FACE RECOGNITION USING MARGIN-ENHANCED

CLASSIFIER IN GRAPH-BASED SPACE

Ju-Chin Chen, Shang-You Shi and Jenn-Jier James Lien

Department of Computer Science and Information Engineering, National Cheng Kung University, Tainan, Taiwan

Keywords: Dimensionality Reduction, Distance Metric Learning, Face Recognition.

Abstract: In this paper, we develop a face recognition system with the derived subspace learning method, i.e.

classifier-concerning subspace, where not only the discriminant structure of data can be preserved but also

the classification ability can be explicitly considered by introducing the Mahalanobis distance metric in the

subspace. Most of graph-based subspace learning methods find a subspace with the preservation of certain

geometric and discriminant structure of data but not explicitly include the classification information from

the classifier. Via the distance metric, which is constrained by k-NN classification rule, the pairwise

distance relation can be locally adjusted and thus the projected data in the classifier-concerning subspace

are more suitable for k-NN classifier. In addition, an iterative procedure is derived to get rid of the

overfitting problem. Experimental results show that the proposed system can yield the promising

recognition results under various lighting, pose and expression conditions.

1 INTRODUCTION

Face recognition has been an important issue over

the last decades, which has created a wide range of

applications, such as surveillance, security systems,

etc. Among those appearance-based methods

(Murase et al., 1999; Turk et al., 1991), the most

well-known algorithms are Eigenface (Turk et al.,

1991) and Fisherface (

Belhumeur et al., 1997). The

former is based on principal component analysis

(PCA) to obtain the linear transformation by

maximizing the variance of training images but the

class information is excluded; while the latter

applied linear discriminant analysis (LDA) in

(Etemad et al., 1997) which includes the class label

information and the discriminant projection is

obtained by maximizing the ratio of the between-

class and within-class distance.

For non-linearly distributed data such as those

associated with non-frontal facial images and under

different lighting conditions, the classification

performances of the PCA and LDA are somewhat

limited due to their assumption of an essentially

linear data structure. To resolve the problem, the

graph-based dimensionality reduction methods are

recently developed by investigating the local

information and the essential structure of data

manifold which are important for classification

purpose, including isometric feature mapping

(ISOMAP) (Tenebaum et al., 2000), locally linear

embedding (LLE) (Roweis et al., 2000), and

Laplacian eigenmap (LE) (Belkin et al., 2003). In

(Yan et al., 2007), a unified framework for

dimensionality reduction algorithms, i.e. graph

embedding, is proposed and most dimensionality

reduction algorithms such as PCA, LDA, LPP (He et

al., 2003), ISOMAP, LLE and LE can be re-

formulated via specific graph Laplacian matrix

design. Moreover, it provides a platform such that

the new algorithm for dimensionality reduction can

be developed with a specific motivation.

Mostly, after obtaining the desired low-

dimensional subspace via the above dimensional

reduction algorithm, k-NN classifier, which is

simpler and more flexible for the multiple-class

extension, are often applied based on Euclidean

distance metric for recognition. However, Euclidean

distance metric ignores the statistical properties of

data (Weinberger et al., 2009). Instead of Euclidean

distance metric, several distance metric algorithms

(Bar-Hillel et al., 2005; Goldberger et al., 2004;

Weinberger et al., 2009; Xing et al., 2003) are

proposed to obtain a new distance metric to

investigate the data properties from class labels in

order that the classification accuracy can be

improved for k-NN classifier. Among them, Mahala-

382

Chen J., Shi S. and James Lien J. (2010).

FACE RECOGNITION USING MARGIN-ENHANCED CLASSIFIER IN GRAPH-BASED SPACE.

In Proceedings of the International Conference on Computer Vision Theory and Applications, pages 382-388

DOI: 10.5220/0002831903820388

 SciTePress

Figure 1: Schematic illustration of proposed subspace learning method, classifier-concerning subspace. (a) The distribution

of training images in original input space. (b) Projecting face images to graph-based subspace using the linear

transformation matrix A. (c) The classifier-concerning subspace is obtained by applying Large Margin Nearest Neighbour

in subspace (b). The pairwise distance relation are locally adjusted to be more suitable for k-NN classifier.

nobis distance metric is learned based on various

object function. Relevant Component Analysis

(RCA) (Bar-Hillel et al., 2005) learns a full ranked

Mahalanobis distance metric using equivalence

constraints and the linear transformation can be

obtained via solving eigen-problem. In (Xing et al.,

2003), a Mahalanobis metric based on the unimodel

assumption is proposed. Different from these

studies, Large Margin Nearest Neighbour (LMNN)

(Weinberger et al., 2009) learn a Mahalanobis

distance metric by the constraint on the distance

metric imposed by accurate k-NN classification.

