Journal of Machine Learning Research 8 (2007) 1165-1195 Submitted 8/06; Revised 1/07; Published 5/07
Local Discriminant Wavelet Packet Coordinates for Face Recognition
Chao-Chun Liu
Dao-Qing Dai∗ Center for Computer Vision and Department of Mathematics
Sun Yat-Sen (Zhongshan) University
Guangzhou, 510275 China
Hong Yan †
Department of Electric Engineering
City University of Hong Kong
83 Tat Chee Avenue
Kowloon, Hong Kong, China
Editor: Donald Geman
Abstract
STSDDQ@MAIL.SYSU.EDU.CN
H.YAN@CITYU.EDU.HK
Face recognition is a challenging problem due to variations in pose, illumination, and expression.
Techniques that can provide effective feature representation with enhanced discriminability are crucial.
Wavelets have played an important role in image processing for its ability to capture localized
spatial-frequency information of images. In this paper, we propose a novel local discriminant coordinates
method based on wavelet packet for face recognition to compensate for these variations.
Traditional wavelet-based methods for face recognition select or operate on the most discriminant
subband, and neglect the scattered characteristic of discriminant features. The proposed method selects
the most discriminant coordinates uniformly from all spatial frequency subbands to overcome
the deficiency of traditional wavelet-based methods. To measure the discriminability of coordinates,
a new dilation invariant entropy and a maximum a posterior logistic model are put forward.
Moreover, a new triangle square ratio criterion is used to improve classification using the Euclidean
distance and the cosine criterion. Experimental results show that the proposed method is robust for
face recognition under variations in illumination, pose and expression.
Keywords: local discriminant coordinates, invariant entropy, logistic model, wavelet packet, face
recognition, illumination, pose and expression variations
1. Introduction
Face recognition (Zhao et al., 2003; Jain et al., 2004) has become one of the most active research
areas in pattern recognition. It plays an important role in many application areas, such as humanmachine
interaction, authentication and surveillance. However, the wide-range variations of human
face, due to pose, illumination, and expression, result in a highly complex distribution and deteriorate
the recognition rate. It seems impractical to collect sufficient prototype images covering all the
possible variations. Therefore, how to construct a small-size-training face recognizer robust to environmental
variations is a challenging research issue. Wavelets have been successfully used in image
∗. Also Department of Electric Engineering, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong,
China. Dao-Qing Dai is the corresponding author.
†. Also School of Electrical and Information Engineering, University of Sydney, NSW 2006, Australia.
c○2007 Chao-Chun Liu, Dao-Qing Dai and Hong Yan.
LIU, DAI AND YAN
processing. Their ability to capture localized spatial-frequency information of image motivates us
to use them for feature extraction. In this study, we investigate a new approach by extracting the
features not sensitive to environmental changes from a wavelet packet dictionary.
Generally, feature extraction, discriminant analysis and classifying criterion are the three basic
elements of a face recognition system. The performance and robustness of face recognition could
be enhanced by improving these elements. Feature extraction in the sense of some linear or nonlinear
transform of the data with subsequent feature selection is commonly used for reducing the
dimensionality of facial image so that the extracted features are as representative as possible. A
lot of work on face recognition has been carried out based on similarities analysis (P. Howland
and Park, 2006; Belhumeur et al., 1997; Jiang et al., 2006; Martinez and Zhu, 2005; Vaswani and
Chellappa, 2006; Xiang et al., 2006; Zhao et al., 2003). A well-known feature extraction method is
called FisherFace, based on linear discriminant analysis (LDA), which linearly projects the image
space to a low-dimensional subspace so as to discount environmental variations (Belhumeur et al.,
1997; Fukunaga, 1990). This method is a statistical linear projection method which largely relies on
the representation of the training samples. On the other hand, wavelet-based methods with no special
focus on the training data have been used for feature extraction (Mallat, 1989; Coifman et al.,
1992). The decomposition of the data into different frequency ranges allows us to isolate the frequency
components introduced by intrinsic deformations due to expression or extrinsic factors (like
illumination) into certain subbands. Wavelet-based methods prune away these variable subbands,
and focus on the subbands that contain the most relevant information to better represent the data.
WaveletFace (Chien and Wu, 2002) only uses the low-frequency subband to present the basic figure
of an image, and ignores the efficacy of high-frequency components. Our previous study (Dai and
Yuen, 2006) uses a wavelet enhanced regularized discriminant analysis after dimensionality reducing
with low-pass filter to solve the small sample size problem, which is also a method based on the
low frequency subband. Similarly, some other studies (Feng et al., 2000; Ekenel and Sanker, 2005;
Zhang et al., 2004, 2005) employ the traditional transform (e.g., ICA, PCA, Neural Networks) to
enhance the discriminant power in one or several special subbands, the latter always fuse the discriminant
power in these different subbands for final classification (Ekenel and Sanker, 2005; Zhang
et al., 2005). Moreover, as a generalization of the wavelet transform, the wavelet packet not only
offers an attractive tool for reducing the dimensionality by feature extraction, but also allows us
to create localized subbands of the data in both space and frequency domains. Saito and Coifman
introduced the local discriminant basis (LDB) algorithm based on a best-basis paradigm to search
for the most discriminant subbands (basis) that illuminates the dissimilarities among classes from
the wavelet packet dictionary (Coifman and Saito, 1994; Saito and Coifman, 1994, 1995). Some
studies (Saito et al., 2002; Stranss et al., 2003) constructed the modified LDB later. In Kouzani
et al. (1997), the best-basis algorithm of Coifman and Wicherhauser (1992) is used to search for
the wavelet packet basis for face representation. In Bhagavatula and Savvides (2005), PCA is performed
in wavelet packet subbands and the subbands which generalize better across illumination
variations for face recognition are sought. All the methods on these studies are based on the whole
discriminant subband.
It is known that a good feature extractor for a face recognition system is claimed to select as
many the best discriminant features as possible, which are not sensitive to arbitrary environmental
variations. Nastar and Ayach (1996) investigated the relationships between variations in facial appearance
and their deformation spectrum. They found that facial expressions and small occlusions
affect the intensity manifold locally. Under frequency-based representation, only high-frequency
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LOCAL DISCRIMINANT WAVELET PACKET COORDINATES FOR FACE RECOGNITION
spectrum is affected. Moreover, changes in pose or scale of a face and most illumination variations
affect the intensity manifold globally, in which only their low-frequency spectrum is affected. Only
a change in face will affect all frequency components (Zhang et al., 2004). So there are no special
subbands whose all coordinates are not sensitive to these variations. In each subband, there may
be only segmental coordinates which have enough discriminant power to distinguish different person,
the remainder may be sensitive to environmental changes, but the methods based on the whole
subband will also extract these sensitive features. Moreover, there may be no special subbands
containing all the best discriminant features, because the features not sensitive to environmental
variations are always distributed in different coordinates of different subbands locally. The methods
based on the segmental subbands will lose some good discriminant features. Furthermore, in different
subbands, the amount and distribution of best discriminant coordinates are always different.
Many less discriminant coordinates in one subband may add up to a larger discriminability than
another subband whose discriminability is added up with few best discriminant coordinates and
residual small discriminant coordinates (Saito et al., 2002), then the few best discriminant coordinates
will be discarded by the methods which search for the best discriminate subbands, but only
the few best discriminant coordinates are needed. So the best discriminant information selection
should be independent of their seated subbands, and only depends on their discriminability for face
recognition. However, the methods based on the whole subband neglect the distribution of features,
they are deficient to select the best discriminant features sometimes.
Moreover, how to measure the discriminability of coordinate is one crucial element of the whole
algorithm. We translate it into the separability of each coordinate-loading ensemble, and propose
a new dilation invariant entropy which is independent of the order of magnitude (OM), instead of
deficient absolute “distance” measures. Furthermore, we construct a maximum a posterior (MAP)
logistic model to produce a separability measure function which presents factually the separability
of each coordinate-loading ensemble, that is, discriminability of each coordinate. Based on the new
dilation invariant entropy and its derived separability measure function, any two coordinates are
comparable for their discriminability, either they locate in the same subband or different subbands.
To solve the “small sample size” (SSS) problem, we use the complete linear discriminant analysis
(CLDA) idea (Yang et al., 2005) which captures both regular and irregular discriminant information
and makes a more powerful discriminator. For classifying criterion, the traditional Euclidean
distance cannot measure the similarity very well when there exist illumination variations on facial
images, and the cosine criterion is unsatisfactory when there exist pose and expression changes.
Thus, we propose a new triangle square ratio criterion. Experimental results show that it can overcome
the deficiency of the Euclidean distance and cosine criterion very well.
In this paper, to deal with illumination, pose and expression problems, we propose a new local
discriminant coordinates (LDC) algorithm to select uniformly the most discriminant independent
coordinates in all spatial frequency subbands for face recognition, in order to overcome the limitation
of the methods based on whole subband. Experimental results show that our LDC feature
extraction has almost overcome the shortcomings of the methods based on subband and improves
the effect of feature extraction for face recognition under different environmental variations.
The contribution of this paper consists of the following:
• Further extension of wavelets to face recognition to deal with illumination, pose and expression
problems.
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LIU, DAI AND YAN
• Introduction of a dilation invariant entropy and a maximum a posterior logistic model for
selection of wavelet packet coordinates.
• Use of a new similarity criterion coupled with the nearest neighbor classifier.
• Design of a face recognition system, which solves the small sample size problem and is robust
to variations in illumination, pose and expression.
