Digital Signal Processing Reference
In-Depth Information
adaptively using fuzzy transformations of a distance criterion at each image
position.
In this framework the weights are determined by fuzzy transformations
based on features from local data. The fuzzy module extracts information
without any
a priori
knowledge about noise characteristics. The weighting
coefficients are transformations of the distance between the vector under con-
sideration (center of the processing window
W
) and all other vector samples
inside the processing window
W
. This transformation can be considered as
a membership function with respect to a specific window component. The
adaptive algorithm evaluates a membership function based on a given vector
signal and then uses the membership values to calculate the filter output.
Adaptive fuzzy algorithms utilize features extracted from local data, here in
the form of a sum of distances, as inputs to the fuzzy weights. In this case,
the distance functions are not used to order input vectors. Instead, they pro-
vide selected features in reduced space, features used as inputs for the fuzzy
membership function.
Several candidate functions, such as triangular, trapezoidal, piecewise lin-
ear, or Gaussian-like functions, can be used as a membership function. If the
distance criterion described by Equation 12.37 is used as a distance measure,
a sigmoidal membership function can be selected
5
,
110
w
i
=
β(
A
i
}
)
−
r
,
1
+
exp
{
(12.42)
where
A
i
is a cumulative distance from Equation 12.37, while
and
r
are
parameters to be determined. The
r
value is used to adjust the weighting
effect of the membership function and
β
β
is a weight-scale threshold. If the
Minkowski
L
p
metric is used as the distance function, the fuzzy membership
function with exponential form gives good results
exp
,
R
i
β
w
i
=
−
(12.43)
where
R
i
is a cumulative distance associated with the
i
th vector in the process-
ing window
W
using generalized Minkowski norm,
r
is a positive constant,
and
is a distance threshold.
Within the general fuzzy adaptive filter framework, numerous filters may
be constructed by changing the form of the nonlinear function
f
β
,aswell as
the way the fuzzy weights are calculated. The choice of these two parameters
determines the filter characteristics.
(
·
)
12.4.2.1 Fuzzy Weighted Average Filter
The first class of filters derived from the general nonlinear fuzzy algorithm is
the
fuzzy weighted average filters
(FWAF). In this case, the output of the filter is
a fuzzy weighted sum of the input set. The form of the filter is given as
n
n
1
Z
F
0
=
0
w
i
F
i
,
Z
=
0
w
.
(12.44)
i
i
=
i
=
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