Invariant Representative Cocycles of Cohomology Generators using Irregular Graph Pyramids

TL;DR

This paper introduces an efficient algorithm for computing topological cohomology invariants, specifically cocycles, in 2D graph pyramids, with extensions for invariance to scanning and rotation, enhancing structural pattern recognition.

Contribution

It presents a novel algorithm for computing cohomology cocycles in 2D graph pyramids, incorporating invariance to scanning and rotation for improved pattern recognition.

Findings

Efficient computation of cohomology cocycles in 2D graph pyramids.

Extension for rotation and scanning invariance.

Application to structural pattern recognition.

Abstract

Structural pattern recognition describes and classifies data based on the relationships of features and parts. Topological invariants, like the Euler number, characterize the structure of objects of any dimension. Cohomology can provide more refined algebraic invariants to a topological space than does homology. It assigns `quantities' to the chains used in homology to characterize holes of any dimension. Graph pyramids can be used to describe subdivisions of the same object at multiple levels of detail. This paper presents cohomology in the context of structural pattern recognition and introduces an algorithm to efficiently compute representative cocycles (the basic elements of cohomology) in 2D using a graph pyramid. An extension to obtain scanning and rotation invariant cocycles is given.