2000/09/18 by Marco Grandis, Grandis, Marco · 2 citations
Computer Science · Mathematics · #18G55 #54G99 #55Q05 #55U10 #68U10 #Algebraic Topology (math.AT) #Category Theory (math.CT) #Combinatorics (math.CO) #Digital Image Processing Techniques #FOS: Mathematics #Medical Image Segmentation Techniques #Topological and Geometric Data Analysis #math.AT #math.CO #math.CT #msc:18G55 #msc:54G99 #msc:55Q05 #msc:55U10 #msc:68U10
paper · pdf · doi:10.48550/arxiv.math/0009166
46 pages
arxiv created 2000/09/18 · openalex publication_date 2000/09/18 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.