Discrete Approximate Circle Bundles

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.

Figures (19)

• Figure 1: The Klein bottle $K$ as a circle bundle over $\mathbb{S}^{1}$. The highlighted region $\pi^{-1}(U)$ in $K$ shows all fibers above the neighborhood $U$ in the base space and is homeomorphic to $U\times\mathbb{S}^{1}$. Globally, however, $K$ is not homeomorphic to $\mathbb{S}^{1}\times\mathbb{S}^{1}$.
• Figure 2: A noisy sample $X$ from a torus whose fiber radius continuously oscillates. The persistence diagrams show only a single significant class in dimension 1 (recall that the non-zero Betti numbers for the torus are $\beta_{0} = 1$, $\beta_{1}=2$, $\beta_{2}=1$). Figure (b) shows the dataset colored according to circular coordinates computed with DREiMac DREiMac. Figure (c) shows the same dataset colored according to the fiber coordinates computed using the software package accompanying this paper.
• Figure 3: Various stages of the weights filtration on the nerve of an open cover. Each edge is labeled with its associated weight and the value assigned by a $\mathbb{Z}_{2}$-simplicial cochain $\omega$.
• Figure 4: A frame from the Sintel video with sample (scaled) optical flow vectors shown
• Figure 5: A sample of mean-centered and normalized high-contrast optical flow patches labeled by predominant direction (represented as an angle in $[0,\pi)$)

Theorems & Definitions (110)

• Definition 2.1
• Definition 2.2
• Proposition 2.3
• Definition 2.4
• Proposition 2.5
• proof
• Definition 2.6
• Remark 2.7
• Definition 2.8
• Proposition 2.9