Image Processing Reference
In-Depth Information
as probability distribution. This means that we assume two normally distributed,
independent real parameters that will determine our uncertain field. In this simple
case, the two random variables
W
1
:
Ω
→ R
,
W
1
(ω)
=
ω
1
,
W
2
:
Ω
→ R
,
W
2
(ω)
=
ω
2
determine the values at the two given positions
x
1
and
x
2
, respectively. It is natural
to define the linearly interpolated uncertain field
f
on the real line by
f
:
Ω
→
(
R
), (
f
(ω))
x
:=
ω
1
(
1
−
x
)
+
ω
2
x
.
With the notation
f
ω
(
x
)
=
ω
1
(
1
−
x
)
+
ω
2
x
,
it becomes pretty clear that we are really defining a linear interpolation of the values
at 0 and 1 on the real line for each given
. However, the whole point of the chapter
is that we are really defining a Gaussian process! The short argument is that this
follows from slightly more abstract arguments of Adler and Taylor [
3
, pp. 17-19].
However, some basic computations might improve understanding of this point: At
every position
x
ω
∈
D
, we have the random variable
f
x
(ω)
=
ω
1
(
1
−
x
)
+
ω
2
x
.
ω
1
,ω
2
are independent Gaussian variables, this is a Gaussian variable with
expectation
As
μ(
x
)
=
E
(
f
x
(ω))
=
μ
1
(
1
−
x
)
+
μ
2
x
and variance
2
2
2
2
2
x
2
2
σ
(
x
)
=
E
((
f
x
(ω)
−
μ(
x
))
)
=
σ
1
(
1
−
x
)
+
σ
.
For the covariance function
C
:
D
×
D
→ R
,wehave
C
(
x
,
y
)
=
E
((
f
x
(ω)
−
μ(
x
))(
f
y
(ω)
−
μ(
y
)))
=
E
(((ω
1
−
μ
1
)(
1
−
x
)
+
(ω
2
−
μ
2
)
x
)((ω
1
−
μ
1
)(
1
−
y
)
+
(ω
2
−
μ
2
)
y
))
2
2
=
(
1
−
x
)(
1
−
y
)
E
((ω
1
−
μ
1
)
)
+
xyE
((ω
2
−
μ
2
)
)
2
1
2
2
=
(
1
−
x
)(
1
−
y
)σ
+
xy
σ
because of the independence of
ω
1
,ω
2
, i.e.
E
((ω
1
−
μ
1
)(ω
2
−
μ
2
))
=
0. For
σ
1
=
σ
2
,
this coincides with the construction by Pöthkow and Hege [
5
].
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