Image Processing Reference
In-Depth Information
Based on a t-norm t and a complementation c , another operator T , referred to as
the t-conorm, can be defined by duality:
[0 , 1] 2 ,T ( x, y )= c t c ( x ) ,c ( u ) .
( x, y )
[8.47]
Therefore, a t-conorm is a function T :[0 , 1]
×
[0 , 1]
[0 , 1] such that:
[0 , 1] 2 ,T ( x, y )= T ( y, x );
- T is commutative:
( x, y )
[0 , 1] 3 ,T [ T ( x, y ) ,z ]= T [ x, T ( y, z )];
- T is associative:
( x, y, z )
- 0 is an identity element:
[0 , 1] ,T ( x, 0) = T (0 ,x )= x ;
- T is increasing with respect to the two variables:
x
( x, x ,y,y )
[0 , 1] 4 , ( x
x and y
y )=
T ( x ,y ) .
T ( x, y )
Furthermore, we have: T (0 , 1) = T (1 , 1) = T (1 , 0) = 1, T (0 , 0) = 0 and 1 is a
zero element (
x
[0 , 1] ,T ( x, 1) = 1).
The
most
common
examples
of
t-conorms
are: max( x, y ), x + y
xy ,
min(1 ,x + y ).
T-conorms generalize to fuzzy sets the concept of union or of the logical “or”.
For any t-conorm, we have:
[0 , 1] 2 ,T ( x, y )
( x, y )
max( x, y ) .
[8.48]
This shows that the “max” is the smallest t-conorm and that any t-conorm has a
disjunctive behavior.
On the other hand, any t-conorm is smaller than T 0 , which is the highest t-conorm,
defined by:
x
if y =0
[0 , 1] 2 ,T 0 ( x, y )=
( x, y )
[8.49]
y
if x =0
1
otherwise
Furthermore, we have the following inequalities between the most common t-
conorms 3 :
[0 , 1] 2 ,T 0 ( x, y )
( x, y )
min(1 ,x + y )
x + y
xy
max( x, y ) .
[8.50]
3. Again, there is no complete order for all of the t-conorms.
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