Image Processing Reference
InDepth Information
Based on a tnorm
t
and a complementation
c
, another operator
T
, referred to as
the tconorm, can be defined by duality:
[0
,
1]
2
,T
(
x, y
)=
c
t
c
(
x
)
,c
(
u
)
.
∀
(
x, y
)
∈
[8.47]
Therefore, a tconorm 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
)
∈
 0 is an identity element:
[0
,
1]
,T
(
x,
0) =
T
(0
,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
)
≤
Furthermore, we have:
T
(0
,
1) =
T
(1
,
1) =
T
(1
,
0) = 1,
T
(0
,
0) = 0 and 1 is a
zero element (
∀
x
∈
[0
,
1]
,T
(
x,
1) = 1).
The
most
common
examples
of
tconorms
are: max(
x, y
),
x
+
y
−
xy
,
min(1
,x
+
y
).
Tconorms generalize to fuzzy sets the concept of union or of the logical “or”.
For any tconorm, we have:
[0
,
1]
2
,T
(
x, y
)
∀
(
x, y
)
∈
≥
max(
x, y
)
.
[8.48]
This shows that the “max” is the smallest tconorm and that any tconorm has a
disjunctive behavior.
On the other hand, any tconorm is smaller than
T
0
, which is the highest tconorm,
defined by:
⎧
⎨
x
if
y
=0
[0
,
1]
2
,T
0
(
x, y
)=
∀
(
x, y
)
∈
[8.49]
y
if
x
=0
⎩
1
otherwise
Furthermore, we have the following inequalities between the most common t
conorms
3
:
[0
,
1]
2
,T
0
(
x, y
)
∀
(
x, y
)
∈
≥
min(1
,x
+
y
)
≥
x
+
y
−
xy
≥
max(
x, y
)
.
[8.50]
3. Again, there is no complete order for all of the tconorms.
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