The paper is organized as follows. In Section 2, the wavelet packet decomposition and the
local discriminant basis algorithm will be introduced. Our proposed algorithm and the whole procedure
will be presented in Section 3. In Section 4, experimental results are presented, followed by
discussions and conclusion in Section 5.
2. Feature Extraction by Local Discriminant Basis
In this section, we first make a review on the wavelet packet decomposition, then the local discriminant
basis (LDB) algorithm and the modified LDB algorithm are introduced.
2.1 The Wavelet Packet Decomposition
Wavelets are functions that satisfy certain mathematical requirements and are used as basis functions
in representing data at different scales and time-frequency locations. Wavelets (Kouzani et al., 1997;
Vaidyanathan, 1993) can be generated from a two-channel filter bank method which uses repeated
filtering and downsampling to decompose signals into time-frequency subbands. The two-channel
filter bank has a lowpass filter which removes the high frequencies and a highpass filter which
removes the low frequencies. For the wavelet transform, only the lowpass filtered subband is further
iterated. As a generalization of the wavelet transform, the two-channel filter banks are iterated over
the lowpass and the highpass subbands in the wavelet packet decomposition. This generates a tree
structure which provides many possible wavelet packet bases, accordingly, signals are decomposed
into a time-frequency dictionary.
When dealing with images, the wavelet decomposition or the wavelet packet decomposition
is first applied along the rows of the images, then their results are further decomposed along the
columns. This results in four decomposed subimages L1, H1, V1 and D1. These subimages represent
different frequency localizations of the original image which refer to Low-Low, Low-High,
High-Low and High-High respectively. Their frequency components comprise the original frequency
components but now in distinct ranges. While the process being iterated, only L1 is further
decomposed in the wavelet decomposition, but all L1, H1, V1 and D1 are further decomposed in the
wavelet packet decomposition. Figure 1 shows a two-dimensional examples of a facial image for
the wavelet decomposition and the wavelet packet decomposition with depth 2.
2.2 The Local Discriminant Basis (LDB) Algorithm
The local discriminant bases algorithm (Coifman and Saito, 1994; Saito and Coifman, 1994, 1995)
uses an adjustment of dictionary, or a wavelet packet decomposition tree which offers a library of
orthonormal basis localized both in space and in frequency. Before proceeding further, let us set our
notations. Let X = {x1,x2,··· ,x N} be an ensemble of training samples with K classes, X = S K
and Xy = {x y
1 ,xy
2 ,···xy
y=1Xy,
Ny}, where Ny is the number of samples belong to class y, and N = ∑ K
y=1 Ny.
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LOCAL DISCRIMINANT WAVELET PACKET COORDINATES FOR FACE RECOGNITION
Figure 1: (Top) The two-dimensional wavelet decomposition of facial image with depth 2. (Bottom)
The two-dimensional wavelet packet decomposition of facial image with depth 2.
We use D to represent the space-frequency dictionary consisting of a collection of wavelet packet
subbands {B j}, j = 1,··· ,(4L+1 − 1)/3, where B j = {b j1,bj2,··· ,bjn j }, b ji(i = 1,2,··· ,nj) are
wavelet packet coefficients and n j is the size of wavelet packet subband B j, L is the decomposition
level of wavelet packet.
The LDB algorithm first decomposes the training samples in the dictionary D, then sample
energies at the basis coordinates are accumulated for each sample class separately to form a spacefrequency
energy distribution per class. Let Γ (y)
(B j) be a normalized energy of class y samples
presented on the subbands B j:
Γ (y)
(B j) = (Γ (y)
(b j1),Γ (y)
(b j2),··· ,Γ (y)
(b jn j )) ∀B j ⊂ D, (1)
i=1
Γ (y)
(b jt) Δ = ∑Ny
�
�b jt · x y�
�
i
2
∑ Ny
i=1 �xy
i �2
where · denotes the standard inner product in the Euclidean space. The loss function φ1 is used to
measure “distances” among K vectors Γ (1)
(B j),Γ (2)
(B j),··· ,Γ (K)
(B j):
φ1(B j) = φ1(Γ (1)
(B j),Γ (2)
(B j),··· ,Γ (K)
(B j)) Δ =
(2)
K
∑ d
m,n=1
m�=n
∗ (Γ (m)
(B j),Γ (n)
(B j)) (3)
where d ∗ (·,·) is a “distance” measure, it can be the l 2 distance, the relative entropy, or the J-
Divergence. Then φ1(B j) will be a measure of efficacy of the subband B j for classification, and
local discriminant basis are selected by the best-basis algorithm (Coifman and Wicherhauser, 1992)
using the following criterion:
Ψ = argmax
B j ∈D φ1(B j). (4)
The final step is to construct traditional discriminant analysis (e.g., LDA, CT) with features derived
from the LDB feature extraction.
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2.3 The Modified LDB (MLDB) Algorithm
LIU, DAI AND YAN
In Saito et al. (2002), a modified version of the LDB algorithm is introduced using the empirical
probability distributions instead of the space-frequency energy distribution as their selection strategy
to eliminate some less discriminant coordinates in each subband locally. Let
δ jt Δ = φ2(Γ (1)
(b jt),Γ (2)
(b jt),···Γ (K)
(b jt)) =
K
∑ d
m,n=1
m�=n
∗ (Γ (m)
(b jt),Γ (n)
(b jt)) (5)
that is, the discriminability of coordinate b jt (t = 1,2,··· ,nj). Then the measure of the discriminability
of B j is obtained by summing only the n0(< n j) largest terms, that is,
φ2(B j) Δ =
n0
∑ δ j(t)
t=1
where {δ j(t)} is the decreasing rearrangement of {δ jt}, and local discriminant basis are selected by
the best-basis algorithm using the criterion (4) as LDB. The final step is the same as LDB.
Although the MLDB algorithm may overcome some limitations of LDB, the selection of coordinates
is only limited to each subband so that coordinates in different subbands are still incomparable.
3. The General Framework of the LDC Algorithm
Our LDC algorithm uses a ternary architecture similar to LDB. We use the wavelet packet feature
extraction at the first step. The main difference between LDB and our LDC algorithm is the nature
of “distance” measure and feature selection strategy. We propose a new dilation invariant entropy to
take the place of traditional traditional absolute “distance” measures. This ensures that the comparison
of discriminability among all coordinates is independent of spatial frequency subbands. Thus,
our selection can be based on all coordinates of the dictionary, but not the subbands themselves.
Moreover, LDB uses only the between-class difference, and ignores the within-class difference.
This may lead to an unsatisfactory discriminability. The solution presented in this paper makes use
of the maximum a posterior (MAP) logistic model. Its derived separability measure function will
get a contrastive term to ensure not only the within-class difference is low, but also the betweenclass
difference is large. Our LDC algorithm does not need the best-basis algorithm (Coifman and
Wicherhauser, 1992) used in LDB, it ensures that we can select the most discriminant features without
any impact of the best-basis algorithm. Subsequently, the LDC algorithm uses the complete
linear discriminant analysis (CLDA) to solve the “small sample size” (SSS) problem, instead of
the traditional LDA or CT in LDB. Finally, we modify the Euclidean distance and the cosine criterion
in the nearest neighbor classifier, and replace them with the triangle square ratio criterion for
classification.
3.1 The Wavelet Packet Decomposition in our LDC Algorithm
In the LDC algorithm, the wavelet packet technique is used to decompose an image into subbands
that are localized in both space and frequency domains, and offers a choice of optimal coordinates
for the representation of a human face. Therefore, it is possible to seek the most discriminative
coordinates for classification. Because each child subband is derived from its parent subband at
the above level, the coordinates in the two levels are linearly dependent. In the first experiment in
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Level 1
Level 2
Level 3
LOCAL DISCRIMINANT WAVELET PACKET COORDINATES FOR FACE RECOGNITION
Original Image
L1 H1
V1 D1
LL2 LH2 LV2 LD2
LLL3 LLH3 LLV3 LLD3 LHL3 LHH3 LHV3 LHD3 LVL3 LVH3 LVV3 LVD3 LDL3 LDH3 LDV3 LDD3
Figure 2: The wavelet decomposition tree used in this study. The dashed part is the spatialfrequency
dictionary D in the LDC algorithm
Section 4, we will search for the most discriminant level by the best performance of its selected
coordinates. Because it is more time-consuming when the decomposition level L is larger than 4,
L = 4 will be used in the experiment. In the first level, four subband images—L1, H1, V1, D1—are
obtained. However, the high frequency H1, V1, D1 are sensitive to noises in facial images, and
Ekenel and Sanker (2005) claimed that they have low performance for classification. Moreover, the
results in Table 4 show that our dilation invariant entropy used in the LDC algorithm may extract
few high frequency components which may slightly affect the performance, also for the sake of
computational efficiency, the H1, V1, D1 components are not further decomposed. Our experimental
results show that Level 3 has better performance than Level 1, 2, 4, and the same results are also
presented in Chien and Wu (2002). In fact, with the further wavelet packet decomposition, more
fine scale information which may have good discriminant power is generated, however, the resolution
of subband images becomes lower so that less information exists for the purpose of object
localization (Grewe and Brooks, 1997). Neither little scale information nor little localization information
can generate a judicious combination which has best discriminate power, so Level 3 which
may give a suitable tradeoff between scale information and localization information is used in some
studies (Chien and Wu, 2002; Feng et al., 2000). We also use Level 3 in the LDC algorithm, and
our spatial-frequency dictionary D consists of 16 subbands in Level 3 (a subset of the dictionary in
the LDB algorithm) (see Figure 2). The Daubechies db4 wavelet will be used for image decomposition
(Daubechies, 1990), if the sizes of facial images are not the dyadic numbers, we will apply
zero-padding extension to create the smallest dyadic images for the wavelet packet decomposition.
