Digital Signal Processing Reference
In-Depth Information
or sample spread, weighting is introduced in the weighted sum output. The
second class of filters established is the fuzzy generalization of weighted me-
dian filters.
7
,
8
Fuzzy weighted median filters are realized by simply inserting
fuzzy samples into the standard weighted median filter formulation. As is
demonstrated in Section 3.2, the affinity and fuzzy generalizations lead to
improved performance.
2.4.1
Affine Filters
Affine filters are realized by including an affinity weighting, to a specified
reference point, in a standard weighted sum filter. Two important subclasses
of affine filters are the median affine and center affine filter classes. The median
affine filter class is established by utilizing the median sample as the reference
point and introducing the affinity weighting into the standard weighted sum
filter. The center affine filter class is realized by setting the central observation
sample as the reference point and introducing the affinity weighting into the
weighted sum of order statistics filter.
2.4.1.1 Median Affine Filters
The standard linear FIR filter has been successfully applied to many problems.
Indeed, by formulating the filter output as a weighted sum of spatially, or
temporally, ordered samples, important filter characteristics are realized, such
as frequency selectivity. Linear FIR filters, however, perform poorly in the
presence of outliers. This performance can be improved if a simple valid
reference point can be established and the validity of each sample measured
in reference to the reference point. This is the motivation behind the median
affine filter.
The median operator is robust, and can thus serve to establish a valid
reference point for many signal statistic cases. Thus, the standard weighted
sum FIR filter given in Equation 2.9 can be made more robust by weighting
each sample according to its affinity to the median reference point. Utilizing
again
δ
=
(
N
+
1
)/
2 as the central index, the median affine filter is defined as
i
=
1
w
R
i,
(δ)
x
i
i
=
MAFF[
x
]
i
=
1
|
w
,
(2.34)
R
i,
(δ)
|
i
w
i
are the filter weights and
R
i,
(δ)
where the
is the affinity of the
i
th observation
with respect to the median reference point, MED
.
The filter structure in Equation 2.34 weights each observation twice: first,
according to its reliability, and second, according to its natural order, Figure
2.8. Median affine estimates are therefore based on observations that are both
reliable and favorable due to their natural order. Observations that fail to meet
either criterion have only a limited influence on the estimate.
By varying the dispersion of the affinity (or membership) function, cer-
tain properties of the median affine filter can be stressed: a large value of
(
x
)
=
x
(δ)
σ
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