Generic Axiomatized Digital Surface-Structures

Abstract

In digital topology, Euclidean n-space R n is usually modeled either by the set of points of a discrete grid, or by the set of n-cells in a convex cell complex whose union is R n. For commonly used grids and complexes in the cases n = 2 and 3, certain pairs of adjacency relations (κ, λ) on the grid points or n-cells (such as (4,8) and (8,4) on Z 2 are known to be “good pairs”. For these pairs of relations (κ, λ), many results of digital topology concerning a set of grid points or n-cells and its complement (such as Rosenfeld's digital Jordan curve theorem) have versions in which κ-adjacency is used to define connectedness on the set and λ-adjacency is used to define connectedness on its complement. At present, results of 2D and 3D digital topology are usually proved for one good pair of adjacency relations at a time — so for each result there are different (but analogous) theorems for different good pairs of adjacency relations. In this paper we take the first steps in developing an alternative approach to digital topology based on very general axiomatic definitions of “well-behaved digital spaces”. This approach gives the possibility of stating and proving results of digital topology as single theorems which apply to all spaces of the appropriate dimensionality that satisfy our axioms. Specifically, this paper introduces the notion of a generic axiomatized digital surface-structure (GADS) — a general, axiomatically defined, type of discrete structure that models subsets of the Euclidean plane and of other surfaces. Instances of this notion include GADS corresponding to all of the good pairs of adjacency relations that have previously been used (by ourselves or others) in digital topology on planar grids and boundary surfaces. We define basic concepts for a GADS (such as homotopy of paths and the intersection number of two paths), give a discrete definition of planar GADS (which are GADS that model subsets of the Euclidean plane) and present some fundamental results including a Jordan curve theorem for planar GADS.