Convergence behavior and N-roots of stack filters

The convergence behavior of two types of stack filters is investigated. Both types are shown to possess the convergence property and to exhibit nontrivial behavior. The first type of stack filter has the erosive property; it erodes any input signal to a root after a sufficient number of passes. The second type of stack filter has the dilative property; it dilates any input signal to a root after a sufficient number of passes. For each type of stack filter, an algorithm is presented which can determine a filter that has any specific signal or set of signals as roots. These two algorithms are efficient in that their execution time is a linear function of the length of the input signal, the width of the filter window, and the number of signals to be preserved. Since some stack filters have the phenomenon of oscillations when they filter some input signals successively, a partial ordering is defined over the set of stack filters which makes it possible to determine upper and lower bounds for these oscillations