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.
Abstract (translated)
在这篇论文中，我们为Sintel数据集中采样的高对比度$3\times 3$光学流补丁的空间中的密集核心子集识别出了低维模型。具体来说，我们利用近似和离散圆束理论来识别出一个三维流形，其边界是一个先前提出的光学流环面，并且与二进制阶跃边缘范围图像补丁对相对应的分离圆圈相交。 我们引入的三维流形模型解释了为什么之前的环面模型不能通过直接方法（例如，简单的持久同调计算）来验证。此外，我们也展示了几乎所有按对比度范数排名前1%的光学流补丁都靠近上述二进制阶跃边缘圆家族，而非光学流环面上。并且，这些频繁出现的补丁集中在运动边界附近（这对于如目标分割和跟踪等计算机视觉任务特别重要）。 我们的发现提供了关于在视觉数据推断中拓扑与几何之间微妙相互作用的理解。
URL
https://arxiv.org/abs/2603.06853
