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Representing Simplicial Complexes with Mangroves

  • Conference paper
Proceedings of the 22nd International Meshing Roundtable

Summary

Simplicial complexes are extensively used for discretizing digital shapes in two, three, and higher dimensions within a variety of application domains. There have been many proposals of topological data structures, which represent the connectivity information among simplices. We introduce the Mangrove Topological Data Structure (Mangrove TDS) framework, a tool which supports the efficient implementation of data structures for simplicial complexes of any dimension under the same application interface. Our framework is based on a graph-based representation of connectivity relations, that we call the mangrove. It can be customized in order to simulate the content of any topological data structure with a negligible overhead. Thus, the Mangrove TDS framework is extensible, and supports the most diverse modeling needs. We also provide implicit representations of those simplices, which are not directly encoded in a specific topological data structure. Our tests show that these representations, that we call ghost simplices, improve the expressive power and the efficiency of topological queries. In order to prove the validity of our approach, we design two topological data structures, specific for non-manifold complexes, within our framework. We perform comparisons with some widely-used representations in the literature as well as with libraries available in the public domain.

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References

  1. Alumbaugh, T.J., Jiao, X.: Compact Array-based Mesh Data Structures. In: Proc. of the 14th Int. Mesh. Rnd., pp. 485–503 (2005)

    Google Scholar 

  2. Arvind, V., Bireswar, D., Kobler, J.: The Space Complexity of k-Tree Isomorphism. In: Proc. of the 18th Int. Conf. on Alg. and Comp. (2007)

    Google Scholar 

  3. Attali, D., Lieutier, A., Salinas, D.: Efficient Data Structure for Simplifying Simplicial Complexes in High Dimensions. Int. Journal of Comp. Geom. and Appl. 22(4), 279–303 (2012)

    Article  MathSciNet  Google Scholar 

  4. Blandford, D., Blelloch, G., Cardoze, D., Kadow, C.: Compact Representations of Simplicial Meshes in Two and Three Dimensions. In: Proc. of the 12th Int. Mesh. Rnd., pp. 135–146 (2003)

    Google Scholar 

  5. Boissonnat, J.-D., Maria, C.: The simplex tree: An Efficient Data Structure for General Simplicial Complexes. In: Epstein, L., Ferragina, P. (eds.) ESA 2012. LNCS, vol. 7501, pp. 731–742. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  6. Botsch, M., Steinberg, S., Bischoff, S., Kobbelt, L.: Open-Mesh: a Generic and Efficient Polygon Mesh Data Structure. In: Proc. of the OpenSG Symp. (2002)

    Google Scholar 

  7. Brewer, M., Freitag, L., Knupp, P.M., Leurent, T., Melander, D.: The Mesquite Mesh Quality Improvement Toolkit. In: Proc. of the 12th Int. Mesh. Rnd., pp. 239–250 (2003)

    Google Scholar 

  8. Brisson, E.: Representing Geometric Structures in d-dimensions: Topology and Order. In: Proc. of the Symp. on Comp. Geom., pp. 218–227 (1989)

    Google Scholar 

  9. Campagna, S., Kobbelt, L., Seidel, H.-P.: Directed Edges: a Scalable Representation for Triangle Meshes. Jour. of Graph. Tools 3(4) (1998)

    Google Scholar 

  10. Canino, D.: Tools for Modeling and Analysis of Non-Manifold Shapes. PhD thesis, Department of Computer Science, University of Genova, Italy (2012)

    Google Scholar 

  11. Canino, D., De Floriani, L., Weiss, K.: IA*: an Adjacency-based Representation for Non-Manifold Simplicial Shapes in Arbitrary Dimensions. Comp. & Graph. 35(3), 747–753 (2011)

    Article  Google Scholar 

  12. Cardoze, D., Miller, G., Phillips, T.: Representing Topological Data Structures using Cell-Chains. In: Proc. of the 4th Int. Conf. on Geom. Mod. and Proc., pp. 248–266 (2006)

    Google Scholar 

  13. Cazier, D., Kraemer, P.: X-Maps: an Efficient Model for Non-manifold Modeling. In: Proc. of the Shape Mod. Int. Conf., pp. 226–230 (2010)

