Application of 2DPCA Based Techniques in DCT Domain
for Face Recognition
Messaoud Bengherabi1, Lamia Mezai1, Farid Harizi1, Abderrazak Guessoum2,
and Mohamed Cheriet3
1
Centre de Développement des Technologies Avancées- Algeria
Division Architecture des Systèmes et MultiMédia
Cité 20 Aout, BP 11, Baba Hassen, Algiers-Algeriabengherabi@yahoo.com, l_mezai@yahoo.fr, harizihourizi@yahoo.fr
2
Université Saad Dahlab de Blida – Algeria
Laboratoire Traitement de signal et d’imagerie
Route De Soumaa BP 270 Blida
guessouma@hotmail.com
3
École des Technologies Supérieur –Québec- CanadaLaboratoire d’Imagerie, de Vision et d’Intelligence Artificielle
1100, Rue Notre-Dame Ouest, Montréal (Québec) H3C 1K3 Canada
mohamed.cheriet@gpa.etsmtl.ca
Abstract. In this paper, we introduce 2DPCA, DiaPCA and DiaPCA+2DPCA in DCT domain
for the aim of face recognition. The 2D DCT transform has been used as a preprocessing step,
then 2DPCA, DiaPCA and DiaPCA+2DPCA are applied on the upper left corner block of the
global 2D DCT transform matrix of the original images. The ORL face database is used to
compare the proposed approach with the conventional ones without DCT under Four matrix
similarity measures: Frobenuis, Yang, Assembled Matrix Distance (AMD) and Volume Measure (VM). The experiments show that in addition to the significant gain in both the training and
testing times, the recognition rate using 2DPCA, DiaPCA and DiaPCA+2DPCA in DCT domain is generally better or at least competitive with the recognition rates obtained by applying
these three 2D appearance based statistical techniques directly on the raw pixel images; especially under the VM similarity measure.
Keywords: Two-Dimensional PCA (2DPCA), Diagonal PCA (DiaPCA), DiaPCA+2DPCA,
face recognition, 2D Discrete Cosine Transform (2D DCT).
1 Introduction
Different appearance based statistical methods for face recognition have been proposed in literature. But the most popular ones are Principal Component Analysis
(PCA) [1] and Linear Discriminate Analysis (LDA) [2], which process images as 2D
holistic patterns. However, a limitation of PCA and LDA is that both involve eigendecomposition, which is extremely time-consuming for high dimensional data.
Recently, a new technique called two-dimensional principal component analysis
2DPCA was proposed by J. Yang et al. [3] for face recognition. Its idea is to estimate
the covariance matrix based on the 2D original training image matrices, resulting in a
E. Corchado et al. (Eds.): CISIS 2008, ASC 53, pp. 243–250, 2009.
© Springer-Verlag Berlin Heidelberg 2009
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covariance matrix whose size is equal to the width of images, which is quite small
compared with the one used in PCA. However, the projection vectors of 2DPCA
reflect only the variations between the rows of images, while discarding the variations
of columns. A method called Diagonal Principal Component Analysis (DiaPCA) is
proposed by D. Zhang et al. [4] to resolve this problem. DiaPCA seeks the projection
vectors from diagonal face images [4] obtained from the original ones to ensure that
the correlation between rows and those of columns is taken into account. An efficient
2D techniques that results from the combination of DiaPCA and 2DPCA
(DiaPCA+2DPCA) is proposed also in [4].
Discrete cosine transform (DCT) has been used as a feature extraction step in various studies on face recognition. This results in a significant reduction of computational complexity and better recognition rates [5, 6]. DCT provides excellent energy
compaction and a number of fast algorithms exist for calculating it.
In this paper, we introduce 2DPCA, DiaPCA and DiaPCA+2DPCA in DCT domain for face recognition. The DCT transform has been used as a feature extraction
step, then 2DPCA, DiaPCA and DiaPCA+2DPCA are applied only on the upper left
corner block of the global DCT transform matrix of the original images. Our proposed
approach is tested against conventional approaches without DCT under Four matrix
similarity measures: Frobenuis, Yang, Assembled Matrix Distance (AMD) and Volume Measure (VM).
