Convergence properties of multi-dimensional stack filters

Stack filters are generalizations of median filters; a stack filter is a composition of local minimum and maximum operators. A major question in the study of these filters is whether a particular stack filter will make any signal of finite extent converge to an invariant signal - i.e. whether it will 'filter out all the noise.' Here, we consider the class of stack filters that reduce to symmetric threshold functions for binary inputs. We show that, if we surround an n - dimensional signal with an arbitrary fixed boundary, then any n - dimensional stack filter from the class will make the signal converge to an invariant signal, or a cycle of period 2, in a finite number of iterations. If we make the stack filter recursive it will always filter the signal to an invariant signal, no matter how the filter moves over the signal. Our results follow from similar theorems on the convergence of neural networks. Many known useful filters are governed by our results. They include all 1-dimensional ranked-order filters with symmetric windows, and all 2-dimensional ranked-order filters with windows that are invariant under a 180° rotation.