Thus, via the metric, k-nearest neighbours always

belong to the same class while examples from

different classes are separated by a large margin.

Inspired from above studies, we propose a

subspace learning method with the goal that in the

obtained subspace, defined as classifier-concerning

subspace, not only the local geometric and

discriminative structure of data can be preserved but

also the projected data are suitable for using k-NN

classifier. Fig. 1 shows the schematic illustration for

our proposed method. Firstly, in order to analyze the

high-dimensional input images in a compact low-

dimensional subspace, the graph-based subspace

learning method are applied that the discriminant

structure of data can be preserved. As shown in Fig.

1(b), most data in the subspace can be well-

separated by preserving the local discriminant

structure but there exists the data that can not be

correctly classified by the k-NN classifier, i.e. y

and

in Fig. 1(b). Then, in order to make them

separable, the Mahalanobis distance metric is

learned with the constraint on k-NN classification

rule. Thus, via the learned distance metric, the

pairwise distance of bad-separated points can be

locally adjusted while relations of well-separated

ones are kept separable (Fig. 1(c)). Moreover, in

order to cope with the overfitting problem, an

iterative optimization process is derived to obtain a

more general subspace than overfitting one for k-NN

classifier.

2 GRAPH-BASED SUBSPACE

LEARNING

The essential task of subspace learning is to find a

mapping function F: x → y that transforms each

image x

∈

into the desired low-dimensional

representation y

∈

(d<<D) such that y can

represent x well in terms of various optimal criteria

and each of which corresponds to a specific graph

design (Yan et al., 2007). Suppose we have the face

image set X=[x

, x

,…, x

]. and each image x

has

the corresponding subject (class label)

{}

,...,2,1∈

where N is number of training images and c is the

number of subjects.

2.1 Dimensionality Reduction using

Graph Embedding

Given the set X, which could be represented by an

intrinsic graph

{

}

WX,

, where the vertices X

corresponds to all facial data and each element W

similarity matrix

∈W

measures the pairwise

similarity between vertex x

and x

. The diagonal

matrix D and the Laplacian matrix L of graph G are

defined as

∑

∀=−=

ijii

iWD,WDL

(1)

(a) Training face images

Test data

(b)Graph-based Subspace

(c)LMNN in subspace

Test data, subject

Train data, subject 3

Train data, subject 1

Train data, subject 2

FACE RECOGNITION USING MARGIN-ENHANCED CLASSIFIER IN GRAPH-BASED SPACE

383

Let

{}

CX,=

be a constraint graph with the same

vertex set X as that of intrinsic graph G and

constraint matrix C. Note that the similarity matrix

W and the constraint matrix C are designed to

capture certain geometric or statistical properties of

the data set.

The purpose of graph embedding is to find the

low-dimensional representation, i.e. Let

Y=[y

,…,y

] for data set X such that the pairwise

similarities measured by W can be preserved and the

similarities measured by C can be suppressed.

Therefore, the optimal Y can be obtained by

)(minarg)(2min arg

min arg*

CYY

LYY

ICYY

ijji

trtr

Wyy

−=

∑

(2)

In order to minimize the object function, for larger

similarity between samples x

and x

, the distance

between y

and y

should be smaller; while smaller

similarity between x

and x

should lead to larger

distances between y

and y

(Yan et al., 2007). The

optimization of Eq. (2) can be solved by a

generalized eigenvalue problem as LY=λCY.

2.2 Linear Transformation

Assume that the low-dimensional representation y

can be obtained from a linear transformation vector

a, i.e. y

. Thus the Eq. (2) can be rewritten as

)( minarg)( 2minarg*

aXCXa

aXLXa

aXLXaa

IaXCXa

trtr

(3)

The transformation A=[a

,…,a

] can be solved

aXCXaXLX

= by selecting the eigenvectors

corresponding to d smallest eigenvalue. Different

design on intrinsic and constraint graphs will lead to

many popular linear dimensionality reduction

algorithms (Cai et al., 2007; Etemad et al., 1997; He

et al., 2003; Yan et al., 2007). For example, the

graphs of the locality sensitive discriminant analysis

(LSDA) are defined by

⎩

⎨

⎧

∈∈

otherwise,0

)( and )( if,1

iwjjwi

ijw

xNxxNx

⎩

⎨

⎧

∈∈

otherwise,0

)( and )( if,1

ibjjbi

ijb

xNxxNx

(4)

where the subject of each element in N

) is as

same as x

and N

) is as different as x

. The

detailed graph design and objective function can

refer to (Cai et al., 2007).

