Image Processing Reference
In-Depth Information
1.2 Intuitionistic Fuzzy Set
A fuzzy set
A
in a finite set
X
= {
x
1
,
x
2
, …,
x
n
} may be represented mathemati-
cally as
AxxxX
A
=
{( , ())|}
μ
∈
where the function μ
A
(
x
) :
X
→ [0, 1] is the measure of the degree of belong-
ingness or the membership function of an element
x
in the finite set
X
, and
the measure of non-belongingness is 1
−
μ
A
(
x
).
An IFS
A
in a finite set
X
may be mathematically represented as
Axx
=
{( , (), ())|
μν
x xX
∈
}
(1.1)
A
A
where the functions μ
A
(
x
), ν
A
(
x
) :
X
→ [0, 1] are, respectively, the membership
function and the non-membership function of an element
x
in a finite set
X
with the necessary condition
0
≤
μ
() ()
x
+
ν
x
≤
1
A
A
It is clear that every fuzzy set is a particular case of IFS:
Axx
=
{( , (),
μ
1
−
μ
())|
x xX
∈
}
A
A
Atanassov [1] also stressed the necessity of taking into consideration a third
parameter π
A
(
x
), known as the intuitionistic fuzzy index or hesitation degree,
which arises due to the lack of knowledge or 'personal error' in assigning the
membership degree. Therefore, with the introduction of a hesitation degree,
an IFS
A
in
X
may be represented as
Axx
=
{( , (), (), ())|}
μν
x xxX
π
∈
A
A
A
with the condition
π
() () ()
x
+
μ
x
+
ν
x
=
1
(1.2)
A
A
A
It is obvious that
0
≤
π
A
x
() , or each
≤
1
x X
∈
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