This paper derives the joint distribution of medians over moving windows in a two dimensional noisy image. The general formulation presented allows derivation of the probability distribution, needed to evaluate the probability of failing to detect an edge when present ("edge miss probability") and the probability of falsely detecting a nonexistent edge.

In this paper, we define generalized real and generalized root signals of two-dimensional median filtering. A new edge detection method based on generalized root signals of two-dimensional median filtering is proposed. Simulation results are given and compared with the Sobel and Laplace edge detectors.

Generalized max/median filtering is defined as an extension of max/median filtering. Properties for these extensions are given. The output distribution of generalized max/median filtering with independent but not identical inputs is derived and applied to the special case of regular max/median filtering thereby providing a new result. Based on these distribution results, it is shown that max/median and generalized max/median filtering can preserve image details.