3 CLASSIFIER-CONCERNING

SUBSPACE LEARNING

After finding a subspace, k-NN classifier is widely

applied for classification in the desire low-

dimensional subspace. Inspired by this situation, in

Section 3.1, our face recognition system attempts to

find a subspace, i.e. classifier-concerning subspace,

where not only the structure of data can be preserved

but also the classification ability can be explicitly

considered by introducing the Mahalanobis distance

metric in the subspace. In addition, an iterative

optimization for obtaining the subspace is derived in

Section 3.2.

3.1 Margin Enhancement in Subspace

As known, the face images under different poses,

lighting conditions and facial expressions are non-

linearly distributed in high-dimensional input space.

Hence, at the first step of our proposed method (Fig.

1), an projection matrix A=[a

,…,a

], learned by

the graph-based subspace learning method, such that

most of the desired low-dimensional data

{

}

Ry ∈

, i.e. Y=A

X, can be well-separated in

that subspace.

Although the discriminant structure has been

discovered in the subspace A, there exists some bad-

separated data, i.e. the distance of y

and y

of the

same subject (class) is larger than the distance y

and

of different subjects (shown in Fig. 1(b)).

Therefore, in the second step, a Mahalanobis

distance metric is applied to enhance the margin

between data in the subspace A by applying LMNN

(Weinberger et al., 2009) which is according to the

k-NN classification rule. Thus, via the learned

distance metric, the distances for those bad-

separated data can be locally adjusted and

meanwhile the distance for well-separated data can

be kept in the resulting subspace, named as

classifier-concerning subspace (Fig. 1(c)). As

shown, the k-nearest neighbours always belongs to

the same class while data from different classes are

separated by a large margin and thus the projected

data in the classifier-concerning subspace would

have better distance relation for the k-NN classifier

for recognition.

Let each data y

in the subspace A with the class

label

{

}

,...,2,1

∈

. The Mahalanobis distance metric

∈M

can be expressed in terms of the square

matrix

EEM

, where

∈E

represents a

linear transformation. The square distance between

VISAPP 2010 - International Conference on Computer Vision Theory and Applications

384

two low-dimensional embeddings y

and y

computed as:

)(

)()(),(

jiji

yyyyyyD

−=

−−=

(5)

According to the k-NN classification rule, the cost

function can be defined as (Weinberger et al., 2009):

−−−+−+

−−=

∑

])()(1)[1(

)()1()(

ijk

jiikij

jiij

yyyy

ηα

ηαε

(6)

where

}1,0{∈

indicate whether y

is a target

neighbour of y

}1,0{∈

indicate whether y

and

share the same class label, and

)0,max(][ zz =

denotes the standard hinge loss function. Note that

the first term of cost function only penalizes large

distances between each input y

and its target

neighbour. While the second term penalizes small

distances between each input and all other inputs

with different class label. The scalar

can be tuned

the importance between two terms. By introducing a

nonnegative slack variable

ijk

, Eq.(6) can be

reformulate to a semi-definite programming (SDP)

problem, as well as SVM (Weinberger et al., 2009).

1)()( )()( s.t.

)1()()(min

≥

−≥−−−−−

−+−−

∑∑

ijk

ijkji

jiki

ijkik

ij ijk

ijji

jiij

yyyyyyyy

lyyyy

δηαη

(7)

Thus, for the input image x

, the desired low-

dimensional representation contained the

classification information,

, can be obtained by

XPXAEEYY

=== )(

(8)

Note that via the transformation matrix P, the

pairwise distance between low-dimensional data

in the obtained subspace, named as classifier-

concerning subspace, is constraint on the k-NN

classification rule. Thus, not only the desired data

structure in the high-dimensional space can be

preserved but also the low-dimensional data are

suitable for using k-NN classifier for further

classification.

3.2 Iterative Optimization

In this subsection, a procedure is derived to optimize

the classifier-concerning subspace P in Eq. (8) via

iteratively optimizing the graph-embedding

projection A and the distance metric M. From Eq.

(3), learning a subspace via specific graph Laplacian

matrix, the better graph embedding projection can be

obtained if classifier-concerning information could

be acquired from distance metric M. And thus the

desire low-dimensional subspace not only preserve

the data structure but are also suitable for k-NN

classifier, i.e. the ideal case in classifier-concerning

subspace learning is P=A, and M is an identity

metric. However, it is not intuitive to know the

classification results in advance but via the

Mahalanobis distance metric learned in Eq. (7), we

can obtain the pairwise data relation, which embeds

the classification information, in the low-

dimensional subspace.

