Analysis of polynomial weighted median filters

Summary form only given. This paper extends weighted median (WM) filters to the class of polynomial weighted median (PWM) filters. Traditional polynomial filtering theory, based on linear combinations of polynomial terms, is able to approximate important classes of nonlinear systems. The linear combination of polynomial terms, however, yields poor performance in environments characterized by heavy tailed distributions. Weighted median filters, in contrast, are well-known for their outlier suppression and detail preservation properties. The weighted median sample selection methodology is naturally extended to the polynomial sample case, yielding a filter structure that exploits the higher order statistics of the observed samples while simultaneously being robust to outliers. A presented probability density function analysis shows that cross and square terms have heavier tails than the observed samples, indicating that robust combination methods should be utilized to avoid undue influence of outliers. Weighted median processing of polynomial terms is justified from a maximum likelihood perspective. The established PWM filter class is statistically analyzed through the determination of the breakdown probability. Filter parameter optimization procedures are also presented. Finally, the effectiveness of PWM filters is demonstrated through simulations that include temporal and spectrum analysis.