Document Open Access Logo

Multilevel Skeletonization Using Local Separators

Authors J. Andreas Bærentzen , Rasmus Emil Christensen, Emil Toftegaard Gæde , Eva Rotenberg

PDF
Thumbnail PDF

File

LIPIcs.SoCG.2023.13.pdf
  • Filesize: 6.51 MB
  • 18 pages

Document Identifiers

Author Details

J. Andreas Bærentzen
  • Department of Applied Mathematics and Computer Science, Technical University of Denmark, Lyngby, Denmark
Rasmus Emil Christensen
  • Department of Applied Mathematics and Computer Science, Technical University of Denmark, Lyngby, Denmark
Emil Toftegaard Gæde
  • Department of Applied Mathematics and Computer Science, Technical University of Denmark, Lyngby, Denmark
Eva Rotenberg
  • Department of Applied Mathematics and Computer Science, Technical University of Denmark, Lyngby, Denmark

Funding

Emil Toftegaard Gæde and Eva Rotenberg: supported by Eva Rotenberg’s Carlsberg Foundation Young Researcher Fellowship CF21-0302 - "Graph Algorithms with Geometric Applications".
  • Bærentzen, J. Andreas: Partially supported by the Villum Foundation through Villum Investigator Project InnoTop.

Cite AsGet BibTex

J. Andreas Bærentzen, Rasmus Emil Christensen, Emil Toftegaard Gæde, and Eva Rotenberg. Multilevel Skeletonization Using Local Separators. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 13:1-13:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.SoCG.2023.13

Abstract

In this paper we give a new, efficient algorithm for computing curve skeletons, based on local separators. Our efficiency stems from a multilevel approach, where we solve small problems across levels of detail and combine these in order to quickly obtain a skeleton. We do this in a highly modular fashion, ensuring complete flexibility in adapting the algorithm for specific types of input or for otherwise targeting specific applications. Separator based skeletonization was first proposed by Bærentzen and Rotenberg in [ACM Tran. Graphics'21], showing high quality output at the cost of running times which become prohibitive for large inputs. Our new approach retains the high quality output, and applicability to any spatially embedded graph, while being orders of magnitude faster for all practical purposes. We test our skeletonization algorithm for efficiency and quality in practice, comparing it to local separator skeletonization on the University of Groningen Skeletonization Benchmark [Telea'16].

Subject Classification

ACM Subject Classification
  • Computing methodologies → Computer graphics
  • Theory of computation → Computational geometry
  • Software and its engineering → Software design engineering
Keywords
  • Algorithm engineering
  • experimentation and implementation
  • shape skeletonization
  • curve skeletons
  • multilevel algorithm

