Discrete Approximate Circle Bundles

TL;DR

This paper introduces discrete approximate circle bundles, establishing their relation to true circle bundles, and provides algorithms for their analysis, with applications in computer vision and an open-source software package.

Contribution

It defines discrete approximate circle bundles, links them to true circle bundles, and develops algorithms for their invariants and applications in data analysis.

Findings

Algorithms successfully compute cohomology invariants.

Applications demonstrate utility in computer vision tasks.

Open-source software enables reproducibility.

Abstract

In this paper, we introduce discrete approximate circle bundles, a class of objects designed to serve as the data science analog of circle bundles from algebraic topology. We show that, under appropriate conditions, one can meaningfully and stably identify a discrete approximate circle bundle with an isomorphism class of true circle bundles. We also describe two cohomology invariants which uniquely determine the isomorphism class of a circle bundle, and provide algorithms to compute them given a discrete approximate representative. Finally, we propose a novel methodology for coordinatization and dimensionality reduction of circle bundle data. To illustrate the practical utility and viability of our algorithms, we present applications to both real and synthetic datasets from computer vision (e.g., modeling optical flow). The paper is accompanied by an open-source software package, with full documentation and tutorials, enabling reproducible implementation of the proposed algorithms and experiments, including those used to generate the figures in this paper.