Image Processing Reference
In-Depth Information
served signal into the mask range to give the clipped observations
X
j
*
if
X
j
is ini-
tially outside of the amplitude range of the mask shape.
X
,
−≤
k
X
≤
k
,
j
j
*
X
=
k
,
X
>
k
,
(7.16)
j
j
−
k
,
X
< −
k
.
j
This allows the variables around the offset of the pattern to be unchanged by the
quantification, therefore allowing more of the original detail to be retained. Further
reduction of the configuration space could be achieved by quantification within the
aperture range [-
k
,
k
]. This method allows larger apertures to be used to cover areas
in the signal where the gray level changes are large.
The aperture can be regarded as the product between the range [
-k, k
] and the
domain [
-w, w
]. Aperture filters were originally known as W
K
filters.
10,11
The filter
output is estimated by considering the conditional probabilities of the true signal
given the set of observations within the filter window. This is a generalization of
the method outlined in Chapter 2, but now the output can take a number of values.
The optimal constrained filter is given by
E
[
Y
|
X
*
] where
Y
is the ideal output
value. There is an assumption in this analysis that all values including the ideal
Y
used in this estimation fall or are clipped inside the mask range [
-k, k
]. Under this
assumption,
Y
*
is similarly defined by Eq. 7.17. This means that the optimal MSE
estimator uses a constrained ideal
Y
*
. Based on the constrained vector
X
*
the opti-
mal operator is given by Eq. 7.17:
A
=
E
[
Y
*
|
X
*
].
ψ
(7.17)
As with any constraint, there is an associated cost. In this case, there is an error (re-
sulting from range constraint) for using the aperture filter
ψ
A
instead of a window
W
. This error, in terms of mean-absolute
error, is given by Eq. 7.18. Further details are given in Hirata.
12
filter that is not constrained in amplitude
ψ
W
)=
E
[|
Y
-
E
[
Y
*
|
X
*
]|] -
E
[|
Y
-
E
[
Y
|
X
]|].
∆
(
ψ
A
,
ψ
(7.18)
In order to estimate the conditional probabilities, the aperture has to be positioned
in the signal space. Placement of the window can be done in various ways, but the
most important consideration is the reduction of the number of points falling outside
the window range. Examples of placement are explored in Hirata et al.
13
These meth-
ods involve referencing the aperture to the observed value or the median of the ob-
served pattern in the domain window. In general, the best aperture placement strategy
is the one that gives the closest estimate of the output. This will vary depending on the
