Image Processing Reference
In-Depth Information
This can be extended to intuitionistic fuzzy case: if
A
and
B
are two
IFS
s, and due to the hesitation in defining the membership function,
the expert is hesitant to the extent π
A
(
x
) = 1 − μ
A
(
x
) − ν
A
(
x
), then the
intervals of the membership degree for
A
and
B
are
[
]
=
[
]
μμ π
(), ()
x
x
+
( )
x
μ
( ),
x
1
−
ν
()
x
A
A
A
A
A
[μ
B
(
x
),
μ
B
(
x
) + π
B
(
x
)] = [μ
B
(
x
), 1 − ν
B
(
x
)]
respectively.
Using the Hausdorff metric, the distance may be written as
{
}
(
)
dAB
(,)max
=
μ
() (),
x
−
μ
x
1
−
ν
( )
x
−
1
−
ν
( )
x
(4.3)
A
B
A
B
3. A different type of distance measure suggested by Wang and Xin
[21] is given as
⎡
μ
() () () ()
x
−
μ
x
+
ν
x
−
ν
x
⎤
Ai Bi Ai Bi
⎢
⎢
⎢
⎢
⎢
⎥
⎥
⎥
⎥
⎥
n
4
1
∑
dAB
(,)
=
(4.4)
IFS
(
)
n
max(
μ
x
)
−
μ
( ),
x
ν
ν
Ai Bi
() ()
x
−
x
i
=
1
Ai Bi
+
2
⎣
⎦
4. Song and Zhou [16] suggested some modifications on Wang's dis-
tance measure. In Wang's measure, the weights of |μ
A
(
x
i
) − μ
B
(
x
i
)|,
|ν
A
(
x
i
) − ν
B
(
x
i
)|, m a x (|μ
A
(
x
i
) − μ
B
(
x
i
)| and |ν
A
(
x
i
) − ν
B
(
x
i
)|) are fixed as
(1/4), (1/2). But in practice, the weights are not fixed, so Song intro-
duced a weight different from the distance measure
⎡
αμ
() () () ()
x
−
μ
x
+
β ν
x
−
ν
x
⎤
n
Ai Bi
Ai Bi
1
∑
1
⎢
⎢
⎥
⎥
dAB
(,)
=
(4.5)
IFS
(
)
n
+⋅
γ
max(
μ
x
)
−
μ
(
x
), () ()
ν
x
−
ν
x
i
=
⎣
Ai
B
i Ai
B
i
⎦
with α
+ β
+ γ
= 1 and α, β, γ
∈ [0, 1].
But in Equation 4.5, element
x
i
in set
X
= {
x
1
,
x
2
,
x
3
, …,
x
n
} has the same
weight parameters α, β and γ. In some cases, there may be some ele-
ments that affect the distance between the
IFS
s. In that case, a weight
w
i
> 0,
i
∈ {1, 2, 3, …,
n
} is introduced. Then the distance measure becomes
dAB
(,)
IFS
⎡
⎤
αμ
() () () ()
x
−
μ
x
+
β ν
x
−
ν
x
n
1
w
Ai Bi
Ai Bi
∑
⎢
⎢
⎥
⎥
i
=
(4.6)
∑
n
n
(
)
+⋅
γ
max
μ
(() (),
x
−
μ
x
ν
(
x
)
−
ν
(
x
)
w
i
=
1
⎣
A
i
B
i Ai
B
i
⎦
i
1
i
=
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