An Extended Topological Model For High-Contrast Optical Flow

TL;DR

This paper introduces a novel topological 3-manifold model for high-contrast optical flow patches, explaining previous verification issues and highlighting the significance of motion boundaries for computer vision tasks.

Contribution

It develops an extended topological model using circle bundles to better understand the structure of optical flow patches and their relation to motion boundaries.

Findings

Nearly all high-contrast patches are near binary step-edge circles.

The 3-manifold model explains verification difficulties of the torus model.

Motion boundaries are key regions for optical flow analysis.

Abstract

In this paper, we identify low-dimensional models for dense core subsets in the space of $3\times 3$ high-contrast optical flow patches sampled from the Sintel dataset. In particular, we leverage the theory of approximate and discrete circle bundles to identify a 3-manifold whose boundary is a previously proposed optical flow torus, together with disjoint circles corresponding to pairs of binary step-edge range image patches. The 3-manifold model we introduce provides an explanation for why the previously-proposed torus model could not be verified with direct methods (e.g., a straightforward persistent homology computation). We also demonstrate that nearly all optical flow patches in the top 1 percent by contrast norm are found near the family of binary step-edge circles described above, rather than the optical flow torus, and that these frequently occurring patches are concentrated near motion boundaries (which are of particular importance for computer vision tasks such as object segmentation and tracking). Our findings offer insights on the subtle interplay between topology and geometry in inference for visual data.