3.2 The Dilation Invariant Entropy
In this subsection, we first point out the deficiency of absolute “distance” measures in wavelet-based
methods, then introduce our dilation invariant entropy and its property.
3.2.1 DEFICIENCY OF ABSOLUTE “DISTANCE” MEASURES IN WAVELET-BASED METHODS
To introduce our new dilation invariant entropy, we first list several traditional discriminant measures.
Given two nonnegative sequences w = (w1,w2,··· ,wn), z = (z1,z2,··· ,zn), the square ℓ 2 -
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norm is defined by
d sℓ2
LIU, DAI AND YAN
(w,z) Δ = �w − z� 2 n
2 = ∑
i=1
(wi − zi) 2 . (7)
Suppose ∑ n i=1 wi = 1,∑ n i=1 zi = 1, then the Kullback-Leibler divergence (Kullback and Leibler,
1951), also known as relative entropy, is defined by
d KLD (w,z) Δ =
n
∑ wi log
i=1
wi
zi
with the convention that log0 = −∞, logγ/0 = ∞ for γ > 0 and 0(±∞) = 0. A symmetric version
of d KLD is the J-Divergence (Kullback and Leibler, 1951) given by
d JDIV (w,z) Δ = dKLD (w,z) + dKLD (z,w)
. (9)
2
It is easy to show that measures in Equations (7)-(9) are additive discriminant measure, that is,
d ∗ (w,z) =
n
∑ d
i=1
∗ (wi,zi) (∗=sℓ2 ,KLD,JDIV ). (10)
From Equations (3) and (10), we know that the discriminant measure of subband B j in the LDB
algorithm can be written as
φ1(B j) =
K
∑
m,n=1
m�=n
n j
∑
t=1
(8)
d ∗ (Γ (m)
(b jt),Γ (n)
(b jt)). (11)
Also from Equations (5) and (6), we know that the discriminant measure of subband B j in the
MLDB algorithm can be written as
φ2(B j) =
K
∑
m,n=1
m�=n
n0
∑
t=1
d ∗ (Γ (m)
(b j(t)),Γ (n)
(b j(t))), (n0 < n j) (12)
where b j(t), (t = 1,··· ,n0) are the first n0 coordinates with largest discriminability in subband B j.
However, there are no normalized conditions imposed in each subband when the decomposition
level L > 0, because for each subband B j( j > 1) (B1 is the original image)
n j
∑
t=1
Γ (y)
(b jt),
n0
∑
t=1
Γ (y)
(b j(t)) < 1 and
n j
∑
t=1
Γ (y)
(b jt) �≡ C0,
n0
∑
t=1
Γ (y)
(b j(t)) �≡ C1 ∀y
where C0,C1 are constants independent of y and B j. Without the normalized conditions, the absolute
“distance” measures (7)-(9) will lead to a jeopardy that φ1(B j) and φ2(B j) depend absolutely on the
order of magnitude (OM) of Γ (m)
(b jt), Γ (n)
(b jt) and Γ (m)
(b j(t)), Γ (n)
(b j(t)) respectively.
Unfortunately, we find that the OM of coordinate loadings make much difference between lower
spatial frequency subbands and higher spatial frequency subbands. For example, the coordinate
loadings in the first spatial frequency subband B6 (=LL2 in Figure 2) of the second level may vary
from 0.1 to 10, and the coordinate loadings in the second spatial frequency subband B7 (=LH2 in
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LOCAL DISCRIMINANT WAVELET PACKET COORDINATES FOR FACE RECOGNITION
B6
B7
Γ (1)
(b6t) Γ (2)
(b6t) Γ (1)
(b7t) Γ (2)
(b7t)
1.47e-04 3.86e-04 6.72e-08 4.90e-07
8.04e-05 3.57e-04 4.13e-07 5.04e-06
2.07e-04 3.91e-04 1.32e-06 1.10e-07
1.06e-04 3.57e-04 5.45e-07 7.02e-08
1.07e-04 3.83e-04 1.43e-07 5.32e-08
2.05e-04 3.57e-04 5.20e-09 7.14e-08
Table 1: Values of Γ (y)
(b6t),Γ (y)
(b7t) (y = 1,2; t = 1,··· ,6)
Figure 2) of the second level may vary from 0.001 to 0.01. Table 1 lists an example of some values
of Γ (y)
(b6t),Γ (y)
(b7t) computed by Equation (2) in latter experiment.
From Equations (11),(12) and (2), we can deduce a bad result that the coordinates in lower
spatial frequency subbands have more discriminability because of the larger OM of theirs loadings,
and the coordinates in higher spatial frequency subbands have less discriminability because of the
smaller OM of theirs loadings. So the low spatial frequency subbands are dominant in the LDB
and MLDB algorithm. However, it is unreasonable to neglect the middle and high spatial frequency
components merely for small OM of their loadings. Our experimental results also show that not
only low spatial frequency components, but also middle spatial frequency components are useful
for face recognition.
3.2.2 THE DILATION INVARIANT ENTROPY AND ITS PROPERTY
First, we define that the separability of sample ensemble X is the probability of classifying all samples
into their genuine classes by certain discriminant functions. It is well-known that the separability
of X does not depend on the absolute distances based on the OM of sample values, but depends
on the relative distances among all the samples in X. For each coordinate c, the coordinate loadings
from all the training samples can induce a sample ensemble X c in R 1 , and the discriminability of c
is equivalent to the separability of X c , so it is independent of the OM of coordinate loadings, and
only depends on the relative distances among the coordinate loadings from all the training samples.
Obviously, the “distance” measures used in LDB do not take this fact into account. So we propose
a new “distance” measure derived from the J-Divergence. We call it the dilation invariant entropy :
d DIE (w,z) Δ =
n
∑
i=1
1
2(wi + zi) (wi
zi
logwi + zi
logzi) =
wi
n
∑
i=1
(wi − zi)(logwi − logzi)
2(wi + zi)
where w = (w1,w2,··· ,wn), z = (z1,z2,··· ,zn) are two nonnegative sequences, with the convention
that log0 = −∞, logγ/0 = ∞ for γ > 0 and 0(±∞) = 0.
Proposition 1 The new relative entropy defined by Equation (13) is dilation invariant.
Proof Suppose the dilation transform f : w ∈ R n → f (w) = aw ∈ R n , a(> 0) is a dilation constant,
then
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LIU, DAI AND YAN
d DIE ( f (w), f (z)) = d DIE n
(aw,az) = ∑
i=1
n
= ∑
i=1
n
= ∑
i=1
(awi − azi)(logawi − logazi)
2(awi + azi)
a(wi − zi)(logwi + loga − logzi − loga)
2a(wi + zi)
(wi − zi)(logwi − logzi)
2(wi + zi)
= d DIE (w,z).
In fact, the new dilation invariant entropy is the generalization of the J-Divergence (when the
J-Divergence satisfies the constraint: w + z = 1) because
where w ′ i
= wi
wi+zi ,z′ i
d DIE (w,z) =
n
∑
(wi − zi)(logwi − logzi)
2(wi + zi)
i=1
n
= ∑
i=1
= 1
2 (
n
∑ w
i=1
′ i log w′ i
z ′ i
= d JDIV (w ′ ,z ′ )
1 wi
( −
2 wi + zi
zi
)(log
wi + zi
wi
− log
wi + zi
+
n
∑
i=1
z ′ i log z′ i
w ′ )
i
zi
wi + zi
)
(14)
= zi
wi+zi , w′ i +z′ i = 1 and w′ = (w ′ 1 ,··· ,w′ n),z ′ = (z ′ 1 ,··· ,z′ n). Equation (14) shows
that the dilation invariant entropy normalizes the sample ensembles into unit sample ensembles with
the sum formalism, so different sample ensembles are comparable for their separability. Similarly,
we can define the dilation invariant ℓ 2 norm as
d DIℓ2
(w,z) =
�
n
∑(
i=1
wi
−
wi + zi
zi
)
wi + zi
2
� 1
2
=
�
n
∑(w
i=1
′ i − z ′ i) 2
� 1
2
= d ℓ2
(w ′ ,z ′ ). (15)
In Section 4, we will conduct an experiment to test the performance of the LDC algorithm using
both dilation invariant entropy and dilation invariant ℓ 2 norm.
The dilation-invariance of the new relative entropy ensures that the separability of each coordinateloadings
ensemble X c is independent of its OM. Accordingly, the discriminability of each coordinate
c can be independent of its corresponding subband. It offers a benefit that any two coordinates in
the dictionary D are comparable for their discriminability. So all the coordinates in the dictionary
can be uniformly selected by a criterion.
3.3 Feature Selection Criterion
Sometimes, maximizing the between-class difference or minimizing the within-class difference
alone leads to a bad result. So in the LDC algorithm, our separability measure function will contain
a term to maximize the between-class difference and minimize the within-class difference simultaneously.
