By Lawrence R.J.

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**Example text**

The difÞculty for adaptive disjunctive granulometries is that the arrival/service rates are not constant at a given queue and therefore the product solution does not apply. Given these theoretical difÞculties, we now proceed numerically, throughout assuming signal and noise parameters to be uniformly distributed over the square regions from [7, 7] to [16, 16] and from [5, 5] to [14, 14], respectively. The large overlap between uniform distributions will result in optimal Þlters with relatively large errors, but it will serve our purposes by creating rather dispersed steady-state distributions whose geometry can easily be apprehended.

24) occur. There are two parameters, r1 (n) and r2 (n), to adapt. We have the following generic adaptation rules: i. r1 → r1 + 1 and/or r2 → r2 + 1 ii. r1 → r1 − 1 and/or r2 → r2 − 1 iii. r1 → r1 and r2 → r2 if condition a occurs if condition b occurs if condition c or d occurs (105) The adaptation rules are designed so that when a signal grain erroneously does not pass, the passband is expanded; when a noise grain erroneously passes, the passband is reduced; and when a grain is correctly passed or not passed, the passband is not changed.

28 EDWARD R. DOUGHERTY AND YIDONG CHEN 2. Type-[I, 1] Model For this model, again consider a two-parameter opening, but now assume that if a signal grain is erroneously not passed, it is known which parameter has caused the erroneous decision. There are three possibilities: (1) r1 , but not r2 , causes the structuring element not to Þt inside the grain; (2) r2 , but not r1 , causes the structuring element not to Þt; (3) both r1 and r2 cause the structuring element not to Þt. Given a signal grain G, the three conditions can be rigorously stated in the following manner: (1) r (G) = л and there exists r1′ such that, for r′ = (r1′ , r2 ), r′ (G) = G; (2) r (G) = л and there exists r2′ such that, for r′ = (r1 , r2′ ), r′ (G) = G; (3) there does not exist an r1′ or an r2′ satisfying the preceding conditions.