Metrics

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

Related Versions

Supplementary Materials

References

  1. Amine Abou-Rjeili and George Karypis. Multilevel algorithms for partitioning power-law graphs. In 20th International Parallel and Distributed Processing Symposium (IPDPS 2006), Proceedings, 25-29 April 2006, Rhodes Island, Greece. IEEE, 2006. URL: https://doi.org/10.1109/IPDPS.2006.1639360.
  2. Nina Amenta, Sunghee Choi, and Ravi Krishna Kolluri. The power crust. In David C. Anderson and Kunwoo Lee, editors, Sixth ACM Symposium on Solid Modeling and Applications, Sheraton Inn, Ann Arbor, Michigan, USA, June 4-8, 2001, pages 249-266. ACM, 2001. URL: https://doi.org/10.1145/376957.376986.
  3. Kasra Arnavaz, Oswin Krause, Kilian Zepf, Jakob Andreas Bærentzen, Jelena M. Krivokapic, Silja Heilmann, Pia Nyeng, and Aasa Feragen. Quantifying topology in pancreatic tubular networks from live imaging 3d microscopy. Machine Learning for Biomedical Imaging, 1, 2022. URL: https://melba-journal.org/papers/2022:015.html.
  4. Oscar Kin-Chung Au, Chiew-Lan Tai, Hung-Kuo Chu, Daniel Cohen-Or, and Tong-Yee Lee. Skeleton extraction by mesh contraction. ACM Trans. Graph., 27(3):44, 2008. URL: https://doi.org/10.1145/1360612.1360643.
  5. Andreas Bærentzen and Eva Rotenberg. Skeletonization via local separators. ACM Trans. Graph., 40(5):187:1-187:18, 2021. URL: https://doi.org/10.1145/3459233.
  6. Silvia Biasotti, Leila De Floriani, Bianca Falcidieno, Patrizio Frosini, Daniela Giorgi, Claudia Landi, Laura Papaleo, and Michela Spagnuolo. Describing shapes by geometrical-topological properties of real functions. ACM Comput. Surv., 40(4):12:1-12:87, 2008. URL: https://doi.org/10.1145/1391729.1391731.
  7. Silvia Biasotti, Daniela Giorgi, Michela Spagnuolo, and Bianca Falcidieno. Reeb graphs for shape analysis and applications. Theor. Comput. Sci., 392(1-3):5-22, 2008. URL: https://doi.org/10.1016/j.tcs.2007.10.018.
  8. Angela Brennecke and Tobias Isenberg. 3d shape matching using skeleton graphs. In Thomas Schulze, Stefan Schlechtweg, and Volkmar Hinz, editors, Simulation und Visualisierung 2004 (SimVis 2004) 4-5 März 2004, Magdeburg, pages 299-310. SCS Publishing House e.V., 2004. URL: http://wwwisg.cs.uni-magdeburg.de/graphik/pub/files/Brennecke_2004_3SM.pdf.
  9. Aydin Buluç, Henning Meyerhenke, Ilya Safro, Peter Sanders, and Christian Schulz. Recent advances in graph partitioning. In Lasse Kliemann and Peter Sanders, editors, Algorithm Engineering - Selected Results and Surveys, volume 9220 of Lecture Notes in Computer Science, pages 117-158. Springer International Publishing, 2016. URL: https://doi.org/10.1007/978-3-319-49487-6_4.
  10. J. Andreas Bærentzen. Gel. https://github.com/janba/GEL, 2022.
  11. Junjie Cao, Andrea Tagliasacchi, Matt Olson, Hao Zhang, and Zhixun Su. Point cloud skeletons via laplacian based contraction. In SMI 2010, Shape Modeling International Conference, Aix en Provence, France, June 21-23 2010, pages 187-197. IEEE Computer Society, 2010. URL: https://doi.org/10.1109/SMI.2010.25.
  12. Nicu D. Cornea, Deborah Silver, and Patrick Min. Curve-skeleton properties, applications, and algorithms. IEEE Trans. Vis. Comput. Graph., 13(3):530-548, 2007. URL: https://doi.org/10.1109/TVCG.2007.1002.
  13. Tamal K. Dey and Jian Sun. Defining and computing curve-skeletons with medial geodesic function. In Alla Sheffer and Konrad Polthier, editors, Proceedings of the Fourth Eurographics Symposium on Geometry Processing, Cagliari, Sardinia, Italy, June 26-28, 2006, volume 256 of ACM International Conference Proceeding Series, pages 143-152. Eurographics Association, 2006. URL: https://doi.org/10.2312/SGP/SGP06/143-152.
  14. William Harvey, Yusu Wang, and Rephael Wenger. A randomized O(m log m) time algorithm for computing reeb graphs of arbitrary simplicial complexes. In David G. Kirkpatrick and Joseph S. B. Mitchell, editors, Proceedings of the 26th ACM Symposium on Computational Geometry, Snowbird, Utah, USA, June 13-16, 2010, pages 267-276. ACM, 2010. URL: https://doi.org/10.1145/1810959.1811005.
  15. M. Sabry Hassouna and Aly A. Farag. Variational curve skeletons using gradient vector flow. IEEE Trans. Pattern Anal. Mach. Intell., 31(12):2257-2274, 2009. URL: https://doi.org/10.1109/TPAMI.2008.271.
