Document Open Access Logo

SCARST: Schnyder Compact and Regularity Sensitive Triangulation Data Structure

Authors Luca Castelli Aleardi , Olivier Devillers

PDF
Thumbnail PDF

File

LIPIcs.SoCG.2024.32.pdf
  • Filesize: 2.59 MB
  • 19 pages

Document Identifiers

Author Details

Luca Castelli Aleardi
  • LIX, Ecole Polytechnique, Institut Polytechnique de Paris, Palaiseau, France
Olivier Devillers
  • Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France

Funding

  • Castelli Aleardi, Luca: This work is supported by ANR grant “3DMaps” ANR-20-CE48-0018.

Cite AsGet BibTex

Luca Castelli Aleardi and Olivier Devillers. SCARST: Schnyder Compact and Regularity Sensitive Triangulation Data Structure. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 32:1-32:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SoCG.2024.32

Abstract

We consider the design of fast and compact representations of the connectivity information of triangle meshes. Although traditional data structures (Half-Edge, Corner Table) are fast and user-friendly, they tend to be memory-expensive. On the other hand, compression schemes, while meeting information-theoretic lower bounds, do not support navigation within the mesh structure. Compact representations provide an advantageous balance for representing large meshes, enabling a judicious compromise between memory consumption and fast implementation of navigational operations. We propose new representations that are sensitive to the regularity of the graph while still having worst case guarantees. For all our data structures we have both an interesting storage cost, typically 2 or 3 r.p.v. (references per vertex) in the case of very regular triangulations, and provable upper bounds in the worst case scenario. One of our solutions has a worst case cost of 3.33 r.p.v., which is currently the best-known bound improving the previous 4 r.p.v. [Castelli et al. 2018]. Our representations have slightly slower running times (factors 1.5 to 4) than classical data structures. In our experiments we compare on various meshes runtime and memory performance of our representations with those of the most efficient existing solutions.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatoric problems
  • Theory of computation → Computational geometry
Keywords
  • Meshes
  • compression
  • triangulations
  • compact representations

