Image Processing Reference
In-Depth Information
Figure 5.9
Differencing filter implementation.
5.2 Statistical Relevance
The design methods for morphological and logical filters presented in this chapter
should not be seen as new or ad hoc approach to filter design. They are, in fact,
rooted in standard classical statistics. Consistent with the practical nature of this
text, the explanation in this context has been delayed until after the methods have
been described by representative examples. In designing the original filters, the
process outlined in Chapter 2 involved the compilation of a table of observations
which were then used to determine the optimum filter output
ψ
opt
(
x
value for
each combination of input values
x
i
. This is a variation of the conditional expecta-
tion filter design method.
6
The method is greatly simplified in this case since both
the output and input values are binary. For simplicity, let
y
x
In order to design the filter, it was necessary to estimate the conditional expec-
tation of the output. This was carried out using the training set. The value of the cor-
responding pixel in the ideal image was recorded for every input combination.
Since the value of
y
is binary, the conditional output value may be summarized in
terms of the quantity
P
(
y=
1
|
x
i
). This is the probability that the output value
y
equals 1 for the specific input
x
i
. It should also be noted that
P
(
y=
0
|
x
i
)
=
1-
P
(
y=
1
|
x
i
). The value of
P
(
y=
1
|
x
i
) may therefore be estimated by the counts from
the observation table as
=ψ
opt
().
N
NN
)
(
1
i
Py
=
1
|
x
=
,
(5.12)
i
+
1
i
0
i
where
N
1
i
and
N
0
i
are the counts for
y
= 1 and
y
= 0, respectively, for a specific line
i
in the observation table.
