Document Open Access Logo

High-Dimensional Geometry of Sliding Window Embeddings of Periodic Videos

Author Christopher Tralie



PDF
Thumbnail PDF

File

LIPIcs.SoCG.2016.71.pdf
  • Filesize: 2.69 MB
  • 5 pages

Document Identifiers

Author Details

Christopher Tralie

Cite AsGet BibTex

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)
https://doi.org/10.4230/LIPIcs.SoCG.2016.71

Abstract

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.
Keywords
  • Video Processing
  • High Dimensional Geometry
  • Circular Coordinates
  • Nonlinear Time Series

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Vin De Silva, Dmitriy Morozov, and Mikael Vejdemo-Johansson. Persistent cohomology and circular coordinates. Discrete &Computational Geometry, 45(4):737-759, 2011. Google Scholar
  2. Herbert Edelsbrunner and John Harer. Computational topology: an introduction. American Mathematical Soc., 2010. Google Scholar
  3. Holger Kantz and Thomas Schreiber. Nonlinear time series analysis, volume 7. Cambridge university press, 2004. Google Scholar
  4. Jose A Perea and John Harer. Sliding windows and persistence: An application of topological methods to signal analysis. Foundations of Computational Mathematics, 15(3):799-838, 2013. Google Scholar
  5. Arno Schödl, Richard Szeliski, David H Salesin, and Irfan Essa. Video textures. In Proceedings of the 27th annual conference on Computer graphics and interactive techniques, pages 489-498. ACM Press/Addison-Wesley Publishing Co., 2000. Google Scholar
  6. Neal Wadhwa, Michael Rubinstein, Frédo Durand, and William T Freeman. Phase-based video motion processing. ACM Transactions on Graphics (TOG), 32(4):80, 2013. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail