2011 18th IEEE International Conference on Image Processing

Face Recognition Based On Local Uncorrelated And Weighted Global Uncorrelated Discriminant Transforms Xiaoyuan Jing1,2,3, Sheng Li2, David Zhang4, Jingyu Yang5 State Key Laboratory for Software Engineering, Wuhan University, Wuhan, 430079, China 2 College of Automation, Nanjing University of Posts and Telecommunications, 210046, China 3 State Key Laboratory for Novel Software Technology, Nanjing University, Nanjing, 210093, China 4 Department of Computing, Hong Kong Polytechnic University, Kowloon, Hong Kong 5 College of Computer Science, Nanjing University of Science and Technology, 210094, China 1

ABSTRACT Feature extraction is one of the most important problems in image recognition tasks. In many applications such as face recognition, it is desirable to eliminate the redundancy among the extracted discriminant features. In this paper, we propose two novel feature extraction approaches named local uncorrelated discriminant transform (LUDT) and weighted global uncorrelated discriminant transform (WGUDT) for face recognition, respectively. LUDT and WGUDT separately construct the local uncorrelated constraints and the weighted global uncorrelated constraints. Then they iteratively calculate the optimal discriminant vectors that maximize the Fisher criterion under the corresponding statistical uncorrelated constraints, respectively. The proposed LUDT and WGUDT approaches are evaluated on the public AR and FERET face databases. Experimental results demonstrate that the proposed approaches outperform several representative feature extraction methods. Index Terms—Feature extraction, uncorrelated constraints, local uncorrelated discriminant transform, weighted global uncorrelated discriminant transform, face recognition. 1. INTRODUCTION Feature extraction is an active research area in the field of image processing and pattern recognition. Linear discrimination analysis (LDA) is a widely-used supervised feature extraction method [1], and locality preserving projections (LPP) is a well-known unsupervised feature extraction method [2]. LDA extracts the discriminant features to maximize the between-class scatter and minimize the within-class scatter simultaneously. In recent years, many feature extraction methods have been developed to enhance the classification performance of LDA, such as improved LDA [3], LDA/QR [4], etc. In many real-world applications such as face recognition, it is usually desirable to eliminate the redundancy among the extracted discriminant features. To realize this aim, Foley and Sammon presented the optimal discriminant vectors method (FSODV) [5] that makes each discriminant vector satisfy the orthogonal constraints. Uncorrelated optimal discriminant vectors (UODV) method [6-7] is proved to be more powerful than FSODV. It makes each discriminant vector satisfy both the Fisher criterion and

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the statistical uncorrelated constraints. Liang et al. combined uncorrelated discriminant vectors with the weighted pair-wise Fisher criterion [8]. Ye et al. presented the UODV/GSVD method to calculate the uncorrelated discriminant vectors [9]. Zhang discussed the uncorrelated trace ratio for undersampled problems [10]. Yang and Zhang presented a Gabor Feature based Sparse Representation for Face Recognition [16]. 1.1. Motivation and Contribution In uncorrelated discriminant methods, the total scatter matrix St = E ( xi − Exi )( xi − Exi )T is the key component in

the uncorrelated constraints ϕiT Stϕ j = 0, j = 1,

,i − 1 (i.e.,

the St -orthogonal constraints) [11-12], where xi is a training sample, ϕ j is an optimal discriminant vector. Current uncorrelated methods employ the mean sample m of sample set to estimate the expectation Exi for all samples

such that St = ∑ ( xi − m)( xi − m)T . However, in this paper,

we try to construct St by redefining the expectation Exi for each xi , and construct the reformative uncorrelated constraints. To the best of our knowledge, this paper is the first attempt to change the constraints of uncorrelated discriminant methods. In this paper, we propose two novel feature extraction approaches, which are local uncorrelated discriminant transform (LUDT) and weighted global uncorrelated discriminant transform (WGUDT), to extract uncorrelated discriminant features for face recognition, respectively. LUDT and WGUDT separately construct the local uncorrelated constraints and the weighted global uncorrelated constraints by redefining the expectation Exi in St . Then, LUDT iteratively calculates the optimal discriminant vectors, which maximize the Fisher criterion under the local uncorrelated constraints. Similarly, WGUDT iteratively calculates the optimal discriminant vectors under the weighted global uncorrelated constraints. Experimental results on the AR and FERET face databases demonstrate that the proposed approaches obtain better classification performance than several representative feature extraction methods. 1.2 Organization

