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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