Position paper

International Journal of Computer Vision

, Volume 76, Issue 1, pp 1-12

First online:

On the Local Behavior of Spaces of Natural Images

  • Gunnar CarlssonAffiliated withDepartment of Mathematics, Stanford University Email author 
  • , Tigran IshkhanovAffiliated withDepartment of Mathematics, Stanford University
  • , Vin de SilvaAffiliated withDepartment of Mathematics, Stanford University
  • , Afra ZomorodianAffiliated withDepartment of Mathematics, Stanford University

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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.


Topology Natural images Manifold Filtration Klein bottle Persistent homology