On the Roots of the 3×3 Median Filter

A characterization of the roots of the 3x3 median filter is given. We define the properties of local smoothness and of local roughness for the roots of the 3x3 median filter. Roots that are locally rough everywhere are binary and periodic; otherwise, unlike the 1D case, a root may be non binary or non periodic. This partially generalizes to dimension 2 the results of Brandt [1] and Tyan [2]; in particular, everywhere local smoothness may be interpreted as local monotonicity in dimension 2. We concentrate on the binary roots of the filter with the 3x3 window shape; the complexities of the general problem of characterizing the roots of the 2D median filter makes this an acceptable starting point.