Towards a general theory of robust nonlinear filtering: Selection filters

In this paper we introduce a general framework for edge preserving filters, derived from the powerful class of M-estimators. First, we show that under very general assumptions, any location estimator generates an edge preserving filter if we approximate the estimate by one of the input samples. Based on this premise, we propose the family of S-estimators or S-filters, as a selection-type class of filters arising from a computationally tractable `selectification' of location M-estimators. S-filters inherit the richness of the theory underlying the M-estimators framework, providing a very flexible family of robust estimators with edge preservation capabilities. Several properties of S-filters axe studied. Sufficient and necessary conditions are given for an S-filter to present edge enhancing capabilities, and several novel filters within this framework are introduced and illustrated. Data, figures and source code utilized in this paper are available at http://www.ee.udel.edu/signals/robust/.