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Homological Shape Analysis Through Discrete Morse Theory

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Perspectives in Shape Analysis

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

Abstract

Homology and persistent homology are fundamental tools for shape analysis and understanding that recently gathered a lot of interest, in particular for analyzing multidimensional data. In this context, discrete Morse theory, a combinatorial counterpart of smooth Morse theory, provides an excellent basis for reducing computational complexity in homology detection. A discrete Morse complex, computed over a given complex discretizing a shape, drastically reduces the number of cells of the latter while maintaining the same homology. Here, we consider the problem of shape analysis through discrete Morse theory, and we review and analyze algorithms for computing homology and persistent homology based on such theory.

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Acknowledgements

This work has been partially supported by the US National Science Foundation under grant number IIS-1116747 and by the University of Genova through PRA 2013. The authors wish to thank Davide Bolognini, Emanuela De Negri and Maria Evelina Rossi for their helpful comments and suggestions.

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Authors and Affiliations

  1. Department of Computer Science, Bioengineering, Robotics, and Systems Engineering, University of Genova, Genova, Italy

    Leila De Floriani

  2. Department of Computer Science and UMIACS, University of Maryland, College Park, MD, USA

    Ulderico Fugacci

  3. Department of Geographical Sciences and UMIACS, University of Maryland, College Park, MD, USA

    Federico Iuricich

Authors
  1. Leila De Floriani
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  2. Ulderico Fugacci
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  3. Federico Iuricich
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Corresponding author

Correspondence to Leila De Floriani .

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Editors and Affiliations

  1. Institute for Applied Mathematics and Scientific Computing, Brandenburg University of Technology, Cottbus, Germany

    Michael Breuß

  2. Israel Institute of Technology Technion, Haifa, Israel

    Alfred Bruckstein

  3. School of Electrical and Computer Engineering, National Technical University of Athens, Athens, Greece

    Petros Maragos

  4. INRIA, Grenoble Rhône-Alpes, Saint Ismier, France

    Stefanie Wuhrer

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De Floriani, L., Fugacci, U., Iuricich, F. (2016). Homological Shape Analysis Through Discrete Morse Theory. In: Breuß, M., Bruckstein, A., Maragos, P., Wuhrer, S. (eds) Perspectives in Shape Analysis. Mathematics and Visualization. Springer, Cham. https://doi.org/10.1007/978-3-319-24726-7_9

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