FACIAL PARTS RECOGNITION USING LIFTING WAVELET

FILTERS LEARNED BY KURTOSIS-MINIMIZATION

Koichi Niijima

Department of Informatics, Kyushu University

6-1, Kasuga-koen, Kasuga, Fukuoka 816-8580, Japan

Keywords:

Facial parts recognition, lifting wavelet ﬁlter, learning, kurtosis, ill-posed problem, regularization.

Abstract:

We propose a method for recognizing facial parts using the lifting wavelet ﬁlters learned by kurtosis-

minimization. This method is based on the following three features of kurtosis: If a random variable has

a gaussian distribution, its kurtosis is zero. If the kurtosis is positive, the respective distribution is super-

gaussian. The value of kurtosis is bounded below. It is known that the histogram of wavelet coefﬁcients for a

natural image behaves like a supergaussian distribution. Exploiting these properties, free parameters included

in the lifting wavelet ﬁlter are learned so that the kurtosis of lifting wavelet coefﬁcients for the target facial

part is minimized. Since this minimization problem is an ill-posed problem, it is solved by employing the

regularization method. Facial parts recognition is accomplished by extracting facial parts similar to the target

facial part. In simulation, a lifting wavelet ﬁlter is learned using the narrow eyes of a female, and the learned

lifting ﬁlter is applied to facial images of 10 females and 10 males, whose expressions are neutral, smile,

anger, and scream, to recognize eye part.

1 INTRODUCTION

Facial parts recognition is an important problem for

face expression recognition. Many face recognition

methods have been proposed so far. Principle compo-

nent analysis is a traditional classiﬁcation technique

for face recognition (Pentland et al., 1994). A frame-

work of hidden Markov models has been used for

recognition of eye movement (Jaimes et al., 2001).

Support vector machine is a new tool for solving the

classiﬁcation problems (Vapnik, 1998). Recently, an

approach using AdaBoost, which is one of the ma-

chine learning techniques, has attracted considerable

attention as a method for face recognition (Tieu and

Viola, 2000).

Unlike such recognition techniques, we have pre-

sented a method of person identiﬁcation, which uses

the learned lifting wavelet ﬁlters (Takano et al., 2003;

Takano et al., 2004; Takano and Niijima, 2005). The

learning technique employed therein is to maximize

the cosine of an angle between a vector whose com-

ponents are lifting ﬁlters and a vector consisting of

pixels in the facial part. In person identiﬁcation, a

slight difference of facial parts such as eyes, nose,

and lips must be distinguished. So, we learned sev-

eral lifting wavelet ﬁlters at the center of each of the

facial parts so that they can capture the features of the

objects. However, since the designed ﬁlters are low-

pass ﬁlters, a recognition method using them is not ro-

bust for changing brightness. More recently, we pre-

sented a fast objects detecting method using the lifting

wavelet ﬁlters learned by variance-maximization (Ni-

ijima, 2005). Although this method is fast enough for

online processing, it extracts unnecessary objects as

well as the target one. This suggests that only the use

of variance, which is the second order statistics, is not

sufﬁcient for the exact detection of objects.

In this paper, we propose a method for recogniz-

ing facial parts exploiting the lifting wavelet ﬁlters

learned by kurtosis-minimization. One of the fea-

tures of kurtosis is that if a random variable has a

gaussian distribution, its kurtosis is zero. If the kur-

tosis is positive, the respective distribution is super-

gaussian, which has a sharper peak and longer tails

than the gaussian distribution. This implies that the

variance of gaussian distribution is bigger than that of

supergaussian one. Another very important feature of

kurtosis is that the value of kurtosis is bounded below.

It is known from numerical experiments that the

histogram of wavelet coefﬁcients for a natural image

41

Niijima K. (2006).

FACIAL PARTS RECOGNITION USING LIFTING WAVELET FILTERS LEARNED BY KURTOSIS-MINIMIZATION.

In Proceedings of the First International Conference on Computer Vision Theory and Applications, pages 41-47

DOI: 10.5220/0001367900410047

Copyright

c

SciTePress

behaves like a supergaussian distribution. Therefore,

by learning free parameters contained in the lifting

wavelet coefﬁcients so as to minimize their kurtosis,

we can make the variance of the coefﬁcients large.