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The remainder of this paper is organized as follows: in Section 2, we outline some related work. In section 3, we describe the proposed LUDT and WGUDT approaches, respectively. Experimental results on the public AR and FERET face databases are reported in Section 4 before drawing conclusions in Section 5. 2. RELATED WORK 2.1 Linear Discriminant Analysis (LDA) Assume that the original sample set X = { x1 , x2 ,

, xN } is

the i th class. Linear discriminant analysis (LDA) tries to find a set of optimal projections W = [ w1 , w2 , , wm ] , such that the following Fisher criterion is maximized, i.e., W T SbW , J (W ) = arg max W W T StW

(1)

where Sb and St are the between-class scatter matrix and the total scatter matrix, respectively [1]. W is constructed by the eigenvectors of ( St ) −1 Sb . 2.2 Uncorrelated Optimal Discriminant Vectors (UODV) UODV achieves a group of optimal discriminant vectors which can satisfy both the Fisher criterion and the following statistical uncorrelated constraints: ϕiT Stϕ j = 0, j = 1, 2, , i − 1 , (2)

where ϕ j is the optimal discriminant vector. According to the UODV algorithm, the first optimal discriminant vector ϕ1 is obtained by maximizing Eq. (1). Then, UODV gives the following lemma [7]: Lemma 1. The i th optimal discriminant vector ϕi ( i ≥ 2 ) is the eigenvector corresponding to the maximal eigen-value of the equation: PSbϕi = λ Stϕi , (3) I = diag (1,1,

, ϕi −1 ]

T

and

,1) .

3. LOCAL UNCORRELATED AND WEIGHTED GLOBAL UNCORRELATED DISCRIMINANT TRANSFORMS 3.1 Covariance Analysis of Discriminant Features Suppose Ω is the sample space. For any sample x ∈ Ω , yi = ϕiT x and y j = ϕ Tj x separately denote the extracted

features after projecting x onto the discriminant vectors ϕi and ϕ j . The covariance between yi and y j is [13]: Cov( yi , y j ) = E ( yi − Eyi )(γ j − Ey j ) =ϕ

T i

1 N ∑ j =1 x j N estimate the expectation Ex for all samples such that

methods employ the mean sample m =

{ {E ( x − m)( x − m ) }ϕ

Cov( yi , y j ) = ϕiT E ( x − Ex)( x − Ex )T }ϕ j =ϕ

composed of c classes, and there are ni training samples in

where P = I − St DT ( DSt DT ) −1 D , D = [ϕ1 ,

i.e., Cov( yi , y j ) = 0 . However, present uncorrelated

T i

= ϕiT Stϕ j = 0

T

j

{E ( x − Ex)( x − Ex ) }ϕ

.

methods, i.e., ϕiT St ϕ j = 0 , where St = 1

N

∑

Uncorrelated discriminant methods require that the extracted discriminant features are mutually uncorrelated,

N i =1

( xi − m)( xi − m)T

is the original total scatter matrix. In this paper, we try to reconstruct the total scatter matrix by redefining the expectation Ex for each x , and construct the reformative uncorrelated constraints. Then we put forward two novel uncorrelated discriminant approaches, that is, local uncorrelated discriminant transform (LUDT) and weighted global discriminant transform (WGUDT). 3.2 Local Uncorrelated Discriminant Transform (LUDT) The LUDT approach constructs the total scatter matrix StL by reforming the expectations of xi , i.e., StL =

