Image Processing Reference
In-Depth Information
If μ
B
(
x
i
) = 0 or ν
B
(
x
i
) = 0, the equation becomes undefined. So, in that case the
equation is modified as
n
μ
()
x
ν
(
x
)
∑
Ai
Ai
I AB
(,)
=
μ
(
x
)ln
+
ν
()ln
x
IFS
A
i
Ai
(
1/
)(()
μ
x
+
μ
(
x
))
(
12
/
)(()
ν
ν
Ai Bi
x
+
(
x
))
Ai Bi
i
=
1
(4.20)
Symmetric information of the discrimination measure for
IFS
is
DABI AB I BA
IFS
(,)
=
( ,) (, )
+
IFS
IFS
4.6 Intuitionistic Fuzzy Entropy
Entropy is a measure of fuzziness in a fuzzy set. Zadeh [25] first introduced
the idea of fuzzy entropy in 1969. Kaufmann [10] used the distance measure to
define fuzzy entropy, whereas Yager [24] defined entropy as the distance from
a fuzzy set and its complement. Similarly in the case of
IFS
, intuitionistic fuzzy
entropy (
IFE
) gives the amount of vagueness or ambiguity in a set. Many authors
defined
IFE
in a different manner. Two definitions of entropy of
IFS
were given
by Burillo and Bustince [2] and Szmidt and Kacprzyk [17]. These two defini-
tions have different frameworks. Burillo and Bustince defined entropy for the
first time in terms of the degree of intuitionism of an
IFS
. Szmidt and Kacprzyk
defined entropy in terms of the non-probabilistic type of entropy. In
IFS
, three
parameters are taken into account with μ
A
+ ν
A
+ π
A
= 1, μ
A
≥ 0, ν
A
, π
A
≤ 1.
The properties of
IFE
by Burillo and Bustince [2] are
A real function
IFE
=
IFSs
(
X
) → [0, 1] is called
IFE
on
IFSs
(
X
) if
1.
IFE
(
A
) = 0, ∀
A
∈
FS
(
X
).
2.
IFE
(
A
) = Cardinal(
X
) =
n,
iff μ
A
(
x
i
) = ν
A
(
x
i
) = 0 ∀
x
i
, that is, the entropy
is maximum if the set is totally intuitionistic.
3. If the membership and non-membership of each element increase,
their sum will increase; thereby, the fuzziness will increase and the
entropy will decrease. Mathematically, it can be written as
IFE
(
A
) ≥
IFE
(
B
) if μ
A
(
x
i
) ≤ μ
B
(
x
i
) and ν
A
(
x
i
) ≤ ν
B
(
x
i
).
4.
IFE
(
A
) =
IFE
(
A
C
).
They defined entropy as
n
∑
π
1
IFEA
()
=
Ai
()
i
=
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