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Minimum-Error Triangulations for Sea Surface Reconstruction

Authors Anna Arutyunova, Anne Driemel, Jan-Henrik Haunert, Herman Haverkort, Jürgen Kusche, Elmar Langetepe, Philip Mayer, Petra Mutzel, Heiko Röglin



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Author Details

Anna Arutyunova
  • Institute for Computer Science, Universität Bonn, Germany
Anne Driemel
  • Hausdorff Center for Mathematics, Universität Bonn, Germany
Jan-Henrik Haunert
  • Institute of Geodesy and Geoinformation, Universität Bonn, Germany
Herman Haverkort
  • Institute for Computer Science, Universität Bonn, Germany
Jürgen Kusche
  • Institute of Geodesy and Geoinformation, Universität Bonn, Germany
Elmar Langetepe
  • Institute for Computer Science, Universität Bonn, Germany
Philip Mayer
  • Institute for Computer Science, Universität Bonn, Germany
Petra Mutzel
  • Institute for Computer Science, Universität Bonn, Germany
Heiko Röglin
  • Institute for Computer Science, Universität Bonn, Germany

Acknowledgements

We thank the anonymous reviewers for their insightful comments and suggestions.

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Anna Arutyunova, Anne Driemel, Jan-Henrik Haunert, Herman Haverkort, Jürgen Kusche, Elmar Langetepe, Philip Mayer, Petra Mutzel, and Heiko Röglin. Minimum-Error Triangulations for Sea Surface Reconstruction. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 7:1-7:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.SoCG.2022.7

Abstract

We apply state-of-the-art computational geometry methods to the problem of reconstructing a time-varying sea surface from tide gauge records. Our work builds on a recent article by Nitzke et al. (Computers & Geosciences, 157:104920, 2021) who have suggested to learn a triangulation D of a given set of tide gauge stations. The objective is to minimize the misfit of the piecewise linear surface induced by D to a reference surface that has been acquired with satellite altimetry. The authors restricted their search to k-order Delaunay (k-OD) triangulations and used an integer linear program in order to solve the resulting optimization problem. In geometric terms, the input to our problem consists of two sets of points in ℝ² with elevations: a set 𝒮 that is to be triangulated, and a set ℛ of reference points. Intuitively, we define the error of a triangulation as the average vertical distance of a point in ℛ to the triangulated surface that is obtained by interpolating elevations of 𝒮 linearly in each triangle. Our goal is to find the triangulation of 𝒮 that has minimum error with respect to ℛ. In our work, we prove that the minimum-error triangulation problem is NP-hard and cannot be approximated within any multiplicative factor in polynomial time unless P = NP. At the same time we show that the problem instances that occur in our application (considering sea level data from several hundreds of tide gauge stations worldwide) can be solved relatively fast using dynamic programming when restricted to k-OD triangulations for k ≤ 7. In particular, instances for which the number of connected components of the so-called k-OD fixed-edge graph is small can be solved within few seconds.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Minimum-Error Triangulation
  • k-Order Delaunay Triangulations
  • Data dependent Triangulations
  • Sea Surface Reconstruction
  • fixed-Edge Graph

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