recognition, control systems, signal processing, machine learning, and intelligent systems. Senjian An received the BS degree from Shandong University, Jinan, China, in 1989, the MS degree from the Chinese Academy of Sciences, Beijing, in 1992, and the PhD degree from Peking University, Beijing, in 1996. He was with the Institute of Systems Science, Chinese Academy of Sciences, Beijing, where he was a Postdoctoral Research Fellow from 1996 to 1998. In 1998, he joined the Beijing Institute of Technology, Beijing, and he was an Associate Professor from 1998 to 2001. From 2001 to 2004, he was a Research Fellow with The University of Melbourne, Parkville, Australia. He joined Curtin University of Technology, Perth, WA, Australia, in 2004, where he is currently a Postdoctoral Research Fellow. His research interests include signal processing, machine learning and computer vision.
Contents lists available at ScienceDirect
Neurocomputing journal homepage: www.elsevier.com/locate/neucom
A simplified GLRAM algorithm for face recognition Chong Lu a, Wanquan Liu b,, Senjian An b a b
Yili Normal College, Yili 835000, PR China Curtin University of Technology, Perth 6102, WA Australia
a r t i c l e in f o
a b s t r a c t
Available online 3 September 2008
In this paper we propose a new face recognition method based on the generalized low rank approximations of matrices (GLRAM). First, we investigate the GLRAM and its associated coupled subspace analysis and then propose a new simplified algorithm, which is named as SGLRAM aiming at deriving the projection matrices for GLRAM. We implement all these algorithms (GLRAM SGLRAM) for face recognition on the ORL and YaleB databases and the experiments show that the SGLRAM can produce comparable high performance compared to the approached of two-dimensional principal component analysis (2DPCA) and GLRAM. However, it will cost much less time than the GLRAM in training and save more space than the 2DPCA in testing. & 2008 Published by Elsevier B.V.
Keywords: Two-dimensional principal component analysis Singular value decomposition Low rank approximation of matrices Face recognition
1. Introduction The problem of dimension reduction has recently received broad attentions in areas such as machine learning, computer vision and information retrieval [1,2,8]. The aim of dimensional reduction is to obtain a lower dimensional compact data representation with few loss of information. There are several important algorithms for dimensional reduction based on vectors selection which may be called vector space model [13,20]. With this model, each datum is treated as a vector and the whole data collection will be a single data matrix, where each column corresponds to a data point. Many algorithms have been proposed based on this model for dimensional reduction in different areas such as face recognition [13], machine learning [5] and information retrieval [4]. A well-known technique based on this vector space model for dimensional reduction is the low rank approximation by using singular value decomposition (SVD). An appealing property of this type of low rank approximation via SVD is that it can achieve the smallest reconstruction error among all approximations with the constrain of same rank in Euclidean distance. However, applications of this technique to high dimensional data, such as images and videos, will quickly meet some critical problems like algorithm complexity, high computational cost, etc. Recently, Yang et al. [18] proposed an algorithm called twodimensional principal component analysis (2DPCA) for face recognition, in which the image covariance (scatter) matrix is directly computed from the image matrix representations. Associated with
Corresponding author.
E-mail addresses: [email protected] (C. Lu), [email protected] (W. Liu), [email protected] (S. An). 0925-2312/$ - see front matter & 2008 Published by Elsevier B.V. doi:10.1016/j.neucom.2007.11.046
this problem, a new type of low rank approximation is proposed recently by Ye et al. [19] based on a matrix space model instead of the vector space model. In addition, Lu et al. [10] presented a necessary condition for the optimality of the GLRAM and Xu et al. [16] proposed an algorithm called coupled subspaces analysis (CSA) based on GLRAM. With those techniques, each datum is treated as a matrix instead of a vector and the whole data collection will be a set of data matrices. In this way, the dimension curse in vector space model disappears. After publications [16,18,19], many applications have been investigated by using this matrix model successfully [15,18]. In this paper, we will study three different methods, the 2DPCA [18], the generalized low rank approximations of matrices (GLRAM) [19] and CSA [16] and then investigate their advantages and shortcomings in face recognition in training and testing stages. The advantage of 2DPCA is that it has unique optimal closed-form solution in training stage. Its main foul is that only one side of image is projected and it will use more time in testing stage, specially for images with large dimensions. The GLRAM and CSA can projected the testing images on both sides in result of efficient testing, however, there is no explicit closed formula for the projection matrices as pointed out in [10] and consequently the complexity is very high in training stage. Based on these observations, we then propose a new algorithm to deriving the projection matrices L and R for the GLRAM directly and call this new algorithm as SGLRAM. Though these obtained projection matrices are not optimal theoretically, they can achieve satisfactory performance in face recognition with the same high performance as 2DPCA and the GLRAM. Then the complexity analysis in training and testing stages are given for the SGLRAM. Finally, we implement all these algorithms for face recognition on the ORL, AMP, Yale and YaleB face databases and compared their performance in recognition accuracy and computational cost.
