Digital Signal Processing Reference
In-Depth Information
Determining the ML estimate is thus equivalent to minimizing
N
p
G
p
(β)
=
1
|
x
i
−
β
|
(2.5)
i
=
with respect to
β
. For the Gaussian case (
p
=
2), this reduces to the sample
mean, or average:
N
1
N
ˆ
β
=
arg min
β
G
2
(β)
=
x
i
.
(2.6)
i
=
1
A much more robust estimator is realized if the underlying sample distribu-
tion is taken to be the heavy-tailed Laplacian distribution (
p
=
1). In this case,
the ML estimator of location is given by the value
β
that minimizes the sum
of absolute deviations,
N
G
1
(β)
=
1
|
x
i
−
β
|
,
(2.7)
i
=
which can easily be shown to be the sample median:
ˆ
β
=
arg min
β
G
1
(β)
=
MED[
x
1
,x
2
,
...
,x
N
]
.
(2.8)
The sample mean and median thus play analogous roles in location estima-
tion: While the mean is associated with the Gaussian distribution, the median
is related to the Laplacian distribution, which has heavier tails and provides a
better model for many signals, such as images, as well as those contaminated
with impulsive outliers.
1
-
4
Although the median is a robust estimator that possesses many optimality
properties, the performance of the median filter is limited by the fact that
it is spatially blind. That is, all observation samples are treated equally re-
gardless of their location within the observation window. This limitation is a
direct result of the i.i.d. assumption made in the filter development. A much
richer class of filters is realized if this assumption is relaxed to the case of
independent, but not identically distributed, samples.
Consider the generalized Gaussian distribution case where the observa-
tion samples have a common location parameter
β
, but where each
x
i
has a
(possibly) unique scale parameter
σ
i
. Incorporating the unique scale param-
eters into the ML criteria yields a location estimate given by the value of
β
minimizing
N
1
σ
p
G
p
(β)
=
|
x
i
−
β
|
.
(2.9)
p
i
i
=
1
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