Alpha-trimmed means and their relationship to median filters

Abstract

Suppose that X is a finite set of N numbers, The α-trimmed mean of X is obtained by sorting X into ascending order, removing (trimming) a fixed fraction <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\alpha(0 \leq \alpha \leq 0.5)</tex> from the high and low ends of the sorted set, and computing the average of the remaining values. When applied to a sliding window of length LW, the α-trimming process is called α-trim filtering. For α = 0.5, the α-trimmed mean of a set is the median of the set and the filtering operation is called median filtering. Repeated application of a median filter to the output of a previous median filter of the same length LW eventually produces a signal which is invariant to median filtering. This final signal is called the root signal. This paper explains the relationship between α-trimmed means and median filters, derives a simple straightforward and fast algorithm for applying a median filter, and provides a new explanation of the convergence of repeated median filtering to the root signal. The latter result incorporates an approach which permits generalization of the associated concepts to a larger class of "index map" filters.