COMPUTING THE REEB GRAPH FOR TRIANGLE MESHES WITH ACTIVE CONTOURS

Laura Brandolini, Marco Piastra

Abstract

This paper illustrates a novel method to compute the Reeb graph for triangle meshes. The algorithm is based on the definition of discrete, active contours as counterparts of continuous level lines. Active contours are made up of edges and vertices with multiple presence and implicitly maintain a faithful representation of the level lines, even in case of coarse meshes with higher genus. This approach gives a great advantage in the identification of the nodes in the Reeb graph and also improves the overall efficiency of the algorithm in that at each step only the information local to the contours and their immediate neighborhood needs to be processed. The validation of functional integrity for the algorithm has been carried out experimentally, with real-world data, without mesh pre-processing.

References

  1. Berretti, S., Del Bimbo, A., and Pala, P. (2009). 3d mesh decomposition using reeb graphs. Image and Vision Computing, 27(10):1540 - 1554. Special Section: Computer Vision Methods for Ambient Intelligence.
  2. Biasotti, S., Giorgi, D., Spagnuolo, M., and Falcidieno, B. (2008). Reeb graphs for shape analysis and applications. Theoretical computer science, 392:5-22.
  3. Biasotti, S., Mortara, M., and Spagnuolo, M. (2000). Surface compression and reconstruction using reeb graphs and shape analysis. In Proceedings of 16th Spring Conference on Computer Graphics, pages 175-184. ACM press.
  4. Cole-McLaughlin, K., Edelsbrunner, H., Harer, J., Natarajan, V., and Pascucci, V. (2003). Loops in reeb graphs of 2-manifolds. In Proceedings of the nineteenth annual symposium on Computational geometry, SCG 7803, pages 344-350, New York, NY, USA. ACM.
  5. Dijkstra, E. W. (1959). A note on two problems in connexion with graphs. Numerische Mathematik, 1:269271.
  6. Doraiswamy, H. and Natarajan, V. (2009). Efficient algorithms for computing reeb graphs. Computational Geometry, 42(6-7):606 - 616.
  7. Edelsbrunner, H., Harer, J., Mascarenhas, A., Pascucci, V., and Snoeyink, J. (2008). Time-varying reeb graphs for continuous space-time data. Computational Geometry, 41(3):149 - 166.
  8. Edelsbrunner, H., Harer, J., and Zomorodian, A. (2003). Hierarchical morse-smale complexes for piecewise linear 2-manifolds. Discrete and Computational Geometry, 30(1):87-107.
  9. Falcidieno, B. (2004). Aim@shape project presentation. In Shape Modeling Applications, 2004. Proceedings, page 329.
  10. Hilaga, M., Shinagawa, Y., Kohmura, T., and Kunii, T. L. (2001). Topology matching for fully automatic similarity estimation of 3d shapes. In Proceedings of the 28th annual conference on Computer graphics and interactive techniques, SIGGRAPH 7801, pages 203- 212, New York, NY, USA. ACM.
  11. Katz, S., Leifman, G., and Tal, A. (2005). Mesh segmentation using feature point and core extraction. The Visual Computer, 21:649-658. 10.1007/s00371-005- 0344-9.
  12. Knuth, D. E. (1998). The Art of Computer Programming Vol. 2: Seminumerical Algorithms. Addison Wesley, 3rd edition.
  13. Lazarus, F. and Verroust, A. (1999). Level set diagrams of polyhedral objects. In Fifth Symposium on Solid Modeling, pages 130-140. ACM.
  14. Milnor, J. (1963). Morse Theory. Princeton University Press.
  15. Mortara, M. and Patane, G. (2002). Affine-invariant skeleton of 3d shapes. Shape Modeling and Applications, International Conference on, 0:245-252.
  16. Novotni, M., Klein, R., and Ii, I. F. I. (2002). Computing geodesic distances on triangular meshes. In In Proc. of WSCG2002, pages 341-347.
  17. Pascucci, V., Scorzelli, G., Bremer, P.-T., and Mascarenhas, A. (2007). Robust on-line computation of reeb graphs: simplicity and speed. ACM Trans. Graph., 26.
  18. Patane, G., Spagnuolo, M., and Falcidieno, B. (2009). A minimal contouring approach to the computation of the reeb graph. IEEE Transactions on Visualization and Computer Graphics, 15:583-595.
  19. Reeb, G. (1946). Sur les points singuliers d une forme de pfaff completement integrable ou d une fonction numerique. In Comptes rendus de l'Academie des Sciences 222, pages 847-849.
  20. Safar, M., Alenzi, K., and Albehairy, S. (2009). Counting cycles in an undirected graph using dfs-xor algorithm. In Networked Digital Technologies, 2009. NDT 7809. First International Conference on, pages 132 -139.
  21. Schaefer, S. and Yuksel, C. (2007). Example-based skeleton extraction. In Proceedings of the fifth Eurographics symposium on Geometry processing, pages 153-162, Aire-la-Ville, Switzerland. Eurographics Association.
  22. Sebastian, T., Klein, P., and Kimia, B. (2002). Shock-based indexing into large shape databases. In Computer Vision ECCV 2002, volume 2352 of Lecture Notes in Computer Science, pages 83-98. Springer Berlin / Heidelberg.
  23. Shapira, L., Shamir, A., and Cohen-Or, D. (2008). Consistent mesh partitioning and skeletonization using the shape diameter function. Visual Comput, 24:249-259.
  24. Shinagawa, Y. and Kunii, T. (1991). Constructing a reeb graph automatically from cross sections. Computer Graphics and Applications, IEEE, 11(6):44 -51.
  25. Shinagawa, Y., Kunii, T., and Kergosien, Y. (1991). Surface coding based on morse theory. Computer Graphics and Applications, IEEE, 11(5):66 -78.
  26. Sundar, H., Silver, D., Gagvani, N., and Dickinson, S. (2003). Skeleton based shape matching and retrieval. In Shape Modeling International, 2003, pages 130 - 139.
  27. Tierny, J., Vandeborre, J., and Daoudi, M. (2006). 3d mesh skeleton extraction using topological and geometrical analyses. In 14th Pacific Conference on Computer Graphics and Applications. Pacific Graphics.
Download


Paper Citation


in Harvard Style

Brandolini L. and Piastra M. (2012). COMPUTING THE REEB GRAPH FOR TRIANGLE MESHES WITH ACTIVE CONTOURS . In Proceedings of the 1st International Conference on Pattern Recognition Applications and Methods - Volume 2: ICPRAM, ISBN 978-989-8425-99-7, pages 80-89. DOI: 10.5220/0003745500800089


in Bibtex Style

@conference{icpram12,
author={Laura Brandolini and Marco Piastra},
title={COMPUTING THE REEB GRAPH FOR TRIANGLE MESHES WITH ACTIVE CONTOURS},
booktitle={Proceedings of the 1st International Conference on Pattern Recognition Applications and Methods - Volume 2: ICPRAM,},
year={2012},
pages={80-89},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0003745500800089},
isbn={978-989-8425-99-7},
}


in EndNote Style

TY - CONF
JO - Proceedings of the 1st International Conference on Pattern Recognition Applications and Methods - Volume 2: ICPRAM,
TI - COMPUTING THE REEB GRAPH FOR TRIANGLE MESHES WITH ACTIVE CONTOURS
SN - 978-989-8425-99-7
AU - Brandolini L.
AU - Piastra M.
PY - 2012
SP - 80
EP - 89
DO - 10.5220/0003745500800089