International Journal of Computer Vision

, Volume 76, Issue 1, pp 1–12

On the Local Behavior of Spaces of Natural Images

  • Gunnar Carlsson
  • Tigran Ishkhanov
  • Vin de Silva
  • Afra Zomorodian
Position paper

DOI: 10.1007/s11263-007-0056-x

Cite this article as:
Carlsson, G., Ishkhanov, T., de Silva, V. et al. Int J Comput Vis (2008) 76: 1. doi:10.1007/s11263-007-0056-x

Abstract

In this study we concentrate on qualitative topological analysis of the local behavior of the space of natural images. To this end, we use a space of 3 by 3 high-contrast patches ℳ. We develop a theoretical model for the high-density 2-dimensional submanifold of ℳ showing that it has the topology of the Klein bottle. Using our topological software package PLEX we experimentally verify our theoretical conclusions. We use polynomial representation to give coordinatization to various subspaces of ℳ. We find the best-fitting embedding of the Klein bottle into the ambient space of ℳ. Our results are currently being used in developing a compression algorithm based on a Klein bottle dictionary.

Keywords

Topology Natural images Manifold Filtration Klein bottle Persistent homology 

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Gunnar Carlsson
    • 1
  • Tigran Ishkhanov
    • 1
  • Vin de Silva
    • 1
  • Afra Zomorodian
    • 1
  1. 1.Department of MathematicsStanford UniversityPalo AltoUSA

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