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On occasion, a filter that is optimal over the general class of nonincreasing filters may actually be optimal over the subclass of increasing filters. Were we to know this beforehand, then it would be statistically advantageous to design the filter directly as a union of erosions rather than as a union of hit-or-miss transforms [37]. Even when the optimal filter is not increasing, the optimal increasing filter may be sufficiently close to optimal that it is advantageous from the perspectives of both design precision and combinational-logic cost at implementation. For instance, increasing filters often work well when degradation is either extensive or antiextensive. For both document restoration (in the present chapter and Chapter 8) and resolution enhancement (Chapter 9), the problem setting is defined, or may be defined, so that increasing filters can perform quite well with a very small number of erosions. Moreover, in settings where a single designed filter might not perform sufficiently well, an iterative filter composed of optimally designed increasing filters may be suitable.
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