On the Local Behavior of Spaces of Natural Images

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.

This is a preview of subscription content, log in to check access.

References

  1. Carlsson, G., & de Silva, V. (2004). Topological estimation using witness complexes. In Symposium on point-based graphics.

  2. de Silva, V. (2003). A weak definition of Delaunay triangulation.

  3. Edelsbrunner, H., Letscher, D., & Zomorodian, A. (2000). Topological persistence and simplification. IEEE symposium on foundations of computer science.

  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.

  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

    Article  Google 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.

    Article  MATH  Google 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.

    Article  MATH  Google Scholar 

  10. Silverman, B. W. (1986). Density estimation for statistics and data analysis. London: Chapman & Hall/CRC.

    Google 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.

  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.

    Article  Google Scholar 

  14. Zomorodian, A., & Carlsson, G. (2004). Computing persistent homology. In 20th ACM symposium on computational geometry.

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Gunnar Carlsson.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Carlsson, G., Ishkhanov, T., de Silva, V. et al. On the Local Behavior of Spaces of Natural Images. Int J Comput Vis 76, 1–12 (2008). https://doi.org/10.1007/s11263-007-0056-x

Download citation

Keywords

  • Topology
  • Natural images
  • Manifold
  • Filtration
  • Klein bottle
  • Persistent homology