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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