Mean, median and mode filtering of images

Abstract

If a two‐dimensional image is simplified by repeatedly replacing its values with the mean in an infinitesimal neighbourhood, it evolves according to the diffusion equation L t = Lxx + Lyy (subscripts denote differentiation). This equation can alternatively be written in gauge coordinates as Lt = Lvv + Lww , where the v ‐direction is tangent to the isophote and the w ‐direction is along the gradient. The alternative evolution scheme of Lt = Lvv has also attracted attention. Guichard & Morel showed, in 1996, that this equation describes the operation of repeated infinitesimal median filtering. In this paper it is proved that repeated infinitesimal mode filtering is described by Lt = Lvv ‐ 2 Lww at regular points, and Lt = 0 at critical points. Other new results are (i) an approximate equation for median filtering at critical points, and (ii) a derivation of the equation for median filtering at regular points, which generalizes to mean and mode filtering. Finally, the results of numerical implementation of all three filtering schemes are briefly presented.