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Towards Topological Analysis of Non-symmetric Tensor Fields via Complexification

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Discrete Geometry and Mathematical Morphology (DGMM 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13493))

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Abstract

Fields of asymmetric tensors play an important role in many applications such as medical imaging (diffusion tensor magnetic resonance imaging), physics, and civil engineering (for example Cauchy-Green-deformation tensor, strain tensor with local rotations, etc.). However, such asymmetric tensors are usually symmetrized and then further processed. Using this procedure results in a loss of information. A new method for the processing of asymmetric tensor fields is proposed restricting our attention to tensors of second-order given by a \(2\times 2\) array or matrix with real entries. This is achieved by a transformation resulting in Hermitian matrices that have an eigendecomposition similar to symmetric matrices. With this new idea numerical results for real-world data arising from a deformation of an object by external forces are given. It is shown that the asymmetric part indeed contains valuable information.

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Author information

Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Saarland University, 66041, Saarbrücken, Germany

    Bernhard Burgeth

  2. Jülich Supercomputing Centre, Forschungszentrum Jülich GmbH, Wilhelm-Johnen-Straße, 52425, Jülich, Germany

    Andreas Kleefeld

  3. Faculty of Medical Engineering and Technomathematics, University of Applied Sciences Aachen, Heinrich-Mußmann-Str. 1, 52428, Jülich, Germany

    Andreas Kleefeld

  4. School of Electrical Engineering and Computer Science, Oregon State University, Corvallis, OR, 97331, USA

    Eugene Zhang & Yue Zhang

Authors
  1. Bernhard Burgeth
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  2. Andreas Kleefeld
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  3. Eugene Zhang
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  4. Yue Zhang
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Corresponding author

Correspondence to Andreas Kleefeld .

Editor information

Editors and Affiliations

  1. University of Strasbourg, Strasbourg, France

    Étienne Baudrier

  2. University of Strasbourg, Strasbourg, France

    Benoît Naegel

  3. University of Strasbourg, Strasbourg, France

    Adrien Krähenbühl

  4. University of Strasbourg, Strasbourg, France

    Mohamed Tajine

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Burgeth, B., Kleefeld, A., Zhang, E., Zhang, Y. (2022). Towards Topological Analysis of Non-symmetric Tensor Fields via Complexification. In: Baudrier, É., Naegel, B., Krähenbühl, A., Tajine, M. (eds) Discrete Geometry and Mathematical Morphology. DGMM 2022. Lecture Notes in Computer Science, vol 13493. Springer, Cham. https://doi.org/10.1007/978-3-031-19897-7_5

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  • DOI: https://doi.org/10.1007/978-3-031-19897-7_5

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-19896-0

  • Online ISBN: 978-3-031-19897-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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