Generalized Mean-Median Filtering for Robust Frequency-Selective Applications

Huber proposed the P_epsi family of epsi-contaminated normal distributions to model environments characterized by heavy-tailed distributions. Based on this two-component mixture distribution, mean-median (MEM) filters were proposed. The MEM filter output is a combination of the sample mean and the sample median, where observation samples are weighted uniformly. This property of MEM filters constrains them to the class of smoothers lacking frequency-selective filtering capabilities. This paper extends MEM filtering to the weighted sum-median (WSM) filtering structure admitting real-valued weights, thereby enabling more general filtering characteristics, i.e, bandpass and high-pass filtering. The proposed filter structure is also well motivated from a presented maximum likelihood (ML) estimate analysis under epsi-contaminated statistics. The ML analysis demonstrates the need for a combination of weighted sum (WS) and weighted median (WM) type filters for processing of signals corrupted by epsi-contaminated noise. The WSM filter is statistically analyzed through the determination of filter output variance and breakdown probability. The combination parameter alpha is optimized to minimize the filter output variance, which is a measure of noise attenuation capability. Moreover, filter design procedures that yield a desired spectral response are detailed. Finally, the proposed WSM filter structure is tested utilizing signal processing applications including low-pass, bandpass, and high-pass filtering and image processing applications including image sharpening and denoising, evaluating and comparing the WSM filter performance to that of the WS, WM, and MEM filters