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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