Streaking in median filtered images

This paper presents a probabilistic analysis of the streaking or blotching effect commonly observed in median filtered signals in both one and two dimensions. The effcts are identified as runs of equal or nearly equal values which create visual impressions that have no visual correlate. For one-dimensional discrete iid random signals with continuous input probability densities, the probability of a streak of length L occurring is computed and shown to be independent of the input probability distribution. Expressions for the first and second moments of the streak length are also derived, and certain asymptotic results are given. As the analysis and definition of the analogous effect in two dimensions is less tractable, the probability that medians taken from distinct overlapping windows will take the same value is derived for various filter geometries. The analytic results are supported by examples using both one- and two-dimensional signals.