Joint Distributions of Weighted Medians on Overlapping Windows in Image Analysis

In this paper, we derive the joint distribution function of the output of weighted median lters with independent but not identically distributed inputs on overlapping windows which may contain an image feature such as a line or edge. These results are used to derive the probability distributions of the diierence between the weighted medians calculated on two adjacent overlapping windows. As an illustration , these latter distributions are used to compute the power functions for detecting a step change in the image for a variety of range of existing weighting schemes. These comparisons demonstrate that optimal structure preserving weighting schemes are not optimal in terms of power to detect an edge in the presence of noise.