Image Processing Reference
In-Depth Information
- based on any t-norm
t
and any continuous, strictly increasing function
h
from
[0
,
1] in [0
,
1] such that
h
(0) = 0 and
h
(1) = 1, we can define another t-norm
t
using
the formula [SCH 63]:
[0
,
1]
2
,t
(
x, y
)=
h
−
1
t
h
(
x
)
,h
(
y
)
.
∀
(
x, y
)
∈
[8.52]
This gives us a way of generating families of t-norms based on an example.
There are generic forms for t-norms and t-conorms with specific properties
[DUB 85]. We will now discuss the two most useful groups: archimedean and nilpo-
tent t-conorms.
A strictly monotonic, archimedean t-norm
t
verifies:
∀
x
∈
[0
,
1]
,t
(
x, x
)
<x,
[8.53]
(
x, y, y
)
[0
,
1]
3
,y<y
=
t
(
x, y
)
<t
(
x, y
)
.
∀
∈
⇒
[8.54]
Likewise, a strictly monotonic archimedean t-conorm
T
verifies the two following
properties:
∀
x
∈
[0
,
1]
,T
(
x, x
)
>x,
[8.55]
(
x, y, y
)
[0
,
1]
3
,y<y
=
T
(
x, y
)
<T
(
x, y
)
.
∀
∈
⇒
[8.56]
Any strictly monotonic, archimedean t-norm
t
can be expressed in the following
form:
[0
,
1]
2
,t
(
x, y
)=
f
−
1
f
(
x
)+
f
(
y
)
,
∀
(
x, y
)
∈
[8.57]
where
f
, referred to as the generating function, is a continuous and decreasing bijec-
tion of [0
,
1] into [0
,
+
∞
] such that
f
(0) = +
∞
and
f
(1) = 0.
The associated t-conorms have the following form:
[0
,
1]
2
,T
(
x, y
)=
ϕ
−
1
ϕ
(
x
)+
ϕ
(
y
)
,
∀
(
x, y
)
∈
[8.58]
where the generating function
ϕ
is a continuous and increasing bijection of [0
,
1] into
[0
,
+
∞
] such that
ϕ
(0) = 0 and
ϕ
(1) = +
∞
.
Such t-norms and t-conorms never satisfy the non-contradiction law and the ex-
cluded middle law. These laws are expressed as:
[0
,
1]
,t
x, c
(
x
)
=0
,
∀
x
∈
[8.59]
[0
,
1]
,T
x, c
(
x
)
=1
.
∀
x
∈
[8.60]
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