Image Processing Reference

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These two equations are usually not verified by strictly monotonic, archimedean

t-norms and t-conorms.

Any strictly monotonic, archimedean t-norm (or t-conorm) can be defined based

on multiplicative generators, by equivalence with additive generators [CHE 89]:

[0
,
1]
2
,t
(
x, y
)=
h
−
1
h
(
x
)
h
(
y
)
,

∀

∈

(
x, y
)

[8.61]

where
h
is a strictly increasing function of [0
,
1] into [0
,
1] such that
h
(0) = 0 and

h
(1) = 1. The equivalence with the additive form is obtained simply by defining:

h
=
e
−
f

[8.62]

where
f
is an additive generating function.

The most common t-norms and t-conorms in this class are the product and the

algebraic sum:

[0
,
1]
2
,t
(
x, y
)=
xy,

∀

(
x, y
)

∈

T
(
x, y
)=
x
+
y

−

xy.

[8.63]

The only rational t-norms of this class are the Hamacher t-norms defined by

[HAM 78]:

xy

[0
,
1]
2
,

∀

(
x, y
)

∈

xy
)
,

[8.64]

γ
+(1

−

γ
)(
x
+
y

−

where
γ
is a positive parameter (for
γ
=1we get the product again). They are illus-

trated in Figure 8.5.

Figure 8.5.
Two examples of Hamacher t-norms, for

γ
=0

(left)

and

γ
=0
.
4

(right)

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