Image Processing Reference

In-Depth Information

To do this, this equation λ
v
=
Cv
which is the eigenvalue equation should be solved for eigen-

values λ ≥ 0and eigenvactors
v
∈
F
.

As
Cv
= (1/
M
) ∑
i
= 1
M
(
Φ
(
X
i
) ·
v
)
Φ
(
X
i
), solutions for
v
with
λ
≠ 0 lie within the span of

Φ(X
1
), … Φ (X
M
)
, these coefficients
α
i
(
i
= 1, …,
M
) are obtained such that

(8)

The equations can be considered as follows

(9)

Having
M
×
M
matrix
K
by
K
ij
=
k
(
X
i
,
X
j
) =
Φ
(
X
i
) ·
Φ
(
X
j
), causes an eigenvalue problem.

The solution to this is

(10)

By selecting the kernels properly, various mappings can be achieved. One of these map-

pings can be achieved by taking the
d
-order correlations, which is known as ARG, between the

entries,
X
i
, of the input vector
X
. The required computation is prohibitive when
d
> 2.

(11)

To map the input data into the feature space
F
, there are four common methods such as lin-

ear (polynomial degree 1), polynomial, Gaussian, and sigmoid, which all are examined in this

work in addition to PCA.

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