An Extended Topological Model For High-Contrast Optical Flow

TL;DR

The theory of approximate and discrete circle bundles is used 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.

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.

Figures (31)

• Figure 1: Frames from the Sintel video labeled with sample (scaled) optical flow vectors
• Figure 2: The linear step-edge annulus model for high-contrast $3 \times 3$ optical image patches from Lee_Mumford. Some binary step-edge patches are shown on the boundary circles. The central circle is the primary circle described in Klein_bottle
• Figure 3: The mean-centered contrast-normalized DCT basis for $3 \times 3$ image patches
• Figure 4: Optical flow DCT basis.
• Figure 5: The proposed optical flow torus model. Each column captures patches with the same flow direction. The horizontal flow circle is shown in the first and last column; note however that the two columns differ by a shift by $\pi$. This reflects the fact that $(\theta,\alpha)$ is not a global coordinate system for $\mathcal{T}$. Compare with Figure 7 in opt_flow_torus.

Theorems & Definitions (1)

• proof