Image Processing Reference
In-Depth Information
Likewise, for
A
C
, the least cardinality or min sigma
count
is given as
n
∑
∑
c
min
count A
(
)
=
ν
(
x
)
Ai
i
=
1
The biggest cardinality or max sigma
count
which is due to the hesi-
tation degree, π
A
, is given as
n
∑∑
c
max
count A
(
)
=
( () ( )
ν
x
+
π
x
Ai
Ai
i
=
1
Then cardinality of
IFS
is defined in an interval as
∑
∑
=
⎡
⎣
⎤
⎦
cardA
min
count A
(
),max
count A
(
)
To explain the entropy as the ratio of the distances between an
IFS
and its nearest and farthest crisp sets, Figure 4.1 is used.
A
and
B
correspond to a non-fuzzy/crisp set where point
A
cor-
responds to an element that fully belongs to 1, that is, (μ
A
, ν
A
, π
A
) =
(1, 0, 0), and point
B
corresponds to an element that does not belong
to the set (μ
B
, ν
B
, π
B
) = (0, 1, 0). Fuzzy set corresponds to line
AB
. When
we move away along the line from point
A
to point
B
, the mem-
bership function decreases from 1 at point
A
to 0 at point
B
. At
C
,
that is, at the midpoint, both the membership and non-membership
F
(0, 0, 1)
X
a
b
A
(1, 0, 0)
C
B
(0, 1, 0)
FIGURE 4.1
Graphical representation of
IFS
.
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