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3.3.1 THE MAXIMUM a posteriori (MAP) LOGISTIC MODEL
To select the most discriminant coordinates, we make use of the Bayesian algorithm based on minimizing
the error on the training set. The Bayesian algorithm adopts a probabilistic measure of
similarity based on a Bayesian MAP analysis of face differences. In the traditional methods (Wang
et al., 2006; Chou, 2000), the similarity measure is used to characterize what kind of image variation
is typical for the same person and what is for different persons. In this paper, the MAP similarity
measure is used to choose the coordinates that make the training data set with known class labels
having the minimum error. In this way the selected coordinates can make the known classification
of training data set the most probable:
ĉ = argmax
c
K
∑
y=1
Z
1
#(Xy)
X
Pc(Xy|x)1(x ∈ Xy)dP(x)
where #(·) is the cardinal number, and 1(·) is the indicator function. The posterior probability
Pc(Xy|x) can be rewritten as
Pc(Xy|x) = Pc(x|Xy)Pc(Xy)/Pc(x) = Pc(x|Xy)Pc(Xy)/P(x).
Since P(x) is not a function of the class index and thus has no effect in the MAP decision, the needed
probabilistic knowledge can be represented by the class prior distribution Pc(Xy) and the conditional
probability Pc(x|Xy) which will be modeled by logistic functions.
Definition 1 The prior distribution Pc(Xy) is defined as
Pc(Xy) = 1
T
T =
Definition 2 The conditional probability Pc(x y
Pc(x y
i |Xy) =
1
1 + exp(−d DIE (Γ (y) (c),Γ (0) (c))) ,
K
1
∑
y=1 1 + exp(−d DIE (Γ (y) (c),Γ (0) (c))) .
i |Xy)(x = x y
i
1
) is defined as
1 + exp(d DIE (Γ (y)
i (c),Γ(y) (c))
where Γ (y) (c) is defined by Equation (2), representing the normalized spatial-frequency energy map
of class y on coordinate c, and can be thought of as the center of class y. Similar to that of LDB, we
set
Γ (y)
i (c) Δ |c · xy i = |2
�x y , Γ(0)
i�2 (c) Δ = ∑Ky=1
∑ Ny
i=1 |c · xy i |2
∑ K
y=1 ∑ Ny
(16)
(17)
i=1 �xy
. (18)
i�2 Γ (y)
i (c) represents the normalized spatial-frequency energy map of sample xy i on coordinate c, and
Γ (0) (c) represents the normalized spatial-frequency energy map of all the training samples on coordinate
c, which can be considered as the center of all samples.
The properties of the probability functions Pc(Xy) and Pc(x y
i |Xy) can be made clear by consider-
ing the sigmoid function:
f (d) =
1
1 + exp(−γd + θ)
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LIU, DAI AND YAN
with θ normally set to zero and γ set to 1 for Pc(Xy) and -1 for Pc(x y
i |Xy). When γ = 1, f (d) is
a monotonically increasing function, a larger d DIE
(Γ (y)
(c),Γ (0)
(c))) means that it is more probable
to separate class set Xy. Contrarily, when γ = −1, f (d) is a monotonically decreasing function, a
smaller d DIE
(Γ (y)
i (c),Γ(y)
(c)) means that sample x y
i is more likely to belong to class set Xy. In fact,
the idea of MAP logistic model is derived from the Fisher criterion. Moreover, the sigmoid function
can effectively allay the effect of outliers which have great effect on the Fisher criterion.
3.3.2 SEPARABILITY MEASURE AND FEATURE SELECTION CRITERION
For the given training data set, the empirical probability measure P(x) defined on the training data set
is a discrete probability measure that assigns equal mass at each sample. We define the separability
measure as
SM(c) =
≈
=
K
∑
y=1
K
∑
y=1
K
∑
y=1
Z
1
#(Xy)
1
Ny
X
Pc(Xy|x)1(x ∈ Xy)dP(x)
Ny
∑ Pc(x
i=1
y
i |Xy)Pc(Xy)
�
1
Pc(Xy)
Ny
Ny
∑ Pc(x
i=1
y
i |Xy)
In fact, the separability measure defined by Equation (19) is an empirical measure. If the training
samples are obtained by an independent sampling from a space with a fixed probability distribution
P0(x), the empirical probability distribution P(x) will converge to P0(x) in distribution as N → ∞.
Then the empirical measure defined on the N independent training samples will converge to the
expected measure as the sample size N increases. With sufficient training samples, the empirical
measure is an estimate of the expected measure. The goodness of this estimate is determined by the
training sample size N and the convergence rate of the empirical probability measure P(x) to the
limit distribution P0(x).
Furthermore, we use the following criterion for feature selection:
Criterion: Select uniformly the first N0 coordinates from the dictionary D with largest separability
measure defined by Equation (19).
3.4 Discriminant Analysis
LDA (Fukunaga, 1990) is a linear statistic classification method, which tries to find a linear transform
so that after its application the scatter of sample vectors is minimized within each class and
the scatter of mean vectors around the total mean vector is maximized simultaneously.
Let the between-class scatter operator Sb and the within-class scatter operator Sw be:
Sb = 1
N
K
∑
y=1
Ny(my − m0)(my − m0) T , Sw = 1
N
K
�
Ny
∑ ∑
y=1 i=1
.
(x y
i
y
T
− my)(xi − my)
where my is the mean of the mapped training sample of class y, and m0 is the mean across all the
mapped training samples. Then the Fisher criterion function can be defined by
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LOCAL DISCRIMINANT WAVELET PACKET COORDINATES FOR FACE RECOGNITION
J1(ϕ1) = ϕT1 Sbϕ1
ϕT 1 Swϕ1
, (ϕ1 �= 0,�ϕ1� = 1). (20)
The solution to maximizing J1(ϕ1) can be found by searching for a direction which maximizes the
projected class means (the numerator) while minimizing the class variances in this direction (the
denominator).
However, the LDA algorithm often suffers from the “small sample size” (SSS) problem which
exists in high-dimensional pattern recognition tasks, where the number of available samples is
smaller than the dimensionality of the samples. Many methods (Mika et al., 1999; Baudat and
Anouar, 2000; S. Mika and Müller, 2003; Yang, 2002) discard the discriminant information contained
in the null space of Sw. But a significant result is a finding that there exists crucial discriminative
information in the null space of Sw (Chen et al., 2000; Zhuang and Dai, 2007; Yang and
Yang, 2003; Yu and Yang, 2001). We proposed the use of regularization (Dai and Yuen, 2003), but
it involves a determination of parameters. Yang et al. (2005) proposed a complete kernel Fisher
discriminant analysis algorithm which makes full use of two kinds of discriminant information, regular
and irregular in kernel feature space. Its advantage is that no estimation of parameter is needed.
Based on their idea, we use complete linear discriminant analysis (CLDA) in the LDC algorithm to
solve the SSS problem.
In Equation (20), if the within-class scatter operator Sw is invertible, ϕ T 1 Swϕ1 > 0 always holds
for every nonzero vector ϕ1, and the Fisher criterion can be directly employed to extract a set of
optimal discriminant vectors. If Sw is singular, there always exist vectors satisfying ˜ϕ T Sw ˜ϕ = 0.
These vectors are from the null space of Sw (null(Sw)) and can be very effective if they satisfy
˜ϕ T Sb ˜ϕ > 0 at the same time (Chen et al., 2000; Zhuang and Dai, 2007; Yang and Yang, 2003; Yu
and Yang, 2001). In this case, the Fisher criterion degenerates into the following between-class
scatter criterion:
J2(ϕ2) = ϕ T 2 Sbϕ2, (�ϕ2� = 1). (21)
CLDA uses the between-class scatter criterion defined in Equation (21) to derive the irregular discriminant
vectors from null(Sw), while using the standard Fisher criterion defined in Equation (20)
to derive the regular discriminant vectors from range(Sw).
In our experiments, we capture all the regular discriminant vectors which satisfy J1(ϕ1) > 0
from the range space of Sw, simultaneously, we capture all the irregular discriminant vectors which
satisfy J2(ϕ2) > 0 from the null space of Sw.
3.5 A New Criterion for the Nearest Neighbor (NN) Classifier
After the discriminant features are extracted, a remaining key element of face recognition is to
design a robust classifier. Because there are large numbers of classes in face recognition problems,
we do not use the hyperplane classifier, but the NN classifier which is more suitable for such manyclass
problems. Because the NN classifier forms class boundaries with piecewise linear hyperplanes,
any classifying border can be approximated by a series of hyperplanes defined locally. Moreover,
classifying criterion is the core of the design. The Euclidean distance is the most popular one which
exhibits the distance between two vectors intuitively. However, it ignores the correlation which is
also important for measuring the similarity of two vectors. On the contrary, the cosine criterion
(Zhang and Korfhage, 1999) exhibits the correlation, but ignores the distance between two vectors.
In order to improve the Euclidean distance and cosine criterion, we propose a new triangle square
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ratio which takes into account of both distance and correlation between two vectors. Suppose υ1,
υ2 are two vectors, the triangle square ratio is defined as
T SR(υ1,υ2) = �υ1 − υ2�2 2
�υ1�2 2 + �υ2�2 .
2
The triangle square ratio is a similarity measure based on the argument and modulus of each vector,
as shown in proposition 2.
Proposition 2 Suppose θ is the include angle of υ1 and υ2, then T SR(υ1,υ2) → 0 if and only if
�υ1�2 → �υ2�2 and θ → 0, which implies the correlation between υ1 and υ2 should approach 1.
Proof
T SR(υ1,υ2) = 1 − 2�υ1�2 · �υ2�2
�υ1�2 2 + �υ2�2 cosθ
2
(by the cosine law)
≥ 1 − cosθ (“=” holds if and only if �υ1�2 = �υ2�2)
≥ 0 (“=” holds if and only if θ = 0).
In fact, if υ1 and υ2 are unit vectors, the triangle square ratio is equivalent to Euclidean distance.
Also, Equation (22) shows that triangle square ratio is a modification of cosine criterion by the
. If �υ1�2 = �υ2�2, it is equivalent to cosine criterion. Moreover, we have done
term 2�υ1�2·�υ2�2
�υ1� 2 2 +�υ2� 2 2
large numbers of numerical experiments which exclusively show that � T SR(υ1,υ2) satisfies the
triangle inequality, and the symmetric and positive definitive properties are obvious. So we guess
� T SR(υ1,υ2) is a distance measure, its proof in theory is an open problem.
Experimental results in Section 4 show that the triangular square ratio is more robust against
illumination variations than the Euclidean distance, whilst retaining the robustness against pose and
expression changes as the Euclidean distance. On the other hand, although the triangular square
ratio marginally underperforms the cosine criterion when there are variations of illumination, it can
obviously outperform the cosine criterion when there are changes of pose and expression.
3.6 The Procedure of Proposed LDC Algorithm
We summarize our local discriminant wavelet packet coordinates algorithm as follows:
Step 1: The wavelet packet transform
Expand each training sample x y
i into the spatial-frequency dictionary D (see Figure 2) by
the wavelet packet decomposition, then x y
i will be represented by the loadings of coordinates in D.
Step 2: The LDC selection transform
(2.a) For each coordinate c in the dictionary D, use the Equations (2), (18), (16), and (17) to
compute its prior distribution Pc(Xy) and conditional probability Pc(x y
i |Xy), whereafter, compute its
separability measure defined by Equation (19), that is, its discriminability.
(2.b) Select the first N0 coordinates from D with the largest discriminability. Whereupon, each
training sample x y
i can be represented by a feature vector vyi
which is formed by the loadings of the
selected coordinates. These feature vectors form a new feature space F .
Step 3: The CLDA transform
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Use the Equations (20) and (21) to construct the subspace template by the complete linear discriminant
analysis (CLDA) in F .
Step 4: Testing a new probe sample
For a new probe sample, it will be expanded into the spatial-frequency dictionary D by the
wavelet packet decomposition, and be extracted the loadings of the corresponding coordinates selected
in Step 2 to form a new feature vector v new. Then v new will be projected to the subspace
constructed in Step 3 and classified by the nearest neighbor classifier.
Simply, the LDC algorithm can be represented as:
Out put = T3 · T2 · T1 · Input
where T1 is the wavelet packet transform, T2 is the LDC selection transform. and T3 is the CLDA
transform.
3.7 Computational Complexity Comparison of the LDC and LDB Algorithm
The framework of the LDC algorithm is similar to LDB. We compare their computational complexity
step by step:
Step 1: The wavelet packet transform
The same procedure of the LDC and LDB algorithms cost O(N · nrnc · L), where nr × nc is the
size of facial images, L is the level of the wavelet packet decomposition.
Step 2: The LDC/LDB selection transform
(2.a) For each coordinate in D LDC, the LDC algorithm needs to compute the prior distribution
(16), the conditional probability (17) and its discriminability (19), so the costs of all the coordinates
in D LDC are O(N · nrnc) + O(K · nrnc) + O(N · nrnc). For each subband in D LDB, the LDB algorithm
needs to compute the space-frequency energy distribution (2), (1) and its measure of efficacy (3).
So the costs of all the subbands in D LDB are O(N · nrnc ·L) + O(K 2 · nrnc ·L).
(2.b) The LDC algorithm needs to sort all the coordinates in D LDC by their discriminability,
which costs O(nrnc · log 2 (nrnc)). The LDB algorithm needs to select the local discriminant basis
from D LDB using the best-basis algorithm, which costs O(L · 4 L ). Then both algorithms need to
represent all the training samples by new feature vectors, which cost O(N · N0).
Step 3: The CLDA/LDA transform
CLDA in the LDC algorithm has the same computational complexity O((N0) 3 ) as LDA in the
LDB algorithm in the new feature space F .
Step 4: Testing a new probe sample
A new probe sample should be transformed by T1, T2, T3 with complexity O(nrnc ·L)+O(N0)+
O(N0 · Nev) and classified by the NN classifier with complexity O(N · Nev), where Nev is the number
of eigenvectors extracted by CLDA or LDA.
The computational complexity of Step 2 shows that the LDC algorithm is more efficient than
LDB in many real applications when K is large. Table 11 also validates the fact.
4. Experiment Results
The results presented in this section are divided into five parts. First, we construct a dictionary D
and choose a preferable N0 for our LDC algorithm. As aforementioned, the LDC algorithm consists
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LIU, DAI AND YAN
of the LDC based feature extraction, the complete linear discriminant analysis (CLDA) and the
nearest neighbor (NN) classifier with the triangle square ratio (T SR) criterion. In the second and
third parts, we evaluate the efficacy of the LDC feature extraction and the new classifying criterion
respectively. The fourth part gives the performance of the whole LDC algorithm. Some further
researches of the LDC algorithm are shown in the final part.
4.1 Database
1) FERET Database: The FERET database, distributed by the National Institute of Standards and
Technology, consists of 14051 eight-bit grayscale images of human heads with different expressions,
poses, occlusion and illuminations (Phillips et al., 2000). Two data sets of the database are
used in our experiments, one is a small data set, which contains 432 images of 72 people and each
individual has six images, the other is a large data set with 255 individuals, and each person has four
frontal images, the datas are extracted from four different sets, namely, Fa, Fb, Fc, and duplicate
(Phillips et al., 2000). There are 1020 images in this data set. All the images are aligned by the
centers of eyes and mouth, and then normalized with the resolution 92 × 112. Some images from
both data sets of the FERET database are shown in Figure 3.
2) ORL Database: The Olivetti-Oracle Research Lab (ORL) database has 40 subjects and each
subject has 10 different facial views representing various expressions, small occlusion (by glasses),
different scales and orientations. So there are totally 400 facial images in the database. Each image
has 92 × 112 pixels in gray scale. Some samples are shown in Figure 4.
3) Hybrid Database: As aforementioned, the variations of the ORL database and the FERET
database are very different, which lead to unequal covariance distribution. So we blend the small
FERET data set and the ORL database together, in order to test the performance of the LDC algorithm
when facial images have larger illumination variations and pose, expression changes (Loog
and Duin, 2004). The hybrid database has 832 images of 112 persons.
4) CMU PIE Database: The CMU Pose, Illumination, and Expression (PIE) (Sim et al., 2003)
database consists of 41368 images of 68 people. Each person has images captured under 13 different
poses and 43 different illumination conditions and with four different expressions. In this paper, we
use a subset that focuses on illumination variations with pose and expression variations in frontal
Figure 3: Facial images of the FERET database. (Top) A person from the small data set. (Bottom)
A person from the large data set.
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LOCAL DISCRIMINANT WAVELET PACKET COORDINATES FOR FACE RECOGNITION
Figure 4: Segmental facial images of one person. (Top) From the ORL database. (Bottom) From
the CMU PIE database.
Database
Total number
of images
Number of images
per person
Number of
classes
ORL 400 10 40
SmallFERET 432 6 72
LargeFERET 1020 4 255
Hybrid 832 6 or 10 112
CMU-lights 2924 43 68
Table 2: Statistics for face images
view. There are 68 persons with each 43 images yielding a total of 2924 images. Each image has
92 × 112 pixels in gray scale. Some samples are shown in Figure 4.
The statistics of each data set is listed in Table 2.
4.2 Parameter Setting
In order to show more comparability with the PCA+CLDA, WaveletFace, LDB and MLDB algorithms
and present the performance of our LDC algorithm more accurately, CLDA is used to capture
the complete discriminant features in the five algorithms. The number of discriminant vectors is obtained
in the same way as the LDC algorithm (see Subsection 3.4).
The FisherFace technique uses the classical PCA+LDA. The Ntrain −K −λ (λ is the critical value
which ensures Sw is non-singular) eigenvectors with largest eigenvalues are preserved on ‘PCA step’
(Belhumeur et al., 1997). For the PCA+CLDA algorithm, we select the first min(N0,M0)(M0 is the
number of non-zero eigenvalues) eigenvectors with largest eigenvalues on ‘PCA step’. For direct
LDA (DLDA), we use all the eigenvectors in their Step 2 (Yu and Yang, 2001).
The third-level lowest frequency subband LLL3 with a matrix of (nr/8)×(nc/8) (where nr ×nc
is the resolution of original image) is referred to as WaveletFace (Chien and Wu, 2002). Because
the LDB algorithm selects a best discriminant subset of the whole basis, we choose four subbands,
and the number of selected coordinates is closest to N0. For the MLDB algorithm, we choose N0
coordinates as LDC on the scheme—five subbands with each 260 coordinates.
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Methods
LIU, DAI AND YAN
Parameters
number of features for discriminant analysis classifier (criterion)
FisherFace Ntrain − K − λ NN(l 2 )
PCA+CLDA min(N0,M0) NN(l 2 )
DLDA all eigenvectors NN(l 2 )
WaveletFace subband LLL3 in D NN(l 2 )
LDB four subbands NN(l 2 )
MLDB 5×260 NN(l 2 )
LDC N0 NN(T SR)
Table 3: Parameters of aforementioned methods
In the ‘Decision Step’, we use the nearest neighbor classifier with the Euclidean distance for the
aforementioned methods as their original forms. For our LDC algorithm, we use the new triangle
square ratio criterion, the Euclidean distance and the cosine criterion are used for comparison. Most
parameters are listed in Table 3.
