Threefold Symmetry Detection in Hexagonal Images Based on Finite Eisenstein Fields

This paper considers an algebraic method for symmetry analysis of hexagonally sampled images, based on the interpretation of such images as functions on "Eisenstein fields". These are finite fields GF(p(2)) of special characteristics p = 12k + 5, where k > 0 is an integer. Some properties of such fields are studied; in particular, it is shown that its elements may be considered as "discrete Eisenstein numbers" and are in natural correspondence with hexagons in a (p x p)-diamond-shaped fragment of a regular plane tiling. The concept of logarithm in Eisenstein fields is introduced and used to define a "log-polar"-representation of hexagonal images. Next, an algorithm for threefold symmetry detection in gray-level images is proposed.