An Extended Topological Model For High-Contrast Optical Flow

2026-03-06

AI Summary

Existing torus models struggle to accurately capture the low-dimensional topological structure of high-contrast optical flow data and lack direct validation mechanisms. This work constructs the first three-dimensional manifold model based on 3×3 high-contrast optical flow patches from the Sintel dataset, explicitly elucidating the failure modes of torus-based representations. The boundary of this manifold corresponds to the known optical flow torus, and it incorporates a family of disjoint circles associated with binary step-edge images to characterize its topological geometry. By integrating discrete circle bundle theory, persistent homology analysis, and sampling of high-contrast optical flow patches, the study reveals that over 99% of such flows concentrate near motion boundaries along these binary step-edge circle families, offering a novel topological foundation for vision tasks such as object segmentation and tracking.

Technology Category

• Computer Vision: Low Level & Physics-based Vision
• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• optical flow
• topological model
• high-contrast
• motion boundaries
• low-dimensional structure

Innovation

• optical flow
• topological model
• circle bundles
• persistent homology
• motion boundaries