Document

# Minimum-Error Triangulations for Sea Surface Reconstruction

## File

LIPIcs.SoCG.2022.7.pdf
• Filesize: 1.88 MB
• 18 pages

## Acknowledgements

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

## Cite As

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

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. M. Ablain, A. Cazenave, G. Larnicol, M. Balmaseda, P. Cipollini, Y. Faugère, M. J. Fernandes, O. Henry, J. A. Johannessen, P. Knudsen, O. Andersen, J. Legeais, B. Meyssignac, N. Picot, M. Roca, S. Rudenko, M. G. Scharffenberg, D. Stammer, G. Timms, and J. Benveniste. Improved sea level record over the satellite altimetry era (1993–2010) from the Climate Change Initiative project. Ocean Science, 11(1):67-82, 2015. URL: https://doi.org/10.5194/os-11-67-2015.
2. Pankaj K. Agarwal and Subhash Suri. Surface approximation and geometric partitions. In Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '94, pages 24-33, USA, 1994. Society for Industrial and Applied Mathematics.
3. Lyuba Alboul, Gertjan Kloosterman, Cornelis Traas, and Ruud van Damme. Best data-dependent triangulations. Journal of Computational and Applied Mathematics, 119(1):1-12, 2000. URL: https://doi.org/10.1016/S0377-0427(00)00368-X.
4. Efthymios Anagnostou and Derek Corneil. Polynomial-time instances of the minimum weight triangulation problem. Computational Geometry, 3(5):247-259, 1993. URL: https://doi.org/10.1016/0925-7721(93)90016-Y.
5. 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. 2022. URL: http://arxiv.org/abs/2203.07325v1.
6. Marshall Bern and David Eppstein. Mesh generation and optimal triangulation. In Computing in Euclidean Geometry, 1992. URL: https://doi.org/10.1142/9789814355858_0002.
7. Magdalene Borgelt, Christian Borgelt, and Christos Levcopoulos. Fixed parameter algorithms for the minimum weight triangulation problem. Int. J. Comput. Geometry Appl., 18:185-220, June 2008. URL: https://doi.org/10.1142/S0218195908002581.
8. Jeffrey L. Brown. Vertex based data dependent triangulations. Computer Aided Geometric Design, 8(3):239-251, 1991. URL: https://doi.org/10.1016/0167-8396(91)90008-Y.
9. Siu-Wing Cheng, Mordecai J. Golin, and Jeffrey Tsang. Expected case analysis of 221-skeletons with applications to the construction of minimum-weight triangulations. Master’s thesis, Hong Kong University of Science and Technology, 1995.
10. John A. Church, Neil J. White, Richard Coleman, Kurt Lambeck, and Jerry X. Mitrovica. Estimates of the Regional Distribution of Sea Level Rise over the 1950-2000 Period. Journal of Climate, 17(13):2609-2625, July 2004. URL: https://doi.org/10.1175/1520-0442(2004)017<2609:EOTRDO>2.0.CO;2.
11. Thierry de Kok, Marc van Kreveld, and Maarten Löffler. Generating realistic terrains with higher-order Delaunay triangulations. Computational Geometry, 36(1):52-65, 2007. Special Issue on the 21st European Workshop on Computational Geometry. URL: https://doi.org/10.1016/j.comgeo.2005.09.005.
12. Nira Dyn, David Levin, and Samuel Rippa. Data Dependent Triangulations for Piecewise Linear Interpolation. IMA Journal of Numerical Analysis, 10(1):137-154, January 1990. URL: https://doi.org/10.1093/imanum/10.1.137.
13. ESA. Sea level CCI ECV dataset: Time series of gridded sea level anomalies(sla), 2021. European Space Agency (ESA). URL: https://catalogue.ceda.ac.uk/uuid/142052b9dc754f6da47a631e35ec4609.
14. S.P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, and C. Scheffer. On hard instances of the minimum-weight triangulation problem. In 36th European Workshop on Computational Geometry (EuroCG 2020), March 2020.
