Spectral design of weighted median filters admitting negative weights

A closed-form spectral optimization method for the design of weighted median (WM) filters admitting negative weights is presented. The algorithm is a generalization of Mallows´ (1980) theory for nonlinear smoothers that consists of first finding a set of positive weights for a WM filter whose sample selection probabilities are as close as possible to the coefficients of a corresponding finite impulse response (FIR) filter with the desired spectral response. The signs of the weights associated with general WM filtering structures are then coupled with the input samples prior to replication by the weight magnitudes. The spectral characteristics of these WM filters, designed under the proposed method, are shown to be very similar to those of the equivalent linear FIR filters and arbitrary spectral behavior can be achieved. Unlike their FIR filter counterparts, WM filters are robust to impulsive noise, as demonstrated by simulations.