A characterization of the roots of the 3x3, 5+, 5x is given. Then we present results regarding the roots of the 13-pt. cross shaped median filter and of the 5x5 median filter. We define the properties of smoothness and roughness. Roots that are rough are binary and periodic. This partially generalizes to dimension 2 the results of Tyan [1] and Brandt [2]. We concentrate on the binary roots of the 3x3, 5+, 5x, 13pt. and 5x5 window shape median filter; the complexities of the general problem of characterizing the roots of the 2D median filter makes this an acceptable starting point.