Image Processing Reference
In-Depth Information
which enables us to construct the following complementation:
[0 , 1] ,c ( x )= 1
x n 1 /n .
x
[8.39]
x . The higher
n becomes (for n> 1), or the lower it becomes (for n< 1), the more binary the
resulting form. In the first case, most of the values, except for those close to 1, have
a complement that is close to 1 and in the second case, most of the values, except for
those close to 0, have a complement close to 0.
For n =1, we find the most common form used, i.e. c ( x )=1
If, for a real number a in ]0 , 1], ϕ has the following form:
ax
x
[0 , 1] ( x )=
a ) x +1 ,
[8.40]
(1
then the corresponding complementation is:
x
1+ a 2 x .
1
x
[0 , 1] ,c ( x )=
[8.41]
Here is another example, depending on four parameters a , b , c and n such that
0
a<b<c
1:
1
if 0
x
a
x
n
1
2
a
1
if a
x
b
b
a
x
[0 , 1] ,c ( x )=
[8.42]
c
n
1
2
x
if b
x
c
c
b
0
if c
x
1
A few examples are illustrated in Figure 8.2.
8.5.2. Triangular norms and conorms
In the context of stochastic geometry [MEN 42, SCH 83], a triangular norm, or
t-norm, 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 )
- 1 is a neutral element:
[0 , 1] ,t ( x, 1) = t (1 ,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 )
 
Search WWH ::




Custom Search