Comparative Assessment of Independent Component Analysis (ICA) for Face Recognition
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Appears in the Second International Conference on Audio- and Video-based Biometric Person Authentication, AVBPA’99, Washington D. C. USA, March 22-24, 1999.
Comparative Assessment of Independent Component
Analysis (ICA) for Face Recognition
Chengjun Liu and Harry Wechsler
Department of Computer Science, George Mason University,
4400 University Drive, Fairfax, VA 22030-4444, USA
cliu, wechsler @cs.gmu.edu
Abstract fication problems. A successful face recognition method-
ology depends heavily on the particular choice of the fea-
This paper addresses the relative usefulness of Inde- tures used by the (pattern) classifier [4], [17], [3]. Feature
pendent Component Analysis (ICA) for Face Recognition. selection in pattern recognition involves the derivation of
Comparative assessments are made regarding (i) ICA sen- salient features from the raw input data in order to reduce
sitivity to the dimension of the space where it is carried out, the amount of data used for classification and simultane-
and (ii) ICA discriminant performance alone or when com- ously provide enhanced discriminatory power.
bined with other discriminant criteria such as Bayesian One popular technique for feature selection and di-
framework or Fisher’s Linear Discriminant (FLD). Sensi- mensionality reduction is Principal Component Analysis
tivity analysis suggests that for enhanced performance ICA (PCA) [8], [6]. PCA is a standard decorrelation technique
should be carried out in a compressed and whitened Prin- and following its application one derives an orthogonal
cipal Component Analysis (PCA) space where the small projection basis which directly leads to dimensionality re-
trailing eigenvalues are discarded. The reason for this duction, and possibly to feature selection. PCA was first
finding is that during whitening the eigenvalues of the co- applied to reconstruct human faces by Kirby and Sirovich
variance matrix appear in the denominator and that the [10] and to recognize faces by Turk and Pentland [19]. The
small trailing eigenvalues mostly encode noise. As a con- recognition method, known as eigenfaces, defines a feature
sequence the whitening component, if used in an uncom- space, or “face space”, which drastically reduces the di-
pressed image space, would fit for misleading variations mensionality of the original space, and face detection and
and thus generalize poorly to new data. Discriminant anal- identification are then carried out in the reduced space.
ysis shows that the ICA criterion, when carried out in the Independent Component Analysis (ICA) has emerged
properly compressed and whitened space, performs bet- recently as one powerful solution to the problem of blind
ter than the eigenfaces and Fisherfaces methods for face source separation [5], [9], [7] while its possible use for
recognition, but its performance deteriorates when aug- face recognition has been shown by Bartlett and Sejnowski
mented by additional criteria such as the Maximum A Pos- [1]. ICA searches for a linear transformation to express a
teriori (MAP) rule of the Bayes classifier or the FLD. The set of random variables as linear combinations of statisti-
reason for the last finding is that the Mahalanobis distance cally independent source variables [5]. The search crite-
embedded in the MAP classifier duplicates to some extent rion involves the minimization of the mutual information
the whitening component, while using FLD is counter to expressed as a function of high order cumulants. Basically
the independence criterion intrinsic to ICA. PCA considers the 2nd order moments only and it uncor-
relates data, while ICA accounts for higher order statistics
and it identifies the independent source components from
1. Introduction their linear mixtures (the observables). ICA thus provides
a more powerful data representation than PCA [9]. As
Face recognition is important not only because it has a PCA derives only the most expressive features for face re-
lot of potential applications in research fields such as Hu- construction rather than face classification, one would usu-
man Computer Interaction (HCI), biometrics and security, ally use some subsequent discriminant analysis to enhance
but also because it is a typical Pattern Recognition (PR) PCA performance [18].
problem whose solution would help solving other classi- This paper makes a comparative assessment on the useof ICA as a discriminant analysis criterion whose goal is faces and the Most Discriminant Features (MDF) method)
to enhance PCA stand alone performance. Experiments for face recognition. Methods that combine PCA and the
in support of our comparative assessment of ICA for face standard FLD, however, lack in their generalization ability
recognition are carried out using a large data set consisting as they overfit to the training data. To address the overfit-
of 1,107 images and drawn from the FERET database [16]. ting problem Liu and Wechsler [11] introduced Enhanced
The comparative assessment suggests that for enhanced FLD Models (EFM) to improve on the generalization ca-
face recognition performance ICA should be carried out pability of the standard FLD based classifiers such as Fish-
in a compressed and whitened space, and that ICA per- erfaces [2].
formance deteriorate when it is augmented by additional
decision rules such as the Bayes classifier or the Fisher’s 3. Independent Component Analysis (ICA)
linear discriminant analysis.
As PCA considers the 2nd order moments only it lacks
2. Background information on higher order statistics. ICA accounts for
higher order statistics and it identifies the independent
PCA provides an optimal signal representation tech- source components from their linear mixtures (the observ-
nique in the mean square error sense. The motivation be- ables). ICA thus provides a more powerful data represen-
hind using PCA for human face representation and recog- tation than PCA [9] as its goal is that of providing an inde-
nition comes from its optimal and robust image compres- pendent rather than uncorrelated image decomposition and
sion and reconstruction capability [10] [15]. PCA yields representation.
projection axes based on the variations from all the training ICA of a random vector searches for a linear trans-
samples, hence these axes are fairly robust for representing formation which minimizes the statistical dependence be-
both training and testing images (not seen during training).
PCA does not distinguish, however, the different roles of
tween its components [5]. In particular, let
random vector representing an image, where be a
is the di-
mensionality of the image space. The vector is formed by
the variations (within- and between-class variations) and
it treats them equally. This leads to poor performance concatenating the rows or the columns of the image which
when the distributions of the face classes are not separated may be normalized to have a unit norm and/or an equalized
"!$#
by the mean-difference but separated by the covariance- histogram. The covariance matrix of is defined as
difference [6].
&%
(1)
()(*,+- '
Swets and Weng [18] point out that PCA derives only
the most expressive features which are unrelated to actual where is the expectation operator, denotes the trans-
face recognition, and in order to improve performance one pose operation, and . The ICA of factor-
.,/0. !
needs additional discriminant analysis. One such discrim- izes the covariance matrix into the following form
inant criterion, the Bayes classifier, yields the minimum
/ .
classification error when the underlying probability density (2)
functions (pdf) are known. The use of the Bayes classifier
is conditioned on the availability of an adequate amount where
1
is diagonal real positive and transforms the
2. 1
of representative training data in order to estimate the pdf. original data into
As an example, Moghaddam and Pentland [13] developed
(3)
1
a Probabilistic Visual Learning (PVL) method which uses
the eigenspace decomposition as an integral part of esti-
.
such that the components of the new data are indepen-
mating the pdf in high-dimensional image spaces. To ad-
dent or “the most independent possible” [5].
dress the lack of sufficient training data Liu and Wechsler
To derive the ICA transformation , Comon [5] de-
[12] introduced the Probabilistic Reasoning Models (PRM)
veloped an algorithm which consists of three operations:
where the conditional pdf for each class is modeled using
whitening, rotation, and normalization. First, the whiten-
3
the pooled within-class scatter and the Maximum A Pos-
ing operation transforms a random vector into another
teriori (MAP) Bayes classification rule is implemented in
254 687&9$: 3
one that has a unit covariance matrix.
the reduced PCA subspace.
4 6
The Fisher’s Linear Discriminant (FLD) is yet another (4)
popular discriminant criterion. By applying first PCA
4 6(4 !
for dimensionality reduction and then FLD for discrimi- where and are derived by solving the following eigen-
nant analysis, Belhumire, Hespanha, and Kriegman [2] and value equation.
Swets and Weng [18] developed similar methods (Fisher- (5)
24 7 26 :
where
tor matrix and
7 :
is an orthonormal
eigenvec-
is a diagonal # small trailing eigenvalues are excluded so that we can ob-
tain more robust whitening results. Toward that end, dur-
eigenvalue matrix of . One notes that whitening, an in- ing the whitening transformation we should keep a balance
tegral ICA component, counteracts the fact that the Mean between the need that the selected eigenvalues account for
Square Error (MSE) preferentially weighs low frequencies most of the spectral energy of the raw data and the require-
[14]. The rotation operations, then, perform source sepa- ment that the trailing eigenvalues of the covariance matrix
ration (to derive independent components) by minimizing are not too small. As a result, the eigenvalue spectrum of
the mutual information approximated using higher order the training data supplies useful information for the deci-
cumulants. Finally, the normalization operation derives sion of the dimensionality of the subspace.
unique independent components in terms of orientation,
unit norm, and order of projections [5].
4. Sensitivity Analysis of ICA
3 6 79:4 !
Eq. 4 can be rearranged to the following form
4 6
(6)
where and are eigenvector and eigenvalue matrices,
respectively (see Eq. 5). Eq. 6 shows that during whiten-
ing the eigenvalues appear in the denominator. The trailing
eigenvalues, which tend to capture noise as their values are
fairly small, thus cause the whitening step to fit for mis-
leading variations and make the ICA criterion to generalize
poorly when it is exposed to new data. As a consequence, if
the whitening step is preceded by a dimensionality reduc-
tion procedure ICA performance would be enhanced and
computational complexity reduced. Specifically, space di-
mensionality is first reduced to discard the small trailing
! "$#&%('*)!+-,/.!01'2' 3 & 4 06587 '9:0;4to decreased ICA performance (see Fig. 3 and Fig. 4) as 0.96
they amplify the effects of noise. So we set the dimension
W 0.95
of the reduced space as .
0.94
In order to assess the sensitivity of ICA in terms of the
dimension of the compressed and whitened space where it 0.93
correct retrieval rate
is implemented, we carried out a comparative assessment 0.92
for different dimensional ( ) whitened subspace. Fig. 3
0.91
and Fig. 4 show the ICA face recognition performance
when different is chosen during the whitening transfor- 0.9
mation (see Eq. 9) and as the number of features used can
0.89
range up to the dimension ofW the compressed space. The
curve corresponding to performs better than all the 0.88 m = 30
m = 40
other curves obtained for different values. Fig. 4 shows 0.87
m = 50
m = 70
that ICA performance deteriorates quite severely as is
0.86
getting larger. The reason for this deterioration is that for 20 25 30 35 40 45 50 55 60 65 70
number of features
the large values of , more small trailing eigenvalues (see
! "$#&%(' $+ .!065&4% 4 <
. % 0;4 P'
. >
! "
Fig. 2 ) are included in the whitening step and this leads
I!T= P%@:06 4 A P%@ AK'C%@T. B '
C W
#
to an unstable transformation. Note that smaller values for
lead to decreased ICA performance as well because the %(' =5!' PA 'consequence overfitting takes place. To improve the gen- 0.96
*)
eralization ability of Fisherfaces, we implemented it in the
PCA subspace. The comparative performance 0.94
of eigenfaces, Fisherfaces and ICA is W plotted in Fig. 5
when ICA was implemented in the PCA subspace.
Fig. 5 shows that ICA criterion performs better than both
correct retrieval rate
0.92
eigenfaces and Fisherfaces.
We also assessed the ICA discriminant criterion against
0.9
two other popular discriminant criteria: the MAP rule of
the Bayes classifier and the Fisher’s linear discriminant
as they are embedded within the Probabilistic Reasoning 0.88
Models (PRM) [12] and the Enhanced FLD Models (EFM) ICA
[11]. Fig. 6 plots the comparative performances of PRM-1 PRM−1
0.86
and EFM-1 against the ICA method with again being set EFM−1
to 40. ICA is shown to have comparable face recognition 18 20 22 24 26 28 30 32 34 36 38
number of features
performance with the MAP rule of the Bayes classifier and
the Fisher’s linear discriminant as embedded within PRM
! "$#&%(' $+ . 065&4% 4 .C% 0;4 P' .C> '< "&'
" "
> 4CP'= !T=BC'(4CP' = 4 &I +
! "$#&%(' $+ .!0 54%@4 A ' 5!'(.C% 0;4&P'.C> 4 I
JP.!06L8 C'0.95 the 4th Annual Joint Symposium on Neural Computation,
Pasadena, CA, May 17, 1997.
0.9
[2] P. Belhumeur, J. Hespanha, and D. Kriegman. Eigenfaces
0.85 vs. fisherfaces: Recognition using class specific linear pro-
jection. IEEE Trans. Pattern Analysis and Machine Intelli-
0.8 gence, 19(7):711–720, 1997.
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0.75
templates. IEEE Trans. Pattern Analysis and Machine In-
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0.6
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