Unification of order-statistics based filters to piecewise-linear filters

Order-statistics based filters that were originally provided by the robust estimation theory have proved to be efficient in image/signal filtering in the presence of additive white noise or impulsive noise. Their algorithms are simple and easy to implement. Their analysis, however, is not straightforward. In this paper, we show that filters based on order statistics can be explained by using the theory of piecewise-linear (PWL) functions which was established originally for circuit analysis and has recently been applied to nonlinear filtering. We also prove that an L-filter is a PWL filter defined on IRⁿ and a median filter by threshold decomposition is a piecewise-constant (PWC) filter on [0,M-1]ⁿ. The main results lead to the unification of order-statistics based filters with the PWL filter class. Based on the fact that PWL functions are a general class of approximation functions which are uniformly dense in the domain concerned, it is expected that the results obtained can provide a new way to the extension, as well as further study of, order-statistics based filters