Stack filters

The median and other rank-order operators possess two properties called the threshold decomposition and the stacking properties. The first is a limited superposition property which leads to a new architecture for these filters; the second is an ordering property which allows an efficient VLSI implementation of the threshold decomposition architecture. Motivated by the success of rank-order filters in a wide variety of applications and by the ease with which they can now be implemented, we consider in this paper a new class of filters called stack filters. They share the threshold decomposition and stacking properties of rank-order filters but are otherwise unconstrained. They are shown to form a very large class of easily implemented nonlinear filters which includes the rank-order operators as well as all compositions of morphological operators. The convergence properties of these filters are investigated using techniques similar to those used to determine root signal behavior of median filters. The results obtained include necessary conditions for a stack filter to preserve monotone regions or edges in signals. The output distribution for these filters is also found. All the stack filters of window width 3 are determined along with their convergence properties. Among these filters are found two which we have named asymmetric median filters. They share all the properties of median filters except that they remove impulses of one sign only; that is, one removes only positive going edges, the other removes only negative going edges, while the median filter removes impulses of both signs. This investigation of the properties of stack filters thus produces several new, useful, and easily implemented filters.