Image Processing Reference
In-Depth Information
a value.
4
If the parameter
ω
∈
Ω
ω
is known, the random variable has a fixed value.
P
Ω
From the probability measure
on
, we can derive a probability distribution of X
v
: For any set
A
v
v
on
R
⊂ R
in the Borelalgebra
B
(
R
)
,weset
X
−
1
P
(
X
∈
A
)
:= P
(
(
A
)).
As final step, we will define uncertain fields now. Basically, we need to define
a random variable at every position
x
D
. However, there has to be some strong
correlation between the random variables at close positions because, in visualization,
we are usually dealing with continuous or even differentiable fields. Following Adler
and Taylor [
2
], we define an
uncertain field
depending on our uncertain parameter
ω
∈
Ω
∈
over the domain
D
as a measurable, separable
5
map
v
D
f
:
Ω
→
(
R
)
.
In perfect analogy to random variables, each parameter
ω
∈
Ω
gets assigned a deter-
v
, here denoted as an element of
v
D
. Furthermore,
ministic function
f
ω
:
D
→ R
(
R
)
v
for each position
x
∈
D
, we have a random variable
f
x
:
Ω
→ R
that assigns a
fixed value at point
x
∈
D
to the parameter
ω
∈
Ω
. We will use the notations
v
f
ω
(
x
)
:=
f
x
(ω)
:=
f
(ω,
x
)
:=
(
f
(ω))
x
∈ R
∈
ω
∈
Ω
for the value of the uncertain field
f
at position
x
D
given parameter
.
D
can be defined by a consistent description of distributions on
arbitrary finite subsets of positions in D.
6
v
(
R
)
The measure on
4
The case
v
=
1 means a scalar,
v
=
d
,
d
=
d
2
=
2
,
3 means a vector and the case
v
=
d
×
d
,
d
=
2
,
3 describes a second order tensor.
5
This condition removes subtle measurement problems without imposing restrictions of practical
relevance, see Adler and Taylor [
2
, p. 8]. The concept was originally introduced by Doob [
4
]inhis
book on stochastic processes. In essence, it demands a dense countable subset
D
⊂
P
,andafixed
null set
N
∈S
with
P
(
N
)
=
d
and open
I
0 such that for any closed
B
⊂ R
⊂
P
{
ω
|
f
(
x
,ω)
∈
B
∀
x
∈
I
}
Δ
{
ω
|
f
(
x
,ω)
∈
B
∀
x
∈
I
∩
D
}⊂
N
with symmetric set difference
.
6
According to Doob [
4
, I.5, II.1] and going back to theorems by Kolmogorov, one needs to define
probability distribution functions
Δ
F
x
1
,...,
x
n
(
a
1
,...,
a
n
)
=
P
(
|
x
1
|≤
a
1
,...,
|
x
n
|≤
a
n
)
for arbitrary finite tuples
of points in D, such that the following rather obvious
two consistency conditions hold for all finite subsets of points
(
x
1
,...,
x
n
)
{
x
1
,...,
x
n
}
and value bounds
a
1
,...,
a
n
∈
:
F
x
1
,...,
x
n
(
a
1
,...,
a
n
)
=
F
x
α
1
,...,
x
α
n
(
a
α
1
,...,
a
α
n
)
∀
α
permutations
and
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