Abstract
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.
This research is partially supported by the Defense Advanced Research Projects Agency (DARPA), under grants HR0011-05-1-0057 and HR0011-09-0065, as well as the National Science Foundation (NSF), under grant DBI-0820624.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bajaj, C.L., Pascucci, V., Schikore, D.R.: The contour spectrum. In: Proc. 8th IEEE Conf. Visualization, pp. 167–173 (1997)
Carlsson, G., Collins, A., Guibas, L.J., Zomorodian, Z.: Persistence barcodes for shapes. Internat. J. Shape Modeling 11, 149–187 (2005)
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)
Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Extending persistence using Poincaré and Lefschetz duality. Found. Comput. Math. 9, 79–103 (2009)
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)
Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discrete Comput. Geom. 28, 511–533 (2002)
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)
Edelsbrunner, H., Morozov, D., Patel, A.: Quantifying transversality by measuring the robustness of intersections. Dept. Comput. Sci., Duke Univ., Durham, North Carolina (2009) (Manuscript)
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)
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)
Munkres, J.R.: Elements of Algebraic Topology. Perseus, Cambridge (1984)
Newman, T.S., Yi, H.: A survey of the marching cube algorithm. Computers and Graphics 30, 854–879 (2006)
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)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bendich, P., Edelsbrunner, H., Morozov, D., Patel, A. (2010). The Robustness of Level Sets. In: de Berg, M., Meyer, U. (eds) Algorithms – ESA 2010. ESA 2010. Lecture Notes in Computer Science, vol 6346. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15775-2_1
Download citation
DOI: https://doi.org/10.1007/978-3-642-15775-2_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15774-5
Online ISBN: 978-3-642-15775-2
eBook Packages: Computer ScienceComputer Science (R0)