Ghrist Barcoded Video Frames. Application in Detecting Persistent Visual Scene Surface Shapes captured in Videos

Abstract

This article introduces an application of Ghrist barcodes in the study of persistent Betti numbers derived from vortex nerve
complexes found in triangulations of video frames. A Ghrist barcode (also called a persistence barcode) is a topology of data pictograph
useful in representing the persistence of the features of changing shapes. The basic approach is to introduce a free Abelian
group representation of intersecting filled polygons on the barycenters of the triangles of Alexandroff nerves. An Alexandroff nerve
is a maximal collection of triangles of a common vertex in the triangulation of a finite, bounded planar region. In our case, the
planar region is a video frame. A Betti number is a count of the number of generators is a finite Abelian group. The focus here
is on the persistent Betti numbers across sequences of triangulated video frames. Each Betti number is mapped to an entry in a
Ghrist barcode. Two main results are given, namely, vortex nerves are Edelsbrunner-Harer nerve complexes and the Betti number
of a vortex nerve equals k + 2 for a vortex nerve containing k edges attached between a pair of vortex cycles in the nerve.