    Google Scholar 

  14. Computational Geometry Algorithms Library (2011), http://www.cgal.org

  15. Čomić, L., De Floriani, L., Iuricich, F.: Multi-resolution Cell Complexes based on Homology-Preserving Euler Operators. In: Gonzalez-Diaz, R., Jimenez, M.-J., Medrano, B. (eds.) DGCI 2013. LNCS, vol. 7749, pp. 323–334. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  16. De Floriani, L., Greenfieldboyce, D., Hui, A.: A Data Structure for Non-mani-fold Simplicial d-complexes. In: Proc. of the Symp. on Geom. Proc., pp. 83–92 (2004)

    Google Scholar 

  17. De Floriani, L., Hui, A.: A Scalable Data Structure for Three-dimensional Non-manifold Objects. In: Proc. of the Symp. on Geom. Proc., pp. 72–82 (2003)

    Google Scholar 

  18. De Floriani, L., Hui, A.: Data Structures for Simplicial Complexes: an Analysis and a Comparison. In: Proc. of the Symp. on Geom. Proc., pp. 119–128 (2005)

    Google Scholar 

  19. De Floriani, L., Hui, A., Panozzo, D., Canino, D.: A Dimension-independent Data Structure for Simplicial Complexes. In: Proc. of the 19th Int. Mesh. Rnd., pp. 403–420 (2010)

    Google Scholar 

  20. De Floriani, L., Magillo, P., Puppo, E., Sobrero, D.: A Multi-resolution Topological Representation for Non-manifold Meshes. CAD Jour. 36(2), 141–159 (2004)

    Google Scholar 

  21. Di Carlo, A., Milicchio, F., Paoluzzi, A., Shapiro, V.: Solid and Physical Modeling with Chain Complexes. In: Proc. of the Symp. on Sol. and Phys. Mod., pp. 73–84 (2007)

    Google Scholar 

  22. Dobkin, D., Laszlo, M.: Primitives for the Manipulation of Three-dimensional Subdivision. Algor. 5(4), 3–32 (1989)

    Article  MathSciNet  Google Scholar 

  23. Ebeida, M.S., Knupp, P.M.: LBMD: a Layer-based Mesh Data Structure tailored for Generic Implementations. In: Proc. of the 20th AIAA Comp. Fluid Dyn. Conf. (2011)

    Google Scholar 

  24. Edelsbrunner, H.: Algorithms in Combinatorial Geometry. Springer (1987)

    Google Scholar 

  25. Garimella, R.V.: Mesh Data Structure Selection for Mesh Generation and FEA Applications. Int. Jour. of Num. Meth. in Eng. 55(4), 451–478 (2002)

    Article  MATH  Google Scholar 

  26. Non-Manifold Meshes Repository. Department of Computer Science, Genova, Italy (2009), http://indy.disi.unige.it/nmcollection

  27. Gurung, T., Laney, D., Lindstrom, P., Rossignac, J.: SQuad: a Compact Representation for Triangle Meshes. Comp. Graph. For. 30(2), 355–364 (2011)

    Google Scholar 

  28. Gurung, T., Luffel, M., Lindstrom, P., Rossignac, J.: LR: Compact Connectivity Representation for Triangle Meshes. ACM Trans. on Graph. 30(4) (2011)

    Google Scholar 

  29. Gurung, T., Luffel, M., Lindstrom, P., Rossignac, J.: Zipper: a Compact Connectivity Data Structure for Triangle Meshes. Comp.-Aid. Des. 45(2), 262–269 (2013)

    Article  Google Scholar 

  30. Gurung, T., Rossignac, J.: SOT: Compact Representation for Tetrahedral Meshes. In: Proc. of the Symp. on Sol. and Phys. Mod., pp. 79–88 (2009)

    Google Scholar 

  31. Kettner, L.: Using Generic Programming for Designing a Data Structure for Polyhedral Surfaces. Comp. Geom. - Theory and Appl. 1(13), 65–90 (1999)

    Article  Google Scholar 

  32. Kremer, M., Bommes, D., Kobbelt, L.: OpenVolumeMesh – A Versatile Index-based Data Structure for 3D Polytopal Complexes. In: Jiao, X., Weill, J.-C. (eds.) Proceedings of the 21st International Meshing Roundtable, vol. 123, pp. 531–548. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  33. Lage, M., Lewiner, T., Lopes, H., Velho, L.: CHF: a Scalable Topological Data Structure for Tetrahedral Meshes. In: Proc. of the 18th Braz. Symp. on Comp. Grap. and Img. Proc., pp. 349–356 (2005)

    Google Scholar 

  34. Ledoux, F., Weill, J.-C., Bertrand, Y.: GDMS: a Generic Mesh Data Structure. In: Proc. of the 17th Int. Mesh. Rnd. (2008)

