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Chapter 5:
Increasing Filters and Mathematical Morphology
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 independent 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 feasible. However, for larger windows the number of inputs was huge and it was impossible 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 unknown. 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 anything 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 undefined points. A common approach is to limit the filter to a particular type of function known as an increasing function. An increasing function is one that can be expressed without the use of negation.
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