Information Technology Reference
In-Depth Information
sample of a stochastic process, the PCA basis vectors (eigenfaces) are defined as the
orthonormal eigenvectors of the covariance matrix of image vectors. They describe a
set of axis within the image space and most of the variances of images are along these
axis. The eigenvalues associated with each eigen vector define the degree of spread
(variance) of the image population in the direction of that eigenvector (along these
axis). The eigenvectors associated with largest eigenvalue is the axis of maximum
variance. The eigenvectors associated with second largest eigenvalue is the orthogo-
nal axis, with second largest variance. Thus, the eigenvectors corresponding to higher
eigenvalues carry significant information for representation, which best accounts for
the distribution of images within the entire image space and so on. For most appli-
cations, the eigen vectors corresponding to very small eigenvalues are considered
as noise and not taken into account during identification. All images in the training
set are projected onto reduced subspace (set of eigenvectors corresponding to higher
eigenvalues) to find a set of weights (new representation of images) that describes
the contribution of each vector in the image space. By comparing the weights of the
test image with the set of weights of the images in training set, the image in the test
data can be identified.
Basic steps in
R
PCA
algorithm for feature extraction can be summarized as
follows:
1. Collect the images in data matrix X (M by N). Find the mean subtracted data
matrix,
A
avg
.
2. Calculate the covariance matrix
C
=
X
−
A
T
A
; where
A
T
denotes the transpose of
=
A.
3. Find the eigenvectors (basis images)
A
T
v
i
(N by M) of
C
, such that
A
T
AA
T
v
i
=
e
i
A
T
v
i
.
4. In finding the eigenvectors of
C
, however for the moderate size of an image
(
N
=
p
×
q
), the dimension of
C
will be
pq
×
pq
. Hence, calculations become
quite unmanageable.
5. Turk [
24
] then circumnavigated the problem by proposing following scheme.
Consider the eigenvectors
v
i
of
AA
T
, such that
•
AA
T
v
i
=
e
i
v
i
(7.3)
Each eigenvector
v
i
will be of size M and there will be such M eigenvectors.
Thus, the calculations are greatly reduced from the order of number of pixels
(N) in the image to the order of number of images (M) in the training set,
M
N
.
Multiply Eq.
7.3
both side by
A
T
, hence
•
A
T
AA
T
v
i
e
i
A
T
v
i
=
(7.4)
CA
T
v
i
e
i
A
T
v
i
=
(7.5)
The eigenvectors (basis images) of covariance matrix
C
will be
A
T
v
i
of size
N by M. Thus eigenvectors of
A
T
A
(a N by N matrix) can be found from
Search WWH ::
Custom Search