The large values of the obtained lifting wavelet coef-

ﬁcients have the features of the facial part in question,

and the learned ﬁlter can be considered as a recog-

nizer of the target facial part. The facial part is called

positive data, and the training image except for it neg-

ative data. For the lifting wavelet coefﬁcients of the

negative data, the free parameters are learned so that

their variance becomes small.

Such a minimization problem is a kind of inverse

problem and ill-conditioned. So, we apply the regu-

larization method to solve the problem. The solutions

of the problem can be found by exploiting various

gradient methods such as the steepest descent method

and the conjugate gradient method. However, these

techniques usually need a lot of time to obtain conver-

gence results. In this paper, we replace the problem

by a problem of seeking stationary points of the cor-

responding functional, and obtain them by Newton’s

method. The stationary points are local minima of

the functional. Different local minima can be found

depending on the starting values of Newton’s itera-

tion. We seek the solution by starting the iteration

from zero-vector.

We extract facial parts from a query image similar

to the target facial part by applying the learned lifting

wavelet ﬁlter to the query image. The extracted facial

part is recognized as the target one.

In simulation, the anger face of a female is used as

a training image. The target facial part is her narrow

eyes. A lifting ﬁlter including the learned parameters

is applied to a variety of human faces whose expres-

sions are standard, smile, anger, and scream. It is also

checked whether the proposed method is robust for

changing brightness for some illuminated facial im-

ages.

The remainder of this paper is organized as fol-

lows. Section 2 describes the relation between a lift-

ing dyadic wavelet ﬁlter and an elliptic-type of partial

differential operator. Our learning algorithm is pre-

sented in Section 3. We describe an extraction method

in Section 4, and a recognition method in Section 5.

Section 6 is simulation. Finally, we conclude with

Section 7.

2 LIFTING WAVELET FILTERS

Let {h

o

n

,g

o

n

,

˜

h

o

n

, ˜g

o

n

} be a set of dyadic wavelet ﬁlters

(Mallat, 1998). The ﬁlters h

o

n

and g

o

n

are called low-

pass and high-pass analysis ﬁlters, respectively, and

the ﬁlters

˜

h

o

n

and ˜g

o

n

are low-pass and high-pass syn-

thesis ﬁlters, respectively. A lifting scheme for the

dyadic wavelet is described as follows:

h

n

= h

o

n

,

g

n

= g

o

n

−

k

λ

k

h

o

n−k

, (1)

˜

h

n

=

˜

h

o

n

+

k

λ

−k

˜g

o

n−k

,

˜g

n

=˜g

o

n

.

This scheme generalizes Sweldens’ biorthogonal lift-

ing scheme (Sweldens, 1996). We proved that the

lifted ﬁlters {h

n

,g

n

,

˜

h

n

, ˜g

n

} also become a set of

dyadic wavelet ﬁlters (Abdukirim et al., 2005). Here

λ

k

’s denote free parameters. In this paper, we only

use the lifted ﬁlter (1).

We denote an image by u

i,j

. By applying the low-

pass analysis ﬁlter h

o

n

in vertical direction to u

i,j

,we

can get

C

col

m,k

=

j

h

o

j

u

m,k+j

.

Next, an application of the lifted ﬁlter (1) in hori-

zontal direction to C

col

m,k

yields the following lifting

wavelet coefﬁcients

D

m,k

=

i

g

d

i

C

col

m+i,k

. (2)

Here g

d

i

’s are given by

g

d

i

= g

o

i

−

L

l=−L

λ

d

l

h

o

i−l

,i= −L−M, ..., L+M +1,

where λ

d

l

’s represent free parameters in horizontal di-

rection and we assumed that the index i of the ﬁlter

h

o

i

moves from −M to M +1. Similarly, we obtain

lifting wavelet coefﬁcients in vertical direction

E

m,k

=

j

g

e

j

C

row

m,k+j

. (3)

Here C

row

m,k

is given by

C

row

m,k

=

i

h

o

i

u

m+i,k

,

and g

e

j

’s are determined as follows:

g

e

j

= g

o

j

−

L

l=−L

λ

e

l

h

o

j−l

,j= −L−M, ..., L+M+1,

where λ

e

l

’s represent free parameters in vertical direc-

tion.

