# A torus model for optical flow

## TL;DR

This paper introduces a novel torus model for high-contrast optical flow patches, utilizing computational topology tools to reveal the underlying manifold structure in a computer-generated video dataset.

## Contribution

It demonstrates that high-contrast optical flow patches can be effectively modeled by a torus, and reveals the fiber bundle structure related to range image statistics.

## Key findings

- High-contrast optical flow patches are well-modeled by a torus.
- Persistent homology confirms the torus structure in the data.
- The optical flow torus has a natural fiber bundle structure.

## Abstract

We propose a torus model for high-contrast patches of optical flow. Our model is derived from a database of ground-truth optical flow from the computer-generated video \emph{Sintel}, collected by Butler et al.\ in \emph{A naturalistic open source movie for optical flow evaluation}. Using persistent homology and zigzag persistence, popular tools from the field of computational topology, we show that the high-contrast $3\times 3$ patches from this video are well-modeled by a \emph{torus}, a nonlinear 2-dimensional manifold. Furthermore, we show that the optical flow torus model is naturally equipped with the structure of a fiber bundle, related to the statistics of range image patches.