Image Processing Reference
In-Depth Information
Let us note that auto-duality differs from the duality mentioned between t-norms
and t-conorms. For those operators, inverting the scale changes the type of operator.
Here, the scale of values can be inverted without changing the way they are combined.
This property was used in particular for combining expert opinions. It could also be
expressed with other complementations.
From these basic properties, we can infer that:
-
σ
(1
,
1) = 1,
-
x
)=1
/
2,
- the only symmetric sum that is both associative and a mean is the median with
the parameter 1
/
2.
∀
x
∈
]0
,
1[,
σ
(
x,
1
−
The general form of symmetric sums is given by:
g
(
x, y
)
g
(
x, y
)+
g
(1
σ
(
x, y
)=
y
)
,
[8.73]
−
x,
1
−
where
g
is a continuous, positive, increasing function of [0
,
1]
[0
,
1] into [0
,
1], such
that
g
(0
,
0) = 0. Typically, a continuous t-norm or t-conorm can be chosen as
g
.
×
If
∀
x
∈
[0
,
1],
g
(0
,x
)=0, then
σ
(0
,
1) is not defined; otherwise
σ
(0
,
1) = 1
/
2.
The general form of strictly increasing, associative, symmetric sums is given by:
[0
,
1]
2
,σ
(
x, y
)=
ψ
−
1
ψ
(
x
)+
ψ
(
y
)
,
∀
(
x, y
)
∈
[8.74]
where
ψ
is a strictly monotonic function such that
ψ
(0) and
ψ
(1) are not bounded
and
x
)+
ψ
(
x
)=0. From this, we infer that 0 and 1 are identity
elements and that 1
/
2 is a zero element.
∀
x
∈
[0
,
1]
,ψ
(1
−
Table 8.2 shows a few typical examples of symmetric sums. They are obtained by
using various t-norms and t-conorms as generating function
g
.
g
(
x, y
)
σ
(
x, y
)
property
xy
1
−
x
−
y
+2
xy
xy
σ
0
(
x, y
)=
associative
x
+
y
−
xy
1+
x
+
y
−
2
xy
x
+
y
−
xy
σ
+
(
x, y
)=
non-associative
σ
min
(
x, y
)=
min(
x,y
)
1
−|
x
−
y
|
min(
x, y
)
mean
σ
max
(
x, y
)=
max(
x,y
)
1+
|
x
−
y
|
max(
x, y
)
mean
Table 8.2.
Examples of symmetric sums, defined based on t-norms and t-conorms. For
σ
0
,
we adopt the convention that
σ
0
(0
,
1) =
σ
0
(1
,
0) = 0
and for
σ
min
, we assume that
σ
min
(0
,
1) =
σ
min
(1
,
0) = 0
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