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The rest of this paper is organized as follows. We give a brief overview of the 2DPCA [18], GLRAM [19] and CSA based on the matrix space model. The difference between CSA and GLRAM is that CSA subtracts the mean in objective criterion function and the original GLRAM does not. The experiment shows that the accuracy of face recognition using CSA is a little higher than that of GLRAM. A new algorithm is proposed in Section 3 via investigating 2DPCA, GLRAM and CSA. We then will analyze the computational cost and storage request on the three algorithms, respectively. In Section 4, extensive experiments are conducted on image database ORL, AMP, Yale and YaleB. Conclusions and directions for future work are summarized in Section 5.
2. 2DPCA and GLRAM In this section we will briefly outline the 2DPCA, the GLRAM and the CSA in order to present the contribution of this paper clearly. 2.1. 2DPCA Yang et al. [18] proposed the 2DPCA algorithm for face recognition and we briefly describe this technique here in order to clearly present the proposed algorithm later. Given input image matrices Ai 2 Rmn , i ¼ 1 . . . N, where mn is the number of pixels in the image, and N is the number of the images. The following criterion is adopted for 2DPCA in [18]: JðXÞ ¼ trðSX Þ ¼ trfX > ½EðA EAÞ> ðA EAÞXg
(1)
where SX is the covariance matrix of Ai ði ¼ 1; 2; . . . ; MÞ with the projection matrix X and E is the expectation of a stochastic variable. In fact, the covariance matrix G 2 Rmm with finite N images is given by G¼
N 1X ðA AN ÞðAj AN Þ> N j¼1 j
(2)
P where AN ¼ ð1=NÞ N i¼1 Ai is the mean image matrix of the N training samples. Alternatively the criterion in (1) can be expressed by the following: JðXÞ ¼ trðX > GXÞ
(3)
where X is a unitary column vector. The matrix X that maximizes the criterion is called the optimal projection axis. The optimal projection X opt is a set of unitary vectors that maximize JðXÞ, i.e., the eigenvectors of G corresponding to the large eigenvalues [18]. We usually choose a subset of only d eigenvectors corresponding to largest d eigenvalues to be included in the model, i.e., fX 1 . . . X d g ¼ arg max JðXÞ satisfying X > i X j ¼ 0(iaj; i; j ¼ 1; 2; . . . ; d). With the above mentioned solution, each image can thus be optimally approximated in the least-squares sense up to a predefined reconstruction error. Every input image Ai will project into a point in the d-dimensional subspace spanned by the selected eigenmatrix X opt [18]. In face recognition, an optimal projection matrix X opt will be derived with a given dimension d and training samples. This optimal projection matrix will be used for face recognition in testing stage using the nearest neighbor approach as described in [18]. It should be noted that the testing image is only reduced in one side (column or row) in the testing stage. 2.2. GLRAM The traditional low rank approximation of matrices was formulated as follows: given A 2 RWn , find a matrix B 2 RWn
213
with rank(B)¼ k, such that B ¼ arg
min kA BkF
rankðBÞ¼k
(4)
where the Frobenius norm kMkF of a matrix M ¼ ðM ij Þ is defined as sffiffiffiffiffiffiffiffiffiffiffiffiffiffi X 2ffi kMkF ¼ M ij ij
