Image Processing Reference
In-Depth Information
(5)
where
is the mean matrix of input images and finally we have
(6)
First, the
n
×
n
matrix of
G
t
is calculated from all of the training images. Then, the unitary
vectors
X
are obtained by geting the eigenvector matrix of
G
t
. This stage decides how many
eigenvectors are to be used in the projection of data. To achieve this, the eigenvalues of the cor-
responding eigenvectors are arranged in a descending order, and a subset of the higher values
is selected. Assuming
d
eigenvectors (with optimal projection axes
X
1
,
X
2
, …,
X
d
) are selected,
then how to achieve feature extraction and classification stages are explained in the next sec-
tion.
4 Kernel mapping along row and column direction
4.1 Two-Dimensional KPCA
The main idea of using kernel function in PCA is that the data are first mapped into another
space using a mapping function and then PCA is performed on the nonlinearly mapped data.
2DPCA is beter than 1D PCA in terms of speed and accuracy. The idea of using kernel func-
tion in 2DPCA is to improve the accuracy of the system. With
N
input images, let
A
i
be the
i
th image, where
i
= 1, 2, …,
N
, and
A
i
j
be the
j
th row of the matrix
A
i
, where
j
= 1, 2, …,
n
. The
nonlinear mapping is defined as follows:
(7)
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