Proximal Nerve Complexes. A Computational Topology Approach

Abstract

This article introduces a theory of proximal nerve complexes and nerve spokes, restricted to the triangulation of finite regions in the Euclidean plane. A nerve complex is a collection of filled triangles with a common vertex, covering a finite region of the plane. Structures called $k$-spokes, $k\ge 1$, are a natural extension of nerve complexes. A $k$-spoke is the union of a collection of filled triangles that pairwise either have a common edge or a common vertex. A consideration of the closeness of nerve complexes leads to a proximal view of simplicial complexes. A practical application of proximal nerve complexes is given, briefly, in terms of object shape geometry in digital images.

arXiv comment16 pages, 9 figures

References in corpus (1)

• Strong Proximities on Smooth Manifolds and Vorono\" i Diagrams

Cited by in corpus (4)

• Proximal Planar Shape Signatures. Homology Nerves and Descriptive Proximity
• Delta Complexes in Digital Images. Approximating Image Object Shapes
• Geodesics of Triangulated Image Object Shapes. Approximating Image Shapes via Rectilinear and Curvilinear Triangulations
• Proximal Planar Shapes. Correspondence between Shapes and Nerve Complexes