This problem has a closed-form solution via SVD. However, the computational cost is high when W is large as stated in [19]. In order to overcome the computational complexity in the above rank approximation problem, Ye [19] proposed a matrix space model for low rank approximation of matrices called the GLRAM, which can be stated as follows. Let Ai 2 Rmn , for i ¼ 1; 2; . . . ; N; be the N data points in the training set. The aim is to compute two matrices L 2 Rmr and R 2 Rnc with orthogonal columns and N matrices M i 2 Rrc for i ¼ 1; 2; . . . ; N such that LM i R> approximates Ai well for all i. Mathematically, this can be formulated as an optimization problem min >
N X
L> L¼Ir ;R R¼Ic i¼1
kAi LM i R> k2F
(5)
Until now, no closed-form solution has been found for the proposed GLRAM. In order to find an optimal solution for this GLARM, Ye in [19] has proved that the proposed GLRAM is equivalent to the following optimization problem. max JðL; RÞ ¼ L;R
N X
kL> Ai Rk2F
i¼1
¼
N X
Tr½ L> Ai RR> A> i L
(6)
i¼1
where L and R satisfy LL> ¼ Ir ; RR> ¼ Ic : Actually, this is a nonlinear optimization problem which is difficult to solve as pointed out in [10]. In order to simplify the problem, Ye [19] observed the following fact in Theorem 1 and proposed an Algorithm GLRAM stated below instead of solving the above optimization problem directly. Theorem 1. Let L, R and fM i gN i¼1 be the optimal solution to the minimization problem in GLRAM. Then (1) For a given R, L consists of the r eigenvectors of matrix M L ðRÞ ¼
N X
Ai RR> A> i
(7)
i¼1
corresponding to the largest r eigenvalues. (2) For a given L, R consists of the c eigenvectors of the matrix M R ðLÞ ¼
N X
> A> i LL Ai
(8)
i¼1
corresponding to the largest c eigenvalues. Based on the above observations, the following algorithm for solving the GLRAM is proposed in [19], which is called the Algorithm GLRAM. Algorithm GLRAM [19] Input: matrices fAi g, r and c. Output: matrices L, R and fM i g 1. Obtain initial L0 for L and set i ¼ 1. 2. While not convergent
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3. From the matrix M R ðLÞ ¼ eigenvectors
PN
> > j¼1 Aj Li1 Li1 Aj
compute the c
R ffj gcj¼1
of M R corresponding to the largest c
R
R
eigenvalues. 4. Let Ri ¼ ½f1 ; . . . ; fc
PN
> > j¼1 Aj Ri Ri Aj L r eigenvectors ffj gj¼1 corresponding to the L L 6. Let Li ¼ ½f1 ; ; fr
5. From the matrix M L ðRÞ ¼
compute the r largest r eigenvalues.
7. i ¼ i þ 1 8. Endwhile 9. L ¼ Li1 10. R ¼ Ri1 11. Mj ¼ L> Aj R It should be noted here that the convergence of this algorithm cannot be guaranteed theoretically in general [10] and therefore the implementation from steps 2 to 8 may never end. In order to investigate the convergence issue of the proposed algorithm, the following criteria are defined in [19]: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u1 X RMSRE ¼ t kA LM i R> k2F N i¼1 i where RMSRE stands for the root mean square reconstruction error. It is easy to show that each iteration in the Algorithm GLRAM will decrease the RMSRE value and also it is obvious that the sequences of RMSRE is bounded. Mathematically, this will guarantee that the sequences of RMSRE converge. This iterative algorithm GLRAM is proposed to obtain the optimal L, R in [19] and experiments showed the proposed algorithm is very time costive in general.
