Statistical analysis of two dimensional median filtered images

Median filters have been used with success in a number of image processing applications where they are often used to smooth noisy images without perturbing edge structures. In particular, square window median filters and separable median filters have been popular. This paper addresses the statistical properties of these operations. Specifically, the output distribution for the separable two dimensional median filter and the square window two dimensional median filter have been derived for several cases, including first order and multivariate output distributions with white noise, signals plus white noise, and general input images with multivariate distributions. These are then used to compute some specific statistics in several illustrative examples.