Image Processing Reference
In-Depth Information
The new kernel function distance function is written as
2
ϕϕ
() () (,)
x
−
v
=
KxxKvv Kx v
+
(
, ) (,)
−
2
( 7. 6 )
i
k
i
i
kk i k
where
i
= 1, 2, …,
n
k
= 1, 2, …,
c
The kernel function may be a radial basis function (RBF), hypertangent,
Gaussian or polynomial kernel.
⎛
2
⎞
−−
xv
i
k
Hypertangent kernel
Hx v
(, )=−
1
tanh
⎜
⎟
i
k
2
σ
⎝
⎠
Using this function, the distance function becomes
2
ϕϕ
() () (,)
x
−
v
=
HxxHvv Hx v xx Hv
+
(
, ) (,)
−
2
as
( ,) , (,
=
1
v
k
)
=
1
i
k
i
i
kk
i k
i
i
k
so,
2
ϕϕ
() () (
x
−
v
=
21
−
Hx v
(
, )
( 7. 7 )
i
k
i k
⎛
−−
xv
⎞
i
k
Gaussian kernel
Gx v
(,) xp
=
( 7. 8)
⎜
⎟
i k
σ
2
⎝
⎠
In this case,
G
(
x
i
,
x
i
) = 1,
G
(
v
k
,
v
k
) = 1, so ||φ(
x
i
) − φ(
v
k
) ||
2
= 2(1 −
G
(
x
i
,
v
k
)):
⎛
a
b
⎞
a
xv
−
⎜
⎜
i
k
⎟
⎟
Radial basiskernel
Rx v
(,) xp
=
−
i k
σ
2
⎝
⎠
where σ,
a
and
b
are the adjustable parameters.
In this case also,
2
ϕϕ
() () (
x
−
v
=
21
−
Rx v
(
, )
( 7. 9)
i
k
i k
To show the application of kernel clustering on images, a few examples are
given.
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