Image Processing Reference
In-Depth Information
any
x
,
D
(
x
) ∈ [0, 1] denotes the degree to which
x
satisfies the desired require-
ment. The problem becomes
Dx FAxAxAx Ax
n
() {(), (), (),
=
1
…
, ()}
2
3
Suppose we desire to model the interrelationship between the criteria. At one
extreme, one can desire that 'all' the criteria are satisfied. So,
x
must satisfy
A
1
and
A
2
and
A
n
. Thus, the requirement is 'anding' the values. At another
extreme, we may desire that at least one of the criteria is satisfied. So,
x
must
satisfy
A
1
or
A
2
or
A
n
. Thus, the requirement is 'oring' the values.
There exist a class of operators called
t
-norms that quantitatively implement
'anding' aggregation which implies 'all' the criteria are satisfied. Similar to
t
-norms, there also exist
t
-conorms where 'oring' aggregation is applied which
implies that 'at least' one of the criteria is satisfied. Recently, many authors pro-
posed
t
-norms and
t
-conorms.
t
-norm serves as a union operator, and
t
-conorm
serves as an intersection operator which can be used to handle multiple rules.
In this section, different
t
operators are discussed along with their properties.
3.3.1
t
-Norm
A
t
-norm
T
: [0, 1] → [0, 1] is a kind of binary operation used in the framework
of fuzzy logic and probabilistic metric space. It represents the intersection in
fuzzy set theory or an 'ANDing' operator. The four basic
t
-norms are
1. Zadeh's intersection:
T
M
(
x
,
y
) = min(
x
,
y
)
2. Product intersection:
T
P
(
x
,
y
) =
x
⋅
y
3. Lukasiewicz intersection:
T
L
(
x
,
y
) = max(
x
+
y
− 1, 0)
4. Nilpotent
T
-norm:
Txy
n
(,) min( , ,
,
=
xy xy
if
otherwise
+>
1
0
Definition:
:
T
: [0, 1] × [0, 1] → [0, 1] is a
t
-norm iff it satisfies the following prop-
erties where
x
,
y
,
z
∈ [0, 1] [12,15]:
1. Boundary condition:
T
(0, 0) =
T
(0, 1) =
T
(1, 0) = 0 and
T
(1, 1) = 1.
2. Associativity:
T
(
T
(
x
,
y
),
z
) =
T
(
x
,
T
(
y
,
z
)).
3. Commutativity:
T
(
x
,
y
) =
T
(
y
,
x
).
4. Monotonicity:
T
(
x
,
y
) ≤
T
(
x
,
z
) if
y
≤
z
.
5. One identity:
T
(
x
, 1) =
x
.
6. A
t
-norm,
T
, is called Archimedean iff.
7.
T
is continuous.
8.
T
(
x
,
x
) <
x
for all
x
∈ (0, 1).
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