To obtain a virtual high-dimensional distance

metric which can pass the low-dimensional pairwise

distance relation into the original input space, we

first inspect the pairwise distance in the desire low-

dimensional subspace

)(

)()(

jiij

xxxx

yyyyd

−−=

AMA

(9)

From Eq. (9), we know Mahalanobis distance metric

M can be re-projecting to the original space via the

projection A. i.e.

AMAM =

.As the metric can be

represented as

EEM

, the metric M

can be as

EEM

ˆˆˆ

2/12/1 TTT

SVD

UUUU =ΛΛ=Λ=

(10)

RR →:

is a linear transformation. Thus,

through

, each original input data x

can be

represented as

xEx

such that the all input data

XEX

ˆˆ

= would implicitly have k-NN classification

information. Therefore, by using the data with

classification information, i.e.

, the new graph-

embedding projection A=[a

,…,a

] can be

obtained based on the data structure as:

)

ˆˆ

( minarg*

∑

−=

ijji

wxxaa

(11)

By introducing the Laplacian matrix L and

constraint matrix C according to the original data

structure, Eq. (11) can be derived as

FACE RECOGNITION USING MARGIN-ENHANCED CLASSIFIER IN GRAPH-BASED SPACE

385

)

ˆˆ

(minarg

)

ˆˆ

(minarg*

aEXCXEa

aEXLXEa

aXCXa

aXLXa

TTT

(12)

The optimal projection A in Eq. (12) can be solved

by the generalized eigenvalue problem, and the

newly low-dimensional embedding are

XAY

.Though the iterative process, the new low-

dimensional distance metric M and projection A are

obtained alternatively by using the information from

the other until

IM ≈

, e.g. all the classification

information can be embedded to the project A, i.e.

P=A. The proposed iteratively updating procedure is

summarized in Fig. 2.

4 EXPERIMENTAL RESULTS

In this section, we investigate the performance of the

proposed subspace learning method for face

recognition under different lightings, poses, and

expressions. Subspace learning methods: supervised-

LPP (Zheng et al., 2007) (designated as SLPP) and

LSDA (Cai et al., 2007) are compared with our

proposed model, e.g. supervised-LPP cascaded by

LMNN model (designated as SLPP+LMNN) and

LSDA cascaded by LMNN model (designated as

LSDA+LMNN), respectively. In addition, Eigenface

(PCA) (Murase et al., 1995) and RLDA (Ye et al.,

2006) which give impressive results (Cai et al. 2007)

are compared as well. Note that k-nearest

neighbours (k=3) are applied in the following

experiments. We use the source code kindly proved

the authors of (Cai et al., 2007; Weinberger et al.,

2009).

4.1 Database and Image Preprocess

Both Extended Yale-B

and CMU PIE

database are

considered as a tough task for face recognition

problem because they are in a complex

environmental setting. Hence, we use both databases

to evaluate our proposed method. The Extended

Yale-B face database contains 16128 images of 38

human subjects under 9 poses and 64 illumination

conditions. We choose the frontal pose and use all

the images under different illumination, thus we get

64 images for each person. The CMU PIE face data

base contains 68 human subjects with 41,368 face

images. The face images were acquired across

different poses, illumination conditions, and

expressions. In our experiment, five near frontal po-

Input: All training images

R,...,xxx ∈= } ,{

with the class label

l }{

and the graph for L and C.

Output: Subspace P

1. Initialize: IE =

)1(

2. For t=1: iter

3. XEX

)()(

ˆˆ

4. Compute

)

ˆˆ

(minarg

)()(

)(

aXCXa

aXLXa

tr=

5. For all data i,

)(

)( t

xy A=

6. Compute

)}{,}({LMNN

)(

)( N

)()()( t

EEM =

8. Compute high-dimensional metric

)(t

tttt )()()()(

AMAM

9. Perform SVD on

)(

M :

**)(

ˆˆˆ

EEM

10.

)(*)1(

ˆˆˆ

EEE ←

11. Check convergence condition:

<− IM

)(t

12. End

13.

tt )()(

EAP =

Figure 2: Procedure of iteratively optimizing the subspace

ses (C05, C07, C09, C27, C29) and all the images

under different illuminations and expressions are

used, thus we get 170 images for each individual.

For both databases, all these face images are

manually aligned and cropped to 32x32 pixels, and

the pixel values are then normalized to the range [0,

1] (divided by 256). Each database is then

partitioned into the gallery and probe set, denoted as

PG / ,where p images per person are randomly

selected for training and the remaining q images are

used for test. Note that in order to reduce the noise,

the data are processed by PCA and 98% energy is

saved. The dimensionality of feature subspace is set

to c-1 for all subspace learning methods, where c is

the number of individuals.

