Image Processing Reference
In-Depth Information
3.4.1 Weighted Averaging Operator
A WA operator of dimension '
n
' is a mapping WA :
R
n
→
R
that is associated
with the '
n
' vector of weights:
w
= (
w
1
,
w
2
, …,
w
n
)
T
to the vector
a
i
(
i
= 1, 2, 3, …,
n
)
such that
n
∑
=
w
1
and
w
∈
[ ,]
0 1
i
i
i
=
1
n
∑
So,
Faaa wa wa wa
(,,
)
=
=
+
+
+
wa
…
12
n
i
i
11 2
2
nn
i
=
1
The weighted operator first weights all the arguments and then aggregates
all these weighted arguments.
3.4.2 Ordered Weighted Averaging Operator
An OWA operator is almost the same as the WA operator, but reordering
is given to all arguments. First, it reorders all the arguments in descend-
ing order; next, weights are given to these arguments; and finally, the OWA
operator aggregates all the ordered arguments.
An OWA operator of dimension '
n
' is a mapping OWA :
R
n
→
R
that has an
associated set of weights:
w
= (
w
1
,
w
2
, …,
w
n
)
T
to the vector
a
i
(
i
= 1, 2, 3, …,
n
)
such that
n
∑
=
w
1
and
w
∈
[ ,]
0 1
i
i
i
=
1
n
∑
So,
Fa a
(,,
…
,
a
)
=
wa
12
n
i
σ
()
i
i
=
1
where
σ
= 1, 2, 3, …,
n
a
σ(
i
)
is the
i
ith largest value in the set (
a
1
,
a
2
, …,
a
n
) such that
a
σ(
i
)
≥
a
σ(
i
−1)
3.5 Intuitionistic Fuzzy Weighted Averaging Operator
The averaging operator, as described in fuzzy sense, is extended using
intuitionistic fuzzy set (IFS) where both membership and non-membership
functions are taken into account. Intuitionistic fuzzy aggregation operators
are studied by many authors [10,17,20]. An IFS in X is given as
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