Image Processing Reference
In-Depth Information
∈
If we consider the situation at a single point
x
D
and a measurable subset of
v
, e.g. V is closed or open, we have the probability
⊂ R
values
V
P
(
f
(ω,
x
)
∈
V
)
= P
(
{
ω
|
f
(ω,
x
)
∈
V
}
).
As an example for the probability space
(Ω,
S
,
P
)
, we assume that we have a set
p
1
p
N
2
in the plane. At these positions, we have uncertain
of positions
{
,...,
}∈R
v
1
v
N
with normal distributions
7
W
i
scalar values
{
,...,
}∈R
∼
N
(μ
i
,σ
i
)
.Wemay
assume that these values are not independent with covariances
C
ii
.
C
ij
=
E
((
v
i
−
μ
i
)(
v
j
−
μ
j
))
with
σ
i
=
N
n
Then, we have
. This means that our proba-
bility space is N-dimensional real space with an
N
-dimensional normal distribution
with mean vector
(Ω,
S
,
P
)
=
(
R
,
B
(
R
),
N
(μ,
C
))
N
×
N
.Itshall
be noted that it is possible to derive a space with independent Gaussian variables with
potentially smaller dimension
M
N
and (symmetric) covariance matrix
C
μ
∈ R
∈ R
N
by spectral decomposition of
C
and using the
eigenvectors with eigenvalue different from 0. In the following sections, we will see
how we can define an uncertain scalar field from these data.
<
9.3 Gaussian Processes
The previous section introduced stochastic processes without referring to a specific
type of distribution at every position. A careful look at the footnotes or intuition
tells that the distributions at the different points have to be somehow consistent,
and that a simple solution might be to use distributions of the same type every-
where. Looking at the literature, it can be said that Gaussian distributions are the
most often used case. If one uses them, one arrives at the special topic of Gaussian
processes. They have been analyzed in detail with respect to geometric proper-
ties by Adler and Taylor [
1
-
3
] in a mathematically rigorous fashion. But Gaussian
processes have also been applied in other areas of computer science. A nice ex-
ample is provided by machine learning as described in the topic by Rasmussen
F
x
1
,...,
x
m
(
a
1
,...,
a
m
)
=
lim
n
F
x
1
,...,
x
n
(
a
1
,...,
a
n
)
∀
m
<
n
λ
j
→∞
,
j
=
m
+
1
,...,
We will use multivariate Gaussian distributions for this purpose in the next sections. This footnote
illustrates that other distributions are possible.
7
A normal distribution on
is defined by a probability density function
exp
−
(
x
−
μ)
2
1
√
2
φ(
x
)
=
2
.
2
σ
πσ
μ
is the mean of the distribution and
σ
the standard deviation.
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