Group Theoretic Properties of Polygons, Polyhedrons and Hyper Polyhedrons Defined Over Discrete Lattices

Abstract

Coding of n dimensional image is a critical task in image compression. There exist a number of image coding methods based on symbolic or number coding. This paper discusses shape coding or pextral coding, in which the coding is done based on morphological analysis of shapes. This shape based coding method uses the permutation and cyclic groups of polygons, polyhedrons and hyper polyhedrons for image coding called pextral coding. This paper discusses the generation of permutation groups for 2-D, 3-D and multidimensional digital images. There are 16 convex polygons in a 3X3 two dimensional array of cells and 256 convex polyhedrons in a 3X3X3 three dimensional array of cells. These two arrays of cells are sub lattices of discrete infinite lattices Z2 and Z3 respectively. Similarly, one can construct 216 convex hyper polyhedrons in a sub lattice of the discrete infinite lattice Z4. In the abstract sense, convex hyper polyhedrons could be constructed in a sub lattice of the discrete infinite lattice Zn. Thus, given an n-dimensional discrete infinite lattice, one can build a finite or a potentially infinite set of convex geometric shapes which could be treated as the ground set of a permutation group. This paper describes the results of a study carried out on permutations groups of convex geometric shapes defined over n-dimensional discrete lattices Zn, where n varies from 2 to infinity. Also an application of pextral coding using permutation and cyclic groups, for 2-D image is explained.