Axiomatic Digital Topology

TL;DR

This paper introduces a new set of axioms for digital topology, defining locally finite spaces with properties that relate to common adjacency relations in digital images, aiming to improve understanding and application in computer imagery.

Contribution

It proposes a new axiomatic framework for digital topology that simplifies application development and clarifies the relationship between digital spaces and abstract cell complexes.

Findings

LF spaces satisfying the axioms are isomorphic to abstract cell complexes

Only certain (a, b)-adjacencies have analogs in LF spaces, with limitations

LF spaces can be easily used on standard orthogonal grids in digital imagery

Abstract

The paper presents a new set of axioms of digital topology, which are easily understandable for application developers. They define a class of locally finite (LF) topological spaces. An important property of LF spaces satisfying the axioms is that the neighborhood relation is antisymmetric and transitive. Therefore any connected and non-trivial LF space is isomorphic to an abstract cell complex. The paper demonstrates that in an n-dimensional digital space only those of the (a, b)-adjacencies commonly used in computer imagery have analogs among the LF spaces, in which a and b are different and one of the adjacencies is the "maximal" one, corresponding to 3n\"i1 neighbors. Even these (a, b)-adjacencies have important limitations and drawbacks. The most important one is that they are applicable only to binary images. The way of easily using LF spaces in computer imagery on standard orthogonal grids containing only pixels or voxels and no cells of lower dimensions is suggested.