Image Processing Reference
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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
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 )|,
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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