    Google Scholar 

  35. Lee, S., Lee, K.: The Partial-Entity Structure: a Fast and Compact Non-manifold Boundary Representation based on Partial Topological Entities. In: Proc. of the Symp. on Sol. and Phys. Mod., pp. 159–170 (2001)

    Google Scholar 

  36. Lienhardt, P.: N-dimensional Generalized Combinatorial Maps and Cellular Quasi-manifolds. Int. Jour. of Comp. Geom. and Appl. 4(3), 275–324 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  37. Lopes, H., Tavares, G.: Structural Operators for Modeling 3-manifolds. In: Proc. of the Symp. on Sol. Mod. and Appl., pp. 10–18 (1997)

    Google Scholar 

  38. Mangrove Topological Data Structure Library (2012), http://mangrovetds.sourceforge.net

  39. Mantyla, M.: An Introduction to Solid Modeling. Computer Science Press (1987)

    Google Scholar 

  40. Mc-Mains, S.: Geometric Algorithms and Data Representation for Solid Free-form Fabrication. PhD thesis, University of California, Berkeley, USA (2000)

    Google Scholar 

  41. Nabutovsky, A.: Geometry of the Space of Triangulations of a Compact Manifold. Comm. in Math. Phys. 181, 303–330 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  42. OpenMesh Library (2002), http://openmesh.org

  43. OpenVolumeMesh Library (2012), http://openvolumemesh.org

  44. Paoluzzi, A., Bernardini, F., Cattani, C., Ferrucci, V.: Dimension-Independent Modeling with Simplicial Complexes. ACM Trans. on Graph. 12(1), 56–102 (1993)

    Article  MATH  Google Scholar 

  45. Remacle, J.-F., Shephard, M.S.: An Algorithm Oriented Mesh Database. Int. Jour. of Comp. Geom. and Appl. 58(2), 393–374 (2003)

    Google Scholar 

  46. Rossignac, J., Safonova, A., Szymczak, A.: 3D Compression made Simple: Edge-breaker on a Corner Table. In: Proc. of the Shape Mod. Int. Conf., pp. 278–283 (2001)

    Google Scholar 

  47. Seol, E.S., Shephard, M.S.: Efficient Distributed Mesh Data Structure for Parallel Automated Adaptative Analysis. Eng. with Comp. 22(3), 197–213 (2006)

    Article  Google Scholar 

  48. Sieger, D., Botsch, M.: Design, implementation, and evaluation of the surface_mesh data structure. In: Quadros, W.R. (ed.) Proceedings of the 20th International Meshing Roundtable, vol. 90, pp. 533–550. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  49. Tautges, T.: MOAB-SD: Integrated Structured and Unstructured Mesh Representation. Eng. with Comp. 20(3) (2004)

    Google Scholar 

  50. Visual Computing Graphics Library (2004), http://vcg.isti.cnr.it/

  51. Weiler, K.: The Radial-Edge data structure: a Topological Representation for Non-manifold Geometric Boundary Modeling. In: Geom. Mod. for CAD Appl., pp. 3–36. Elsevier (1998)

    Google Scholar 

  52. Wright, P.K.: 21st Century Manifacturing. Prentice Hall (2001)

    Google Scholar 

  53. Zeng, L., Liu, Y.-J., Lee, S.H., Yuen, M.: Q-Complex: Efficient Non-Manifold Boundary Representation with Inclusion Topology. Comp.-Aid. Des. 44(11), 1115–1126 (2012)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, Bioengineering, Robotics and Systems Engineering (DIBRIS), University of Genova, Genova, Italy

    David Canino & Leila De Floriani

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  1. David Canino
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  2. Leila De Floriani
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Correspondence to David Canino .

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Editors and Affiliations

  1. Departament de Matemàtica Aplicada III, Polytechnic University of Catalonia, Barcelona, Spain

    Josep Sarrate

  2. Computational Simulation Infrastructure, Sandia National Laboratories, Albuquerque, New Mexico, USA

    Matthew Staten

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Canino, D., De Floriani, L. (2014). Representing Simplicial Complexes with Mangroves. In: Sarrate, J., Staten, M. (eds) Proceedings of the 22nd International Meshing Roundtable. Springer, Cham. https://doi.org/10.1007/978-3-319-02335-9_26

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  • DOI: https://doi.org/10.1007/978-3-319-02335-9_26

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-02334-2

  • Online ISBN: 978-3-319-02335-9

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