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Enhanced Topology-Sensitive Clustering by Reeb Graph Shattering

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Topological Methods in Data Analysis and Visualization II

Part of the book series: Mathematics and Visualization ((MATHVISUAL))

Abstract

Scalar-valued functions are ubiquitous in scientific research. Analysis and visualization of scalar functions defined on low-dimensional and simple domains is a well-understood problem, but complications arise when the domain is high-dimensional or topologically complex. Topological analysis and Morse theory provide tools that are effective in distilling useful information from such difficult scalar functions. A recently proposed topological method for understanding high-dimensional scalar functions approximates the Morse-Smale complex of a scalar function using a fast and efficient clustering technique. The resulting clusters (the so-called Morse crystals) are each approximately monotone and are amenable to geometric summarization and dimensionality reduction. However, some Morse crystals may contain loops. This shortcoming can affect the quality of the analysis performed on each crystal, as regions of the domain with potentially disparate geometry are assigned to the same cluster. We propose to use the Reeb graph of each Morse crystal to detect and resolve certain classes of problematic clustering. This provides a simple and efficient enhancement to the previous Morse crystals clustering. We provide preliminary experimental results to demonstrate that our improved topology-sensitive clustering algorithm yields a more accurate and reliable description of the topology of the underlying scalar function.

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Acknowledgements

We would like to thank Peter G. Lindstrom for providing us with the optimization dataset and his help and insight into the problem. This work was funded by the National Science Foundation (NSF) under grants CCF-0747082, DBI-0750891, and CCF-1048983. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. We would like to thank the Livermore Elks for their scholarship support. Publication number: LLNL-CONF-468782.

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Authors and Affiliations

  1. Ohio State University, 487 Dreese Lab, 2015 Neil Ave., Columbus, OH, 43210, USA

    W. Harvey & Y. Wang

  2. Lawrence Livermore National Laboratory, 422, 808, Livermore, CA, 94551-0808, USA

    O. Rübel & P.-T. Bremer

  3. Scientific Computing and Imaging Institute, School of Computing, University of Utah, 72 South Central Campus Drive, Salt Lake City, UT, 84112, USA

    V. Pascucci & P.-T. Bremer

Authors
  1. W. Harvey
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  2. O. Rübel
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  3. V. Pascucci
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  4. P.-T. Bremer
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  5. Y. Wang
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Corresponding author

Correspondence to W. Harvey .

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Editors and Affiliations

  1. Inst. Computational Science, CAB G 65.1, ETH Zürich, Universitätsstraße 6, Zürich, 8092, Switzerland

    Ronald Peikert

  2. , Dept. of Informatics, University of Bergen, Bergen, Norway

    Helwig Hauser

  3. , School of Computing, University of Leeds, Woodhouse Lane, Leeds, LS2 9JT, United Kingdom

    Hamish Carr

  4. , Computational Science, ETH Zürich, Universitätsstraße 6, Zürich, 8092, Switzerland

    Raphael Fuchs

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© 2012 Springer-Verlag Berlin Heidelberg

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Harvey, W., Rübel, O., Pascucci, V., Bremer, PT., Wang, Y. (2012). Enhanced Topology-Sensitive Clustering by Reeb Graph Shattering. In: Peikert, R., Hauser, H., Carr, H., Fuchs, R. (eds) Topological Methods in Data Analysis and Visualization II. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23175-9_6

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