Yanushkevich S.N., Wang P.S.P., Gavrilova M.L., Srihari S.N. (eds.) Image Pattern Recognition. Synthesis and Analysis in Biometrics
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April 2, 2007 14:42 World Scientific Review Volume - 9in x 6in Main˙WorldSc˙IPR˙SAB
208 Synthesis and Analysis in Biometrics
three parts: first, by perform 2-level wavelettransform with a new
nontensor product wavelet filter, a face images are represented by the
lowest resolution subbands after decomposition. Second, the Principal
Component Analysis (PCA) feature selection scheme is adopted to
reduce the computational complexity of feature representation. Finally,
to test the robustness of the proposed facial feature representation,
the Support Vector Machines (SVM) is applied for classification. The
experimental results show that our method is superior to other methods
in terms of recognition accuracy and efficiency.
Contents
8.1. Introduction ................................... 208
8.2. Construction of Nontensor Product Wavelet Filters Banks . . . . . . . . . 211
8.2.1. Characteristics of Centrally Symmetric Orthogonal Matrix . . . . 212
8.2.2. NontensorProductWaveletFilter.................. 214
8.2.3. Examples................................ 216
8.3. ExperimentalResults.............................. 217
8.4. Conclusions ................................... 221
Bibliography ....................................... 222
Glossary
DNWT — Discrete Nontensor product Wavelet Transform
DSWT — Discrete tensor product Wavelet Transform
KFDA — Kernel Fisher’s discriminant Analysis
KPCA — Kernel Principle Component Analysis
LDA — Linear discriminant Analysis
MRA — Multiresolution analysis
PCA — Principal Component Analysis
NFL — Nearest Feature Line algorithm
NN — Nearest Neighbor algorithm
SVM — Support Vector Machines
8.1. Introduction
Face recognition is a active research area, and they can be used in wide
range of applications such as surveillance and security, telecommunication
and digital libraries, human-computer intelligent interaction, and smart
environments. Compared to classical pattern recognition problems such
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Nontensor-Product-Wavelet-Based Facial Feature Representation 209
as fingerprint recognition, face recognition is much more difficult because
there are usually many individuals(classes), only a few images (samples) per
person, so a face recognition system must recognize faces by extrapolating
from the training samples. Various changes in face images also present a
great challenge, and a face recognition system must be robust with respect
to the many variabilities of face images such as viewpoint, illumination,
and facial expression conditions.
Many novel attempts have been made to face recognition research
since the late 1970s
[
25,26
]
. There are two major approaches for vision
research: geometrical local feature-based (e.g. relative positions of eyes,
nose, and mouth.) schemes and holistic template-based systems and their
variations
[
27
]
. The geometrical-based approach performs successfully in
accurate facial feature detection scheme. However, it remains limited
applications because of its difficult implementation and its unreliability
in some cases. Compared to this approach, template-based approach is
more promising due to its ease of implementation and robustness. In
holistic template-matching systems, attempts are made to capture the
most appropriate representation of face images as a whole and exploit the
statistical regularities of pixel intensity variations. Principal Component
Analysis (PCA)
[
22
]
and Linear discriminant Analysis (LDA)
[
13,22
]
are
the two most classical and popular methods. The PCA is a typical
method, which faces are represented by a linear combination of weighted
eigenvectors, known as eigenfaces
[
1
]
. The LDA obtain features through
eigenvector analysis of scatter matrices with the objective of maximizing
between-class variations and minimizing within-class variations. These two
methods both provides a small set of features that carry the most relevant
information for classification purposes. However, the PCA usually give
high similarities indiscriminately for two images from a single person or
from two different persons and the LDA is also complex as there is a lot of
within-class variation due to differing facial expressions, head orientations,
lighting conditions, etc. Although many improving approaches have been
proposed based on the two methods such as kernel PCA (KPCA)
[
16
]
and
kernel Fisher’s discriminant analysis (KFDA)
[
8,12,24
]
whichusedkernel
skills, the essence problem has not been solved.
