Image Processing Reference

In-Depth Information

(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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