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

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 

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References

  1. Carlsson, G., & de Silva, V. (2004). Topological estimation using witness complexes. In Symposium on point-based graphics. Google Scholar
  2. de Silva, V. (2003). A weak definition of Delaunay triangulation. Google Scholar
  3. Edelsbrunner, H., Letscher, D., & Zomorodian, A. (2000). Topological persistence and simplification. IEEE symposium on foundations of computer science. Google Scholar
  4. Field, D. J. (1987). Relations between the statistics of natural images and the response properties of cortical cells. Journal of the Optical Society of America, 4(12), 2379–2394. Google Scholar
  5. Geman, D., & Koloydenko, A. (1999). Invariant statistics and coding of natural microimages. In Proceedings of the IEEE workshop on statistical and computational theories of vision. Google Scholar
  6. Hatcher, A. (2001). Algebraic topology. Cambridge: Cambridge University Press. Google Scholar
  7. Knill, D. C., Field, D. J., & Kersten, D. (1990). Human discrimination of fractal images. Journal of the Optical Society of America A, 7(6), 1113–1123 CrossRefGoogle Scholar
  8. Lee, A. B., Pedersen, K. S., & Mumford, D. (2003). The non-linear statistics of high-contrast patches in natural images. International Journal of Computer Vision, 54(1–3), 83–103. CrossRefMATHGoogle Scholar
  9. Reinagel, P., & Zador, A. M. (1999). Natural scene statistics at the center of gaze. Network: Computation in Neural Systems, 10(4), 341–350. CrossRefMATHGoogle Scholar
  10. Silverman, B. W. (1986). Density estimation for statistics and data analysis. London: Chapman & Hall/CRC. MATHGoogle Scholar
  11. Singh, G., Mémoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., & Ringach, D., (2007, submitted). Topological structure of population activity in primary visual cortex. PLoS. Google Scholar
  12. van Hateren, J. H. (1992). Theoretical predictions of spatiotemporal receptive fields of fly LMCs, and experimental validation. Journal of Computational Physiology A, 171, 157–170. Google Scholar
  13. van Hateren, J. H., & van der Schaaf, A. (1998). Independent component filters of natural images compared with simple cells in primary visual cortex. Proceedings of the Royal Society of London Series B, 265, 359–366. CrossRefGoogle Scholar
  14. Zomorodian, A., & Carlsson, G. (2004). Computing persistent homology. In 20th ACM symposium on computational geometry. Google Scholar

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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