Digital Signal Processing Reference
In-Depth Information
This filter provides a vector-valued signal, which is not included in the orig-
inal set of inputs. The weighted average form of the filter provides a com-
promise between a nonlinear order-statistics filter and an adaptive filter with
data-dependent coefficients. Depending on the form of the distance criterion
and the corresponding fuzzy transformation, different fuzzy filters can be de-
signed. If the distance criterion selected is the sum of vector angles, the
fuzzy
vector directional filter
(FVDF) is obtained. If an
L
1
norm is used as the distance
criterion, a fuzzy generalization of the VMF is constructed.
12.4.2.2 Maximum Fuzzy Vector Directional Filters
Another possible choice of the nonlinear function
f
is the maximum se-
lector. In this case, the output of the nonlinear function is the input vector
that corresponds to the maximum fuzzy weight. Using the maximum selec-
tor concept, the output of the filter is a part of the original input set. The form
of this filter is
(
·
)
F
0
=
F
i
with
i
=
arg max
w
i
,
i
=
0
,
...
,n
.
(12.45)
In other words, as an output the input vector associated with the maximum
fuzzy weight is selected. It must be emphasized that through the fuzzy mem-
bership function, the maximum fuzzy weight corresponds to the minimum
distance. If the vector angle criterion is used to calculate distances, the fuzzy
filter delivers the same output as the BVDF.
5
,
110
If the
L
1
or
L
2
is adopted
as the distance criterion, the filter provides the same output as the VMF. By
utilizing the appropriate distance function, different filters can be obtained.
Thus, filters such as VMF or BVDF can be seen as special cases of this specific
class of fuzzy filters.
12.4.2.3 Fuzzy Ordered Vector Directional Filters
In many cases it is favorable not to use all the inputs inside the operational
window to produce the final output of the nonlinear filter. Instead, only a part
of the vector-valued input signals can be used. The input vectors are ordered
according to their respective fuzzy membership strengths. The form of the
fuzzy ordered vector directional filter is given as
τ
τ
1
Z
F
∗
=
0
w
(
i
)
F
(
i
)
,
Z
=
0
w
(
i
)
,
(12.46)
i
=
i
=
where
w
(
i
)
represents the
i
th ordered fuzzy membership function and
w
(τ )
≤
w
(τ
−
1
)
≤
...
≤
w
(
0
)
, with
w
(
0
)
the fuzzy coefficient with the largest membership
strength.
The above form of the filter constitutes a fuzzy generalization of the
-
trimmed filters (Equation 12.34).
4
Through the fuzzy transformation, the
weights to be sorted are scalar values. In this way the nonlinear ordering pro-
cess does not introduce any significant computational burden. Depending on
α
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