Mean , median and mode ̄ ltering of images

If a two-dimensional image is simpli­ ed by repeatedly replacing its values with the mean in an in­ nitesimal neighbourhood, it evolves according to the di¬usion equation Lt = Lxx + Lyy (subscripts denote di¬erentiation). 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 in­ nitesimal median ­ ltering. In this paper it is proved that repeated in­ nitesimal mode ­ ltering is described by Lt = Lvv ¡ 2Lww at regular points, and Lt = 0 at critical points. Other new results are (i) an approximate equation for median ­ ltering at critical points, and (ii) a derivation of the equation for median ­ ltering at regular points, which generalizes to mean and mode ­ ltering. Finally, the results of numerical implementation of all three ­ ltering schemes are brie®y presented.