  16. Bruce Hendrickson and Robert W. Leland. A multi-level algorithm for partitioning graphs. In Sidney Karin, editor, Proceedings Supercomputing '95, San Diego, CA, USA, December 4-8, 1995, page 28. ACM, 1995. URL: https://doi.org/10.1145/224170.224228.
  17. Jacob Holm, Kristian de Lichtenberg, and Mikkel Thorup. Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity. J. ACM, 48(4):723-760, 2001. URL: https://doi.org/10.1145/502090.502095.
  18. Hui Huang, Shihao Wu, Daniel Cohen-Or, Minglun Gong, Hao Zhang, Guiqing Li, and Baoquan Chen. L_1-medial skeleton of point cloud. ACM Trans. Graph., 32(4):65:1-65:8, 2013. URL: https://doi.org/10.1145/2461912.2461913.
  19. Andrei C. Jalba, Jacek Kustra, and Alexandru C. Telea. Surface and curve skeletonization of large 3d models on the GPU. IEEE Trans. Pattern Anal. Mach. Intell., 35(6):1495-1508, 2013. URL: https://doi.org/10.1109/TPAMI.2012.212.
  20. Wei Jiang, Kai Xu, Zhi-Quan Cheng, Ralph R. Martin, and Gang Dang. Curve skeleton extraction by coupled graph contraction and surface clustering. Graph. Model., 75(3):137-148, 2013. URL: https://doi.org/10.1016/j.gmod.2012.10.005.
  21. George Karypis and Vipin Kumar. Analysis of multilevel graph partitioning. In Sidney Karin, editor, Proceedings Supercomputing '95, San Diego, CA, USA, December 4-8, 1995, page 29. ACM, 1995. URL: https://doi.org/10.1145/224170.224229.
  22. Mattia Natali, Silvia Biasotti, Giuseppe Patanè, and Bianca Falcidieno. Graph-based representations of point clouds. Graph. Model., 73(5):151-164, 2011. URL: https://doi.org/10.1016/j.gmod.2011.03.002.
  23. Valerio Pascucci, Giorgio Scorzelli, Peer-Timo Bremer, and Ajith Mascarenhas. Robust on-line computation of reeb graphs: simplicity and speed. ACM Trans. Graph., 26(3):58, 2007. URL: https://doi.org/10.1145/1276377.1276449.
  24. Diane Perchet, Catalin I. Fetita, and Françoise J. Prêteux. Advanced navigation tools for virtual bronchoscopy. In Edward R. Dougherty, Jaakko Astola, and Karen O. Egiazarian, editors, Image Processing: Algorithms and Systems III, San Jose, California, USA, January 18, 2004, volume 5298 of SPIE Proceedings, pages 147-158. SPIE, 2004. URL: https://doi.org/10.1117/12.533096.
  25. Dennie Reniers, Jarke J. van Wijk, and Alexandru C. Telea. Computing multiscale curve and surface skeletons of genus 0 shapes using a global importance measure. IEEE Trans. Vis. Comput. Graph., 14(2):355-368, 2008. URL: https://doi.org/10.1109/TVCG.2008.23.
  26. Andrea Tagliasacchi, Ibraheem Alhashim, Matt Olson, and Hao Zhang. Mean curvature skeletons. Comput. Graph. Forum, 31(5):1735-1744, 2012. URL: https://doi.org/10.1111/j.1467-8659.2012.03178.x.
  27. Andrea Tagliasacchi, Thomas Delamé, Michela Spagnuolo, Nina Amenta, and Alexandru C. Telea. 3d skeletons: A state-of-the-art report. Comput. Graph. Forum, 35(2):573-597, 2016. URL: https://doi.org/10.1111/cgf.12865.
  28. Alexandru C. Telea. 3d skeletonization benchmark. https://webspace.science.uu.nl/~telea001/Shapes/SkelBenchmark, 2016. Accessed: 2022-10-31.
  29. Alexandru C. Telea and Andrei C. Jalba. Computing curve skeletons from medial surfaces of 3d shapes. In Hamish A. Carr and Silvester Czanner, editors, Theory and Practice of Computer Graphics, Rutherford, United Kingdom, 2012. Proceedings, pages 99-106. Eurographics Association, 2012. URL: https://doi.org/10.2312/LocalChapterEvents/TPCG/TPCG12/099-106.
  30. Ming Wan, Frank Dachille, and Arie E. Kaufman. Distance-field-based skeletons for virtual navigation. In Thomas Ertl, Kenneth I. Joy, and Amitabh Varshney, editors, 12th IEEE Visualization Conference, IEEE Vis 2001, San Diego, CA, USA, October 24-26, 2001, Proceedings, pages 239-246. IEEE Computer Society, 2001. URL: https://doi.org/10.1109/VISUAL.2001.964517.
  31. Yu-Shuen Wang and Tong-Yee Lee. Curve-skeleton extraction using iterative least squares optimization. IEEE Trans. Vis. Comput. Graph., 14(4):926-936, 2008. URL: https://doi.org/10.1109/TVCG.2008.38.
  32. Yajie Yan, Kyle Sykes, Erin W. Chambers, David Letscher, and Tao Ju. Erosion thickness on medial axes of 3d shapes. ACM Trans. Graph., 35(4):38:1-38:12, 2016. URL: https://doi.org/10.1145/2897824.2925938.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact