Image Processing Reference
In-Depth Information
the similarity measure is defined as [22] from the entropy in Equation 4.27,
which is written as
SAB
(,)
IVIFS
{
}
−
{
}
−
−
−
−
μ
+
+
+
+
2
−
min(
μ
x
)
−
μ
( ), () () min
x
ν
x
−
ν
x
() (),
x
−
μ
x
ν
(
x
)
−
ν
(
x
)
n
1
∑
Ai Bi Ai Bi
A
i
B
i Ai
B
i
=
{
}
+
{
}
n
−
−
−
−
+
+
+
+
2
+
max(
μ
x
)
−
μ
( ),
x
ν
(
x
)
−
ν
(
x
) max () (),
μ
x
−
μ
x
ν
(
x
)
−
ν
(
x
)
Ai Bi
A
i
B
i
A
i
B
i Ai
B
i
i
=
1
∀
A
,
BB VIFS A
∈
()
The normalized Euclidean similarity measure induced by the Hausdorff
metric [15] is given as
n
1
2
=−
⎩
∑
(
)
2
⎡
⎢
(,)
1
−
(
)
−
(
)
+
(
)
+
(
)
SAB
μ
x
−
μ
x
∨
μ
x
−
μ
x
IVIFS
A
i
B
i
A
i
B
i
n
i
=
1
12
/
(
)
⎥
⎭
2
−
() () () ()
−
+
+
+
ν
x
−
ν
x
∨
ν
x
−
ν
x
Ai Bi Ai Bi
The distance measure using Hamming and Euclidean distance [15] is given as
da b
(,)
IVIFS
n
1
4
∑
⎣
⎦
−
−
+
+
−
−
+
+
=
μ
() () () () () (
x
−
μ
x
+
μ
x
−
μ
x
+
ν
x
−
ν
x
)
+
ν
(
x
)
−
ν
(
x
)
Ai Bi Ai Bi Ai
B
i
A
i
B
i
1
i
=
n
1
4
∑
⎡
⎢
(
)
+
2
(
)
2
−
−
+
+
da b
(,)
=
μ
(
x
)
−
μ
(
x
)
μ
(
x
)
−
μ
(
x
)
IVIFS
A
i
B
i
A
i
B
i
i
=
1
2
12
/
(
)
+
2
(
)
⎥
−
−
+
+
+
ν
() ()
x
−
ν
x
ν
() ()
x
−
ν
x
Ai Bi
Ai Bi
4.9 Summary
As is well known that entropy and similarity measures are two important
issues in fuzzy set theory, they are widely used in image processing, pat-
tern recognition, cluster analysis and so on. But in real-life situations, there
are many uncertainties where human negation does not satisfy logical nega-
tion, and so in this chapter, the definition, properties and different types of
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