Chain Homotopies for Object Topological Representations

TL;DR

This paper introduces algebraic tools called AM-models for extracting topological features from simplicial complexes and digital images, including algorithms for computing homology, cohomology, and updating models after image operations.

Contribution

It presents a novel algebraic-topological framework (AM-models) for analyzing topological features of objects and digital images, with algorithms for computation and updates.

Findings

Algorithm for computing AM-models and cohomological invariants in any dimension.

Extension of AM-models to 3D digital images and voxel set operations.

Introduction of generators with 'nice' representative cycles.

Abstract

This paper presents a set of tools to compute topological information of simplicial complexes, tools that are applicable to extract topological information from digital pictures. A simplicial complex is encoded in a (non-unique) algebraic-topological format called AM-model. An AM-model for a given object K is determined by a concrete chain homotopy and it provides, in particular, integer (co)homology generators of K and representative (co)cycles of these generators. An algorithm for computing an AM-model and the cohomological invariant HB1 (derived from the rank of the cohomology ring) with integer coefficients for a finite simplicial complex in any dimension is designed here. A concept of generators which are "nicely" representative cycles is also presented. Moreover, we extend the definition of AM-models to 3D binary digital images and we design algorithms to update the AM-model information after voxel set operations (union, intersection, difference and inverse).