remove the difficulty for finding the optimal solutions in the GLRAM and can use its advantage on reduced dimensions on both sides for testing. Of course, we expect the performance in face recognition with the approximate solution will not decline significantly compared to the performance with the optimal projection matrices. Such expectation is based on the fact that each side solution is optimal with explicit closed-form solution. Next, we will describe this new algorithm in detail. In order to utilize the idea in the CSA, the samples Ai (i ¼ 1; 2; . . . ; NÞ are assumed to subtract the mean A as M i . In this case, Eqs. (7) and (8) can be rewritten as ML ðRÞ ¼
N X
M i RR> M > i
(9)
i¼1
MR ðLÞ ¼
N X
> M> i LL M i
(10)
i¼1
We intend to obtain the one side optimal solution for the GLRAM. For such purpose, we assumed that R and L are the identity matrices first. With this assumption, we can obtain P PN > > ML ðRÞ ¼ N i¼1 M i M i and M R ðLÞ ¼ i¼1 M i M i . Further, treat M L ðRÞ and MR ðLÞ as covariance matrices in the 2DPCA. Then two couples of eigenmatrices U and V can be obtained by SVD corresponding to the ML ðRÞ and M R ðLÞ, respectively. Now, we have the following Algorithm SGLRAM: Algorithm SGLRAM Input: matrices fAi g, r and c, i ¼ 1; 2; . . . ; N: Output: matrices L, R and fQ i g P 1. Calculate M i ¼ Ai 1=N N i¼1 Ai . 2. Calculate M L ðRÞ and M R ðLÞ as following: N X ML ðRÞ ¼ Mi M> i i¼1
2.3. Coupled subspace analysis Xu et al. [16] proposed the CSA algorithm which is actually based on the GLRAM. The only difference is that in CSA Ai is P replaced by Ai ð1=NÞ N i¼1 Ai . Our preliminary experiments show that CSA performs slightly better than the GLRAM in face recognition due to the average value difference.
3. The proposed algorithm: SGLRAM We have presented the algorithms of the 2DPCA, the GLRAM and the CSA, and found that the GLRAM and CSA are actually the same with double sides in matrix optimization and the 2DPCA is one side matrix optimization. Consequently, the advantage of the 2DPCA is that it has a closed-form solution, which results in less expensive training time, while the advantage of the GLRAM is that the testing image will have lower dimension on both sides of the projected image and lead to less testing time. However, the GLRAM has no explicit closed-form solution, which needs an iterative algorithm to derive its optimal solution and this is usually difficult or costive due to two reasons: one is the convergence issue has not been solved theoretically [10] and even the GLRAM converges, the other problem is that its convergence speed is unknown. Similarly, in testing stage, the 2DPCA will have a high dimension in one side and will use a long time for testing. Based on these observations, we propose a new algorithm called the SGLRAM. The idea is very explicit. We will use the idea in 2DPCA algorithm to obtain one side optimization, respectively, for both left and right sides and then we use these two direct solutions as approximating solutions for the GLRAM. This will
MR ðLÞ ¼
N X
M> i Mi
i¼1
3. Calculate the left eigenmatrix U for M L ðRÞ and the right eigenmatrix V for M R ðLÞ. These two matrices can be obtained directly by doing SVD on the M L ðRÞ and M R ðLÞ, respectively. 4. Using the eigenmatrices U ¼ L and V ¼ R to obtain Q i ¼ LAi R. From the above algorithm we can use eigenmatrices U, V obtained directly in the Algorithm SGLRAM instead of L and R which were derived iteratively in the GLRAM (CSA). We expect the corresponding proposed face recognition algorithm will produce as same good performance as the GLRAM and the 2DPCA. Obviously, the Algorithm SGLRAM will save time in training compared to the GLRAM and save time in testing compared to the 2DPCA as demonstrated in theoretical analysis in next subsection and experimental results in next section. 