Image Processing Reference
In-Depth Information
The second useful family of t-norms and t-conorms 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
t-conorms is inferred from it by duality.
These operators satisfy the excluded middle law and the non-contradiction law.
The most common t-norms and t-conorms 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 t-norms and t-conorms 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.
Search WWH ::

Custom Search