The rest of this paper is organized as follows. In Section 2 we give a review of
2DPCA, DiaPCA and DiaPCA+2DPCA approaches and also we review different
matrix similarity measures. In section 3, we present our contribution. In section 4 we
report the experimental results and highlight a possible perspective of this work. Finally, in section 5 we conclude this paper.
2 Overview of 2DPCA, DiaPCA, DiaPCA+2DPCA and Matrix
Similarity Measures
2.1 Overview of 2D PCA, DiaPCA and DiaPCA+2DPCA
2.1.1 Two-Dimensional PCA
Given M training face images, denoted by m×n matrices Ak (k = 1, 2… M), twodimensional PCA (2DPCA) first uses all the training images to construct the image
covariance matrix G given by [3]
G=
1
M
∑ (A
M
k
−A
k =1
) (A
T
k
−A
)
(1)
Where A is the mean image of all training images. Then, the projection axes of
2DPCA, Xopt=[x1… xd] can be obtained by solving the algebraic eigenvalue problem
Gxi=λixi, where xi is the eigenvector corresponding to the ith largest eigenvalue of G
[3]. The low dimensional feature matrix C of a test image matrix A is extracted by
C = AX opt
(2)
In Eq.(2) the dimension of 2DPCA projector Xopt is n×d, and the dimension of
2DPCA feature matrix C is m×d.
Application of 2DPCA Based Techniques in DCT Domain
245
2.1.2 Diagonal Principal Component Analysis
Suppose that there are M training face images, denoted by m×n matrices Ak(k = 1, 2,
…, M). For each training face image Ak, we calculate the corresponding diagonal face
image Bk as it is defined in [4].
Based on these diagonal faces, diagonal covariance matrix is defined as [4]:
G DIAG =
Where B =
1
M
1
M
∑ (B
M
k =1
k
−B
) (B
T
k
−B
)
(3)
M
∑B
k
k =1
is the mean diagonal face. According to Eq. (3), the projection
vectors Xopt=[x1, …, xd] can be obtained by computing the d eigenvectors corresponding to the d biggest eigenvalues of GDIAG. The training faces Ak’s are projected onto
Xopt, yielding m×d feature matrices.
C k = Ak X opt
(4)
Given a test face image A, first use Eq. (4) to get the feature matrix C = AX opt , then a
matrix similarity metric can be used for classification.
2.1.3 DiaPCA+2DPCA
Suppose the n by d matrix X=[x1, …, xd] is the projection matrix of DiaPCA. Let
Y=[y1, …, yd] the projection matrix of 2DPCA is computed as follows: When the
height m is equal to the width n, Y is obtained by computing the q eigenvectors corresponding to the q biggest eigenvalues of the image covarinace matrix
1 M
(A − A)T (A − A) . On the other hand, when the height m is not equal to the width
M
∑
k =1
k
k
n, Y is obtained by computing the q eigenvectors corresponding to the q biggest ei-
∑ (A
M
genvalues of the alternative image covariance matrix 1
M
k =1
k
)(
)
T
− A Ak − A .
Projecting training faces Aks onto X and Y together, yielding the q×d feature matrices
D k = Y T Ak X
(5)
Given a test face image A, first use Eq. (5) to get the feature matrix D = Y T AX , then a
matrix similarity metric can be used for classification.
2.2 Overview of Matrix Similarity Measures
An important aspect of 2D appearance based face recognition approaches is the similarity measure between matrix features used at the decision level. In our work, we
have used four matrix similarity measures.
2.2.1 Frobenius Distance
Given two feature matrices A = (aij)m×d and B = (bij)m×d, the Frobenius distance [7]
measure is given by:
⎛ m
d F ( A, B ) = ⎜⎜ ∑
⎝ i =1
∑ (a
12
d
j =1
ij
2⎞
− bij ) ⎟⎟
⎠
(6)
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M. Bengherabi et al.
