Image Processing Reference

In-Depth Information

(4)

which means the first
M
− 1 eigenvectors
λ
and eigenvalues
v
can be obtained by calculat-

ing
WW
T
.

When we have
M
eigenvectors and eigenvalues, the images could be projected onto
L M

dimensions by computing

(5)

where
Ω
is the projected value. Finally, to determine which face provides the best description

of an input image, the Euclidean distance is calculated using Equation
(6)
.

(6)

And finally, the minimum ∈
k
will decide the unknown data into
k
class.

4 Kernel principal component analysis

4.1 KPCA Algorithm

Unlike PCA, KPCA extracts the features of the data nonlinearly. It obtains the principal com-

ponents in
F
which is a high-dimensional feature space that is related to the feature spaces

nonlinearly. The main idea of KPCA is to map the input data to the feature space
F
first us-

ing a nonlinear mapping
Φ
when input data have nonlinearly been mapped, the PCA will be

where
M

is the number of input data. The covariance matrix of
F
can be defined as

(7)

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