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Tropical Geometry, Mathematical Morphology and Weighted Lattices

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Mathematical Morphology and Its Applications to Signal and Image Processing (ISMM 2019)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 11564))

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Abstract

Mathematical Morphology and Tropical Geometry share the same max/min-plus scalar arithmetic and matrix algebra. In this paper we summarize their common ideas and algebraic structure, generalize and extend both of them using weighted lattices and a max-\(\star \) algebra with an arbitrary binary operation \(\star \) that distributes over max, and outline applications to geometry, image analysis, and optimization. Further, we outline the optimal solution of max-\(\star \) equations using weighted lattice adjunctions, and apply it to optimal regression for fitting max-\(\star \) tropical curves on arbitrary data.

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Acknowledgements

I wish to thank V. Charisopoulos and E. Theodosis for insightful discussions and help with the figures. This work was co-financed by the European Regional Development Fund of the EU and Greek national funds through the Operational Program Competitiveness, Entrepreneurship and Innovation, under the call ‘Research – Create – Innovate’ (T1EDK-01248, “i-Walk”).

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Authors and Affiliations

  1. National Technical University of Athens, Athens, Greece

    Petros Maragos

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  1. Petros Maragos
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Correspondence to Petros Maragos .

Editor information

Editors and Affiliations

  1. Universität des Saarlandes, Saarbrücken, Germany

    Bernhard Burgeth

  2. Forschungszentrum Jülich GmbH, Jülich, Germany

    Andreas Kleefeld

  3. Université de Strasbourg, Illkirch Cedex, France

    Benoît Naegel

  4. Université de Reims Champagne-Ardenne, Reims Cedex 2, France

    Nicolas Passat

  5. Université Paris-Est, Noisy le Grand Cedex, France

    Benjamin Perret

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Maragos, P. (2019). Tropical Geometry, Mathematical Morphology and Weighted Lattices. In: Burgeth, B., Kleefeld, A., Naegel, B., Passat, N., Perret, B. (eds) Mathematical Morphology and Its Applications to Signal and Image Processing. ISMM 2019. Lecture Notes in Computer Science(), vol 11564. Springer, Cham. https://doi.org/10.1007/978-3-030-20867-7_1

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  • DOI: https://doi.org/10.1007/978-3-030-20867-7_1

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-20866-0

  • Online ISBN: 978-3-030-20867-7

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