Image Processing Reference
In-Depth Information
A function
T
: [0, 1] × [0, 1] → [0, 1] is called an Archimedean
t
-norm iff there
exists a decreasing and continuous function
f
: [0, 1] → [0, ∞] with
f
(1) = 0:
−
1
Txyf fx fy
(,)
=
( () ())
+
(3.1)
3.3.2
t
-Conorm
A
t
-conorm
T
*
: [0, 1] → [0, 1] is a kind of binary operation used in the frame-
work of fuzzy logic and probabilistic metric spaces. It represents union in
fuzzy set theory or an 'ORing' operator. The four basic
t
-conorms are
*
(,)max(,)
1. Zadeh's union:
Txy
=
x y
M
*
(,)
=+−⋅
2. Product union:
Txyxyxy
P
*
(,)min(
3. Lukasiewicz union:
Txy
=
x
+
y
, )
1
L
*
(,) max( , ,
,
4. Nilpotent
T
-conorm:
Txy
=
xy xy
if
otherwise
+<
1
n
0
Definition:
:
T
*
: [0, 1] × [0, 1] → [0, 1] is a
t
-conorm iff it satisfies the following
properties where
x
,
y
,
z
∈ [0, 1] [12,15]:
1. Boundary condition:
T
*
(0, 0) = 0 and
T
*
(0, 1) =
T
*
(1, 0) =
T
*
(1, 1) = 1.
2. Commutativity:
T
*
(
x
,
y
) =
T
*
(
y
,
x
).
3. Monotonicity:
T
*
(
x
,
y
) ≤
T
*
(
x
,
z
) if
y
≤
z
.
4. Associativity:
T
*
(
T
*
(
x
,
y
),
z
) =
T
*
(
x
,
T
*
(
y
,
z
)).
5. Zero identity:
T
*
(
x
, 0) =
x
.
6. A
T
-conorm,
T
*, is called Archimedean iff.
7.
T
* is continuous.
8.
T
*
(
x
,
x
) >
x
for all
x
∈ (0, 1).
A function
T
*
: [0, 1] × [0, 1] → [0, 1] is called an Archimedean
t
-conorm iff
there exists an increasing and continuous function
g
: [0, 1] → [0, ∞] with
g
(0) = 0:
*
(,)
−
1
Txyggx gy
=
( () ())
+
(3.2)
f
and
g
are additive generators.
There are many triangular operators, and they are classified as a class of
(1) conditional operators and (2) algebraic operators [7,12,15,18]. Conditional
operators contain min or max operators or the combination of min and
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