Abstract
We explore the high dimensional geometry of sliding windows of periodic videos. Under
a reas- onable 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 in- dependent 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 mo- tions 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.
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