## On the Roots of the 3x3, 5+, 5x, 13-pt and 5x5 2D Median Filters

A. Cordoba Arenas and A. Restrepo Palacios (Columbia)

### Keywords

Median filter, root signals, nonlinear discrete filters, nonlinear signals and systems.

### Abstract

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. Fig. 1. Window shapes considered. An atomic tile is a window of data with an invariant central pixel. A version of an atomic tile is a negated, rotated or reflected version tile. The weight of a binary atomic tile is given by the number of pixels with value equal to the value of the central pixel; for a window size of 2k+1, the minimal weight is k+1. The set of neighbors of a pixel are those in a window centered at the pixel; thus, the relation connectivity for pixels depends on the window at hand and for the 3x3 window, 8-neighbor connectivity results, for the 5+ window, 4-neighbor connectivity and so on; also, for the 5x5 window, each pixel has 24 neighbors. A path between two pixels is a sequence {pi} of pixels such that for each pixel pi, the window centered at the pixel contains the previous pixel pi-1 and the following pixel pi+1, unless pi is the first or last of the pixels in the sequence. A block is a set of pixels of the same value and path-connected. A sister of a pixel is a neighbor pixel of the same value. Each pixel of a binary root signal of a median filter of window size 2k+1 has least k sisters.