Image Processing Reference
In-Depth Information
The interval
[(), ()][ () (),
is the hesita-
ππ μ
−
x
+
x
=− −
1
−
x
ν
−
x
1
−
μ
+
( )
x
−
ν
+
( )]
x
A
A
A
A
A
A
tion degree.
A real function
E
:
IVIFS
(
X
) → (0, 1) is called the entropy of
IVIFS
set if it
satisfies the following properties [22]:
1.
E
(
A
) = 0 if
A
is a crisp set.
2.
E
(
A
) = 1 if
[(), ()][(), ()]
μμ ν
−
x
+
x
=
−
x
ν
+
x
for all
x
i
∈
X
.
A
A
A
A
3.
E
(
A
) ≤
E
(
B
) if
A
⊆
B
when
μ
−
() ()
x
≤
ν
−
x
and
μ
+
() ()
x
≤
ν
+
x
or
B
⊆
A
if
B
B
B
B
μ
−
() ()
x
≥
ν
−
x
and
μ
+
() ()
x
≥
ν
+
x
for any
x
i
∈
X
.
B
B
B
B
As in the entropy of
IFS
, the entropy of
IVIFS
is given as [22]
{
}
+
{
}
+
−
−
+
+
−
+
n
min(
μ
x
), () min(
ν
x
μ
x
), () (
ν
x
π
x
)
+
π
(
x
)
1
∑
Ai Ai
Ai Ai
A
i
A
i
()
EA
n
=
{
}
+
{
}
+
IVIFS
−
−
+
+
−
+
max(
μ
x
), () max(
ν
x
μ
x
), () (
ν
x
π
x
)
+
π
(
x
)
Ai Ai
Ai Ai
A
i
A
i
i
=
1
(4.28)
Another type of entropy by Zhang and Jiang [26] is given as
∑
n
(
)
−
−
+
+
μ
() () () ()
x
∧
ν
x
+
μ
x
∧
ν
x
Ai Ai Ai Ai
()
1
EA
=
i
=
(4.29)
IVIFS
∑
n
(
)
−
−
+
+
μ
()
x
∨
ν
() () ()
x
+
μ
x
∨
ν
x
Ai
A
i
A
i
A
i
i
=
1
n
1
∑
(
)
()
1
−
() () () ()
−
+
+
EA
n
=−
μ
x
−
ν
x
∨
μ
x
−
ν
x
IVIFS
A
i
B
i
A
i
B
i
i
=
1
4.8 Similarity Measure and Distance Measures of
IVIFS
For any two
IVIFS
s
A
and
B
,
{
}
IVIFS
,
⎣
−
( ), (),
+
⎦
⎣
−
( ), ()
+
⎦
A
=
x
μμ ν
x
x
x
ν
x xX
∈
A
A
A
A
{
}
IVIFS
,
⎣
−
( ), (),
+
⎦
⎣
−
( ), ()
+
⎦
B
=
x
μμ νν
x
x
x
x xX
∈
B
B
B
B
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