We choose the initial high-pass ﬁlters g

o

n

as g

o

0

=

g

o

2

= −0.25

√

2, g

o

1

=0.5

√

2 and g

o

i

=0other-

wise. Such dyadic wavelet ﬁlters have been provided

in (Mallat, 1998). We put

w

m,k

= D

m,k

+ E

m,k

. (4)

VISAPP 2006 - IMAGE UNDERSTANDING

42

From (2) and (3), the sum w

m,k

can be expressed as

w

m,k

=

i

g

o

i

C

col

m+i,k

+

j

g

o

j

C

row

m,k+j

−

L

l=−L

λ

d

l

C

m+l,k

+

L

l=−L

λ

e

l

C

m,k+l

(5)

with C

m,k

=

i,j

h

o

i

h

o

j

u

m+i,k+j

. Therefore, a lift-

ing wavelet ﬁlter deﬁned by w

m,k

approximates a par-

tial differential operator L(λ

d

,λ

e

) deﬁned by

L(λ

d

,λ

e

)u = −

∂

2

∂x

2

(I

y

u)+

∂

2

∂y

2

(I

x

u)

− I(λ

d

,λ

e

)u. (6)

Here I

y

u, I

x

u and I(λ

d

,λ

e

)u represent the integral

versions of C

col

m,k

, C

row

m,k

and the last term of (5), re-

spectively, and

λ

d

=(λ

d

−L

, ..., λ

d

L

),λ

e

=(λ

e

−L

, ..., λ

e

L

).

3 KURTOSIS-MINIMIZATION

LEARNING

We start with the deﬁnition of kurtosis. Kurtosis is

deﬁned in the zero-mean case by the equation

kurt(w)=<w

4

> −3 <w

2

>

2

, (7)

where w is a random variable and <w>denotes

the expectation of w. In case of kurt(w) > 0, the

distribution of w is said to be supergaussian. If w

has a gaussian distribution, then the kurtosis is zero,

i.e., kurt(w)=0. A typical supergaussian probability

density has a shaper peak and longer tails than the

gaussian probability density function (pdf). There-

fore, the variance of gaussian distribution is bigger

than that of supergaussian pdf. It is known that the

value of kurtosis is bounded below. Such properties

of kurtosis are useful for our analysis.

Let us denote a training image also by u

i,j

, and its

domain by Ω

d

. We extract a facial part such as eyes,

nose, and lips from the training image. The region

of the extracted facial part is denoted by ω

d

, and the

number of pixels in ω

d

by P . The facial part is called

positive data, and the image in the region Ω

d

\ ω

d

negative data. The number of pixels in Ω

d

\ ω

d

is

denoted by N. Using these positive and negative data,

we learn free parameters λ

d

l

’s and λ

e

l

’s appeared in

(5). Our learning method is to minimize the kurtosis

of the lifting wavelet coefﬁcients for the positive data,

and to minimize the variance of those for the negative

data.

We extend the facial part periodically in horizon-

tal and vertical directions, and compute the wavelet

coefﬁcients w

m,k

, deﬁned by (4), of the facial part.

Assume that λ

d

l

’s and λ

e

l

’s satisfy

L

l=−L

(λ

d

l

+ λ

e

l

)=0. (8)

Then, we can prove

<w>=

1

P

(m,k)∈ω

d

w

m,k

=0. (9)

Therefore, the kurtosis of the lifting wavelet coefﬁ-

cients w

m,k

is given by

J

1

=

1

P

(m,k)∈ω

d

w

4

m,k

− 3

⎛

⎝

1

P

(m,k)∈ω

d

w

2

m,k

⎞

⎠

2

,

(10)

which is minimized.

For the negative data in the region Ω

d

\ω

d

, we com-

pute their lifting wavelet coefﬁcients w

m,k

, and mini-

mize the variance

J

2

=

1

N

(m,k)∈Ω

d

\ω

d

w

2

m,k

. (11)

Thus, our learning algorithm of free parameters λ

d

l

’s

and λ

e

l

’s is a process of minimizing the sum of (10)

and (11) under the condition (8).

On the other hand, the continuous version of this

minimization problem is to minimize

ω

L(λ

d

,λ

e

)u(x, y)

4

dxdy

−3

ω

L(λ

d

,λ

e

)u(x, y)

2

dxdy

2

and

Ω\ω

L(λ

d

,λ

e

)u(x, y)

2

dxdy

subject to the constraint (8). Here Ω is a region corre-

sponding to Ω

d

, and ω a region corresponding to ω

d

.

