Image Processing Reference
In-Depth Information
4.3 Different Types of Distance and Similarity Measures
1. Szmidt and Kacprzyk [18] introduced some popular distance mea-
sures between two
IFS
s.
Let
A
and
B
be two
IFS
s where
A
= {(
x
, μ
A
(
x
), ν
A
(
x
))|
x
∈
X
} and
B
= {(
x
, μ
B
(
x
), ν
B
(
x
))|
x
∈
X
}, and μ
A
(
x
) and ν
A
(
x
) are the membership
and non-membership functions, respectively.
Intuitionistic Euclidean Distance
DAB
(,)
IFS
⎛
n
⎞
i
⎣
2
2
)
2
⎦
=
⎜
(()
μ
x
−
μ
(
x
)) (()
+
ν
x
−
ν
(
x
)) (()
+
π
x
−
π
(
x
)
⎟
Ai Bi
Ai Bi
Ai Bi
⎝
⎠
=
1
Intuitionistic Normalized Euclidean Distance
DAB
(,)
IFS
⎛
n
⎞
1
∑
⎣
2
2
))
2
⎦
=
⎜
(()
μ
x
−
μ
(
x
)) (()
+
ν
x
−
ν
(
x
)) (()
+
π
x
−
π
(
x
⎟
Ai Bi
Ai Bi
Ai
B
i
n
⎝
⎠
i
=
1
Intuitionistic Hamming Distance
n
∑
μ
(
)
DAB
(,)
=
(
x
)
−
μ
(
x
)
+
ν
(
x
)
−
ν
(
x
)
+
π
(
x
)
−
π
(
x
)
IFS
A
i
B
i
A
i
B
i
A
i
B
i
i
=
1
2. Distance measure based on the Hausdorff metric
Grzegorzewski [6] introduced the distances based on the Hausdorff
metric.
The Hausdorff distance on any two non-empty subsets
A
and
B
of
a compact metric space is defined as [8]
⎭
⎩
dAB
(,)max supinf
=
a b
−
,supinf
a b
−
(4.1)
aA
bB
∈
bB
aA
∈
∈
∈
Now, if the intervals of sets
A
and
B
are
A
= [
a
1
,
a
2
] and
B
= [
b
1
,
b
2
]
with
a
1
<
a
2
and
b
1
<
b
2
, then the Hausdorff metric becomes
{
}
dAB
(,)max
=
⎣
a bab
−
,
−
⎦
(4.2)
1
1
2
2
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