Image Processing Reference
InDepth Information
The second useful family of tnorms and tconorms is comprised of the nilpotent
operators, which have the following general form:
[0
,
1]
2
,t
(
x, y
)=
f
∗
f
(
x
)+
f
(
y
)
,
∀
(
x, y
)
∈
[8.67]
where
f
is a decreasing bijection of [0
,
1] into [0
,
1], such that
f
(0) = 1,
f
(1) = 0 and
f
∗
(
x
)=
f
−
1
(
x
) if
x
[0
,
1], with
f
∗
(
x
)=0if
x
∈
≥
1. The general form of nilpotent
tconorms is inferred from it by duality.
These operators satisfy the excluded middle law and the noncontradiction law.
The most common tnorms and tconorms in this class are the Lukasiewicz opera
tors:
[0
,
1]
2
,t
(
x, y
) = max(0
,x
+
y
∀
∈
−
(
x, y
)
1)
,T
(
x, y
)=min(1
,x
+
y
)
.
Examples of generating functions
f
have been suggested by Schweizer and Sklar
[SCH 63], as well as by Yager [YAG 80].
Operators combining tnorms and tconorms have been suggested, for example, by
[ZIM 80]:
[0
,
1]
2
,C
γ
(
x, y
)=
t
(
x, y
)
1
−
γ
T
(
x, y
)
γ
,
∀
(
x, y
)
∈
[8.68]
where
γ
is a parameter in [0
,
1].
8.5.3.
Mean operators
A mean operator is a function
m
:[0
,
1]
[0
,
1] such that:
 the result of the combination is always included between the min and the max:
×
[0
,
1]
→
[0
,
1]
2
,
min(
x, y
)
∀
(
x, y
)
∈
≤
m
(
x, y
)
≤
max(
x, y
),but
m
=minand
m
=max;

m
is commutative;

m
is increasing with respect to the two variables:
(
x, x
,y,y
)
[0
,
1]
4
,
(
x
x
and
y
y
)=
m
(
x
,y
)
.
∀
∈
≤
≤
⇒
m
(
x, y
)
≤
The first property entails that
m
is always an idempotent operation:
∀
x
∈
[0
,
1]
,m
(
x, x
)=
x.
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