Image Processing Reference
In-Depth Information
which enables us to construct the following complementation:
[0
,
1]
,c
(
x
)=
1
x
n
1
/n
.
∀
x
∈
−
[8.39]
x
. The higher
n
becomes (for
n>
1), or the lower it becomes (for
n<
1), the more binary the
resulting form. In the first case, most of the values, except for those close to 1, have
a complement that is close to 1 and in the second case, most of the values, except for
those close to 0, have a complement close to 0.
For
n
=1, we find the most common form used, i.e.
c
(
x
)=1
−
If, for a real number
a
in ]0
,
1],
ϕ
has the following form:
ax
∀
∈
x
[0
,
1]
,ϕ
(
x
)=
a
)
x
+1
,
[8.40]
(1
−
then the corresponding complementation is:
x
1+
a
2
x
.
1
−
∀
x
∈
[0
,
1]
,c
(
x
)=
[8.41]
Here is another example, depending on four parameters
a
,
b
,
c
and
n
such that
0
≤
a<b<c
≤
1:
⎧
⎨
1
if 0
≤
x
≤
a
x
n
1
2
−
a
1
−
if
a
≤
x
≤
b
b
−
a
∀
∈
x
[0
,
1]
,c
(
x
)=
[8.42]
c
n
⎩
1
2
−
x
if
b
≤
x
≤
c
c
−
b
0
if
c
≤
x
≤
1
A few examples are illustrated in Figure 8.2.
8.5.2.
Triangular norms and conorms
In the context of stochastic geometry [MEN 42, SCH 83], a triangular norm, or
t-norm, is a function
t
:[0
,
1]
×
[0
,
1]
→
[0
,
1] such that:
[0
,
1]
2
,t
(
x, y
)=
t
(
y, x
);
-
t
is commutative:
∀
(
x, y
)
∈
[0
,
1]
3
,t
[
t
(
x, y
)
,z
]=
t
[
x, t
(
y, z
)];
-
t
is associative:
∀
(
x, y, z
)
∈
- 1 is a neutral element:
[0
,
1]
,t
(
x,
1) =
t
(1
,x
)=
x
;
-
t
is increasing with respect to the two variables:
∀
x
∈
(
x, x
,y,y
)
[0
,
1]
4
,
(
x
x
and
y
y
)=
t
(
x
,y
)
.
∀
∈
≤
≤
⇒
≤
t
(
x, y
)
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