A generalized weighted median filter structure admitting real-valued weights

Weighted median filters (smoothers) have been shown to be analogous to normalized FIR linear filters constrained to have only positive weights. In this paper, it is shown that much like the mean is generalized to the rich class of linear FIR filters, the median can be generalized to a richer class of weighted median (WM) filters admitting positive and negative weights. The generalization follows naturaly and is surprisingly simple. In order to analyze and design this class of WM filters, a new threshold decomposition theory admitting real-valued input signals is developed which, in turn, is used to develop fast adaptive algorithms to optimally design the real-valued filter coefficients. The new WM filter formulation leads to significantly more powerful estimators capable of effectively addressing a number of fundamental problems in signal processing which could not adequately be addressed by prior WM filter (smoother) structures.