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