Image Processing Reference
In-Depth Information
Furthermore, the t-norms mentioned above verify:
[0
,
1]
2
,t
0
(
x, y
)
∀
(
x, y
)
∈
≤
max(0
,x
+
y
−
1)
≤
xy
≤
min(
x, y
)
.
[8.45]
Let us note, however, that there is no complete order for all of the t-norms.
Parametric forms allow some variations between certain of these operators. For
example, the t-norm defined in [YAG 80] by:
min
1
,
(1
y
)
p
1
/p
[0
,
1]
2
,t
(
x, y
)=1
x
)
p
+(1
∀
(
x, y
)
∈
−
−
−
[8.46]
varies from the Lukasiewicz t-norm max(0
,x
+
y
−
1) for
p
=1to the min for
p
=+
∞
.
Examples of t-norms are shown in Figure 8.3.
Figure 8.3.
Four examples of t-norms. First line:
t
0
(lowest t-norm) Lukasiewicz t-norm.
Second line: product and minimum (highest t-norm)
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