Image Processing Reference
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(1-x^4)^0.25
(1-x^2)^0.5
1-x
a = 0.3
b = 0.6
c = 0.9
n = 2
(1-x)/(1+3x)
(1-x^0.5)^2
(1-x^0.25)^4
Figure 8.2. A few examples of fuzzy complementation
Additionally, we have: t (0 , 1) = t (0 , 0) = t (1 , 0) = 0, t (1 , 1) = 1 and 0 is a zero
element (
x
[0 , 1] ,t ( x, 0) = 0).
Continuity is often added to this list of properties.
The operators min( x, y ), xy , max(0 ,x + y
1) are examples of t-norms, which
are by far the most commonly used.
T-norms generalize to fuzzy sets the concept of intersection as well as the logical
“and”.
The following result is easy to prove. For any t-norm t ,wehave:
[0 , 1] 2 ,t ( x, y )
( x, y )
min( x, y ) .
[8.43]
This shows that the “min” is the highest t-norm and that any t-norm has a conjunc-
tive behavior.
On the other hand, any t-norm is always higher than t 0 , which is the smallest t-
norm, defined by:
x
if y =1
[0 , 1] 2 ,t 0 ( x, y )=
( x, y )
y
if x =1
[8.44]
0
otherwise
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