Fast algorithms for weighted myriad computation by fixed-point search

This paper develops fast algorithms to compute the output of the weighted myriad filter. Myriad filters form a large and important class of nonlinear filters for robust non-Gaussian signal processing and communications in impulsive noise environments. Just as the weighted mean and the weighted median are optimized for the Gaussian and Laplacian distributions, respectively, the weighted myriad is based on the class of (alpha)-stable distributions, which can accurately model impulsive processes. The weighted myriad is an M-estimator that is defined in an implicit manner; no closed-form expression exists for it, and its direct computation is a nontrivial and prohibitively expensive task. In this paper, the weighted myriad is formulated as one of the fixed points of a certain mapping. An iterative algorithm is proposed to compute these fixed points, and its convergence is proved rigorously. Using these fixed point iterations, fast algorithms are developed for the weighted myriad. Numerical simulations demonstrate that these algorithms compute the weighted myriad with a high degree of accuracy at a relatively low computational cost.