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Fig. 3.
Contourlet coecients of a sample face
[19]. Wavelets provide a time-frequency representation of signals and are good
at analyzing point (or zero dimensional) discontinuities. Therefore, Wavelets are
suitable for analyzing one dimensional signals. On the other hand, images are
inherently two dimensional and can have one dimensional discontinuities such
as curves. These discontinuities can be captured by Contourlets [8]. The Con-
tourlet transform performs multi-resolution and multi-directional decomposition
of images allowing for different number of directions at each scale [8].
Let
represent the vector of Contourlet coe
cients of the
i
th image (where
i
=1
...
23) at scale
s
and orientation
k
. The Contourlet transform has 33%
inherent redundancy [8]. Moreover, the Contourlet coecients (at the same scale
and orientation) of many faces can be approximated by a much smaller linear
subspace. Therefore, the Contourlet coecients of all training images calculated
at the same scale and orientation are projected separately to PCA subspaces.
Let
a
sk
i
A
sk
a
sk
ij
,and
j
=1
,
2
,...G
)representthe
matrix of Contourlet coecients of
N
training images (under different illumina-
tions) of
G
subjects in the training data at the same scale
s
and same orientation
k
. Note that only a subset of the 23 images under different illuminations are used
for training. Each column of
=[
](where
i
∈{
1
,
2
...
23
}
A
sk
contains the Contourlet coecients of one im-
age. The mean of the matrix is given by
N×G
1
µ
sk
=
n
=1
A
s
n
,
(1)
N
×
G
and the covariance matrix by
N×G
1
C
sk
=
A
s
n
−
µ
sk
)(
A
s
n
−
µ
sk
)
T
(
.
(2)
N
×
G
n
=1
C
sk
are calculated by Singular Value Decomposition
U
sk
S
sk
(
The eigenvectors of
V
sk
)
T
=
C
sk
,
(3)
U
sk
contains the eigenvectors sorted according to the decreas-
ing order of eigenvalues and the diagonal matrix
where the matrix
S
sk
contains the respective
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