Smoothing of discontinuous signals: the competitive approach

Discontinuous signals buried in noise cannot be recovered by linear filtering methods. This paper presents a new class of nonlinear filters in which sets of forward and backward linear predictors and smoothers compete with each other at each timestep. The winner of each competition is granted the right to produce the smoothed estimate at that timestep. This conceptually simple approach to nonlinear filtering, called the competitive smoothing approach, is justified by considering sets of Kalman filters (corresponding to the hypotheses used in the Bayesian framework) which are used to derive model credibility coefficients. These are shown to essentially "switch" between the various models. We argue that the concept of competitive smoothing is considerably more general than just the Kalman setting, and can be used with almost any predictors and/or smoothers. Several examples are presented which demonstrate the efficacy of the approach at both smoothing and preserving jump discontinuities. Comparisons are made with the other main nonlinear filtering approach, the median filter.