The operator L(λ

d

,λ

e

) has been given in (6). This

continuous problem is an inverse problem which is

ill-conditioned. Therefore, the discrete version is also

ill-conditioned. To overcome this difﬁculty, we em-

ploy the regularization method. Thus, a functional to

be minimized is provided by

J =

1

4

J

1

+

K

0

4

J

2

+

K

1

2

L

l=−L

(λ

d

l

+ λ

e

l

)

2

+

δ

2

L

l=−L

(λ

d

l

)

2

+(λ

e

l

)

2

. (12)

Here the third term of the right hand side comes from

(8), and K

i

,i =0, 1 are penalty constants. The last

FACIAL PARTS RECOGNITION USING LIFTING WAVELET FILTERS LEARNED BY

KURTOSIS-MINIMIZATION

43

term means regularization and δ is a sufﬁciently small

positive number.

Although the functional (12) is a polynomial of

fourth degree with respect to the free parameters λ

d

l

’s

and λ

e

l

’s, it has a possibility of having many local

minima. Since it is difﬁcult to obtain a global min-

imum, we seek local minima. Various gradient meth-

ods are often used for computing local minima. How-

ever, these methods are slow in convergence. In this

paper, we employ Newton’s method to solve the prob-

lem fast. Newton’s method is applied to a system of

simultaneous nonlinear equations:

∂J

∂λ

d

l

=0,l= −L, ..., L, (13)

∂J

∂λ

e

l

=0,l= −L, ..., L. (14)

The process of solving (13) and (14) by Newton’s

method gives our algorithm for learning the free para-

meters λ

d

l

’s and λ

e

l

’s.

4 FACIAL PARTS EXTRACTION

The learned parameters λ

d

l

’s and λ

e

l

’s make the vari-

ance of the lifting wavelet coefﬁcients for the posi-

tive data large, and that for the negative data small.

Therefore, we can extract facial parts of a query im-

age similar to the training facial part by selecting the

large wavelet coefﬁcients, which are computed using

the learned lifting ﬁlter. Our facial extraction algo-

rithm involves the following steps.

1. Compute wavelet coefﬁcients in horizontal and

vertical directions by applying the initial dyadic

wavelet ﬁlters to a query image u

i,j

.

2. By combining them with the parameters λ

d

l

’s and

λ

e

l

’s learned for the training image, compute the

lifting wavelet coefﬁcients D

m,k

and E

m,k

deﬁned

by (2) and (3), respectively.

3. Calculate the sum w

m,k

= D

m,k

+ E

m,k

.

4. Find the locations (m, k) such that |w

m,k

|≥Rσ,

where σ is the standard deviation of lifting wavelet

coefﬁcients computed for the training facial part

and R denotes some constant.

5. Extract an image region, in which the detected lo-

cations are concentrated, as an object similar to the

training facial part.

5 RECOGNIZER

The learned ﬁlter can extract only a facial part sim-

ilar to the training one. For example, if the train-

ing pattern is narrow eyes, it does not extract closed

eyes. Therefore, the learned ﬁlter is a recognizer of

the training facial part. It is important to indicate that

a lifting ﬁlter has to be learned per facial part. The

positive data may be constructed by combining the

same type of several facial parts such as large eyes,

narrow eyes, and closed eyes.

6 SIMULATION

We conducted our experiments on the AR face data-

base (Martinez and Benavente, 1998). The initial ﬁl-

ters we use are the cubic spline dyadic wavelet ﬁl-

ters listed in Table 1 (Mallat, 1998). The number of

Table 1: Cubic spline dyadic wavelet ﬁlters (only low-pass

and high-pass analysis ﬁlters).

n h

o

n

/

√

2 g

o

n

/

√

2

-2 0.03125

-1 0.15625

0 0.31250 -0.25

1 0.31250 0.50

2 0.15625 -0.25

3 0.03125

free parameters in each direction is 15, i.e., L =7.

Therefore, 30 free parameters are determined exploit-

ing our learning algorithm. The gray scales of images

are converted into the range [0, 1] for stable computa-

tion.

The training pattern is the facial image of a female

having 120 ×120 size, which is shown in Figure 1(a).

The positive data are her narrow eyes and the nega-

tive data are the image except for the eye region. The

positive data are illustrated in Figure 1(b).

(a) (b)

Figure 1: (a) Training facial image, (b) Positive data of nar-

row eyes.

The penalty constants K

i

,i=0, 1 appeared in (12)

are chosen as K

0

=50and K

1

= 1000, respec-

tively. The regularization constant δ is selected as

δ =0.0001. Newton’s iteration for solving (13) and

(14) was started from zero-vector. We list the learned

parameters in Table 2.

