The Robustness of Level Sets

  • Paul Bendich
  • Herbert Edelsbrunner
  • Dmitriy Morozov
  • Amit Patel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6346)


We define the robustness of a level set homology class of a function \(f: {\mathbb X} \to {\mathbb R}\) as the magnitude of a perturbation necessary to kill the class. Casting this notion into a group theoretic framework, we compute the robustness for each class, using a connection to extended persistent homology. The special case \({\mathbb X} = {\mathbb R}^3\) has ramifications in medical imaging and scientific visualization.


Topological spaces continuous functions level sets perturbations homology extended persistence well groups well diagrams robustness 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bajaj, C.L., Pascucci, V., Schikore, D.R.: The contour spectrum. In: Proc. 8th IEEE Conf. Visualization, pp. 167–173 (1997)Google Scholar
  2. 2.
    Carlsson, G., Collins, A., Guibas, L.J., Zomorodian, Z.: Persistence barcodes for shapes. Internat. J. Shape Modeling 11, 149–187 (2005)zbMATHCrossRefGoogle Scholar
  3. 3.
    Carlsson, G., de Silva, V., Morozov, D.: Zigzag persistent homology and real-valued functions. In: Proc. 25th Ann. Sympos. Comput. Geom., pp. 247–256 (2009)Google Scholar
  4. 4.
    Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Extending persistence using Poincaré and Lefschetz duality. Found. Comput. Math. 9, 79–103 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Edelsbrunner, H., Harer, J.: Persistent homology — a survey. In: Goodman, J.E., Pach, J., Pollack, R. (eds.) Surveys on Discrete and Computational Geometry. Twenty Years Later. Contemporary Mathematics, vol. 453, pp. 257–282. Amer. Math. Soc., Providence (2008)Google Scholar
  6. 6.
    Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discrete Comput. Geom. 28, 511–533 (2002)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Edelsbrunner, H., Morozov, D., Patel, A.: The stability of the apparent contour of an orientable 2-manifold. In: Pascucci, V., Tierny, J. (eds.) Topological Methods in Data Analysis and Visualization: Theory, Algorithms, and Applications. Springer, Heidelberg (to appear)Google Scholar
  8. 8.
    Edelsbrunner, H., Morozov, D., Patel, A.: Quantifying transversality by measuring the robustness of intersections. Dept. Comput. Sci., Duke Univ., Durham, North Carolina (2009) (Manuscript)Google Scholar
  9. 9.
    Fang, S., Biddlecome, T., Tuceryan, M.: Image-based transfer function design for data exploration in volume visualization. In: Proc. 9th IEEE Conf. Visualization, pp. 319–326 (1998)Google Scholar
  10. 10.
    van Krefeld, M., van Oostrum, R., Bajaj, C.L., Pascucci, V., Schikore, D.R.: Contour trees and small seed sets for isosurface traversal. In: Proc. 13th Ann. Sympos. Comput. Geom., pp. 212–220 (1997)Google Scholar
  11. 11.
    Munkres, J.R.: Elements of Algebraic Topology. Perseus, Cambridge (1984)zbMATHGoogle Scholar
  12. 12.
    Newman, T.S., Yi, H.: A survey of the marching cube algorithm. Computers and Graphics 30, 854–879 (2006)CrossRefGoogle Scholar
  13. 13.
    Wittenbrink, C.M., Malzbender, T., Goss, M.E.: Opacity-weighted color interpolation for volume sampling. In: Proc. IEEE Proc. Volume Visualization, pp. 135–142 (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Paul Bendich
    • 1
    • 2
    • 3
  • Herbert Edelsbrunner
    • 1
    • 2
    • 3
    • 5
  • Dmitriy Morozov
    • 4
  • Amit Patel
    • 1
    • 2
  1. 1.IST Austria (Institute of Science and Technology Austria)KlosterneuburgAustria
  2. 2.Dept. Comput. Sci.Duke Univ.Durham
  3. 3.Dept. MathematicsDuke Univ.Durham
  4. 4.Depts. Comput. Sci. and Math.Stanford Univ.Stanford
  5. 5.Geomagic, Research Triangle Park

Personalised recommendations