15. P. Gilbert. New results on planar triangulations. Master’s thesis, University of Illinois, Coordinated Science Lab, Urbana, IL, USA, 1979.
16. Joachim Gudmundsson, Mikael Hammar, and Marc van Kreveld. Higher order Delaunay triangulations. Computational Geometry, 23(1):85-98, 2002. URL: https://doi.org/10.1016/S0925-7721(01)00027-X.
17. Andreas Haas. Solving large-scale minimum-weight triangulation instances to provable optimality. In Bettina Speckmann and Csaba D. Tóth, editors, 34th International Symposium on Computational Geometry, SoCG 2018, volume 99 of LIPIcs, pages 44:1-44:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.SoCG.2018.44.
18. Simon J. Holgate, Andrew Matthews, Philip L. Woodworth, Lesley J. Rickards, Mark E. Tamisiea, Elizabeth Bradshaw, Peter R. Foden, Kathleen M. Gordon, Svetlana Jevrejeva, and Jeff Pugh. New Data Systems and Products at the Permanent Service for Mean Sea Level. Journal of Coastal Research, 29(3):493-504, December 2012. URL: https://doi.org/10.2112/JCOASTRES-D-12-00175.1.
19. G.T. Klincsek. Minimal triangulations of polygonal domains. In Peter L. Hammer, editor, Combinatorics 79, volume 9 of Annals of Discrete Mathematics, pages 121-123. Elsevier, 1980. URL: https://doi.org/10.1016/S0167-5060(08)70044-X.
20. Donald E. Knuth and Arvind Raghunathan. The problem of compatible representatives. SIAM Journal on Discrete Mathematics, 5(3):422-427, 1992.
21. Charles L. Lawson. Software for C¹ surface interpolation. In John R. Rice, editor, Mathematical Software, pages 161-194. Academic Press, 1977. URL: https://doi.org/10.1016/B978-0-12-587260-7.50011-X.
22. David Lichtenstein. Planar formulae and their uses. SIAM Journal on Computing, 11:329-343, 1982.
23. Wolfgang Mulzer and Günter Rote. Minimum-weight triangulation is NP-hard. Journal of the ACM, 55(2):1-29, May 2008. URL: https://doi.org/10.1145/1346330.1346336.
24. Alina Nitzke, Benjamin Niedermann, Luciana Fenoglio-Marc, Jürgen Kusche, and Jan-Henrik Haunert. Reconstructing the dynamic sea surface from tide gauge records using optimal data-dependent triangulations. Computers & Geosciences, 157:104920, 2021. URL: https://doi.org/10.1016/j.cageo.2021.104920.
25. Marco Olivieri and Giorgio Spada. Spatial sea-level reconstruction in the Baltic Sea and in the Pacific Ocean from tide gauges observations. Annals of Geophysics, 59(3), 2016. URL: https://doi.org/10.4401/ag-6966.
26. Permanent service for mean sea level (PSMSL), 2021. Retrieved 19 Apr 2021 from URL: http://www.psmsl.org/data/obtaining/.
27. Natalia Rodríguez and Rodrigo I. Silveira. Implementing data-dependent triangulations with higher order Delaunay triangulations. ISPRS International Journal of Geo-Information, 6(12), 2017. URL: https://doi.org/10.3390/ijgi6120390.
28. Rodrigo I. Silveira and Marc van Kreveld. Optimal higher order Delaunay triangulations of polygons. Computational Geometry, 42(8):803-813, 2009. Special Issue on the 23rd European Workshop on Computational Geometry. URL: https://doi.org/10.1016/j.comgeo.2008.02.006.
29. Kai Wang, Chor-Pang Lo, George A. Brook, and Hamid R. Arabnia. Comparison of existing triangulation methods for regularly and irregularly spaced height fields. International Journal of Geographical Information Science, 15(8):743-762, 2001. URL: https://doi.org/10.1080/13658810110074492.
X

Feedback for Dagstuhl Publishing

### Thanks for your feedback!

Feedback submitted

### Could not send message

Please try again later or send an E-mail