2.2.2 Yang Distance Measure
Given two feature matrices A = (aij)m×d and B = (bij)m×d, the Yang distance [7] is given by:
12
d
⎛ m
2⎞
dY ( A, B ) = ∑ ⎜ ∑ (aij − bij ) ⎟
j =1 ⎝ i =1
⎠
(7)
2.2.3 Assembled Matrix Distance (AMD)
A new distance called assembled matrix distance (AMD) metric to calculate the distance between two feature matrices is proposed recently by Zuo et al [7]. Given two
feature matrices A = (aij)m×d and B = (bij)m×d, the assembled matrix distance dAMD(A,B)
is defined as follows :
(1 2 ) p
⎞
⎛ d ⎛ m
2⎞
⎟
d AMD ( A, B ) = ⎜ ∑ ⎜ ∑ (aij − bij ) ⎟
⎟
⎜ j =1 ⎝ i =1
⎠
⎠
⎝
12
( p > 0)
(8)
It was experimentally verified in [7] that best recognition rate can be obtained when
p≤0.125 while it decrease as p increases. In our work the parameter p is set equal to 0.125.
2.2.4 Volume Measure (VM)
The VM similarity measure is based on the theory of high-dimensional geometry
space. The volume of an m×n matrix of rank p is given by [8]
∑det
Vol A =
( I ,J )∈N
2
A IJ
(9)
where AIJ denotes the submatrix of A with rows I and columns J, N is the index set of
p×p nonsingular submatrix of A, and if p=0, then Vol A = 0 by definition.
3 The Proposed Approach
In this section, we introduce 2DPCA, DiaPCA and DiaPCA+2DPCA in DCT domain
for the aim of face recognition. The DCT is a popular technique in imaging and video
compression, which was first applied in image compression in 1974 by Ahmed et al
[9]. Applying the DCT to an input sequence decomposes it into a weighted sum of
basis cosine sequences. our methodology is based on the use of the 2D DCT as a
feature extraction or preprocessing step, then 2DPCA, DiaPCA and DiaPCA+2DPCA
are applied to w×w upper left block of the global 2D DCT transform matrix of the
original images. In this approach, we keep only a sub-block containing the first coefficients of the 2D DCT matrix as shown in Fig.1, from the fact that, the most significant information is contained in these coefficients.
2D DCT
c11 c12 … c1w
.
.
cw1 cw2 … cww
Fig. 1. Feature extraction in our approach
Application of 2DPCA Based Techniques in DCT Domain
247
With this approach and inversely to what is presented in literature of DCT-based
face recognition approaches, the 2D structure is kept and the dimensionality reduction
is carried out. Then, the 2DPCA, DiaPCA and DiaPCA+2DPCA are applied to w×w
block of 2D DCT coefficients. The training and testing block diagrams describing the
proposed approach is illustrated in Fig.2.
Training a lgorithm
based on
92DPCA
9Dia PCA
9Dia PCA+2DPCA
Block w*w
of 2D DCT
coefficients
2D
DCT
Tra ined
Model
Training data
2D DCT ima ge
Projection of the DCT bloc
of the test ima ge using the
eigenvectors of
92DPCA
9Dia PCA
9Dia PCA+2DPCA
Block w*w
of 2D DCT
coefficients
2D
DCT
Test ima ge
2D DCT Block
Features
2D DCT ima ge
Compa rison using
9Frobenius
9Yang
9AMD
9VM
2D DCT Block
Fea tures
Decision
Fig. 2. Block diagram of 2DPCA, DiaPCA and DiaPCA+2DPCA in DCT domain
4 Experimental Results and Discussion
In this part, we evaluate the performance of 2DPCA, DiaPCA and DiaPCA+2DPCA
in DCT domain and we compare it to the original 2DPCA, DiaPCA and
DiaPCA+2DPCA methods. All the experiments are carried out on a PENTUIM 4 PC
with 3.2GHz CPU and 1Gbyte memory. Matlab [10] is used to carry out these experiments. The database used in this research is the ORL [11] (Olivetti Research
Laboratory) face database. This database contains 400 images for 40 individuals, for
each person we have 10 different images of size 112×92 pixels. For some subjects, the
images captured at different times. The facial expressions and facial appearance also
vary. Ten images of one person from the ORL database are shown in Fig.3.
In our experiment, we have used the first five image samples per class for training
and the remaining images for test. So, the total number of training samples and test
samples were both 200. Herein and without DCT the size of diagonal covariance
matrix is 92×92, and each feature matrix with a size of 112×p where p varies from 1
to 92. However with DCT preprocessing the dimension of these matrices depends on
the w×w DCT block where w varies from 8 to 64. We have calculated the recognition
rate of 2DPCA, DiaPCA, DiaPCA+2DPCA with and without DCT.
In this experiment, we have investigated the effect of the matrix metric on the performance of the 2D face recognition approaches presented in section 2. We see from
table 1, that the VM provides the best results whereas the Frobenius gives the worst
ones, this is justified by the fact that the Frobenius metric is just the sum of the
248
M. Bengherabi et al.
(a)
(b)
Fig. 3. Ten images of one subject in the ORL face database, (a) Training, (b) Testing
Euclidean distance between two feature vectors in a feature matrix. So, this measure
is not compatible with the high-dimensional geometry theory [8].
Table 1. Best recognition rates of 2DPCA, DiaPCA and DiaPCA+2DPCA without DCT
Methods
2DPCA
DiaPCA
DiaPCA+2DPCA
Frobenius
91.50 (112×8)
91.50 (112×8)
92.50 (16×10)
Yang
93.00 (112×7)
92.50 (112×10)
94.00 (13×11)
AMD p=0,125
95.00 (112×4)
91.50 (112×8)
93.00 (12×6)
Volume Distance
95.00 (112×3)
94.00 (112×9)
96.00 (21×8)
Tables 2, and Table 3 summarize the best performances under different 2D DCT
block sizes and different matrix similarity measures.
Table 2. 2DPCA, DiaPCA and DiaPCA+2DPCA under different DCT block sizes using the
Frobenius and Yang matrix distance
Best Recognition rate (feature matrix dimension)
DiaPCA+2DPCA
2DPCA
DiaPCA
Yang
91.50 (6×6)
93.50 (8×6)
93.50 (8×5)
92.00 (9×5)
93.00 (9×6)
95.00 (9×9)
92.00 (10×5)
94.50 (10×6)
95.50 (10×9)
92.00 (9×5)
94.00 (11×6)
95.50 (11×5)
91.50 (9×5)
94.50 (12×6)
95.50 (12×5)
92.00 (12×11)
94.50 (13×6)
95.00 (13×5)
92.00 (12×7)
94.50 (14×6)
94.50 (14×5)
2D DCT
block
size
8×8
9×9
10×10
11×11
12×12
13×13
14×14
91.50 (8×8)
92.00 (9×9)
91.50 (10×5)
92.00 (11×8)
92.00 (12×8)
91.50 (13×7)
92.00 (14×7)
DiaPCA
Frobenius
91.50 (8×6)
92.00 (9×5)
92.00 (10×5)
91.50 (11×5)
91.50 (12×10)
92.00 (13×11)
91.50 (14×7)
15×15
16×16
32×32
91.50 (15×5)
92.50 (16×10)
92.00 (32×6)
91.50 (15×5)
91.50 (16×11)
91.50 (32×6)
92.00 (13×15)
92.00 (4×10)
92.00 (11×7)
94.00 (15×9)
94.00 (16×7)
93.00 (32×6)
94.50 (15×5)
94.50 (16×5)
93.50 (32×5)
95.50 (12×5)
95.00 (12×5)
95.00 (12×5)
64×64
91.50 (64×6)
91.00 (32×6)
92.00 (14×12)
93.00 (64×7)
93.50 (64×5)
95.00 (12×5)
2DPCA
DiaPCA+2DPCA
93.50 (8×5)
95.00 (9×9)
95.50 (10×9)
95.50 (11×5)
95.50 (12×5)
95.00 (11×5)
95.00 (12×5)
From these four tables, we notice that in addition to the importance of matrix similarity measures, by the use of DCT we have always better performance in terms of
recognition rate and this is valid for all matrix measures, we have only to choose the
DCT block size and appropriate feature matrix dimension. An important remark is
that a block size of 16×16 or less is sufficient to have the optimal performance. So,
this results in a significant reduction in training and testing time. This significant gain
Application of 2DPCA Based Techniques in DCT Domain
249
Table 3. 2DPCA, DiaPCA and DiaPCA+2DPCA under different DCT block sizes using the
AMD distance and VM similarity measure on the ORL database
2D DCT
block size
2DPCA
8×8
9×9
10×10
11×11
12×12
13×13
14×14
15×15
16×16
32×32
64×64
94.00 (8×4)
94.50 (9×4)
94.50 (10×4)
95.50 (11×5)
95.50 (12×5)
96.00 (13×4)
96.00 (14×4)
96.00 (15×4)
96.00 (16×4)