Figure 2 shows the histograms of the wavelet co-

efﬁcients w

m,k

, (m, k) ∈ ω

d

obtained exploiting the

initial and the learned lifting ﬁlters. As seen from Fig-

ure 2, the latter histogram is ﬂatter compared with the

VISAPP 2006 - IMAGE UNDERSTANDING

44

Table 2: Learned parameters for the positive and negative

data of the facial image illustrated in Figure 1.

l λ

d

l

λ

e

l

-7 -0.026973 -0.013236

-6 0.103753 0.107106

-5 -0.166372 -0.291907

-4 0.056374 0.200126

-3 0.222947 0.288220

-2 -0.427482 -0.631153

-1 0.760302 0.906046

0 -2.181300 -2.181300

1 3.417086 3.379362

2 -1.808546 -1.953494

3 -0.912882 -0.740206

4 1.503153 1.561924

5 -0.382691 -0.580599

6 -0.360328 -0.224570

7 0.204865 0.171818

0

100

200

300

400

500

600

-40 -20 0 20 40

"hist_org.dat" using 1:2

0

100

200

300

400

500

600

-40 -20 0 20 40

"hist_lift.dat" using 1:2

(a) (b)

Figure 2: Histograms of the wavelet coefﬁcients

w

m,k

, (m, k) ∈ ω

d

obtained by using (a) the initial ﬁlter,

(b) the learned lifting ﬁlter.

former one. This means that the variance of the latter

histogram is bigger than that of the former one. Ac-

tually, the standard derivations of the former and the

latter distributions for the positive data were 0.0155

and 0.0770, respectively.

Using the learned ﬁlter, we tried to extract the eyes

from the faces of 10 females and 10 males, whose ex-

pressions involve standard, smile, anger and scream.

The constant R in the extraction algorithm was cho-

sen as R =0.4. Figures 3 and 4 show the experimen-

tal results for the females, and for the males, respec-

tively. We see from Figures 3 and 4 that many of the

narrow eyes have been extracted. In several examples,

a part of the large eyes has been detected, because

it is similar to that of the training narrow eyes. The

learned ﬁlter never extracts the closed eyes. However,

the eyes of persons wearing the glasses can not be ex-

tracted enough using the present learned ﬁlter. The

learned ﬁlter is a recognizer of narrow eyes.

We also tested our algorithm for some illuminated

faces, which contain large eyes. The experimental re-

sults are shown in Figure 5 for the females, and in

Figure 6 for males. Since a part of the training narrow

eyes is similar to that of large eyes, almost all large

eyes have been extracted, independent of illumination

change. The learning time was 1 msec and the detec-

tion time was 0.1 msec per face, by using the laptop

computer with Pentium M, 1.1GHz.

For comparison, we carried out numerical experi-

ments using the variance-maximization method pro-

posed in (Niijima, 2005) for the same faces shown in

Figures 3 through 6. Although the same training and

target images were used, eyebrows, lips and teeth as

well as the narrow eyes were extracted for many of the

faces. For some facial images, the target eyes could

not be extracted.

Figure 3: Narrow eyes extracting results for females.

FACIAL PARTS RECOGNITION USING LIFTING WAVELET FILTERS LEARNED BY

KURTOSIS-MINIMIZATION

45

Figure 4: Narrow eyes extracting results for males.

7 CONCLUSION

We have proposed a facial parts recognition method.

The method is based on the kurtosis-minimization

learning of the lifting wavelet ﬁlters. Our learning and

recognition algorithms are very fast, because only one

set of free parameters is learned and only one pair of

lifting wavelet ﬁlters with the learned parameters is

applied to a query image for ﬁnding facial parts simi-

lar to the target facial part.

In simulation, we succeeded to extract and recog-

nize the narrow eyes for almost all facial images. The

learned ﬁlter never extracts the closed eyes. Our ﬁl-

ter extracts large eyes for illuminated facial images.

It is a future work to construct a lifting wavelet ﬁlter

recognizable the eyes of a person wearing glasses.

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VISAPP 2006 - IMAGE UNDERSTANDING

46

Figure 5: Extraction results for the illuminated faces of fe-

males.

Figure 6: Extraction results for the illuminated faces of

males.

FACIAL PARTS RECOGNITION USING LIFTING WAVELET FILTERS LEARNED BY

KURTOSIS-MINIMIZATION

47