1 N

N

∑ (x i =1

i

− mˆ i )( xi − mˆ i )T ,

(6)

where ⎧⎪1, if x j is the k nearest neighbor of xi 1 N mˆ i = ∑αij x j , αij = ⎨ . N j =1 ⎪⎩0, otherwise In Eq. (6), we estimate the expectation of xi by using the mean of its k nearest neighbors. The local uncorrelated constraints can be expressed as: ϕiT StLϕ j = 0, j = 1, , i − 1 . (7) The first discriminant vector ϕ1 of LUDT is same as that of LDA. Then, LUDT calculates other uncorrelated discriminant vectors using the following theorem. Theorem 1. The i th uncorrelated discriminant vector ϕi of LUDT is the eigenvector corresponding to the maximum eigenvalue of the following equation: PSbϕi = λ Stϕi , (8) where P = I − StL DT (DStL St−1StL DT )−1 DStL St−1 , D = [ϕ1 , ϕ2 , , ϕ j ]T and I = diag (1,1, ,1) . Proof. Use the Lagrange multiplier to express the Fisher criterion under the weighted global uncorrelated constraints in Eq. (7). We have L(ϕi ) = ϕiT Sbϕi − λ (ϕiT St ϕi − c) − ∑ μ k ϕiT StLϕ k ,

(4)

j

. (5)

Eq. (5) implies the origin of the so-called St orthogonal constraints employed in current uncorrelated

i −1

T

to

(9)

k =1

where λ and μ k are Lagrange multipliers, and c is a constant.

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zero:

∂ ( L(α i ) ) ∂ (α i )

i −1

= 2Sbϕi − 2λ Stϕi − ∑ μ k StLϕ k = 0 .

(10)

k =1

Left multiplying Eq. (10) by ϕk StL St−1 (k = 1,… , i − 1) allows us to obtain: i −1

2ϕk StL St−1 Sbϕi − ∑ μk ϕk StL St−1 StLϕk = 0, (k = 1,…, i − 1) . (11) k =1

Let U = [ μ1 ,

, μk ]T , D = [ϕ1 ,

, ϕ k ] . Eq. (11) can be T

represented in the form of matrix DStL St−1 StL DU = 2 DStL St−1 Sbϕi . (12) Hence, we obtain U = 2( DStL St−1 StL D) −1 DStL St−1 Sbϕi (13) Then, Eq. (10) can be rewritten in the following form: 2Sbϕi − 2λ Stϕi − StL DT U = 0 , (14) Substituting Eq. (14) into Eq. (13), we have: Sbϕi − λ Stϕi − StL DT ( DStL St−1 StL D) −1 DStL St−1 Sbϕi = 0 , (15) i.e., ( I − StL D ( DStL S S D) DStL S ) Sbϕi = λ Stϕi . −1 t tL

T

−1

−1 t

1 N

N

∑ ( xi − mi )( xi − mi )T ,

(16)

i =1

|| x − x ||2 1 α ij x j , α ij = exp(− j 2 i ) . ∑ 2σ N j =1 We construct the weighted global uncorrelated constraints as: ϕiT StGϕ j = 0, j = 1, , i − 1 . (17)

where mi =

4.1 Experiments Using the AR Face Database The AR face database [14] includes 119 individuals with each one contributing 26 face images. Thus, there are overall 3094(=119×26) samples. Each face image was scaled to 60×60 with 256 gray levels. Fig. 1 shows all of the samples of one individual. The major differences between them are the facial expression, illumination, position, pose and sampling time. We randomly select 6 samples of each person for training and the remainders for testing.

