Hexagonal images processing over finite Eisenstein fields

This paper considers a new algebraic method for analysis and processing of hexagonally sampled images. The method is based on the interpretation of such images as functions on “Eisenstein fields”. These are finite fields GF(p2) 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 (pxp)-diamond-shaped fragment of a regular plane tiling. We show that in some cases multiplications in Eisenstein fields correspond to rotations combined with appropriate scalings, and use this fact for hexagonal images sharpening, smoothering and segmentation. The proposed algorithms have complexity O(p2) and can be used also for processing of square-sampled digital images over finite Gaussian fields.