95.50 (32×4)
95.00 (64×4)
DiaPCA
AMD
95.00 (8×6)
94.50 (9×5)
95.50 (10×5)
96.00 (11×5)
96.50 (12×7)
95.50 (13×5)
95.00 (14×5)
95.00 (15×5)
95.50 (16×5)
95.00 (32×9)
94.50 (64×9)
Best Recognition rate (feature matrix dimension)
DiaPCA+2DPCA
2DPCA
DiaPCA
VM
95.00 (7×5)
96.00 (8×3)
93.50 (8×4)
94.50 (9×5)
95.00 (9×4)
95.00 (9×5)
96.00 (9×7)
95.00 (10×3)
95.00 (10×4)
94.50 (11×3)
95.50 (11×3)
96.50 (9×6)
95.50 (12×5)
96.00 (12×5)
96.50 (9×7)
95.50 (12×5)
96.00 (13×9)
96.00 (13×5)
95.50 (10×5)
95.00 (14×3)
95.50 (14×5)
96.00 (9×7)
96.00 (15×8)
96.00 (15×5)
95.50 (16×8)
96.00 (16×5)
96.50 (12×5)
96.00 (11×5)
95.00 (32×3)
95.50 (32×5)
96.00 (12×5)
95.00 (64×3)
95.00 (64×5)
DiaPCA+2DPCA
93.50 (8×4)
95.00 (9×5)
95.00 (10×4)
95.50 (11×3)
96.00 (11×5)
96.50 (10×5)
96.50 (10×5)
96.50 (10×5)
96.50 (10×5)
96.50 (9×5)
96.50 (21×5)
in computation is better illustrated in table 4 and table 5, which illustrate the total
training and total testing time of 200 persons -in seconds - of the ORL database under
2DPCA, DiaPCA and DiaPCA+2DPCA without and with DCT, respectively. We
should mention that the computation of DCT was not taken into consideration when
computing the training and testing time of DCT based approaches.
Table 4. Training and testing time without DCT using Frobenius matrix distance
Methods
Training time in sec
Testing time in sec
2DPCA
5.837 (112×8)
1.294 (112×8)
DiaPCA
5.886 (112×8)
2.779 (112×8)
DiaPCA+2DPCA
10.99 (16×10)
0.78 (16×10)
Table 5. Training and testing time with DCT using the Frobenius distance and the same matrixfeature dimensions as in Table2
2D DCT
block size
8×8
9×9
10×10
11×11
12×12
13×13
14×14
15×15
16×16
2DPCA
0.047
0.048
0.047
0.048
0.063
0.062
0.079
0.094
0.125
Training time in sec
DiaPCA
DiaPCA+2DPCA
0.047
0.047
0.048
0.124
0.048
0.094
0.047
0.063
0.046
0.094
0.047
0.126
0.062
0.14
0.078
0.173
0.141
0.219
2DPCA
0.655
0.626
0.611
0.578
0.641
0.642
0.656
0.641
0.813
Testing time in sec
DiaPCA
DiaPCA+2DPCA
0.704
0.61
0.671
0.656
0.719
0.625
0.734
0.5
0.764
0.657
0.843
0.796
0.735
0.718
0.702
0.796
0.829
0.827
We can conclude from this experiment, that the proposed approach is very efficient
in weakly constrained environments, which is the case of the ORL database.
5 Conclusion
In this paper, 2DPCA, DiaPCA and DiaPCA+2PCA are introduced in DCT domain.
The main advantage of the DCT transform is that it discards redundant information and
it can be used as a feature extraction step. So, computational complexity is significantly reduced. The experimental results show that in addition to the significant gain in
both the training and testing times, the recognition rate using 2DPCA, DiaPCA and
DiaPCA+2DPCA in DCT domain is generally better or at least competitive with the
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M. Bengherabi et al.
recognition rates obtained by applying these three techniques directly on the raw pixel
images; especially under the VM similarity measure. The proposed approaches will be
very efficient for real time face identification applications such as telesurveillance and
access control.
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