Fig. 1. Demo images of one subject on the AR database. UODV

N

The first discriminant vector ϕ1 of WGUDT is calculated by maximizing Eq. (1). Then, WGUDT calculates the i th uncorrelated discriminant vector ϕi using the following theorem. Theorem 2. The i th uncorrelated discriminant vector ϕi of WGUDT is the eigenvector corresponding to the maximum eigenvalue of the following equation: PSbϕi = λ Stϕi , (18) where P = I − StG DT (DStG St−1StG DT )−1 DStG St−1 , D = [ϕ1, ,ϕi −1 ]

T

and I = diag (1,1, ,1) . The proof is similar to that of Theorem 1. 4. EXPERIMENTS In this section, we compare the classification performance of the proposed LUDT and WGUDT approaches with UODV [7], UODV/GSVD [9] and LPP [2] on the AR and

UODV/GSVD

LPP

LUDT

WGUDT

95

□

3.3 Weighted Global Uncorrelated Discriminant Transform (WGUDT) The WGUDT approach first calculates the weighted mean sample mi for each sample xi , and then compute the total scatter matrix StG as follows: StG =

FERET face databases. For all compared methods, we use the nearest neighbor classifier with the cosine distance to do classification.

Recognition Rates (%)

Let the partial derivatives ∂ ( L(α i ) ) ∂ (α i ) be equal to

90

85

80

75

70 1

2

4

6

8

10

12

Random Testing No.

14

16

18

20

Fig. 2. Recognition rates of LUDT, WGUDT and other methods on the AR face database.

The number of nearest neighbors k in LUDT is set as 30. Fig. 2 shows the recognition rates of 20 random tests of our approaches and other compared methods. The proposed LUDT and WGUDT approaches obviously outperform other compared methods in almost all cases. Table 1. Average recognition rates of compared methods on the AR face database.

Methods UODV UODV/GSVD LPP LUDT WGUDT

Average recognition rates (%) 80.87 80.03 79.73 84.51 84.07

Table 1 shows the average recognition rates of all compared methods on the AR face database. Compared with UODV, UODV/GSVD and LPP, LUDT improves the average recognition rate at least by 3.64% (=84.51%80.87%), and WGUDT boosts the average recognition rate at least by 3.20% (=84.07%-80.87%), respectively. 4.2 Experiments Using the FERET database The FERET face database [15] employed in the experiments includes 2,200 face images of 200 individuals with each one contributing 11 images. The images in this database are captured under various illuminations and display a variety of facial expressions and poses. We crop the images and normalize them with a resolution of 60 × 50. Fig. 3 shows

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2011 18th IEEE International Conference on Image Processing

all samples of one subject. We randomly select 4 samples of each person for training and the remainders for testing. Fig. 3. Demo images of one subject on the FERET database.

The parameter k in LUDT is set as 20. Fig. 4 shows the recognition rates across 20 runs of LPP, UODV, UODV/GSVD and our approaches, respectively. It also demonstrates that LUDT and WGUDT achieve better recognition performance than other compared methods. UODV

UODV/GSVD

LPP

LUDT

WGUDT

Recognition Rates (%)

80

75

70

65

60

55 1

2

4

6

8

10

12

Random Testing No.

14

16

18

20

Fig. 4. Recognition rates of LUDT, WGUDT and other methods on the FERET face database.

Table 2 shows the average recognition rates of all compared methods on the FERET face database. Compared with LPP, UODV and UODV/GSVD, LUDT boosts the average recognition rate at least by 3.31% (=64.98%61.67%). And the average recognition rate of WGUDT is at least 3.57% (=65.24%-61.67%) higher than other compared methods. Table 2. Average recognition rates of all compared methods on the FERET face database.

Methods UODV UODV/GSVD LPP LUDT WGUDT

Average recognition rates (%) 61.26 61.67 60.96 64.98 65.24

5. CONCLUSION To our best knowledge, this paper presents the first study on reforming the constraints of uncorrelated discriminant methods. In this paper, we propose two novel feature extraction approaches, that is, local uncorrelated discriminant transform (LUDT) and weighted global uncorrelated discriminant transform (WGUDT) for face recognition, respectively. LUDT and WGUDT separately construct the local uncorrelated constraints and the weighted global uncorrelated constraints, and iteratively calculate the optimal discriminant vectors under the corresponding uncorrelated constraints. Experimental results on the AR and FERET face databases demonstrate that the LUDT and WGUDT outperform several representative feature extraction methods. LUDT boosts the average rate at least by 3.31% and WGUDT improves the average recognition rate at least by 3.20% in contrast with related methods.