3.1. Analysis of computational complexity In this subsection we will analyze the algorithm complexity for the 2DPCA, the GLRAM (CSA) and the SGLRAM. Here we assume that the image size of Ai is m n with nXm. The number of training images is N and the reduced dimension for both sides of GLRAM is d for simplicity. Also we assume that the iteration number of convergence in the GLRAM is r. In training stage, the complexity of the 2DPCA can be calculated easily to be Oðn3 þ NÞ due to the fact that the complexity of one SVD is of Oðn3 Þ [7]. Further the complexity of the SGLRAM is Oðn3 þ m3 þ NÞ and the complexity of the GLRAM (CSA) is of Oðrn3 þ rm3 þ NÞ. One can see
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0.8
YaleB
d ¼ 10
d ¼ 15
d ¼ 20
d ¼ 25
2DPCA SGLRAM
159.069 119.531
172.128 112.402
206.497 134.219
240.737 136.972
ORL
d¼5
d ¼ 10
d ¼ 15
d ¼ 20
d ¼ 25
2DPCA SGLRAM
7.871 5.198
25.447 11.757
34.65 16.434
47.66 22.462
18.467 8.773
Table 2 The RMSRE value ðE þ 003Þ
Recognition accuracy
Table 1 Computational cost (seconds)
0.75
0.7 CSA SGLRAM 2DPCA GLRAM 0.65 0
GLRAM SGLRAM
215
d ¼ 10
d ¼ 15
d ¼ 20
d ¼ 25
d ¼ 30
14.2 14.311
11.963 12.066
10.48 10.57
9.366 9.4697
8.4769 8.5273
that the computational cost is much higher for the GLRAM when r is large. In the testing stage, when we project the training images into a new space the required storage of the 2DPCA is Nnd and the other 2 two is Nd . Due to the fact d5n in usual, the computational cost by the 2DPCA is much higher than the SGLRAM. In order to illustrate this, the experiments on the two image databases of YaleB and ORL are done and the results are shown in Table 1. The experiment includes projecting 320 training images into the feature spaces and recognizing 560 testing images in the YaleB database, and projecting 160 training images into the feature spaces and recognizing 240 testing images in the ORL database, respectively. We have done all the experiments in this paper on a stand alone PC machine (CPU:PIV 1.3 GHz, RAM:256M) with Matlab. One can see that it is usually much faster and more efficient to use the SGLRAM algorithm in the recognition process when the dimension d is large. The recognition performance in face recognition will be demonstrated in the next section. Here we will evaluate the mathematical comparisons with the proposed criteria in [19]. Recall that vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N X 1u RMSRE ¼ t kA LM i R> k2F N i¼1 i is used in [19] and here we compute this value using the optimal solutions obtained via the SGLRAM and the GLRAM by applying the images in YaleB database as variables. In this experiment, N ¼ 320 is the number of training images, Mi 2 Rrc for i ¼ 1; 2; . . . ; N are obtained matrices through these two algorithms such that LM i R> can approximate Ai well for all i. The experimental results are shown in Table 2. The results in Table 2 implied that the SGLRAM algorithm can produce nearly as same good function values as the GLRAM algorithm and this indicates that the SGLRAM can produce much satisfactory sub-optimal solutions for the GLRAM on the YaleB database. We expect the face recognition performance has the same results, which will be demonstrated in next section.
4. Experimental evaluations In this section we will carry out several experiments to demonstrate the performance of the SGLRAM algorithm. For the chosen databases, we first use two types of input images. One is
5
10 15 Number of feature d
20
25
Fig. 1. Face recognition performance on the ORL database.
