The Community for Technology Leaders
RSS Icon
Subscribe
Issue No.06 - November/December (2009 vol.15)
pp: 1177-1184
J. Tierny , Sci. Comput. & Imaging Inst., Univ. of Utah, Salt Lake City, UT, USA
A. Gyulassy , Sci. Comput. & Imaging Inst., Univ. of Utah, Salt Lake City, UT, USA
E. Simon , Dassault Syst., UT, USA
V. Pascucci , Sci. Comput. & Imaging Inst., Univ. of Utah, Salt Lake City, UT, USA
ABSTRACT
This paper introduces an efficient algorithm for computing the Reeb graph of a scalar function f defined on a volumetric mesh M in Ropf3. We introduce a procedure called "loop surgery" that transforms M into a mesh M' by a sequence of cuts and guarantees the Reeb graph of f(M') to be loop free. Therefore, loop surgery reduces Reeb graph computation to the simpler problem of computing a contour tree, for which well-known algorithms exist that are theoretically efficient (O(n log n)) and fast in practice. Inverse cuts reconstruct the loops removed at the beginning. The time complexity of our algorithm is that of a contour tree computation plus a loop surgery overhead, which depends on the number of handles of the mesh. Our systematic experiments confirm that for real-life data, this overhead is comparable to the computation of the contour tree, demonstrating virtually linear scalability on meshes ranging from 70 thousand to 3.5 million tetrahedra. Performance numbers show that our algorithm, although restricted to volumetric data, has an average speedup factor of 6,500 over the previous fastest techniques, handling larger and more complex data-sets. We demonstrate the verstility of our approach by extending fast topologically clean isosurface extraction to non simply-connected domains. We apply this technique in the context of pressure analysis for mechanical design. In this case, our technique produces results in matter of seconds even for the largest meshes. For the same models, previous Reeb graph techniques do not produce a result.
INDEX TERMS
Surgery, Tree graphs, Data visualization, Isosurfaces, Data mining, Topology, Algorithm design and analysis, Stress, Scalability, Level set, topological simplification, Reeb graph, scalar field topology, isosurfaces
CITATION
J. Tierny, A. Gyulassy, E. Simon, V. Pascucci, "Loop surgery for volumetric meshes: Reeb graphs reduced to contour trees", IEEE Transactions on Visualization & Computer Graphics, vol.15, no. 6, pp. 1177-1184, November/December 2009, doi:10.1109/TVCG.2009.163
REFERENCES
[1] AIM@SHAPE Shape Repository. http://shapes.aim-at-shape.net/, 2006.
[2] P. K. Agarwal, H. Edelsbrunner, J. Harer, and Y. Wang, Extreme elevation on a 2-manifold. In ACM Symp. on Computational Geometry, pages 357–365, 2004.
[3] G. Aujay, F. Hétroy, F. Lazarus, and C. Depraz, Harmonic skeletons for realistic character animation. In Eurographics Symp. on Computer Animation, pages 151–160, 2007.
[4] C. L. Bajaj, V. Pascucci, and D. Schikore, The contour spectrum. In IEEE Visualization, pages 167–174, 1997.
[5] S. Biasotti, B. Falcidieno, and M. Spagnuolo, Extended Reeb graphs for surface understanding and description. In Discrete Geometry for Computer Imagery, pages 185–197, 2000.
[6] S. Biasotti, D. Giorgi, M. Spagnuolo, and B. Falcidieno, Reeb graphs for shape analysis and applications. Theoretical Computer Science, 392: 5– 22, 2008.
[7] H. Carr, J. Snoeyink, and U. Axen, Computing contour trees in all dimensions. In ACM Symp. on Discrete Algorithms, pages 918–926, 2000.
[8] H. Carr, J. Snoeyink, and M. V. de Panne, Simplifying flexible isosurfaces using local geometric measures. In IEEE Visualization, pages 497–504, 2004.
[9] K. Cole-McLaughlin, H. Edelsbrunner, J. Harer, V. Natarajan, and V. Pascucci, Loops in Reeb graphs of 2-manifolds. In ACM Symp. on Computational Geometry, pages 344–350, 2003.
[10] M. de Berg and M. J. van Kreveld, Trekking in the alps without freezing or getting tired. In European Symp. on Algorithms, pages 121–132, 1993.
[11] T. K. Dey and S. Guha, Computing homology groups of simplicial complexes in R3. Journal of the ACM, 45: 266–287, 1998.
[12] H. Doraiswamy and V. Natarajan, Efficient output-sensitive construction of Reeb graphs. In International Symp. on Algorithms and Computation, pages 557–568, 2008.
[13] H. Doraiswamy and V. Natarajan, Efficient algorithms for computing Reeb graphs. Computational Geometry: Theory and Applications, 42: 606–616, 2009.
[14] H. Edelsbrunner and E. P. Mucke, Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms. ACM Trans. on Graphics, 9: 66–104, 1990.
[15] A. Hatcher, Algebraic Topology. Cambridge University Press, 2002.
[16] M. Hilaga, Y. Shinagawa, T. Kohmura, and T. Kunii, Topology matching for fully automatic similarity estimation of 3D shapes. In SIGGRAPH, pages 203–212, 2001.
[17] J. Milnor, Morse Theory. Princeton University Press, 1963.
[18] V. Pascucci and K. Cole-McLaughlin, Parallel computation of the topology of level sets. Algorithmica, 38: 249–268, 2003.
[19] V. Pascucci, G. Scorzelli, P. T. Bremer, and A. Mascarenhas, Robust online computation of Reeb graphs: simplicity and speed. ACM Trans. on Graphics, 26: 58.1–58.9, 2007.
[20] G. Patanè, M. Spagnuolo, and B. Falcidieno, Reeb graph computation based on minimal contouring. In IEEE Shape Modeling International, pages 73–82, 2008.
[21] G. Reeb, Sur les points singuliers d'une forme de Pfaff complètement intégrable ou d'une fonction numérique. Comptes-rendus de l'Académie des Sciences, 222: 847–849, 1946.
[22] Y. Shinagawa, T. L. Kunii, and Y. L. Kergosien, Surface coding based on Morse theory. IEEE Computer Graphics and Applications, 11: 66–78, 1991.
[23] S. Tarasov and M. Vyalyi, Construction of contour trees in 3D in O((n) log (n)) steps. In ACM Symp. on Computational Geometry, pages 68–75, 1998.
[24] M. van Kreveld, R. van Oostrum, L. Bajaj, C, V. Pascucci, and R. Shcikore, D. Contour, trees and small seed sets for isosurface traversal. In ACM Symp. on Computational Geometry, pages 212–220, 1997.
[25] C. T. C. Wall, Surgery on compact manifolds. American Mathematical Society, 1970.
[26] G. H. Weber, P.-T. Bremer, and V. Pascucci, Topological landscapes: a terrain metaphor for scientific data. IEEE Trans. on Visualization and Computer Graphics, 13: 1416–1423, 2007.
[27] G. H. Weber, S. E. Dillard, H. Carr, V. Pascucci, and B. Hamann, Topology-controlled volume rendering. IEEE Trans. on Visualization and Computer Graphics, 13: 330–341, 2007.
[28] J. Wood, Z, H. Hoppe, M. Desbrun, and P. Schrder, Removing excess topology from isosurfaces. ACM Trans. on Graphics, 23: 190–208, 2004.
[29] E. Zhang, K. Mischaikow, and G. Turk, Feature-based surface parametrization and texture mapping. ACM Trans. on Graphics, 24: 1– 27, 2005.
27 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool