Digital Signal Processing Reference
In-Depth Information
In the special case of the standard Gaussian distribution (
p
=
2), the ML
estimate reduces to the normalized weighted average
i
=
1
w
N
1
σ
·
x
i
i
ˆ
2
β
=
arg min
β
i
(
x
i
−
β)
=
i
=
1
w
i
,
(2.10)
2
i
=
1
i
where
w
i
=
1
/σ
>
0. In the heavier-tailed Laplacian distribution special case
(
p
1), the ML estimate reduces to the weighted median (WM), originally
introduced more than 100 years ago by Edgemore,
5
=
and defined as
N
1
σ
i
|
ˆ
β
=
arg min
β
x
i
−
β
|=
MED[
w
1
x
1
,
w
2
x
2
,
...
,
w
N
x
N
]
,
(2.11)
i
=
1
where
w
=
1
/σ
>
0 and
is the replication operator defined as
i
i
w
i
times
w
=
...
.
x
i
x
i
,x
i
,
,x
i
i
A small yet very important, special example of the weighted median filter
(as well as the weighted sum filter) is the identity operator. Assuming the
samples constitute an ordered (temporal or spatial) set from an observed
process, let
δ
=
(
N
+
1
)/
2 be the index of the center observation sample.
c
Then it is easy to see that
x
δ
c
=
MED[
w
x
1
,
w
x
2
,
...
,
w
x
N
]
,
(2.12)
1
2
N
w
δ
c
=
w
=
=
δ
for
c
. Thus the weighted median has two important
special cases: (1) the standard median filter, which operates strictly on rank
order information, and (2) the identity filter, which operates strictly on spatial
order.
To illustrate the importance of these cases, and the corresponding order-
ings on which they are based, consider the filtering of a moving average
(MA) process corrupted by Laplacian noise. Figure 2.1 shows the correlation
between the desired MA process and the filter outputs, for the identity and
median cases, as a function of the signal-to-noise ratio (SNR) in the corrupted
observation. The figure shows that for high SNRs, the identity filter output
(central observation sample) has the highest correlation with the desired out-
put, while for low SNRs the median has the highest correlation. Thus, this
simple example illustrates the importance of spatial order in high-SNR cases
and rank order in low-SNR cases.
The general formulation of the weighted median filter attempts to ex-
ploit both spatial ordering, through repetition of samples, and rank ordering,
through median selection. The filter is thus able to exploit spatial correlations
among neighboring samples and limit the influence of outliers. A more formal
consideration of spatial and rank ordering can be obtained by considering the
full ordering relations between observed samples.
1 and
0 for
i
i
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