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and identification of input image is made on the basis of feature vectors and stored
weights applied to each corresponding component one by one, for all the classes
(subjects) in the test database.
7.2 Multivariate Statistical Techniques in Real
and Complex Domain
A fundamental problem in digital image processing is to find a suitable representa-
tion for image, audio-video or other kind of data. Statistical methods like PCA-ICA
find a set of basis images (Figs.
7.2
and
7.4
) and represent images as a linear com-
bination of these basis images. This reduces the amount of training image data prior
to classification and provide reduction in computational cost while maintaining suf-
ficient information for meaningful analysis. In most of the applications, an image,
x, is available in the form of single or multiple views of 2D (p by q) intensity data
(i.e., pixel values). Thus, inputs to the face recognition system are visuals only.
x
= {
a
k
:
k
∈
S
}
,
where
S
is a square lattice
.
(7.1)
Sometimes, it is more convenient to express an image matrix as a one dimensional
vector of concatenated rows of pixels, then an image vector
= {
a
1
,
a
2
, ...
a
N
}
,
=
×
.
x
where
N
p
q
is the total number of pixels
(7.2)
Such a high dimensional image space is usually inefficient and lacks discrimi-
native power. Therefore, we need to transform 'x' into a feature vector or a new
representation which greatly reduces the image feature dimensions, and yet main-
tains reasonable discriminative power by maximizing the spread of different faces
within the image subspace. Let
X
T
be the M by N matrix of image
data, M is the number of images in the training set. Let vector
avg
= {
x
1
,
x
2
, ...
x
M
}
M
k
=
1
x
k
be
the mean of training images. The goal in following feature extraction techniques is
to find a useful representation of an image by minimizing the statistical dependence
among the basis vectors.
1
=
7.2.1 Feature Extraction with
R
PCA
Principal component analysis in real domain (
R
PCA
) is widely applied for dimen-
sionality reduction and feature extraction by extracting the preferred number of prin-
cipal components of multivariate data [
21
,
24
,
26
]. It captures as much variation as
possible of training data set with fewer number of eigen vector subspace as possible.
If image elements are considered to be random variables and images are seen as a
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