Computing reeb graphs as a union of contour trees

IEEE Transactions on Visualization and Computer Graphics

Volumn 19, Issue 2, 2013, Pages 249-262

  Doraiswamy, Harish ^a   Natarajan, Vijay ^a  

^a INDIAN INSTITUTE OF SCIENCE   (India)

Author keywords

Computational topology; level set topology; out of core algorithm; Reeb graphs; scalar functions

Indexed keywords

COMPUTATIONAL TOPOLOGY; LEVEL SET; OUT-OF-CORE ALGORITHMS; REEB GRAPH; SCALAR FUNCTION;

ALGORITHMS; COMPUTER PROGRAMMING; FORESTRY; PIECEWISE LINEAR TECHNIQUES; TOPOLOGY;

TREES (MATHEMATICS);

ALGORITHMS; COMPUTER PROGRAMING; FORESTRY;

EID: 84871089781     PISSN: 10772626     EISSN: None     Source Type: Journal    
DOI: 10.1109/TVCG.2012.115     Document Type: Article

References (47)

1. F. Lazarus and A. Verroust, "Level Set Diagrams of Polyhedral Objects," Proc. ACM Symp. Solid Modeling and Applications, pp. 130-140, 1999.
2. Y. Shinagawa, T.L. Kunii, and Y.L. Kergosien, "Surface Coding Based on Morse Theory," IEEE Computer Graphics and Applications, vol. 11, no. 5, pp. 66-78, Sept. 1991.
3. S. Takahashi, Y. Shinagawa, and T.L. Kunii, "A Feature-Based Approach for Smooth Surfaces," Proc. Symp. Solid Modeling and Applications, pp. 97-110, 1997.
4. H.S.Y. Shinagawa, T.L. Kunii, and M. Ibusuki, "Modeling Contact of Two Complex Objects: With an Application to Characterizing Dental Articulations," Computers and Graphics, vol. 19, no. 1, pp. 21-28, 1995.
5. M. Hilaga, Y. Shinagawa, T. Kohmura, and T.L. Kunii, "Topology Matching for Fully Automatic Similarity Estimation of 3D Shapes," Proc. ACM SIGGRAPH, pp. 203-212, 2001. (Pubitemid 33047305)
6. I. Guskov and Z. Wood, "Topological Noise Removal," Proc. Graphics Interface, pp. 19-26, 2001. (Pubitemid 34103996)
7. Y.-J. Chiang and X. Lu, "Progressive Simplification of Tetrahedral Meshes Preserving all Isosurface Topologies," Computer Graphics Forum, vol. 22, no. 3, pp. 493-504, 2003.
8. S. Takahashi, G.M. Nielson, Y. Takeshima, and I. Fujishiro, "Topological Volume Skeletonization Using Adaptive Tetrahedralization, " Proc. Geometric Modeling and Processing, pp. 227-236, 2004.
9. Z. Wood, H. Hoppe, M. Desbrun, and P. Schröder, "Removing Excess Topology from Isosurfaces," ACM Trans. Graphics, vol. 23, no. 2, pp. 190-208, 2004.
10. F. Hétroy and D. Attali, "Topological Quadrangulations of Closed Triangulated Surfaces Using the Reeb Graph," Graphical Models, vol. 65, nos. 1-3, pp. 131-148, 2003. (Pubitemid 36873271)
11. M. Mortara and G. Patané, "Affine-Invariant Skeleton of 3D Shapes," Proc. Shape Modeling Int'l, pp. 245, 2002.
12. E. Zhang, K. Mischaikow, and G. Turk, "Feature-Based Surface Parameterization and Texture Mapping," ACM Trans. Graphics, vol. 24, no. 1, pp. 1-27, 2005. (Pubitemid 44849562)
13. M. van Kreveld, R. van Oostrum, C. Bajaj, V. Pascucci, and D.R. Schikore, "Contour Trees and Small Seed Sets for Isosurface Traversal," Proc. 13th Ann. Symp. Computational Geometry, pp. 212-220, 1997.
14. C.L. Bajaj, V. Pascucci, and D.R. Schikore, "The Contour Spectrum," Proc. IEEE Conf. Visualization, pp. 167-173, 1997.
15. H. Carr, J. Snoeyink, and M. van de Panne, "Simplifying Flexible Isosurfaces Using Local Geometric Measures," Proc. IEEE Conf. Visualization, pp. 497-504, 2004. (Pubitemid 40497960)
16. I. Fujishiro, Y. Takeshima, T. Azuma, and S. Takahashi, "Volume Data Mining Using 3D Field Topology Analysis," IEEE Computer Graphics and Applications, vol. 20, no. 5, pp. 46-51, Sept./Oct. 2000.
17. S. Takahashi, Y. Takeshima, and I. Fujishiro, "Topological Volume Skeletonization and Its Application to Transfer Function Design," Graphical Models, vol. 66, no. 1, pp. 24-49, 2004. (Pubitemid 38245838)
18. G.H. Weber, S.E. Dillard, H. Carr, V. Pascucci, and B. Hamann, "Topology-Controlled Volume Rendering," IEEE Trans. Visualization and Computer Graphics, vol. 13, no. 2, pp. 330-341, Mar./Apr. 2007. (Pubitemid 46183078)
19. J. Zhou and M. Takatsuka, "Automatic Transfer Function Generation Using Contour Tree Controlled Residue Flow Model and Color Harmonics," IEEE Trans. Visualization Computer Graphics, vol. 15, no. 6, pp. 1481-1488, 2009.