The recognition rate is calculated as the ratio of the number of successful recognition and the
total number of test samples. All the experiments are repeated 30 times, and the final recognition
rate is the average value of the thirty results. Suppose M is the number of facial images for each
person. On each database, we randomly select i0(< M) images from each person for training, while
the rest M − i0 images of each individual are selected for testing. i0 is a small integer, in order to
show the performance of the LDC algorithm when there are small-size-training samples.
4.3 Construction of the Dictionary D and Choice of N0
In this subsection, we conduct two experiments to construct the dictionary D and select a suitable
N0 for the LDC algorithm.
4.3.1 CONSTRUCTION OF THE DICTIONARY D
In the first experiment, to construct our dictionary D, we search for the most discriminant level
by the best performance of its selected coordinates. Because it is more time-consuming when the
decomposition level L is larger than 4, L = 4 is used in the experiment. In order to test the effect
of high frequency components and show the tolerance of the dilation invariant entropy in LDC to
noise, we design two schemes: Scheme 1 uses all the subbands in the wavelet decomposition tree,
Scheme 2 only uses the left subtree whose root node is L1, that is the H1, V1, D1 components are
not further decomposed. For each level, we select the first 1000 coordinates by the criterion in
Subsection 3.3 for both schemes. Their performances on the small FERET data set and the ORL
database are shown in Table 4.
Table 4 shows that the performance of Scheme 1 is marginally underperform Scheme 2, and
Scheme 1 with all the subbands in the wavelet packet tree is more time-consuming than Scheme
2 which only uses the left subtree whose root node is L1. So Scheme 2 is adopted in the LDC
algorithm. However, the effect of high frequency components is very slight, especially in the first
three levels, which implies that our dilation invariant entropy has good tolerance to noise. Table
4 also shows that the third level has the best performance, it can offer a judicious combination of
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Database Training
samples (i0)
Schemes
1 2
Level
3 4
3
1
2
82.43± 2.40
82.35± 2.54
91.38 ± 2.57
92.19 ± 2.32
92.99 ± 1.60
93.30 ± 1.85
90.43 ± 2.38
92.34 ± 1.78
ORL 4
1
2
85.54± 2.76
85.56± 2.87
94.00 ± 1.56
94.43 ± 1.62
94.47 ± 1.68
95.26 ± 1.46
91.79 ± 2.08
94.18 ± 1.35
5
1
2
87.70± 2.03
87.82± 2.12
95.32 ± 1.45
96.03 ± 1.49
95.42 ± 1.71
96.23 ± 1.55
92.57 ± 1.62
94.12 ± 1.59
3
1
2
92.42± 1.93
92.42± 1.93
91.88 ± 1.69
91.88 ± 1.77
92.33 ± 2.51
92.30 ± 2.43
92.41 ± 2.42
92.53 ± 2.39
FERET
(small)
4
1
2
94.93± 1.40
94.93± 1.40
94.79 ± 2.25
94.72 ± 2.04
95.23 ± 2.40
95.60 ± 2.20
94.31 ± 2.58
95.30 ± 2.94
5
1
2
95.93± 1.41
95.93± 1.41
96.30 ± 1.76
96.07 ± 1.90
97.04 ± 1.92
96.90 ± 2.33
96.39 ± 2.95
97.13 ± 2.13
(∗)±(∗∗): (∗) represents the recognition rate (%), (∗∗) represents standard deviation (%).
Table 4: Effect of high frequency components and performances of level 1,2,3,4
scale information and localization information. So Level 3 is used in the LDC algorithm, and our
spatial-frequency dictionary D consists of the first 16 subbands in the third level (see Figure 2).
4.3.2 CHOICE OF N0
Because all of the top N0 coordinates are used for the classification, a natural way to determine the
best N0 is to select the top N0 coordinates and compute the average recognition rates for various
different N0. Based on the idea, we use a global to local search strategy (Müller et al., 2001) on both
data sets of the FERET database. Because the computational complexity of CLDA is O((N0) 3 ), for
the sake of computational efficiency, we set the range [100,2500] as the original wide range of N0. In
the “global” stage, we compare the performances of the LDC algorithm using the top N0 coordinates
when N0 increases from 100 to 2500 with interval 100 on the small data set, as shown in Figure 5
(Left). It shows that the LDC algorithm has a good and stable performance after N0 = 700 because
more good discriminant features are used. After N0 = 1700, the performance slightly decreases due
to the more redundant information included. So we ascertain a more precise subrange [700, 1700]
where the optimal N0 might exist.
In the “local” stage, we compare the performances of different N0 between 700 and 1700 with
interval 100 on the large data set, as shown in Figure 5 (Right). From the overall comparison of
two stages, we find that when N0 increases from 1300 to 1700 with interval 100, their performances
are very close, and the performance of N0 = 1300 is marginally better than others. Also for the
sake of computational efficiency, we select the first N0 = 1300 best discriminant coordinates in our
following experiments.
However, it should point out that the choice of N0 is not necessarily easy and needs further
research. The best N0 may be not the same for different databases, and it may be ascertained by the
cross validation method. We use the natural method and generalize the same N0 = 1300 to other
databases, in order to show that the LDC algorithm has robustness with respect to N0. Figure 6 also
validates the conclusion.
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Recognition rate
1
0.98
0.96
0.94
0.92
0.9
0.88
0.86
0.84
0.82
N 0 =700 N 0 =1700
LIU, DAI AND YAN
3 training samples
4 training samples
5 training samples
1 3 5 7 9 11 13
N /100
0
15 17 19 21 23 25
Recognition rate
0.98
0.96
0.94
0.92
0.9
0.88
0.86
0.84
2 training samples
3 training samples
N 0 =1300
7 8 9 10 11 12 13 14 15 16 17
N /100
0
Figure 5: (Left) The performances of the LDC algorithm using different N0 between 100 and 2500
with interval 100 on the small FERET data set. (Right) The performances of the LDC
algorithm using different N0 between 700 and 1700 with interval 100 on the large FERET
data set.
Recognition rate
1
0.98
0.96
0.94
3 training samples
4 training samples
5 training samples
0.92
13 14 15
N /100
0
16 17
Recognition rate
1
0.98
0.96
0.94
0.92
3 training samples
6 training samples
9 training samples
0.9
13 14 15
N /100
0
16 17
Figure 6: The performances of the LDC algorithm using different N0 between 1300 and 1700 with
interval 100. (Left) The ORL database. (Right) The CMU-lights database.
4.3.3 RECONFIRMATION OF THE DICTIONARY D USING N0=1300
Moreover, we return to the anterior experiment (the construction of the dictionary D) with the top
N0(= 1300) coordinates. The results prove that the third level has the best performance once again,
as shown in Figure 7.
4.4 Efficacy of the LDC Based Feature Extraction
It is of paramount importance for face recognition to extract most discriminant features that are
less sensitive to environmental variations. In this subsection, we compare the efficacy of feature
1184
Recognition rate
0.97
0.94
0.91
0.88
0.85
0.82
LOCAL DISCRIMINANT WAVELET PACKET COORDINATES FOR FACE RECOGNITION
Level 1
Level 2
Level 3
Level 4
3 4
Training Samples
5
Recognition rate
0.98
0.97
0.96
0.95
0.94
0.93
0.92
Level 1
Level 2
Level 3
Level 4
3 4
Training Samples
5
Figure 7: Performances of the LDC algorithm with N0(=1300) coordinates in four levels. (Left)
The ORL database. (Right) The small FERET data set.
Training Methods with the l2 Database
samples (i0) LDC
criterion
LDB MLDB WaveletFace
3 93.43 ± 1.95 93.30 ± 2.22 92.49 ± 1.91 92.92± 1.60
ORL 4 95.81 ± 1.59 95.60 ± 1.53 95.18 ± 1.39 94.56 ± 1.77
5 96.95 ± 1.59 96.65 ± 1.20 96.43 ± 1.29 94.20 ± 1.82
FERET
(small)
3
4
5
89.63 ± 3.01
92.92 ± 4.57
96.11 ± 3.39
87.56 ± 3.29
91.90 ± 4.48
95.69 ± 3.29
84.23 ± 3.26
88.80 ± 4.11
93.01 ± 3.80
88.80± 3.47
85.74 ± 4.85
87.50 ± 3.28
FERET 2 79.53 ± 3.23 76.05 ± 2.90 67.16 ± 3.36 73.18± 4.77
(large) 3 89.32 ± 1.82 87.45 ± 1.94 79.69 ± 1.99 72.59 ± 4.14
CMUlights
3
6
9
79.46 ± 10.95
93.48 ± 7.63
96.83 ± 4.29
79.92 ± 10.91
93.69 ± 7.52
96.54 ± 4.50
81.30 ± 9.87
94.16 ± 6.89
96.76 ± 4.56
78.44± 10.92
85.08 ± 7.79
54.80 ± 17.52
Table 5: Comparison with wavelet-based methods
extraction in LDC with other wavelet-based methods, such as LDB, MLDB and WaveletFace on the
ORL database, both data sets of the FERET database and the CMU-lights database. The setting of
the feature extractions can be seen in Subsection 4.2. In order to show more comparability, all the
methods use CLDA and the NN classifier with the Euclidean distance.
