Nonrecursive and recursive stack filters and their filtering behavior

It is shown that stack filters that are based on symmetric threshold functions and preserve median-filter roots make all inputs converge to roots or to cycles of period two. This is an important result, since these filters have useful roots (the median-filter roots), and they are time symmetric, i.e. time reversal of an input sequence of such a filter is equivalent to time reversal of the output sequence of the filter. In order to construct stack filters without cycles, the recursive stack filter, which is an extension of the recursive median filter, is introduced. It is shown 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 converge to a root in a finite number of passes