6. ACKNOWLEDGEMENTS The work described in this paper was fully supported by the NSFC under Project No.61073113, Project No.60772059, the New Century Excellent Talents of Education Ministry under Project No. NCET-09-0162, the Doctoral Foundation of Education Ministry under Project No. 20093223110001, the Qing-Lan Engineering Academic Leader of Jiangsu Province, the Foundation of Jiangsu Province Universities under Project No.09KJB510011. 7. REFERENCES [1] P.N. Belhumeur, J.P. Hespanha, and D.J. Kriegman. “Eigenfaces vs. Fisherfaces: Recognition Using Class Specific Linear Projection,” IEEE Trans. Pattern Anal. Mach. Intell., 19(7), pp. 711-720, 1997. [2] X.F. He, S. Yan, Y. Hu, P. Niyogi, and H. Zhang, “Face Recognition Using Laplacianfaces,” IEEE Trans. Pattern Anal. Mach. Intell., 27(3), pp. 328-340, 2005. [3] X.Y. Jing, D. Zhang, and Y.Y. Tang, “An Improved LDA Approach, ” IEEE Trans. Systems, Man and Cybernetics Part B, 34(5), pp. 1942-1951, 2004. [4] J. Ye and Q. Li. “A Two-Stage Linear Discriminant Analysis via QR Decomposition,” IEEE Trans. Pattern Anal. Mach. Intell., 27(6), pp. 929-941, 2005. [5] D.H. Foley, J.W. Sammon, “An optimal set of discriminant vectors,” IEEE Trans Computers, 24(3), pp. 281-289, 1975. [6] X.Y. Jing, D. Zhang, and Z. Jin, “Improvements on the Uncorrelated Optimal Discriminant Vectors”, Pattern Recognition, 36(8), pp. 1921-1923, 2003. [7] X.Y. Jing, D. Zhang, and Z. Jin, “UODV: Improved Algorithm and Generalized Theory,” Pattern Recognition, 36(11), pp. 2593–2602, 2003. [8] Y. Liang, C. Li, W. Gong, and Y. Pan, “Uncorrelated linear discriminant analysis based on weighted pairwise Fisher criterion,” Pattern Recognition, 40(12), pp. 3606-3615, 2007. [9] J. Ye, R. Janardan, Q. Li, and H. Park, “Feature Extraction via Generalized Uncorrelated Linear Discriminant Analysis”, Int. Conf. Machine Learning, pp. 895-902, 2004. [10] L.H. Zhang, “Uncorrelated trace ratio linear discriminant analysis for undersampled problems,” Pattern Recognition Letters, 32(3), pp. 476-484, 2011. [11] Z. Jin, J.Y. Yang, and Z. Lou. “Face recognition based on the uncorrelated discrimination transformation,” Pattern Recognition, 34(7), 1405-1416, 2001. [12] X.Y. Jing, H.S. Wong, D. Zhang, and Y.Y. Tang, “An Uncorrelated Fisherface Approach”, Neurocomputing, 67, pp. 328-334, 2005. [13] J. Yang, J.Y. Yang and D. Zhang, “What’s Wrong with Fisher Criterion?” Pattern Recognition, 35(11), pp. 2665-2668, 2002. [14] A.M. Martinez and R. Benavente. The AR Face Database, CVC Technical Report, 1998. [15] P.J. Phillips, H. Moon, P. Rauss and S.A. Rizvi. “The FERET Evaluation Methodology for Face-Recognition Algorithms,” IEEE Trans. Pattern Anal. Mach. Intell., 22(10), pp. 10901104, 2000. [16] M. Yang and L. Zhang, “Gabor Feature based Sparse Representation for Face Recognition with Gabor Occlusion Dictionary,” ECCV Part VI, pp. 448-461, 2010.

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