Nonrecursive and Recursive Stack Filters and Their Filtering Behavior

Abstract

Stack filters are nonlinear digital filters that are generalizations of median and ranked-order filters. They are, in fact, all filters that can be expressed as compositions of local minimum and maximum operations in a finite window. In this paper, we show that stack filters exist, other than median filters, that preserve the roots (invariant signals) of median filters, and which make any signal of finite length converge to a root, or to a cycle of period 2, in a finite number of passes. For any window width, filters exist in this class, other than median filters, that do not have cycles. However, there are stack filters with cycles; and we give an example of such a filter. In order to construct stack filters without cycles, we introduce the recursive stack filter, which is an extension of the recursive median filter. We show that a recursive stack filter has the same roots as the corresponding nonrecursive stack filter; also, given a nonrecursive filter from the class mentioned above, the corresponding recursive filter will make every input signal of finite length converge to a root in a finite number of passes. © 1990 IEEE