Image Processing Reference
In-Depth Information
Chapter 5
Increasing Filters and Mathematical
Morphology
5.1 Constraints on the Filter Function
In the previous chapter, it was seen that the estimation error of the filters increased
rapidly with window size. This was because the function defining the behavior of
the filter was unconstrained. Referring back to the design process described in
Chapter 2, every line of the table of observations was treated as a separate inde-
pendent entity. It was therefore necessary to see a sufficient number of examples of
every possible input in order to design the filter. For small windows this was feasi-
ble. However, for larger windows the number of inputs was huge and it was impos-
sible to see all of them. In practice it is not necessary for a filter to see all possible
inputs in order to determine the function accurately. This means that an output
value must be assigned to an input pattern that was never seen in training.
Consider the inputs shown in Fig. 5.1. Two of these input patterns were seen in
the training set a sufficient number of times for the output to be allocated a value of
1. The other input patterns were never seen at all and in theory their output is un-
known. However, it can be observed that the unknown patterns sit
between
the
other two patterns and there is no reason to believe that their value should be any-
thing other than 1. In the same way that a linear function may be interpolated with
models such as spline functions, a logic function may be interpolated such that it
fits the data at the known points and provides a good approximation at the unde-
fined points. A common approach is to limit the filter to a particular type of func-
tion known as an
increasing
function. An increasing function is one that can be
expressed without the use of negation, i.e.,
F
inc
()
x
=+
X
XX
+
XXX
is an increasing function
(5.1)
1
2
3
4
5
6
G
non inc
()
x
=+
X
XX
+
XXX
is a nonincreasing function.
(5.2)
1
2
3
4
5
6
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