On the Roots of the 3 × 3 Median Filter

A characterization of the roots of the 3×3 median filter is given. We define the properties of local smoothness and of local roughness for the roots of the 3×3 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.