As we all known, the main challenge in feature representation is to
represent the input data in a reduced low-dimensional feature space, in
which, the most facial features are revealed or kept. Wavelet transform
has much more advantages on this point. Compared to the PCA and LDA
projections, wavelet subband coefficients can efficiently capture substantial
facial features while keeping computational complexity low. It is well known
to all that wavelet transform has a robust multi-resolution capability which
accords well with human visual system. Moreover, it provides a spatial
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210 Synthesis and Analysis in Biometrics
and a frequency decomposition of a image simultaneously. Using wavelet
transforms as multiscale orthogonal representations of face images, different
components of wavelet decomposition capture different visual aspects of a
gray-scale image. This multi-resolution analysis is provided in the form of
coefficient matrix.
Each face is described by a subset of band filtered images containing
waveletcoefficients. At each level of decomposition there are four orthogonal
subimages corresponding to LL, LH, HL, and HH. By spatial frequency
analysis, the image is represented as a weighted combination of basis
functions, in which high frequencies (LH, HL, HH) carry finely detail
information and low frequency (LL) carry coarse, shape-based information.
Only a change in face will affect all frequency components. Earlier
published research
[
27
]
demonstrated that: the effect of different facial
expressions can be attenuated by removing the high-frequency components
and the low-frequency components only are sufficient for recognition.
Subsequently, an appropriate wavelet transform can result in robust
representations with regard to lighting changes and be capable of capturing
substantial facial features while keeping computational complexity low. It
is increasingly popular used in face representation in recent years and good
results are obtained for race and gender classification
[
21,27
]
.
Almost all the literatures of wavelet based face recognition use two-
dimensional tensor product ones, which is the tensor product of one-
dimensional wavelets. However, the property of anisotropic makes tensor
product wavelet not attractive most for the purpose of facial representation
[
9,4
]
. Nontensor product wavelet, which is, the corresponding scaling
function and associated wavelet function can’t be written in the form of
products of one-dimensional ones, can reveal more features than that of the
common used tensor product wavelet transform
[
4,5,10,11,20
]
. Therefore,
we suggest to represent facial features by discrete nontensor product wavelet
transform (DNWT) in this paper.
Many efforts have been spent on constructing nontensor product
wavelets. However, up to now, there is no systematic method to construct
two-dimensional nontensor product wavelets
[
9,4,20
]
. In this chapter, we
present a novel method for constructing nontensor product wavelet filters.
A new nontensor product bivariate waveletfilter banks with linear phase
are constructed from the centrally symmetric matrices. Our investigations
demonstrate that these filter banks have a matrix factorization and they are
capable of describing the features of face image. The new nontensor product
filters derived from our method are applied for the feature representation
of face image. To test the effect of representation based on the nontensor
product wavelet transform, we design a set of scheme for we use Support
Vector Machines (SVM) for classification. The SVM is a newly powerful
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Nontensor-Product-Wavelet-Based Facial Feature Representation 211
machine learning approach, owing to its remarkable characteristics such as
good generalization performance, the absence of local minimal and sparse
representation of solution. It has become a popular research method in
anomaly detection, and good application are reported
[
8,19
]
a
. Experiments
are tested by using popular used ORL face database. The efficiency of our
approach produced a significant improvement which includes a substantial
advance in correctness and in time of processing comparing with those
obtained by the discrete tensor product wavelet transform (DSWT) and
the well-known conventional PCA and LDA methods.
This chapter is organized as follows: The construction of new
nontensor product bivariate wavelet filter banks are briefly described
in Section 8.2. The feature vector selection algorithm and experiment
result are demonstrated in Section 8.3. Finally, conclusions are drawn in
Section 8.4.
8.2. Construction of Nontensor Product Wavelet Filters
Banks
Multiresolution analysis (MRA) theory provides a natural framework for
understanding wavelets and filter banks. According to MRA, refinable
functions (scaling functions) and wavelets are completely determined by a
low-pass filter and high-pass filters, respectively. In subband code schemes,
a low-pass filter and high-pass filters are respectively used as analysis
filter and synthesis filters which form perfect reconstruction filter banks.
Daubechies
[
7
]
designed univariate two-channel perfect reconstruction filter
banks having finite impulse response (FIR) corresponding to a univariate
orthonormal wavelet having a compact support and vanishing moments. It
is well known that there does not exist an orthonormal symmetric wavelet
with a compact support in the univariate dyadic dilation case, that is, two-
channel perfect reconstruction FIR banks having a linear phase are not
available in the univariate case.
Our interest here is in multivariate filter banks
[
23
]
.
A commonly used method builds multivariate filter banks by the tensor
products of univariate filters. This construction of filter banks focuses
excessively on the coordinate direction.
Therefore, nontensor product approaches for construction of multi-
variate filter banks or wavelets are desirable. Much interest has been given
a
The SVM was developed by V. N. Vapnic (V. N. Vapnic, “The Nature of Statistical
Learning Theory”, Springer, 1995), Remarks of Editors.
April 2, 2007 14:42 World Scientific Review Volume - 9in x 6in Main˙WorldSc˙IPR˙SAB
212 Synthesis and Analysis in Biometrics
to the study of nontensor product wavelets in L
2
(R
d
)
[
5,20
]
,andalso
[
4
]
for constructions of multivariate wavelets on invariant sets) as well as to
multiwavelets and corresponding vector-valued filter banks
[
6,17,18
]
.
However, it is not easy to design multivariate filter banks.
At present, no general method is available for designing multivariate
filter banks and vector-valued filter banks. There are two fundamental
difficulties that one encounters in the design of a low-pass filter
and high-pass filters which are used for the construction of refinable
functions and wavelets, respectively.
Most of the current study in multivariate wavelets is given to a dilation
matrix with determinant two
[
11,23
]
, since in this case, only one high-pass
filter is needed to be construct and the matrix extension is the same as the
univariate two-channel case
[
5
]
.
Often, one seeks filter banks leading to smooth wavelets. However, in
the application of filter banks to texture analysis, experiments show that
“smooth” filter banks are not suitable because texture images is not smooth.
Here we describe a general construction of bivariate nontensor product
wavelet filter banks with linear phase by using centrally symmetric matrices
in
[
3,4
]
. The family of filter banks given in this paper are suitable in this
context although it is difficult to achieve smoothness. These filter banks
have a matrix factorization and can be applied to facial representation.
8.2.1. Characteristics of Centrally Symmetric Orthogonal
Matrix
We consider the following n × n centrally symmetric matrix
B =(b
j,l
)
n
j,l=1
,
where
b
j,l
= b
n+1−j,n+1−l
,j,l=1, 2,...,n.
Similarly, centrally anti-symmetric matrix
˜
B =
˜
b
j,l
n
j,l=1
is defined if
˜
b
j,l
= −
˜
b
n+1−j,n+1−l
,j,l=1, 2,...,n.
The centrally symmetric and central anti-symmetric matrices of order
n are closed related to the special matrix H
n
, which is defined by
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Nontensor-Product-Wavelet-Based Facial Feature Representation 213
H
n
:=
00···1
0 ··· 10
.
.
.
.
.
.
.
.
.
.
.
.
10···0
. (8.1)
B is a centrally symmetric matrix of order n if and only if it satisfies the
matrix equation
H
n
BH
n
= B. (8.2)
Similarly,
˜
B is a centrally anti-symmetric matrix if and only if the
following matrix equation holds
˜
B = −H
n
˜
BH
n
. (8.3)
Note that, for centrally anti-symmetric matrix
˜
B =
˜
b
j,l
n
j,l=1
with odd
number n,the
,
[
n+1
2
], [
n+1
2
]
-
elements
˜
b
[
n+1
2
],[
n+1
2
]
= 0. Throughout this
note, we use the notation [x] to denote the integer no more than the real
number x.
To construct two channel filter banks suitable for image processing,
we need to consider the concrete construction of centrally symmetric
orthogonal matrix of order 4, which corresponds the case n =4. Inthis
case, any centrally symmetric orthogonal matrix B has the general form
B =
1
2
!
I
2
−H
2
H
2
I
2
"!
Z
1
0
0 Z
2
"!
I
2
H
2
−H
2
I
2
"
(8.4)
with Z
1
and Z
2
being orthonormal matrices of order 4. Let
S =
10 0−1
01−10
0110
1001
,
more precisely, we have the following equivalent form
B =
1
2
S
ab00
cd00
00ef
00gh
S
T
(8.5)
with real numbers a, b, c, d, e, f, g, h satisfying
a
2
+ b
2
= c
2
+ d
2
=1,ac+ bd =0,
e
2
+ f
2
= g
2
+ h
2
=1,eg+ fh =0.
The parametrization solutions of above equations for these real numbers
are a = d =cosα, b = −sin α, c =sinα, e = h =cosβ, f = −sin β and
April 2, 2007 14:42 World Scientific Review Volume - 9in x 6in Main˙WorldSc˙IPR˙SAB
214 Synthesis and Analysis in Biometrics
g =sinβ for any real numbers α and β. Therefore, any centrally symmetric
orthogonal matrix B has the more simple parametrization representation
B =
1
2
10 0−1
01−10
0110
1001
cos α −sin α 00
sin α cos α 00
00cosβ −sin β
00sinβ cos β
1001
0110
0 −110
−1001
(8.6)
Further, we defined
Q
(α,β)
:= B
=
1
2
cos α +cosβ −sin α +sinβ −sin α − sin β cos α −cos β
sin α − sin β cos α +cosβ cos α − cos β sin α +sinβ
sin α +sinβ cos α −cos β cos α +cosβ sin α − sin β
cos α −cos β −sin α − sin β −sin α +sinβ cos α +cosβ
.
(8.7)
Thus we get an constructive characterization of centrally symmetric
orthogonal matrix of order 4.
In the following,we will offer some examples of construction of centrally
symmetric orthogonal matrix of order 4,which play an crucial role in the
design of nontensor product bivariate filter banks with two channels. The
first case is to let α = β. Letting α = β =
π
2
and α = β =
π
4
respectively,
we have the following 4 order centrally symmetric orthogonal matrix
Q
(
π
2
,
π
2
)
=
00−10
0001
1000
0 −100
,Q
(
π
4
,
π
4
)
=
√
2
2
10−10
0101
1010
0 −101
.
We consider the case with α =0andβ =
π
2
,wegetthat
Q
(0,
π
2
)
=
1
2
11−11
−1111
111−1
1 −111
8.2.2. Nontensor Product Wavelet Filter
Next by using the above, we will develop a general method for constructing
nontensor product bivariate waveletilter banks.
Given a bivariate trigonometric polynomials
m
0
(ξ,η)=
j∈Z
k∈Z
c
j,k
e
−i(jξ+kη)
, (ξ,η) ∈ R
2
,
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Nontensor-Product-Wavelet-Based Facial Feature Representation 215
its polyphase factors are the bivariate trigonometric polynomials m
0,l
defined for l =0, 1, 2, 3as
m
0,0
(ξ,η)=
j∈Z
k∈Z
c
2j,2k
e
−i(jξ+kη)
,
m
0,1
(ξ,η)=
j∈Z
k∈Z
c
2j+1,2k
e
−i(jξ+kη)
,
m
0,2
(ξ,η)=
j∈Z
k∈Z
c
2j,2k+1
e
−i(jξ+kη)
,
m
0,3
(ξ,η)=
j∈Z
k∈Z
c
2j+1,2k+1
e
−i(jξ+kη)
.
Reversing the process, we can construct the bivariate trigonometric
polynomials m
0
from its polyphase factors m
0,l
, j =0, 1, 2, 3 by the formula
m
0
(ξ,η)=m
0,0
(2ξ, 2η)+e
−iξ
m
0,1
(2ξ, 2η)+e
−iη
m
0,2
(2ξ, 2η)
+ e
−i(ξ+η)
m
0,3
(2ξ, 2η),
where (ξ,η) ∈ R
2
.
The construction of multivariate compactly supported orthonormal
multiwavelets using MRA is equivalent to the design of orthogonal FIR
and QMF filter banks, which leads to the following two questions.
(i) Find low-pass filter m
0
(ξ,η) satisfying the orthogonal condition
|m
0
(ξ,η)|
2
+ |m
0
(ξ + π, η)|
2
+ |m
0
(ξ,η + η)|
2
+|m
0
(ξ + π, η + η)|
2
=1, (ξ, η) ∈ R
2
;
(ii) Find 3 high-pass filter m
1
,m
2
,m
3
such that the matrix M =
(α
0
,α
1
,β
0
,β
1
) is unitary, where α
i
and β
i
are the row vector
(m
0
(ξ + iπ, η),m
1
(ξ + iπ, η),m
2
(ξ + iπ, η),m
3
(ξ + iπ, η))
T
,
and (m
0
(ξ+iπ, η+π),m
1
(ξ+iπ, η+π),m
2
(ξ+iπ, η+π),m
3
(ξ+iπ, η+π))
T
,
(i =0, 1).
Given the orthogonal filter banks m
l
,l =0, 1, 2, 3 at hand, one can use
the Pyramid algorithm to decompose and reconstruct the image. It will
benefit us from the point view of polyphase to understand the conditions
(i) and (ii).
Both of the problems (i) and (ii) (equivalently a and b) are nonlinear
problem in mathematics, which is essentially quadratic algebraic equations
with multiple variables. There is no general solution for this problem
presently. Now we will offer a class of solutions of (i) and (ii) starting
from centrally symmetric matrix.
April 2, 2007 14:42 World Scientific Review Volume - 9in x 6in Main˙WorldSc˙IPR˙SAB
216 Synthesis and Analysis in Biometrics
Let
V
0
=(1, 1, 1, 1)
T
,V
1
=(1, −1, 1, −1)
V
2
=(1, 1, −1, −1)
T
,V
3
=(1, −1, −1, 1)
(8.8)
and denote by D(ξ,η) the matrix of trigonometric polynomial
D(ξ,η)=
10 0 0
0 e
−iξ
00
00e
−iη
0
00 0e
−i(ξ+η)
, (ξ,η) ∈ R
2
.
For any fixed positive integer N , arbitrarily chosen real number pairs
(α
k
,β
k
), k =1, 2,...,N (for k = j,(α
k
,β
k
)mayequalto(α
j
,β
j
)), The
low-pass filter m
0
(ξ,η)isdefinedasfollows:
1
4
1,e
−iξ
,e
−iη
,e
−i(ξ+η)
.
N
k=1
Q
(α
k
,β
k
)
D(2ξ, 2η)U
T
(α
k
,β
k
)
/
V
0
, (8.9)
where Q
(α
k
,β
k
)
is centrally symmetric orthogonal matrix defined previously,
and (ξ, η) ∈ R
2
. It is easy to see that m(0, 0) = 1, which means that m
0
is
alow-passfilter.
Correspondingly,three high-pass filters m
j
,j =1, 2, 3 with respect to
the above low-pass filter m
0
(ξ,η) are defined as follows:
m
1
(ξ,η)=
1
4
1,e
−iξ
,e
−iη
,e
−i(ξ+η)
.
N
k=1
Q
(α
k
,β
k
)
D(2ξ, 2η)U
T
(α
k
,β
k
)
/
V
1
,
m
2
(ξ,η)=
1
4
1,e
−iξ
,e
−iη
,e
−i(ξ+η)
.
