Persistent Homology under Non-uniform Error

  • Paul Bendich
  • Herbert Edelsbrunner
  • Michael Kerber
  • Amit Patel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6281)

Abstract

Using ideas from persistent homology, the robustness of a level set of a real-valued function is defined in terms of the magnitude of the perturbation necessary to kill the classes. Prior work has shown that the homology and robustness information can be read off the extended persistence diagram of the function. This paper extends these results to a non-uniform error model in which perturbations vary in their magnitude across the domain.

Keywords

Topological spaces continuous functions level sets perturbations homology extended persistence error models stability robustness 

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References

  1. 1.
    Bendich, P., Edelsbrunner, H., Morozov, D., Patel, A.: Robustness of level and interlevel sets. Manuscript, IST Austria, Klosterneuburg, Austria (2010)Google Scholar
  2. 2.
    Carlsson, G., de Silva, V.: Zigzag persistence. Found. Comput. Math. (2010)Google Scholar
  3. 3.
    Carlsson, G., Ishkhanov, T., de Silva, V., Zomorodian, A.: On the local behavior of spaces of local images. Internat. J. Comput. Vision 76, 1–12 (2008)CrossRefGoogle Scholar
  4. 4.
    Cerri, A., Ferri, M., Giorgi, D.: Retrieval of trademark images by means of size functions. Graphical Models 68, 451–471 (2006)CrossRefGoogle Scholar
  5. 5.
    Chung, M.K., Bubenik, P., Kim, P.T.: Persistence diagrams of cortical surface data. In: Prince, J.L., Pham, D.L., Myers, K.J. (eds.) Information Processing in Medical Imaging. LNCS, vol. 5636, pp. 386–397. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  6. 6.
    Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discrete Comput. Geom. 28, 511–533 (2007)MathSciNetGoogle Scholar
  7. 7.
    Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Extending persistence using Poincaré and Lefschetz duality. Found. Comput. Math. 9, 79–103 (2009)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    de Silva, V., Ghrist, R.: Coverage in sensor networks via persistent homology. Alg. Geom. Topology 7, 339–358 (2007)MATHCrossRefGoogle Scholar
  9. 9.
    Dequèant, M.-L., Ahnert, S., Edelsbrunner, H., Fink, T.M.A., Glynn, E.F., Hattem, G., Kudlicki, A., Mileyko, Y., Morton, J., Mushegian, A.R., Pachter, L., Rowicka, M., Shiu, A., Sturmfels, B., Pourquié, O.: Comparison of pattern detection methods in microarray time series of the segmentation clock. PLoS ONE 3, e2856 (2008), doi:10.1371/journal.pone.0002856CrossRefGoogle Scholar
  10. 10.
    Edelsbrunner, H., Harer, J.L.: Computational Topology. An Introduction. Amer. Math. Soc. (2009)Google Scholar
  11. 11.
    Edelsbrunner, H., Morozov, D., Patel, A.: Quantifying transversality by measuring the robustness of intersections. Manuscript, Dept. Comput. Sci., Duke Univ., Durham, North Carolina (2009)Google Scholar
  12. 12.
    Hatcher, A.: Algebraic Topology. Cambridge Univ. Press, Cambridge (2002)MATHGoogle Scholar
  13. 13.
    Munkres, J.R.: Elements of Algebraic Topology. Perseus, Cambridge, Massachusetts (1984)Google Scholar
  14. 14.
    Thron, W.J.: Topological Structures. Holt, Rinehart & Winston, New York (1966)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Paul Bendich
    • 1
    • 2
    • 3
  • Herbert Edelsbrunner
    • 1
    • 2
    • 3
    • 4
  • Michael Kerber
    • 1
    • 2
  • Amit Patel
    • 1
    • 2
  1. 1.IST Austria (Institute of Science and Technology Austria)KlosterneuburgAustria
  2. 2.Dept.Comput. Sci.Duke Univ.DurhamNorth Carolina
  3. 3.Dept. MathematicsDuke UnivDurhamNorth Carolina
  4. 4.Geomagic, Research Triangle ParkNorth Carolina

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