High-Dimensional Geometry of Sliding Window Embeddings of Periodic Videos

Author Christopher Tralie

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Christopher Tralie

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Christopher Tralie. High-Dimensional Geometry of Sliding Window Embeddings of Periodic Videos. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 71:1-71:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


We explore the high dimensional geometry of sliding windows of periodic videos. Under a reasonable model for periodic videos, we show that the sliding window is necessary to disambiguate all states within a period, and we show that a video embedding with a sliding window of an appropriate dimension lies on a topological loop along a hypertorus. This hypertorus has an independent ellipse for each harmonic of the motion. Natural motions with sharp transitions from foreground to background have many harmonics and are hence in higher dimensions, so linear subspace projections such as PCA do not accurately summarize the geometry of these videos. Noting this, we invoke tools from topological data analysis and cohomology to parameterize motions in high dimensions with circular coordinates after the embeddings. We show applications to videos in which there is obvious periodic motion and to videos in which the motion is hidden.
  • Video Processing
  • High Dimensional Geometry
  • Circular Coordinates
  • Nonlinear Time Series


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