N
k=1
Q
(α
k
,β
k
)
D(2ξ, 2η)U
T
(α
k
,β
k
)
/
V
2
,
m
3
(ξ,η)=
1
4
1,e
−iξ
,e
−iη
,e
−i(ξ+η)
.
N
k=1
Q
(α
k
,β
k
)
D(2ξ, 2η)U
T
(α
k
,β
k
)
/
V
3
,
with V
j
defined in (8.8), where j =1, 2, 3, (ξ, η) ∈ R
2
.Itiseasytocheck
that m
j
(0, 0) = 0,j=1, 2, 3. That is to say, m
j
,j =1, 2, 3 are high-pass
filters.
8.2.3. Examples
Now we will provide two concrete important examples of filter banks
obtained by our approach.
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Nontensor-Product-Wavelet-Based Facial Feature Representation 217
Example 1. Using the previous matrix
Q
(0,
π
2
)
=
1
2
11−11
−1111
111−1
1 −111
and setting N = 1, we get the following filter banks
m
0
(x, y)=1/8(−x + y +1+x
2
y
2
+ x
2
− y
2
+ xy − x
3
y + xy
3
+ x
3
y
3
+xy
2
+ x
3
y
2
+ yx
2
− x
2
y
3
+ x
3
+ y
3
),
m
1
(x, y)=1/8(−x + y +1− x
2
y
2
+ x
2
+ y
2
+ xy − x
3
y − xy
3
− x
3
y
3
−xy
2
− x
3
y
2
+ yx
2
+ x
2
y
3
+ x
3
− y
3
),
m
2
(x, y)=1/8(x − y + x
2
y
2
+ x
2
+ y
2
− xy − 1 − x
3
y − xy
3
+ x
3
y
3
−xy
2
+ x
3
y
2
+ yx
2
− x
2
y
3
+ x
3
− y
3
),
m
3
(x, y)=1/8(−x + y +1+x
2
y
2
− x
2
+ y
2
+ xy + x
3
y − xy
3
+ x
3
y
3
−xy
2
+ x
3
y
2
− yx
2
− x
2
y
3
− x
3
− y
3
),
with x = e
−iξ
,y = e
−iη
.
Example 2. Letting α = − β,andα =
π
2
leads to
Q
(
π
2
,−
π
2
)
=
√
2
2
1 −100
1100
0011
00−11
.
Further, setting N = 2, the matrices Q
(0,
π
2
)
and Q
(
π
2
,−
π
2
)
lead to the
following filter banks
m
0
(x, y)=1/4x
3
y
3
+1/4x
2
y
2
+1/8x
2
+1/8yx
2
− 1/8x
3
y +1/8y
5
x
2
+1/8y
4
x
3
+1/8x
3
y
5
+1/8x
3
− 1/8y
4
x
2
,
m
1
(x, y)=−1/4x
3
y
3
+1/4x
2
y
2
+1/8x
2
+1/8yx
2
+1/8x
3
y +1/8y
5
x
2
−1/8y
4
x
3
− 1/8x
3
y
5
− 1/8x
3
− 1/8y
4
x
2
,
m
2
(x, y)=−1/4x
2
y
3
+1/8x
2
+1/8yx
2
+1/4x
3
y
2
− 1/8x
3
y − 1/8y
5
x
2
−1/8y
4
x
3
− 1/8x
3
y
5
+1/8x
3
+1/8y
4
x
2
,
m
3
(x, y)=−1/4x
2
y
3
+1/8x
2
+1/8yx
2
− 1/4x
3
y
2
+1/8x
3
y − 1/8y
5
x
2
+1/8y
4
x
3
+1/8x
3
y
5
− 1/8x
3
+1/8y
4
x
2
.
8.3. Experimental Results
By using the proposed facial feature representation, we develop a new
nontensor-product-wavelet-based face recognition scheme by combining
the techniques of PCA and SVM. To test the robustness of the
proposed nontensor-product-wavelet-based facial feature representation,
the following experiments are conducted by ORL face database, which