Image Processing Reference
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4. Szmidt and Kacprzyk [17] suggested a similar type of entropy which
is given as
n
1
min{((), ( }
μ
x
ν
x
+
+
π
(
x
)
∑
1
Ai Ai Ai
IFEA
()
=
(4.26)
n
max{((), ( }
μ
x
ν
x
π
(
x
i
)
Ai Ai
A
i
=
5. An entropy suggested by Huang and Liu [7] on a vague set but
extended to
IFS
is defined as
n
1
1
−
μ
() () ()
() () ()
x
−
ν
x
+
π
x
1
∑
Ai Ai Ai
IFEA
()
=
(4.27)
n
+
μ
x
−
ν
x
+
π
x
Ai Ai Ai
i
=
1
6. Vlachos and Sergiadis [19] introduced an entropy which is defined as
n
2
12
μ
() () ()
() () ()
x
⋅
ν
x
+
π
x
∑
Ai Ai Ai
IFEA
()
=
2
2
2
n
μ
x
+
ν
x
+
π
x
Ai Ai Ai
i
=
1
Two different kinds of measures using
IFS
are given by Ye [23]:
EA
()
IFS
n
1
⎡
⎢
π
×+ −
[
1
μ
(
x
)
ν
(
x
)]
π
×− +
[
1
μ
(
x
)
ν
(
x
)]
1
21
⎤
⎥
i
⎩
⎭
Ai Ai
Ai Ai
=
sin
+
sin
−
1
×
n
4
4
−
1
=
EA
()
IFS
n
1
⎡
⎢
π
×+ −
[
1
μ
(
x
)
ν
(
x
)]
π
×− +
[
1
μ
(
x
)
ν
(
x
)]
1
21
⎤
⎥
i
⎩
⎭
Ai Ai
Ai Ai
=
cos
+
cos
−
1
×
n
4
4
−
1
=
4.7 Entropy of Interval-Valued Intuitionistic Fuzzy Set
In
IFS
A
, μ
A
(
x
) and ν
A
(
x
) denote the membership and non-membership
functions, respectively. For convenience, μ
A
(
x
) may be represented as
μ
and ν
A
(
x
) may be represented as
ν
() [(),
x
=
ν
−
x
ν
+
( )].
x
() [(), ()]
x
=
μ
−
x
μ
+
x
A
A
A
A
A
A
So, interval-valued intuitionistic fuzzy set (
IVIFS
) is represented as
Ax x
−
+
−
+
=
{,[(),
μμ ν
( )],[(), ()],
x
x
ν
x xX
∈
}.
A
A
A
A
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