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Modeling and Simplifying Morse Complexes in Arbitrary Dimensions

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Topological Methods in Data Analysis and Visualization

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

Abstract

Ascending and descending Morse complexes, defined by a scalar function f over a manifold domain M, decompose M into regions of influence of the critical points of f, thus representing themorphology of the scalar function f over M in a compact way. Here, we introduce two simplification operators on Morse complexes which work in arbitrary dimensions and we discuss their interpretation as n-dimensional Euler operators. We consider a dual representation of the two Morse complexes in terms of an incidence graph and we describe how our simplification operators affect the graph representation. This provides the basis for defining a multi-scale graph-based model of Morse complexes in arbitrary dimensions.

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Acknowledgements

This work has been partially supported by the National Science Foundation under grant CCF-0541032, by the MIUR-FIRB project SHALOM under contract number RBIN04HWR8, and by the Ministry of Science of the Republic of Serbia through Project 23036.

Author information

Authors and Affiliations

  1. Faculty of Engineering, University of Novi Sad, Novi Sad, Serbia

    Lidija Čomić

  2. Department of Computer Science, University of Genova, Genova, Italy

    Leila De Floriani

Authors
  1. Lidija Čomić
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  2. Leila De Floriani
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Corresponding author

Correspondence to Lidija Čomić .

Editor information

Editors and Affiliations

  1. School of Computing, Scientific Computing and Imaging, University of Utah, S. Central Campus Dr. 50, Salt Lake City, 84132, Utah, USA

    Valerio Pascucci

  2. Department of Computer Science, Purdue University, N. University Street 305, West Lafayette, 47907-2107, Indiana, USA

    Xavier Tricoche

  3. FB Informatik, Universität Kaiserslautern, Kaiserslautern, Germany

    Hans Hagen

  4. LTCI - Telecom ParisTech, CNRS, 46, rue Barrault, Paris Cedex 13, France

    Julien Tierny

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Čomić, L., De Floriani, L. (2011). Modeling and Simplifying Morse Complexes in Arbitrary Dimensions. In: Pascucci, V., Tricoche, X., Hagen, H., Tierny, J. (eds) Topological Methods in Data Analysis and Visualization. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15014-2_7

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