Image Processing Reference

In-Depth Information

This metric is induced from the norm of the Hilbert space
L
2
([
a
,
b
]). Consequently, the kernel

matrix
M
is positive semidefinite, since it is the kernel matrix of the Gaussian kernel for

L
2
([
a
,
b
]). Therefore,
k
is a kernel.

The formula for
d
p
(
F
,
G
) resembles the metric induced by the norm in
L
p
(
R
). However, a CDF

F
cannot be an element of
L
p
(
R
) because . The condition of bounded support

will guarantee the convergence of the integral. In practical applications, this will not likely be a

limitation. Theoretically, the integral could be divergent without this constraint. For example,

let
F
be the step function at 0 and
G
(
x
) =
x
/(
x
+ 1),
x
≥ 0. Then

Given a data sample, (
X
1
,
X
2
, …,
X
n
), an empirical CDF can be constructed as:

which can be used to approximate the distance
d
p
(
F
,
G
).

When
p
= ∞, we have

The distance measures defined above satisfy certain desirable properties of invariance.

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