Image Processing Reference
In-Depth Information
functions are 0.5, that is, the highest degree of fuzziness exists at
this point. It is not known whether point
C
belongs to or does not
belong to the set.
An
IFS
is represented by a triangle
ABF
. Points above line
AB
have
a hesitation degree more than 0. Point
F
is not defined as the hesi-
tation degree is equal to 1, and so it is difficult to say whether this
point belongs/does not belong to the set. For
FC
, the membership
degree μ(
FC
) and non-membership degree ν(
FC
) are equal, but the
hesitation degree is greater than 0, and the condition still follows:
μ
FC
(
x
i
) + ν
FC
(
x
i
) + π
FC
(
x
i
) = 1. So, for every point
x
i
in
FC
,
dist
(
A
,
x
i
) =
dist
(
B
,
x
i
) follows.
The entropy of
IFS
is based on ratio-based measure:
a
b
EX
()
=
(4.23)
where
a
=
dist
(
X
,
X
near
) means the distance from
X
to the nearest point
X
near
from
A
and
B
b
=
dist
(
X
,
X
far
) is the distance from
X
to the farthest point
X
far
from
A
and
B
This equation is the entropy for one point. From this entropy, it can
be said that either
X
fully belongs to point
A
or does not belong to
point
B
.
For '
n
' points, the entropy in Equation 4.23 is redefined as
n
1
∑
IFE
=
EX
i
()
(4.24)
n
i
=
1
3. The generalized entropy measure of set
A
of '
n
' elements in terms of
max sigma
count
is written as [17]
(
)
n
c
max
max
count AA
count AA
∩
1
∑
i
i
IFEA
()
=
(4.25)
(
)
n
c
∪
i
i
i
=
1
where
AA
i
∩=
(
)
(
)
c
c
c
min,
μμ νν
,max
,
i
Ai
Ai
Ai
Ai
∪=
(
)
(
)
i
c
c
c
AA
i
max,
μμ νν
,min
,
Ai
Ai
Ai
Ai
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