Image Processing Reference
In-Depth Information
Figure 5.10
Table of observations from Fig. 2.8 expressed in terms of statistics.
The probability that any particular input will occur is
P
(
x
i
). The prior probability
may also be estimated as
NN
NN
+
P
)
(
1
i
0
i
x
=
.
(5.13)
i
∑
+
1
i
0
i
i
The values of the observation table given in Fig. 2.8 have been reorganized into
probabilities in Fig. 5.10.
In order to minimize the MAE, the output of the filter must be the one which is
correct most often. Therefore,
)
(
)
(
ψ
x
=
1
if
P y
=
1
|
x
≥
0 5
.
and
i
i
i
(5.14)
)
(
)
(
ψ
x
=
0
if
P y
=
1
|
x
<
0 5
. .
i
i
i
This is directly equivalent to selecting the output corresponding to the larger of
the two observation values
N
0
i
and
N
1
i
for each input
i
. Therefore, the method de-
scribed in Chapter 2 corresponds to a practical implementation of the maximum
likelihood approach.
5.3 Summary
This chapter has introduced the idea of filter function constraints. In particular, it
has considered increasing functions and presented some of their properties. It has
shown that the area of mathematical morphology may be put in the context of in-
creasing filters. More importantly, it has provided a methodology by which the
structuring elements of a morphological filter may be designed to implement the