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

Supplementary Materials

References

  1. Tyler J Alumbaugh and Xiangmin Jiao. Compact array-based mesh data structures. In Proceedings of the 14th International Meshing Roundtable, pages 485-503. Springer, 2005. URL: https://doi.org/10.1007/3-540-29090-7_29.
  2. Bruce G Baumgart. Winged edge polyhedron representation. Technical report, DTIC Document, 1972. URL: http://www.dtic.mil/cgi-bin/GetTRDoc?Location=U2&doc=GetTRDoc.pdf&AD=AD0755141.
  3. Bruce G Baumgart. A polyhedron representation for computer vision. In Proceedings National Computer Conference and Exposition, pages 589-596. ACM, 1975. URL: https://doi.org/10.1145/1499949.1500071.
  4. Jean-Daniel Boissonnat, Olivier Devillers, Sylvain Pion, Monique Teillaud, and Mariette Yvinec. Triangulations in CGAL. Computational Geometry: Theory & Applications, 22:5-19, 2002. URL: https://doi.org/10.1016/S0925-7721(01)00054-2.
  5. Luca Castelli Aleardi and Olivier Devillers. Array-based compact data structures for triangulations: Practical solutions with theoretical guarantees. J. Comput. Geom., 9(1):247-289, 2018. URL: https://doi.org/10.20382/jocg.v9i1a8.
  6. Luca Castelli Aleardi, Olivier Devillers, and Abdelkrim Mebarki. Catalog based representation of 2d triangulations. International Journal of Computational Geometry & Applications, 21:393-402, 2011. URL: https://doi.org/10.1142/S021819591100372X.
  7. Luca Castelli Aleardi, Olivier Devillers, and Jarek Rossignac. ESQ: editable squad representation for triangle meshes. In 25th Conference on Graphics, Patterns and Images, SIBGRAPI 2012, pages 110-117, 2012. URL: https://doi.org/10.1109/SIBGRAPI.2012.24.
  8. Luca Castelli Aleardi, Olivier Devillers, and Gilles Schaeffer. Succinct representations of planar maps. Theor. Comput. Sci., 408(2-3):174-187, 2008. URL: https://doi.org/10.1016/j.tcs.2008.08.016.
  9. Luca Castelli Aleardi, Éric Fusy, and Thomas Lewiner. Optimal encoding of triangular and quadrangular meshes with fixed topology. In Proceedings of the 22nd Annual Canadian Conference on Computational Geometry, pages 95-98, 2010. URL: http://cccg.ca/proceedings/2010/paper27.pdf.
  10. Markus Denny and Christian Sohler. Encoding a triangulation as a permutation of its point set. In Canadian Conference on Computational Geometry, 1997. URL: https://api.semanticscholar.org/CorpusID:26767430.
  11. Vincent Despré, Daniel Gonçalves, and Benjamin Lévêque. Encoding toroidal triangulations. Discrete & Computational Geometry, 57(3):507-544, 2017. URL: https://doi.org/10.1007/s00454-016-9832-0.
  12. Leo Ferres, José Fuentes-Sepúlveda, Travis Gagie, Meng He, and Gonzalo Navarro. Fast and compact planar embeddings. Comput. Geom., 89:101630, 2020. URL: https://doi.org/10.1016/j.comgeo.2020.101630.
  13. José Fuentes-Sepúlveda, Diego Seco, and Raquel Viaña. Succinct encoding of binary strings representing triangulations. Algorithmica, 83(11):3432-3468, 2021. URL: https://doi.org/10.1007/s00453-021-00861-4.
  14. Leonidas Guibas and Jorge Stolfi. Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams. ACM transactions on graphics (TOG), 4(2):74-123, 1985. URL: https://doi.org/10.1145/282918.282923.
  15. Stefan Gumhold and Wolfgang Straßer. Real time compression of triangle mesh connectivity. In Proc. of SIGGRAPH 1998, pages 133-140, 1998. URL: https://doi.org/10.1145/280814.280836.
  16. Topraj Gurung, Daniel Laney, Peter Lindstrom, and Jarek Rossignac. SQuad: Compact representation for triangle meshes. Computer Graphics Forum, 30(2):355-364, 2011. URL: https://doi.org/10.1111/j.1467-8659.2011.01866.x.
  17. Topraj Gurung, Mark Luffel, Peter Lindstrom, and Jarek Rossignac. LR: compact connectivity representation for triangle meshes. ACM transactions on graphics (TOG), 30(4), 2011. URL: https://doi.org/10.1145/2010324.1964962.
  18. Topraj Gurung, Mark Luffel, Peter Lindstrom, and Jarek Rossignac. Zipper: A compact connectivity data structure for triangle meshes. Computer-Aided Design, 45(2):262-269, 2013. URL: https://doi.org/10.1016/j.cad.2012.10.009.
  19. Topraj Gurung and Jarek Rossignac. SOT: compact representation for tetrahedral meshes. In 2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling, pages 79-88. ACM, 2009. URL: https://doi.org/10.1145/1629255.1629266.
  20. Marcelo Kallmann and Daniel Thalmann. Star-vertices: a compact representation for planar meshes with adjacency information. Journal of Graphics Tools, 6(1):7-18, 2001. URL: https://doi.org/10.1080/10867651.2001.10487533.
  21. Stephen G. Kobourov. Canonical orders and Schnyder realizers. In Encyclopedia of Algorithms, pages 277-283. Springer, 2016. URL: https://doi.org/10.1007/978-1-4939-2864-4_650.
  22. Dominique Poulalhon and Gilles Schaeffer. Optimal coding and sampling of triangulations. Algorithmica, 46(3-4):505-527, 2006. URL: https://doi.org/10.1007/s00453-006-0114-8.
  23. Jarek Rossignac. Edgebreaker: Connectivity compression for triangle meshes. Visualization and Computer Graphics, IEEE Transactions on, 5(1):47-61, 1999. URL: https://doi.org/10.1109/2945.764870.
  24. Walter Schnyder. Embedding planar graphs on the grid. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, volume 90, pages 138-148, 1990. URL: http://departamento.us.es/dma1euita/PAIX/Referencias/schnyder.pdf.
  25. Jack Snoeyink and Bettina Speckmann. Tripod: a minimalist data structure for embedded triangulations. In Workshop on Comput. Graph Theory and Combinatorics, 1999. URL: https://www.win.tue.nl/~speckman/papers/Tripod.pdf.
  26. Andrzej Szymczak, Davis King, and Jarek Rossignac. An edgebreaker-based efficient compression scheme for regular meshes. Comput. Geom., 20(1-2):53-68, 2001. URL: https://doi.org/10.1016/S0925-7721(01)00035-9.
  27. Costa Touma and Craig Gotsman. Triangle mesh compression. In Proc. of the Graphics Interface 1998 Conference, pages 26-34, 1998. URL: https://doi.org/10.20380/GI1998.04.
  28. Katsuhisa Yamanaka and Shin-ichi Nakano. A compact encoding of plane triangulations with efficient query supports. Information Processing Letters, 110(18):803-809, 2010. URL: https://doi.org/10.1016/j.ipl.2010.06.014.
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact