MUL

TI-DISCRIMINANT CLASSIFICATION ALGORITHM FOR

FACE VERIFICATION

Cheng-Ho Huang and Jhing-Fa Wang

Dept. of Electrical Engineering, National Cheng Kung University, No. 1 University Rd., Tainan City, Taiwan

Keywords:

Linear discriminant analysis, face veriﬁcation, multi-discriminant classiﬁcation.

Abstract:

Linear discriminant analysis (LDA) is a conventional approach for face veriﬁcation. For computing large

amounts of data collected for a given face veriﬁcation system, this study proposes a multi-discriminant classi-

ﬁcation algorithm to classify and verify voluminous facial images. In the training phase, the algorithm extracts

all discriminant features of the training data, and classiﬁes them as the clients’ multi-discriminant sets. The

algorithm veriﬁes a claim to the client’s multi-discriminant set, and then determines whether the claimant is

the client. Comparative results demonstrate that the proposed algorithm reduces the false acceptance rate in

face veriﬁcation.

1 INTRODUCTION

Two primary applications of face recognition are face

identiﬁcation and face veriﬁcation. Face identiﬁca-

tion identiﬁes two similar faces between unknown

user and genuine users; face veriﬁcation compares an

unknown user to a genuine user, and decides whether

the two are the same. Therefore, impostors present

a problem in face veriﬁcation. In particular, impos-

tors are greater in number than clients. Eigenface

(Turk and Pentland, 1991) and Fisherface (Belhumeur

et al., 1997) are two of the best known methods that

adopt feature transformation in order to discriminate

differences in facial features for the purpose of face

veriﬁcation. However, the performance of Eigenface

method is not ideal when numbers of the sample sets

are voluminous. Fisherface, an implementation of lin-

ear discriminant analysis (LDA) (Martinez and Kak,

2001), is often utilized for face veriﬁcation. It em-

ploys both the PCA and Fisher criterion to extract

discriminant information from a set of training data.

Many methods (Liu and Wechsler, 1998; Loog et al.,

2001; Wang and Tang, 2004) have been proposed to

enhance the performance and stability of LDA. Both

classical and modiﬁed LDA methods are efﬁcient for

face recognition.

Although improved LDA approaches are superior

to classical LDA approaches, they still do not provide

adequate discriminant information to permit accurate

discrimination of the highly complex and voluminous

data of facial images. Main reason for this limitation

is given below.

The voluminous data of facial images are not

true Gaussian distributions. Consequently, the clas-

sical linear transform of the “between-class” and the

“within-class” cannot effectively extract the differen-

tial features from the classes.

Therefore, classical LDA is not appropriate for di-

rect analysis of complex and numerous data. As the

amount of data increases, computational loading of

LDA also increases, and the time required for calcu-

lation grows longer, making the method less practi-

cal. To reduce the computations of numerous data,

k-nearest neighbor (KNN) and k-means algorithm are

adopted to classify data into small units. However,

KNN is sensitive to feature mapping; if the feature

mapping is not well distribution, KNN does not ob-

tain robust classiﬁcations. K-means , which is an

unsupervised classiﬁcation algorithm, has problems

with initial centroids and specifying the number of

clusters. Otherwise, if the selected threshold value of

the algorithm is unsuitable, then the false acceptance

rate (FAR) and false rejection rate (FRR) increase; in

particular, the algorithm cannot effectively tune the

threshold parameters for FAR and FRR.

Due to these above-mentioned problems, in order

to avoid the resulting decrease in efﬁciency of the

overall performance caused by the large amounts of

complex data, this study proposes a veriﬁcation al-

gorithm without setting any threshold value to sepa-

rate complex data into simple units and verify face

images. This algorithm splits all of the training data,

299

Huang C. and Wang J. (2008).

MULTI-DISCRIMINANT CLASSIFICATION ALGORITHM FOR FACE VERIFICATION.

In Proceedings of the Third International Conference on Computer Vision Theory and Applications, pages 299-304

DOI: 10.5220/0001082202990304

Copyright

c

SciTePress

enabling each individual’s features to be distinguished

and yield subsets of distinguishable features for each

person. Combining the results obtained by separately

discriminating these subsets is synonymous with ver-

ifying whether an unknown user is the genuine user.

Thus, as evidenced from volumes of face veriﬁcation,

this study has achieved good efﬁciency to avoid im-

postors, and increased the overall robustness of the

method.

The paper is organized as follows. Section 2

presents the multi-discriminant classiﬁcation algo-

rithm on volumes of face veriﬁcation. Experiments

and ﬁnal conclusions are provided in Sections 3 and

4, respectively.

2 THE PROPOSED

MULTI-DISCRIMINANT

CLASSIFICATION

ALGORITHM FOR FACE

VERIFICATION

The proposed multi-discriminant classiﬁcation al-

gorithm (MDCA) consists of two modules, multi-

discriminant classiﬁer and evaluator. Figure 1 shows

the entire framework. Each module is discussed be-

low.

2.1 Multi-discriminant Classiﬁer

The proposed approach using generalized singular

value decomposition LDA (GSVD/LDA) (Howland

and Park, 2004) constructs multi-discriminant sets

(MDS) and performs discriminant analysis to verify

a claimant in the client’s MDS.

Suppose that m-dimensional patterns

A = {x

i

}

i=1,...,n

belong to c different classes

{C

i

}

i=1,...,c

. A ∈ ℜ

m×n

. Let n

k

denote the number of

patterns in class k; thus,

∑

c

k=1

n

k

= n.

µ =

1

n

n

∑

i=1

x

i

, (1)

µ

k

=

1

n

k

∑

x∈C

k

x

k

, (2)

where µ denotes the average of ensemble facial fea-

tures and µ

k

denotes the mean of class C

k

. The

between-class scatter matrix S

B

is deﬁned as

S

B

=

c

∑

k=1

n

k

(µ

k

− µ)(µ

k

− µ)

T

. (3)

x

1

x

2

x

3

x

k

... ...

Result

a claim

... ...

RN

FN

NN

RN

FN

NN

RN

FN

NN

RN

FN

NN

Training Database

Multi-discriminant classifier

Discriminant Feature Extraction

Neighbor Distance Measure

Evaluator

f

x

E

Figure 1: Framework of the proposed algorithm.

The within-class scatter matrix S

W

is deﬁned as

S

W

=

c

∑

k=1

∑

j∈C

k

(x

j

− µ

k

)(x

j

− µ

k

)

T

, (4)

and

S

T

= S

B

+ S

W

. (5)

The transformation matrix G

T

∈ ℜ

l×m

reduces vector

x

i

of A to vector y

i

in the l−dimensional space:

y

i

= G

T

A ∈ ℜ

l×n

,l ¿ m. (6)

The maximum ratio of the between-class to within-

class scatter is obtained by the determinant of the ob-

jective function of the scatter matrices and is deﬁned

as

J(G) = trace((G

T

S

T

G)

†

(G

T

S

B

G)). (7)

When T is full rank,

case 1: l = n

T

†

= (T)

−1

,

case 2: l < n

VISAPP 2008 - International Conference on Computer Vision Theory and Applications

300

T

†

= T

T

(TT

T

)

−1

,

case 3: l > n

T

†

= (T

T

T)

−1

T

T

,

where T = G

T

(S

T

)G and T

†

is the Moore-Penrose

pseudo-inverse can be obtained by GSVD.

The columns of an optimal G comprise the gen-

eralized eigenvectors corresponding to the l largest

eigenvalues in

S

y

B

g

i

= λ

i

(S

y

T

),i = 1,2,...,l (8)

where S

y

B

and S

y

W

are chosen from S

B

and S

W

, respec-

tively; and g

i

is the set of generalized eigenvectors of

S

y

B

and S

y

W

corresponding to the l largest generalized

eigenvalues λ.

The distance measure is derived from the dif-

ferences in features between average face (µ) and

everyone’s (´µ) , and then classiﬁng facial images into

the clients’ MDS. In this case, ´µ = (µ +

1

n

k

∑

n

k

i=1

x

i

)/2.

G

T

is obtained from µ and ´µ in Eq. (7), and the

discriminant feature D is then deﬁned as follows:

if G

T

´µ < G

T

µ

D = −(

°

°

°

G

T

´µ

°

°

°

+

°

°

°

G

T

µ

°

°

°

) (9)

if G

T

´µ > G

T

µ

D

=

°

°

°

G

T

´

µ

°

°

°

+

°

°

°

G

T

µ

°

°

°

(10)

Algorithm 1 illustrates the pseudocode for extract-

ing discriminant features. The clients’ MDS are con-

structed using these differences after extracting the

discriminant features of all faces.

Algorithm 1 The pseudocode of discriminant feature

extraction.

for i=1 to all do

Calculate G

T

i

by Eq. (7).

if G

T

i

´µ < G

T

i

µ then

D

i

← −(

°

°

G

T

i

´µ

i

°

°

+

°

°

G

T

i

µ

°

°

),

else

D

i

←

°

°

G

T

i

´µ

i

°

°

+

°

°

G

T

i

µ

°

°

.

end if

end for

Consider a certain personal set P

S

client

with x mem-

bers. There exist three special subsets NN= {x

i

|i =

P

client

...P

ns

}, FN= {x

j

| j = P

client

...P

f s

} and RN=

{x

m

|m = P

client

...P

rs

}, S = {NN, FN,RN}; where

NN denotes a nearest neighbor subset; FN denotes

a farthest neighbor subset and RN denotes a ran-

dom neighbor subset; ns and f s selections of peo-

ple are similar and non-similar to p

client

, respectively,

and rs denoted the random selections of people to

P

client

. Algorithm 2 illustrates the pseudocode of

multi-discriminant classiﬁer. A subset of t members

is chosen as a subset, where 10 ≤ t ≤ 20.

Algorithm 2 The pseudocode of multi-discriminant

classiﬁer.

for i = 1 to all do

if D

i

∼ D

P

client

and ns ≤ t then

Select x

i

into an NN

P

S

client

.

ns = ns + 1.

end if

if D

i

D

P

client

and f s ≤ t then

Select x

i

into an FN

P

S

client

.

f s = f s + 1.

end if

end for

for rs = 1 to t do

Randomly select x into a RN

P

S

client

.

end for

2.2 The Evaluator of Face Veriﬁcation

The evaluator determines whether a claim is the client

by an evaluation function on the results of discrimina-

tions from NN, FN and RN.

Equation (11) is a similar description described by

the following expression:

f

x

=

1,

i f dist(x

claim

,x

P

client

)

= min(dist(x

claim

,x))

0,

i f dist(x

claim

,x

P

client

)

> min(dist(x

claim

,x))

(11)

where dist is the distance measure function. If x

claim

is similar to x

P

client

, then f

x

= 1 or 0.

Equation (12) is an evaluation function of MDCA,

and is deﬁned as follows:

E(x

claim

,x

P

client

) = ( f

NN

• f

FN

+ f

RN

)

+ ( f

NN

• f

RN

+ f

FN

)

+ ( f

FN

• f

RN

+ f

NN

),

(12)

where E denotes an evaluator; and • and + are AND

and OR Boolean operators, respectively. If x

claim

is

similar to x

P

client

for two out of the three discriminated

MULTI-DISCRIMINANT CLASSIFICATION ALGORITHM FOR FACE VERIFICATION

301

results of subsets NN, FN, and RN, then x

claim

indi-

cates the genuine user x

P

client

. If E is equal to 1, the

result is an acceptance, or a rejection.

Thus, the face veriﬁcation problem can be de-

picted by a multi-identiﬁcation problem. The eval-

uation algorithm is illustrated in Alg. 3.

Algorithm 3 The pseudocode of the evaluator.

Calculate dist(x

claim

,x).

if dist(x

claim

,x

P

client

) = min(dist(x

claim

,x)) then

f

x

P

client

← 1,

else

f

x

← 0.

end if

if E(x

claiim

,x

P

client

) = 1 then

Accept,

else

Reject.

end if

For instance, the statuses of MDS which owns ten

members are described in the Table 1, Table 2 and Ta-

ble 3,respectively. Eq. (12) is used to evaluate x

claim

and x

P

client

, and then obtains E = 1. Therefore, the

result of veriﬁcation is an acceptance.

Table 1: Select top ten nearest neighbors of D into an NN.

Member D dist() f

P2 282 251 0

P4 247 342 0

P6 217 221 0

P7 371 175 0

P10 391 232 0

P11 182 172 0

P12 389 119 0

P15 193 120 0

P20 387 149 0

P

client

120 76 1

3 EXPERIMENTS

The experiments were carried out on the FERET

(Rizvi et al., 1998), XM2VTS (Messer et al., 1999)

and UNDBD-B (Bowyer and Flynn, 2003) face

databases. FERET is a well-known face database pro-

vided by the NIST. The FERET database contains

994 people and over 11,000 face images, including

proﬁles, frontal faces, expressions, and poses. The

XM2VTSDB contains 2560 frontal images, which are

four recordings of 295 people taken over a period

Table 2: Select top ten farthest neighbors of D into an FN.

Member D dist() f

P1 891 240 0

P9 777 310 0

P13 909 130 0

P14 1111 171 0

P17 877 234 0

P18 976 140 0

P21 701 127 0

P24 761 134 0

P25 865 182 0

P

client

120 91 1

Table 3: Select ten random neighbors of D into a RN.

Member D dist() f

P3 617 123 0

P5 489 141 0

P8 412 231 0

P13 909 211 0

P15 193 435 0

P16 435 156 0

P19 430 176 0

P22 600 183 0

P23 533 145 0

P

client

120 83 1

of four months. The UNDBD-B database contains

33,247 visible frontal images of 749 people. This

study adopted only the frontal face images as training

faces, and adopted the other types and frontal images

together as test data.

Therefore, the XM2VTS and UNDBD-B were

adopted as the training and testing databases in Ex-

periments (1) and (2), respectively. In Experiment

(3), the FERET database was considered as outside

data to test proposed algorithm. Two evaluations were

adopted to evaluate the system performance:

- False acceptance rate (FAR): the ratio of the

number of false acceptances to that of impostor ac-

cesses.

- False rejection rate (FRR): the ratio of the num-

ber of false rejections to that of authentic accesses.

The experimental results of the proposed face ver-

iﬁcation using the MDCA are presented below. In

this evaluation, the sizes of the multi-discriminant sets

were 10 and 20. The MDCA was adopted to verify

these cases in the multi-discriminant sets. The re-

sults in Table 4 indicate that as the FAR and FRR

of NN, FN and RN are decreased as the size of an

multi-discriminant set increases from 10 to 20. In Ex-

periment (1), the optimum value of FAR was 0.34%,

while that of FRR was 4.04%, while the results of

VISAPP 2008 - International Conference on Computer Vision Theory and Applications

302

Table 4: Comparison results of FAR and FRR between MDCA and LDA with KNN.

MDCA LDA

Experiment E Evaluation size with

10 20 KNN (K=10)

FAR 6.25% 5.86%

only NN FRR 4.12% 4.03%

FAR 7.56% 6.91% FAR 12.5%

XM2VTS only FN FRR 4.78% 4.60%

FAR 6.70% 6.30%

(1) only RN FRR 4.71% 4.31%

FAR 0.73% 0.34% FRR 8.7%

NN, FN, RN FRR 4.44% 4.04%

FAR 6.36% 5.91%

only NN FRR 4.33% 4.00%

FAR 7.31% 6.89% FAR 14.7%

UNDBD-B only FN FRR 4.96% 4.36%

FAR 7.23% 6.20%

(2) only RN FRR 5.51% 4.24%

FAR 0.69% 0.23% FRR 11.6%

NN, FN, RN FRR 4.91% 4.11%

FAR 6.41% 5.94%

only NN FRR 4.43% 4.08%

XM2VTS FAR 8.32% 7.38% FAR 16.8%

+UNDBD-B only FN FRR 5.11% 4.21%

+FERET

impostors

FRR 6.64% 6.26%

only RN FRR 4.88% 4.18%

(3) FAR 0.72% 0.31% FRR 13.1%

NN, FN, RN FRR 5.08% 4.18%

the NN, FN and RN intersected together. The FAR

and FRR were 0.23% and 4.11%, respectively in Ex-

periment (2), and 0.31% and 4.18%, respectively in

Experiment (3). Regardless of the results of the NN,

FN and RN, their intersection demonstrated the best

performance in each experiment. The proposed per-

formed better overall than LDA with KNN (Lin et al.,

2005).

4 CONCLUSIONS

This study proposes an algorithm to enhance the face

veriﬁcation performance in numerous databases by

using multi-discriminant classiﬁcation. Experimen-

tal results indicate that proposed algorithm elevates

the performance of face veriﬁcation. Moreover, the

proposed method does not require the construction of

any miscellaneous thresholding rule and can actively

solve the veriﬁed problem of face veriﬁcation. The

experimental results reveal that FAR can be decreased

from 8.32% to 0.31% when utilizing evaluation func-

tion E with three discriminant subsets.

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