Breakdown points, breakdown probabilities, midpoint sensitivity curves, and optimization of stack filters

Abstract

Three new concepts -- breakdown points, breakdown probabilities, and midpoint sensitivity curves -- for stack filter analysis are introduced and analyzed in this paper. Breakdown points and probabilities can be used as measures of the robustness of stack filters. Midpoint sensitivity curves in turn give information on how sensitive the output of a stack filter is to the changes of a single value in the input window. The second major contribution of this paper is the extension of the current optimality theory of stack filters. This theory combines noise attenuation and different constraints on the filter's behavior. New constraints are introduced in this paper. A new optimization approach based on breakdown probability as a noise attenuation measure is also derived. In certain special cases it is shown that the optimal stack filter that achieves the best noise attenuation subject to given constraints can be obtained in closed form. An algorithm for finding this form is given in this paper, and its modification for finding a stack filter having (approximately) a required rank selection vector is presented.