Spectral design of polynomial weighted median filters

Summary form only given. A closed-form spectral optimization method for the design of polynomial weighted median (PWM) filters is presented. The algorithm is an extension of the generalized Mallow's theory for nonlinear smoothers that consists of first finding a set of positive weights for each sub-WM filter of the PWM filter, whose sample selection probabilities are as close as possible to the coefficients of a corresponding sub-finite impulse response (FIR) filter of the conventional Volterra filter with the desired spectral response. The signs of the weights associated with the general PWM filtering structure are then coupled with the input sample for each sub-WM filter, prior to replication by weight magnitudes. The spectral characteristics of the PWM filter, designed under the proposed method, are shown to be very similar to those of the equivalent FIR Volterra filter and arbitrary spectral behavior can be achieved. Unlike their FIR Volterra filter counterparts, PWM filters are robust to impulsive noise, as demonstrated by simulations.