4.2 Comparison of Subspace Learning

Methods

The recognition results conducted on PIE and

Extend Yale-B database are listed in Table 1 and 2,

respectively. For each

PG / , we have 35 random

splits but exclude possible over-fitting splits, then

average the remaining splits, report the mean

recognition rate as well as the standard deviation in

the table. Through Table 1 and 2, it can be seen that

our proposed method can further improve the results

VISAPP 2010 - International Conference on Computer Vision Theory and Applications

386

Table 1: Recognition accuracies of different algorithms on

Extended Yale-B database.

G30/P34 G40/P24

PCA (98% energy) 38.63±1.31 33.78±1.03

RLDA 8.60±0.89 6.68±0.84

SLPP 8.60±0.89 6.68±0.84

SLPP+LMNN 8.13±0.64 6.63±0.71

LSDA 8.63±0.85 6.92±0.74

LSDA+LMNN 8.03±0.87 6.62±0.64

than RLDA, SLPP and LSDA, respectively,

especially for the PIE database. This indicates that

embedding the classification information to the

subspace via a trained Mahalanobis distance metric

can discover a more discriminative structure of the

face manifold and hence improve the recognition

rate.

In order to investigate the stability of the

proposed method, we compare the results of

projecting original data X into various d-

dimensionality feature subspaces via LSDA and

LSDA+LMNN subspaces. Fig. 3 shows the results

on Yale-B and PIE database, respectively. As Fig 3

shown, metric learning needs enough desired low-

dimension to learn the Mahalanobis distance metric

for further improvement recognition result. Ideally

desired low-dimension set c-1 gives convenience

and efficiency from the result.

4.3 Recognition Results of

Classifier-Concerning Subspace

with Iterative Optimization

Table 2: Recognition accuracies of different algorithms on

PIE database.

G30/P34 G40/P24

PCA (98% energy)

43.32±0.62 36.02±0.73

RLDA

8.78±0.34 6.45±0.32

SLPP

8.78±0.34 6.45±0.32

SLPP+LMNN

6.60±0.21 4.94±0.20

LSDA

8.87±0.36 6.52±0.32

LSDA+LMNN

6.55±0.25 4.91±0.20

As mentioned in 4.2, some splits cause overfitting

result. The main reason is that the model fit the

training data too much to lose the generality of the

model. From Table 3, it can be seen that the

overfitting problem can be overcome by iterative

way. Interestingly, the number of PIE training data

is enough to cause nearly no over-fitting result.

5 CONCLUSIONS

Table 3: Recognition accuracies of the proposed

iteratively updating framework on extend Yale-B

database.

G30/P34

(train error /

testerror)

G40/P24

(train error /

testerror)

LSDA 6.04±0.49/

8.60±1.0

4.75±0.42/

6.60±0.80

LSDA+LMNN 2.91±0.42/

8.79+0.96

2.54±0.39/

6.70±0.95

LSDA+LMNN+Iter. 5.41±0.6/

8.03+1.06

4.60±0.36/

6.18±0.82

Stability analysis of PIE G30/P140

Dimension

20 30 40 50 60 70 80 90

Error Rate (%)

LSDA + LMNN

LSDA

Stability analysis of Extended YaleB G40/P24

Dimension

25 30 35 40 45 50 55 60

Error Rate(%)

LSDA+LMNN

LSDA

Figure 3: Recognition results on various d-dimensionality

feature subspaces obtained via LSDA and the proposed

method which embed the classification information

additionally.

In this paper, we have shown how to learn a

classifier-concerning subspace for face recognition

and that can provide the promising recognition

results under various lighting, pose and expression

condition. The classifier-concerning subspace

preserves certain data characteristic by specifying a

graph and the low-dimensional projected data are

suitable for the usage of k-NN classifier for

recognition task. Because the proposed method

consists of two learning process, i.e. the initial

subspace and the distance metric learning, the model

FACE RECOGNITION USING MARGIN-ENHANCED CLASSIFIER IN GRAPH-BASED SPACE

387

would suffer from the overfitting to data if the

number of training data is insufficient or ineffective.

Hence, we propose an iterative solution via a virtual

Mahalanobis distance in original high-dimensional

input space that can pass low-dimensional pairwise

distance relation. Based on this virtual Mahalanobis

distance the pairwise distance relation of data in

input space can be adjusted and then the classifier-

concerning subspace can be updated. Ongoing work

is to circumvent overfitting via adding some

mechanics for choosing effective training data or

applying regularization information.

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