Table 5 shows that the LDC based feature extraction has the best result on the ORL database,
both data sets of the FERET database, and it outperforms WaveletFace, though it underperforms
LDB and MLDB marginally on the CMU-lights database. Moreover, LDC is more efficient than
LDB, MLDB because of the lower computational complexity when K is large, especially on the
large FERET data set (K = 255). On the whole, the feature extraction of LDC is more effective than
the kin methods, including LDB, MLDB and WaveletFace.
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LIU, DAI AND YAN
Database
Training
samples (i0)
Methods with the triangle square ratio criterion
LDC LDB MLDB WaveletFace
3 93.49 ± 2.19 92.93 ± 2.17 92.27 ± 1.82 92.80± 1.72
ORL 4 95.54 ± 1.45 95.32 ± 1.51 94.67 ± 1.64 94.13 ± 1.82
5 96.72 ± 1.71 96.65 ± 1.54 96.37 ± 1.41 93.60 ± 1.83
FERET
(small)
3
4
5
92.22 ± 2.18
95.32 ± 1.90
97.27 ± 2.01
90.03 ± 2.39
94.31 ± 2.58
95.56 ± 2.56
87.15 ± 2.91
91.53 ± 3.31
94.17 ± 2.88
90.54± 2.29
87.78 ± 2.92
87.18 ± 2.21
FERET 2 86.54 ± 2.78 84.11 ± 2.82 77.39 ± 3.34 79.01 ± 4.28
(large) 3 94.30 ± 0.37 92.39 ± 0.48 87.87 ± 1.78 74.98 ± 5.17
CMUlights
3
6
9
91.62 ± 6.63
97.82 ± 3.18
98.75 ± 2.09
91.90 ± 6.57
97.82 ± 3.25
98.47 ± 2.30
92.31 ± 6.37
98.14 ± 2.90
98.73 ± 2.14
88.67± 8.28
87.57 ± 6.86
55.96 ± 17.81
Table 6: Efficacy of the triangle square ratio criterion
4.5 Efficacy of the Triangle Square Ratio Criterion
Classifier and its classifying criterion are also important elements for face recognition. Generally,
distance-based criterion is more robust than correlation-based criterion with respect to pose and
expression changes while the contrary result is shown with respect to illumination variations. To
extend the capacity covering variations of pose, expression and illumination, we have proposed
the new triangle square ratio criterion in Subsection 3.5. In this experiment, we use the same four
feature extractions and CLDA as in Subsection 4.4, but the Euclidean distance is replaced by the
triangle square ratio criterion for the NN classifer. The results on the ORL database, both data sets
of the FERET database and the CMU-lights database are shown in Table 6.
Comparing the results on Table 6 with Table 5 which uses the Euclidean distance, it shows
that the triangle square ratio criterion performs better than the Euclidean distance considerably on
both data sets of the FERET database and the CMU-lights database, while its efficacy is very close
to the Euclidean distance on the ORL database. In fact, the FERET database, the CMU-lights
database concern about illumination variations (light intensity and direction respectively), and the
ORL database concerns about expression and pose changes. Comparison results show that the triangle
square ratio criterion is more robust against illumination variations than the Euclidean distance,
whilst retaining the robustness against pose and expression changes as the Euclidean distance.
Furthermore, we replace the triangle square ratio criterion with the cosine criterion, whilst
keeping the other setting, on the ORL database and the small FERET data set. The performance of
the cosine criterion is showed in Table 7. The comparison between Table 6 and Table 7 shows that
the triangle square ratio criterion performs better than the cosine criterion considerably on the ORL
database, although it marginally underperforms the cosine criterion on the small FERET data set.
4.6 Performance of the LDC Algorithm
In this part, we compare the performance of LDC with statistical methods, including FisherFace,
PCA+CLDA, DLDA, and wavelet based methods: WaveletFace, LDB, MLDB on both individual
and hybrid databases. All the methods used for comparison keep their settings as their original
forms (see Subsection 4.2).
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LOCAL DISCRIMINANT WAVELET PACKET COORDINATES FOR FACE RECOGNITION
Training
Database
samples (i0) LDC
Methods with the cosine criterion
LDB MLDB WaveletFace
3 92.54 ± 1.93 91.56 ± 2.12 90.99 ± 2.03 91.88± 1.73
ORL 4 94.78 ± 1.42 94.25 ± 1.50 93.71 ± 1.71 93.46± 1.89
5 96.22 ± 1.60 96.00 ± 1.49 95.53 ± 1.58 93.73 ± 1.76
FERET
(small)
3
4
5
92.41 ± 1.97
95.67 ± 1.96
97.27 ± 2.01
90.15 ± 2.56
94.86 ± 2.31
95.56 ± 2.43
86.99 ± 3.00
91.85 ± 3.29
94.49 ± 2.62
90.82± 2.58
87.94 ± 3.51
87.55 ± 2.59
Table 7: Performance of the cosine criterion
Since our motivation is to compensate for illumination, pose and expression variation, from
the properties of various databases, the ORL database is used to test moderate variations in pose
and expression, the CMU-lights database to test illumination variations, the FERET database for
more generic situation, and the hybrid database to test heteroscedastic class covariance distribution
tolerance.
4.6.1 COMPARISON ON THE INDIVIDUAL DATABASES
The comparison of results are depicted in Table 8. Although some algorithms occasionally have
better performance, LDC shows stable performance for every number of training samples per class
on all databases. Especially on the large FERET data set, it outperforms FisherFace, PCA+CLDA,
DLDA, WaveletFace, LDB, MLDB by 30.94%, 10.49%, 31.85%, 13.36%, 10.49%, 19.38% respectively
when two samples per class are used for training, and by 21.75%, 5.95%, 31.75%, 21.71%,
6.85%, 14.61% respectively when three samples per class are used for training. In particular, LDC
significantly outperforms FisherFace, DLDA, WaveletFace. From the standard derivation, we can
see that LDC has better stability than other algorithms.
4.6.2 COMPARISON ON THE HYBRID DATABASE
On the hybrid database, we randomly select i0 (i0=2 to 4) images from each person for training, the
rest (6−i0) images of each individual in the small FERET data set are tested while the rest (10−i0)
images of each individual in the ORL database are used for testing. The comparison results are
recorded in Table 9.
It shows that when the number of training sample per class increases from 2 to 4, the average
recognition rates of LDC are from 82.20% to 90.28%. The performance is better than Fisher-
Face, PCA+CLDA, DLDA, WaveletFace, LDB and MLDB which increase from 60.38%, 79.06%,
66.86%, 77.43%, 77.99%, 76.00%, to 75.63%, 90.64%, 80.85%, 66.09%, 88.11%, 86.09% respectively
on the hybrid database.
As a whole, these experimental results reveal that LDC has better performance and stability on
the ORL, FERET, CMU-lights databases than other methods, including FisherFace, PCA+CLDA,
DLDA, WaveletFace, LDB and MLDB, especially outperforms FisherFace, DLDA, WaveletFace.
1187
LIU, DAI AND YAN
Database Methods
3
Training samples (i0)
4 5
FisherFace 87.20 ± 1.96 90.53 ± 1.87 92.07 ± 1.97
PCA+CLDA 91.89 ± 1.97 94.81 ± 1.68 96.48 ± 1.58
DLDA 84.17 ± 2.23 87.31 ± 1.92 90.17 ± 1.55
ORL WaveletFace 92.92± 1.60 94.56 ± 1.77 94.20 ± 1.82
LDB 93.30 ± 2.22 95.60 ± 1.53 96.65 ± 1.20
MLDB 92.49 ± 1.91 95.18 ± 1.39 96.43 ± 1.29
LDC(T SR) 93.49 ± 2.19 95.54 ± 1.45 96.72 ± 1.71
FisherFace 84.85 ± 3.64 88.01 ± 4.91 91.94 ± 4.23
PCA+CLDA 89.07 ± 2.88 92.85 ± 4.06 95.60 ± 3.95
FERET
(small)
DLDA
WaveletFace
LDB
80.45 ± 4.81
88.80± 3.47
87.56 ± 3.29
86.34 ± 5.28
85.74 ± 4.85
91.90 ± 4.48
88.61 ± 6.41
87.50 ± 3.28
95.69 ± 3.29
MLDB 84.23 ± 3.26 88.80 ± 4.11 93.01 ± 3.80
LDC(T SR) 92.22 ± 2.18 95.32 ± 1.90 97.27 ± 2.01
Training samples (i0)
2 3
FisherFace 55.60 ± 5.10 72.55 ± 2.57
PCA+CLDA 76.05 ± 3.36 88.35 ± 1.74
FERET
(large)
DLDA
WaveletFace
LDB
54.69 ± 4.86
73.18± 4.77
76.05 ± 2.90
62.55 ± 2.30
72.59 ± 4.14
87.45 ± 1.94
MLDB 67.16 ± 3.36 79.69 ± 1.99
LDC(T SR) 86.54 ± 2.78 94.30 ± 0.37
3
Training samples (i0)
6 9
FisherFace 80.57 ± 8.96 94.46 ± 6.44 97.36 ± 4.08
PCA+CLDA 78.48 ± 10.89 93.91 ± 7.72 97.45 ± 3.70
CMUlights
DLDA
WaveletFace
LDB
76.92 ± 7.48
78.44± 10.92
79.92 ± 10.91
87.49 ± 6.60
85.08 ± 7.79
93.69 ± 7.52
92.65 ± 4.60
54.80 ± 17.52
96.54 ± 4.50
MLDB 81.30 ± 9.87 94.16 ± 6.89 96.76 ± 4.56
LDC(T SR) 91.62 ± 6.63 97.82 ± 3.18 98.75 ± 2.09
Table 8: Comparison on the individual databases
4.7 Some Further Researches of the LDC Algorithm
In this subsection, we keep the experimental settings in Subsection 4.2 and carry out some further
researches of the LDC algorithm, including: effects of different wavelets, effects of different relative
“distance” measures and comparison of CPU time.
