Image Processing Reference
In-Depth Information
4.6.1 Different Types of Entropies
1. Entropy is also defined by Chaira [4] that follows the previous norms
and is given as
For a probability distribution,
p
=
p
1
,
p
2
, …,
p
n
, the exponential
entropy is defined as
H
∑
n
(
1
−
p
)
.
p e
i
=
⋅
i
1
For intuitionistic fuzzy cases, if μ
A
(
x
i
), ν
A
(
x
i
) and π
A
(
x
i
) are the mem-
bership, non-membership and hesitation degrees of the elements of
the set
X
= {
x
1
,
x
2
, …,
x
n
}, respectively, then
IFE
, which denotes the
degree of intuitionism in fuzzy set, may be given as
i
=
n
∑
π
IFEA
()
=
xe
Ai
()
[
⋅
1
−
π
Ai
(
)]
(4.21)
i
=
1
where π
A
(
x
i
) = 1 − (μ
A
(
x
i
) + ν
A
(
x
i
)).
Szmidt and Kacprzyk [17] defined
IFE
in a different manner:
A real function
IFE
=
IFSs
(
X
) → [0, 1] is called
IFE
on
IFSs
(
X
) if
a.
IFE
(
A
) = 0 if
A
is a crisp set, that is, μ
A
(
x
i
) = 0 or μ
A
(
x
i
) = 1 for all
x
i
∈
X
.
b.
IFE
(
A
) = 1, if μ
A
(
x
i
) =
ν
A
(
x
i
) for all
x
i
∈
X
.
c.
IFE
(
A
) ≤
IFE
(
B
) if
A
is less fuzzy than
B
, that is, μ
A
(
x
i
) ≤ μ
B
(
x
i
) and
ν
A
(
x
i
) ≥ ν
B
(
x
i
) for μ
B
(
x
i
) ≤ ν
B
(
x
i
) or μ
A
(
x
i
) ≥ μ
B
(
x
i
) and ν
A
(
x
i
) ≤ ν
B
(
x
i
) for
μ
B
(
x
i
) ≥ ν
B
(
x
i
) for any
x
i
∈
X
.
d.
IFE
(
A
) =
IFE
(
A
C
).
2. Szmidt and Kacprzyk [17] also defined entropy as a ratio of the dis-
tances between an
IFS
and its nearest and farthest crisp sets in terms
of cardinalities of an
IFS
. A brief note on the cardinalities of
IFS
is
given later.
The least cardinality or the sigma
count
is the min sigma
count
of
A
which is the arithmetic sum of the membership grades in a set and
is also called here as min sigma
count
, which is given as
n
∑
∑
min
count A
(
)
=
μ
Ai
(
)
i
=
1
The biggest cardinality or max sigma
count
which is due to the hesi-
tation degree, π
A
, is given as
n
∑
∑
max
count A
(
)
=
( () ( )
μ
x x
Ai Ai
+
π
(4.22)
i
=
1
Search WWH ::
Custom Search