Image Processing Reference
In-Depth Information
If A and B are two fuzzy sets and μ
A
(
x
) and μ
B
(
x
) are the membership val-
ues of the elements, the union operator can be constructed using the OR
operator and is defined as
Union operator: μ
A
∪
B
(
x
) = μ
A
(
x
) ∨ μ
B
(
x
)
Intersection operator: μ
A
∩
B
(
x
) = μ
A
(
x
) ∧ μ
B
(
x
)
Complement: μ
A
c
(
x
) = 1 − μ
A
(
x
)
These operators fulfil the following axioms:
OR operator
1. Monotonicity: If
a
<
a
′ and
b
<
b
′, then
a
∨
b
<
a
′ ∨
b
′
2. Commutativity:
a
∨
b
=
b
∨
a
3. Boundary condition: 0 ∨ 0 = 0, 0 ∨ 1 = 1, 1 ∨ 0 = 1, 1 ∨ 1 = 1
4. Associativity: (
a
∨
b
) ∨
c
=
a
∨ (
b
∨
c
)
The AND operator follows similar types of axioms with the following
boundary condition:
0 ∨ 0 = 0, 0 ∨ 1 = 0, 1 ∨ 0 = 0, 1 ∨ 1 = 1
Complement operator:
a
Boundary condition:
1001
,
Monotonicity: If
a
≤
b
, then
b
≤
a
Involutive:
(
aa
==
=
Fuzzy logic operators are widely used in fuzzy inference systems where if-
then rules follow. Suppose there are two fuzzy sets
A
with
A
= {
a
1
,
a
2
, …,
a
n
}
and
B
with
B
= {
b
1
,
b
2
, …,
b
n
}, the rules may be written as:
R
1
: If (
a
∧
b
), then
c
R
2
: If (
a
∨
b
), then
c
These fuzzy operators can be written using the min(∧) and max(∨) functions.
3.3 Fuzzy Operators Induced by Fuzzy
t
-Norm and
t
-Conorm
Suppose there are
A
1
,
A
2
,
A
3
, …,
A
n
criteria in multi-criteria decision-making
problem. For each
A
j
, let
A
j
(
x
) ∈ [0, 1] be a degree to which
x
satisfies the
criteria. The problem is to formulate an overall decision
D
such that for
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