the ORL face database, which contains 400 images of 40 individuals (each person has 10 different images) under various facial expressions, wearing or not wearing glasses, different lighting conditions with each images being cropped and resized to 112 92 pixels in this experiment. The other database is the YaleB database [6], which contains 650 face images of 10 people, including frontal views of face with different facial expressions, lighting conditions. With these face databases, we will conduct two group experiments using the 2DPCA, CSA and the SGLRAM, respectively. The first one is on the ORL image face database, where the first four images of each person are used as training and the other six images are used as testing. The second experiment is on the YaleB image face database, where the first eight images of each person are used for training and the other 56 images are used for testing. The results are shown in Figs. 1 and 2, respectively. From Fig. 1 we can see that the performance of the SGLRAM is between the 2DPCA and the GLRAM before they achieve the best performance in the ORL database, i.e., with lower dimension d, the performance for the 2DPCA is better than that for the SGLRAM and GLRAM due to the fact that it includes more effective information with lower dimension d since only the dimension in one side of the image matrix is reduced. In this case effective information kept in the other side of the image matrix plays an important role. With the increasing dimension d, the performance of the SGLRAM and the GLRAM are getting better since the effective information is being enriched. Finally, when d is large enough, all their performances are nearly the same. In Fig. 2 we can see the performances are closely the same when dimension increases up to 30 in the YaleB database. The large gap with d ¼ 40 may be due to the fact that the face images in the YaleB database have different backgrounds. From these experiments, we can see that the proposed SGLRAM algorithm can save much more time in training stage compared to the GLRAM algorithm and run much faster in testing stage compared to the 2DPCA. Also their performances are nearly all the same when the reduced dimension is large enough. In order to show the performance on other databases, we also did extensive experiments on the AMP and the Yale databases. The AMP database contains 975 images with 13 individuals (each person has 75 different images) under various facial expressions and lighting conditions with 64 64 pixels. We use the first five images of each person as training and the others are used for testing. The performance is shown in Fig. 3. On the other hand, the Yale database has 15 individuals with each having 11 images under various facial expressions and
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0.72 0.85 Recognition accuracy
Recognition accuracy
0.7 0.68 0.66 0.64
CSA SGLRAM 2DPCA GLRAM
0.62
0.8 0.75 CSA SGLRAM 2DPCA GLRAM
0.7 0.65 0.6
0.6 0
5
10
15
20
25
30
35
2
40
2.5
3
Number of feature d
5
5.5
6
Fig. 4. Face recognition performance on the Yale database.
Fig. 2. Face recognition performance on the YaleB database.
Table 3 The RMSRE values for AMP
1.05
Recognition accuracy
3.5 4 4.5 Number of feature d
GLRAM SGLRAM
1
CSA SGLRAM 2DPCA GLRAM
GLRAM SGLRAM
0.9 2.5
3
3.5
4
4.5
5
5.5
d¼3
d¼4
d¼5
d¼6
3.7295 3.7533
3.5184 3.5167
3.2822 3.2851
3.0698 3.0909
2.9278 2.9484
Table 4 The RMSRE value for the Yale dataset
0.95
2
d¼2
d¼2
d¼3
d¼4
d¼5
d¼6
9.0254 9.0483
8.2 8.3169
7.6847 7.697
7.2255 7.2879
6.9225 6.9561
6
Number of feature d Fig. 3. Face recognition performance on the AMP database.
lighting conditions with 231 195 pixels. The first five images for each person are used for training and the others are used for testing. The performance is shown in Fig. 4. Correspondingly, the RMSRE values for the GLRAM and the SGLRAM for the AMP database are listed in Table 3 and the same result for Yale database is listed in Table 4. Next, we compare the computational time in testing stage with the 2DPCA on these two databases and they are listed in Table 5. Based on these experiments, we can see that the performances for these four algorithms are nearly the same when the reduced dimension d is chosen properly. In general, when d is smaller, the 2DPCA performs better than all other three approaches since it only used one side dimension reduction and contains more effective information on the other side. With d increasing, the performances of all these algorithms are nearly the same as evidenced from Figs. 1 to 4. However, the computational cost in training stage is large for the GLRAM and CSA algorithms and the computing cost is large in testing stage for the 2DPCA as evidenced from the listed tables. In this case, the proposed SGLRAM is the best choice since it can reduce the computational complexity while keeping the same high performance. However, we observed when the dimension d is large enough, the performances of all the four algorithms on the ORL database are deteriorating due to the fact that the extra information brought by the large d is actually becoming noise. The result on the YaleB
Table 5 Computational cost (seconds) Yale
d¼2
d¼3
d¼4
d¼6
2DPCA SGLRAM
2.444 2.003
2.764 2.063
3.435 2.233
3.906 2.534
AMP
d¼2
d¼3
d¼4
d¼6
2DPCA SGLRAM
6.419 5.658
7.741 6.319
8.993 6.92
11.557 8.282
database is interesting with large d, the GLRAM algorithm performs worse than the proposed algorithm SGLRAM, which deserves further investigation in the future.
