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Discrete Poincaré Duality Angles as Shape Signatures on Simplicial Surfaces with Boundary

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Topological Methods in Data Analysis and Visualization V (TopoInVis 2017)

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

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Abstract

We introduce and explore the concept of discrete Poincaré duality angles as an intrinsic measure that quantifies the metric-topological influence of boundary components to compact surfaces with boundary. Based on a discrete Hodge-Morrey-Friedrichs decomposition for piecewise constant vector fields on simplicial surfaces with boundary, the discrete Poincaré duality angles reflect a deep linkage between metric properties of the spaces of discrete harmonic Dirichlet and Neumann fields and the topology of the underlying surface, and may act as a new kind of shape signature. We provide an algorithm for the computation of these angles and discuss them on several exemplary surface models.

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Acknowledgements

The authors thank the anonymous reviewers for their detailed and valuable feedback. This work was supported by the Einstein Center for Mathematics Berlin. The Laurent’s hand model on which the hand model in Fig. 2 is based on is provided courtesy of inria by the aim@shape-visionair Shape Repository.

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Authors and Affiliations

  1. Freie Universität Berlin, Berlin, Germany

    Konstantin Poelke & Konrad Polthier

Authors
  1. Konstantin Poelke
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  2. Konrad Polthier
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Corresponding author

Correspondence to Konstantin Poelke .

Editor information

Editors and Affiliations

  1. University of Leeds, Leeds, UK

    Hamish Carr

  2. Keio University, Yokohama, Kanagawa, Japan

    Issei Fujishiro

  3. Heidelberg University, IWR, Heidelberg, Germany

    Filip Sadlo

  4. University of Aizu, Aizu-Wakamatsu City, Fukushima, Japan

    Shigeo Takahashi

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Poelke, K., Polthier, K. (2020). Discrete Poincaré Duality Angles as Shape Signatures on Simplicial Surfaces with Boundary. In: Carr, H., Fujishiro, I., Sadlo, F., Takahashi, S. (eds) Topological Methods in Data Analysis and Visualization V. TopoInVis 2017. Mathematics and Visualization. Springer, Cham. https://doi.org/10.1007/978-3-030-43036-8_16

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