An intrinsic homotopy theory for simplicial complexes, with applications to image analysis

Abstract

A simplicial complex is a set equipped with a down-closed family of distinguished finite subsets. This structure, usually viewed as codifying a triangulated space, is used here directly, to describe "spaces" whose geometric realisation can be misleading. An intrinsic homotopy theory, not based on such realisation but agreeing with it, is introduced. The applications developed here are aimed at image analysis in metric spaces and have connections with digital topology and mathematical morphology. A metric space X has a structure of simplicial complex at each (positive) resolution e; the resulting n-homotopy group detects those singularities which can be captured by an n-dimensional grid, with edges bound by e; this works equally well for continuous or discrete regions of euclidean spaces. Its computation is based on direct, intrinsic methods.