5. Conclusions In this paper we investigated the 2DPCA, GLRAM and CSA and then derived a new face recognition algorithm called the SGLRAM. We have demonstrated that this new algorithm inherited the advantages of the 2DPCA and the GLRAM while can avoid their corresponding shortcomings in high computational costs, respectively. Experiments have shown that the SGLRAM algorithm needs much less time in training stage compared to the GLRAM or CSA
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and much more efficient in testing compared to the 2DPCA and can still produce high performance as the 2DPCA and the GLRAM. The results in this paper showed that the sub-optimal solution obtained via the SGLRAM can approximate the optimal solution satisfactorily. In the future, we will investigate the incremental learning algorithms for the SGLRAM along the line of incremental learning algorithm for the 2DPCA in [11] and study its performance. The incremental investigation is hard or impossible for the GLRAM due to the complexity in iteration steps in training while it will be feasible for the SGLRAM based on the simplified procedure. One more possible research is to simplify the procedures in the coupled kernel-based subspace learning [17]. Actually, many existing kernel approaches based on subspace analysis for face recognition [9,3,14] can be explored further. Furthermore, the appropriate dimension d for the GLRAM, the SGLRAM and the 2DPCA needs to be investigated for different databases in the future. This problem is very important for the design of automated face recognition systems in selecting optimal model parameters as demonstrated in eigenface approach [12].
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[15] L. Wang, X. Wang, X. Zhang, J. Feng, Equivalence of two dimensional pca to line-based pca, Pattern Recognition Lett. (2005) 57–60. [16] D. Xu, S. Yan, L. Zhang, Z. Liu, H.J. Zhang, Coupled subspaces analysis, in: Proceedings of CVPR’2005, 2005, pp. 204–208. [17] S. Yan, D. Xu, L. Zhang, B. Zhang, H. Zhang, Coupled kernel based subspace learning, in: Proceedings of Computer Vision and Pattern Recognition (CVPR), 2005, pp. 645–650. [18] J. Yang, D. Zhang, A. Frangi, J. Yang, Two-dimensional pca: a new approach to appearance-based face representation and recognition, IEEE Trans. Pattern Anal. Mach. Intell. 5 (1) (2004) 131–137. [19] J. Ye, Generalized low rank approximations of matrices, in: Proceedings of the Twenty-first International Conference on Machine Learning, 2004, pp. 887–894. [20] W. Zhao, R. Chellappa, P. Phillips, Face recognition: a literature survey, ACM Comput. Surv. 35 (2003) 399–458. Chong Lu received the BS degree in Applied Mathematics from YiLi Normal College in 1990 and obtained the certificate of MS in Computer Application in 2002. He worked as Visiting Scholar in Curtin University of Technology from November 2005 to November 2006. His research interests include image processing, computational learning theory and pattern recognition. He became Associate Professor at YiLi Normal College in 2002.
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Wanquan Liu received the BSc degree in Applied Mathematics from Qufu Normal University, PR China, in 1985, the MSc degree in Control Theory and Operation Research from Chinese Academy of Science in 1988, and the PhD degree in Electrical Engineering from Shanghai Jiaotong University, in 1993. He once hold the ARC Fellowship and JSPS Fellowship and attracted research funds from different resources. He is currently a Senior Lecturer in the Department of Computing at Curtin University of Technology. His research interests include large-scale pattern recognition, control systems, signal processing, machine learning, and intelligent systems. Senjian An received the BS degree from Shandong University, Jinan, China, in 1989, the MS degree from the Chinese Academy of Sciences, Beijing, in 1992, and the PhD degree from Peking University, Beijing, in 1996. He was with the Institute of Systems Science, Chinese Academy of Sciences, Beijing, where he was a Postdoctoral Research Fellow from 1996 to 1998. In 1998, he joined the Beijing Institute of Technology, Beijing, and he was an Associate Professor from 1998 to 2001. From 2001 to 2004, he was a Research Fellow with The University of Melbourne, Parkville, Australia. He joined Curtin University of Technology, Perth, WA, Australia, in 2004, where he is currently a Postdoctoral Research Fellow. His research interests include signal processing, machine learning and computer vision.