Digital Signal Processing Reference
In-Depth Information
The minimum variance estimator, which minimizes the above cost, is then
known to be
∞
∞
X
f
(
X
,
F
)
(
F
)
=
X
f
(
X
|
F
)
d
X
=
d
X
,
(12.53)
MV
f
(
F
)
−∞
−∞
with
∞
(
)
=
(
)
(
)
.
f
F
f
X
,
F
f
X
d
X
(12.54)
−∞
If the densities in Equation 12.52 are known and a training record of the sample
pairs
is available, the minimum variance estimator can be derived.
Unfortunately, in a realistic image processing scenarios, no
a priori
knowledge
about the noise process or the image itself is available. Thus, a nonparametric
estimator must be utilized to approximate the probability density functions
(pdf) in Equation 12.52.
Let us assume a window of finite length
n
centered around a noisy vec-
tor
(
X
,
F
)
{
F
}
0
. Through this window, a set of multivariate noisy samples
W
=
(
becomes available. Based on the samples from the filtering
window
W
,anadaptive, data-dependent multivariate kernel estimator can
be devised to approximate the densities in Equation 12.52. The form of the
adaptive kernel estimator selected is as follows:
F
0
,
F
1
,
...
,
F
n
)
K
F
,
n
1
N
1
h
i
−
F
i
f
(
X
,
F
)
=
N
=
n
+
1
,
(12.55)
h
i
i
=
0
where
F
i
is the
i
th training vector, with
i
3isthe dimen-
sionality of the measurement space, and
h
i
is the data-dependent smoothing
parameter, which regulates the shape of the kernel. The variable kernel den-
sity estimator exhibits local smoothing, which depends both on the point
at which the density is evaluated and also on the information on the local
neighborhood in
W
.
The
h
i
can be any function of the sample size
N
.
112
The bandwidths
h
i
(smoothing factors) can be defined as a function of the aggregated distance
between the local observation under consideration and all the other vectors
inside the
W
window. Thus,
=
0
,
1
,
...
,n
,
L
=
n
k
L
A
i
k
L
N
−
N
−
h
i
=
=
0
F
i
−
F
k
,
(12.56)
k
=
where
k
is a design parameter. The choice of the kernel function in Equation
12.55 is not nearly as important as the bandwidth (smoothing factor). For
the applications, the multivariate extension of the exponential kernel
K
(
z
)
=
z
T
z
can be selected.
112
exp
(
−|
z
|
)
or the Gaussian kernel
K
(
z
)
=
exp
(
−|
|
/
2
)
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