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LOCAL DISCRIMINANT WAVELET PACKET COORDINATES FOR FACE RECOGNITION
Methods
2
Training samples (i0)
3 4
FisherFace 60.38 ± 3.24 72.58 ± 2.51 75.63 ± 2.87
PCA+CLDA 79.06 ± 2.22 86.99 ± 1.34 90.64 ± 1.77
DLDA 66.86 ± 2.57 74.55 ± 2.61 80.85 ± 1.89
WaveletFace 77.43 ± 2.28 76.21 ± 1.55 66.09 ± 2.86
LDB 77.99 ± 2.21 85.81 ± 1.73 88.11± 1.78
MLDB 76.00 ± 2.07 83.41 ± 1.85 86.09 ± 2.20
LDC(T SR) 82.20 ± 1.69 88.46 ± 1.44 90.28 ± 1.47
Table 9: Comparison on the hybrid database
Database wavelets
3
Training samples (i0)
4 5
harr 92.49 ± 2.16 94.79 ± 1.55 96.33 ± 1.53
db6 92.86 ± 1.52 95.24 ± 1.53 96.30 ± 1.55
sym2 92.83 ± 2.02 95.29 ± 1.74 96.78 ± 1.77
ORL coif2 93.27± 1.37 94.94 ± 1.39 95.67 ± 1.67
bior2.4 92.38 ± 2.08 94.04 ± 1.37 94.93 ± 1.54
rbio2.4 93.71 ± 2.10 95.19 ± 1.33 95.78 ± 1.64
db4 93.49 ± 2.19 95.54 ± 1.45 96.72 ± 1.71
harr 93.69 ± 2.07 95.86 ± 2.17 97.22 ± 1.96
db6 91.76 ± 2.65 94.58 ± 2.44 95.42 ± 2.42
FERET
(small)
sym2
coif2
bior2.4
92.56 ± 2.41
91.73 ± 2.55
90.88 ± 2.10
95.72 ± 2.60
94.75± 2.15
94.47 ± 2.78
97.22 ± 2.34
96.30 ± 1.94
95.46 ± 1.86
rbio2.4 91.59 ± 2.20 95.12 ± 2.37 96.16 ± 2.55
db4 92.22 ± 2.18 95.32 ± 1.90 97.27 ± 2.01
Table 10: Performances of different wavelet basis functions
4.7.1 EFFECTS OF DIFFERENT WAVELET BASIS FUNCTIONS
We use different wavelet basis functions for the wavelet packet decomposition on the ORL database
and the small FERET data set, including: harr wavelet, Daubechies db6 wavelet, Symlets sym2
wavelet, Coiflets coif2 wavelet, Biorthogonal spline bior2.4 wavelet, Reverse biorthogonal spline
rbio2.4 wavelet. Their performances are depicted in Table 10.
Table 10 shows that the performances of the harr and sym2 wavelets are very close to the db4
wavelet. Although other wavelets a little underperform the db4 wavelet, their performances are also
better than some other methods shown in Table 8. So we can conclude that the changes among
aforementioned different wavelet basis functions have small effects on the performance of the LDC
algorithm. Moreover, the orthogonal wavelets are superior to the biorthogonal wavelets in general.
4.7.2 EFFECTS OF DIFFERENT RELATIVE “DISTANCE” MEASURES
In order to show the effect of different relative “distance” measures on the performance of classification,
we compare the dilation invariant entropy with the dilation invariant l 2 norm (15) on the
1189
Recognition rate
0.97
0.96
0.95
0.94
0.93
Dilation invariant ι 2 norm
Dilation invariant entropy
3 4
Training Samples
5
LIU, DAI AND YAN
Recognition rate
0.98
0.97
0.96
0.95
0.94
0.93
0.92
Dilation invariant ι 2 norm
Dilation invariant entropy
3 4
Training Samples
5
Figure 8: Performances of the LDC algorithm using the dilation invariant entropy and the dilation
invariant l 2 norm. (Left) The ORL database. (Right) The small FERET data set.
Database
LDC FisherFace
Methods
LDB MLDB WaveletFace
ORL (K = 40) 154 8 336 263 23
SmallFERET (K = 72) 164 24 415 400 40
LargeFERET (K = 255) 269 521 2888 3045 123
Table 11: Comparison of training CPU time (seconds)
ORL database and the small FERET data set. The results are shown in Figure 8. It shows that the
dilation invariant l 2 norm marginally underperforms the dilation invariant entropy, which implies
the changes between aforementioned different relative “distance” measures have slight effects on
the performance of the LDC algorithm.
4.7.3 COMPARISON OF CPU TIME
We conduct an experiment to compare the time-consumption of the LDC algorithm with the popular
statistics-based method: FisherFace and the wavelet-based methods: LDB, MLDB, WaveletFace on
the ORL database and both data sets of the FERET database. We randomly select 3 images from
each person for training. The experiments are implemented using MATLAB in a personal computer
with Pentium 4 CPU and 256MB RAM. The time-consumptions are shown in Table 11. Although
LDC is less efficient than WaveletFace, it is considerably more efficient than LDB and MLDB,
especially when K is large. When K increases, the time-consumption of LDC increases more slowly
than that of FisherFace, so that LDC can catch up with and surpass the efficiency of FisherFace.
It should point out that in our experiments, LDB selects the four best discriminant subbands
from the local discriminant basis (see Subsection 4.2) and the number of selected coordinates is
bigger than N0(= 1300). So LDB takes more time than LDC, MLDB in the CLDA transform.
However, MLDB based feature extraction needs to estimate the probability density functions, when
K increases, it takes more and more time than LDB based feature extraction, so the total time in Table
11 shows that LDB is more expensive than MLDB on the ORL database and the small FERET
data set due to their small K, the contrary result is shown on the large FERET data set due to its
large K.
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LOCAL DISCRIMINANT WAVELET PACKET COORDINATES FOR FACE RECOGNITION
5. Discussions and Conclusion
In this paper, we have presented a novel local discriminant coordinates (LDC) method based on
wavelet packet for face recognition to compensate for illumination, pose and expression variations.
The method searches for the most discriminant coordinates from a wavelet packet dictionary, instead
of the most discriminant basis as in the LDB algorithm. The LDC idea makes use of the scattered
characteristic of the best discriminant features. In our method, the feature selection procedure is
independent of subbands, and only depends on the discriminability of all coordinates. We have
shown that the traditional “distance” measures (e.g., the l 2 distance, relative entropy) are deficient
to measure the separability, while comparing the separability of two sample ensembles. We have
proposed a new dilation invariant entropy which is independent of the order of magnitude. We
have used the dilation invariant entropy and a MAP logistic model to measure the separability of
coordinate-loading ensemble accurately. It locates in either low spatial frequency subbands or high
spatial frequency subbands. So any two coordinates in the wavelet packet dictionary are comparable
for their discriminability. Experimental results show that the LDC based feature extraction is more
effective than LDB, MLDB, WaveletFace, PCA for feature extraction.
The LDC based feature extraction not only selects low frequency components, but also middle
frequency components. From its significant improvement upon the WaveletFace method which only
uses low frequency components, we can conclude that middle frequency components are helpful
for face recognition, since their judicious combination with low spatial frequency components can
improve the performance of face recognition greatly.
We have modified the Euclidean distance and the cosine criterion in the nearest neighbor classifier,
and proposed a new triangle square ratio criterion which takes into account of two similarity
measures, distance and correlation. Experimental results show that the triangle square ratio criterion
is more robust against illumination variations than the Euclidean distance, while retaining
the robustness against pose and expression changes as the Euclidean distance. Also, it can obviously
outperform the cosine criterion when there are changes of pose and expression, although it
marginally underperforms the cosine criterion when there are variations of illumination. So it can
well extend the capacity covering variations of pose, expression and illumination.
We have used the ORL database to test moderate variations in pose and expression, the CMU
database to test illumination variations, the FERET database for more generic situation, and the
hybrid to test heteroscedastic class covariance distribution.
In conclusion, experimental results show that our LDC algorithm can well preserve the most
discriminant information of facial image and improve the performance for face recognition under
different variations. Also, experimental results show that our LDC algorithm has robustness with
respect to the number of selected coordinates. The changes among some different wavelet basis
functions and different relative “distance” measures have few effects on its performance. Moreover,
it is an efficient method.
The LDC idea may have numerous applications beyond the one described in this paper. It also
can be applied to feature or variable selection in other dictionaries of basis functions instead of
wavelets, such as the local trigonometric functions.
1191
Acknowledgments
LIU, DAI AND YAN
The authors are grateful to the anonymous reviewers for their valuable comments and suggestions,
which improve the paper. This project is supported in part by NSF of China (60575004, 10231040),
NSF of GuangDong(05101817), the Ministry of Education of China (NCET-04-0791) and the Hong
Kong Research Grant Council (CityU 122506).
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