20. W. Harvey and Y. Wang, "Topological Landscape Ensembles for Visualization of Scalar-Valued Functions," Computer Graphics Forum, vol. 29, pp. 993-1002, 2010.
21. P. Oesterling, C. Heine, H. Jänicke, G. Scheuermann, and G. Heyer, "Visualization of High Dimensional Point Clouds Using Their Density Distribution's Topology," IEEE Trans. Visualization and Computer Graphics, vol. 17, no. 11, pp. 1547-1559, Nov. 2011.
22. S. Biasotti, D. Giorgi, M. Spagnuolo, and B. Falcidieno, "Reeb Graphs for Shape Analysis and Applications," Theoretical Computer Science, vol. 392, pp. 5-22, Feb. 2008. (Pubitemid 351199935)
23. S. Biasotti, L. De Floriani, B. Falcidieno, P. Frosini, D. Giorgi, C. Landi, L. Papaleo, and M. Spagnuolo, "Describing Shapes by Geometrical-Topological Properties of Real Functions," ACM Computing Surveys, vol. 40, pp. 12:1-12:87, Oct. 2008.
24. S. Biasotti, L. Floriani, B. Falcidieno, and L. Papaleo, "Morphological Representations of Scalar Fields," Shape Analysis and Structuring, Mathematics and Visualization, L. Floriani, M. Spagnuolo, G. Farin, H.-C. Hege, D. Hoffman, C.R. Johnson, and K. Polthier, eds., pp. 185-213, Springer, 2008.
25. S. Biasotti, D. Attali, J.-D. Boissonnat, H. Edelsbrunner, G. Elber, M. Mortara, G.S. Baja, M. Spagnuolo, M. Tanase, and R. Veltkamp, "Skeletal Structures," Shape Analysis and Structuring, Mathematics and Visualization, L. Floriani, M. Spagnuolo, G. Farin, H.-C. Hege, D. Hoffman, C.R. Johnson, and K. Polthier, eds., pp. 145-183, Springer, 2008.
26. H. Carr, J. Snoeyink, and U. Axen, "Computing Contour Trees in all Dimensions," Computational Geometry: Theory and Application, vol. 24, no. 2, pp. 75-94, 2003. (Pubitemid 35217991)
27. K. Cole-McLaughlin, H. Edelsbrunner, J. Harer, V. Natarajan, and V. Pascucci, "Loops in Reeb Graphs of 2-Manifolds," Discrete and Computational Geometry, vol. 32, no. 2, pp. 231-244, 2004. (Pubitemid 39129966)
28. H. Doraiswamy and V. Natarajan, "Efficient Algorithms for Computing Reeb Graphs," Computational Geometry: Theory and Applications, vol. 42, nos. 6/7, pp. 606-616, 2009.
29. V. Pascucci, G. Scorzelli, P.-T. Bremer, and A. Mascarenhas, "Robust On-Line Computation of Reeb Graphs: Simplicity and Speed," ACM Trans. Graphics, vol. 26, no. 3, article 58, 2007.
30. W. Harvey, Y. Wang, and R. Wenger, "A Randomized O(m log m) Time Algorithm for Computing Reeb Graphs of Arbitrary Simplicial Complexes," Proc. Symp. Computational geometry, pp. 267-276, 2010.
31. G. Patanè , M. Spagnuolo, and B. Falcidieno, "A Minimal Contouring Approach to the Computation of the Reeb Graph," IEEE Trans. Visualization and Computer Graphics, vol. 15, no. 4, pp. 583-595, July/Aug. 2009.
32. H. Doraiswamy and V. Natarajan, "Output-Sensitive Construction of Reeb Graphs," IEEE Trans. Visualization and Computer Graphics, vol. 18, no. 1, pp. 146-159, Jan. 2012.
33. J. Tierny, A. Gyulassy, E. Simon, and V. Pascucci, "Loop Surgery for Volumetric Meshes: Reeb Graphs Reduced to Contour Trees," IEEE Trans. Visualization and Computer Graphics, vol. 15, no. 6, pp. 1177-1184, Nov./Dec. 2009.
34. T.H. Cormen, C.E. Leiserson, and R.L. Rivest, Introduction to Algorithms. MIT Press, 2001.
35. Y.-J. Chiang, T. Lenz, X. Lu, and G. Rote, "Simple and Optimal Output-Sensitive Construction of Contour Trees Using Monotone Paths," Computational Geometry: Theory and Application, vol. 30, no. 2, pp. 165-195, 2005.
36. Y. Shinagawa and T.L. Kunii, "Constructing a Reeb Graph Automatically from Cross Sections," IEEE Computer Graphics and Application, vol. 11, no. 6, pp. 44-51, Nov. 1991.
37. H. Edelsbrunner and J. Harer, Computational Topology: An Introduction. Am. Math. Soc., 2009.
38. A. Hatcher, Algebraic Topology. Cambridge Univ. Press, 2002.
39. J. Milnor, Morse Theory. Princeton Univ. Press, 1963.
40. T.F. Banchoff, "Critical Points and Curvature for Embedded Polyhedral Surfaces," Am. Math. Monthly, vol. 77, pp. 475-485, 1970.
41. H. Edelsbrunner, J. Harer, V. Natarajan, and V. Pascucci, "Morse-Smale Complexes for Piecewise Linear 3-Manifolds," Proc. 19th Ann. Symp. Computational Geometry, pp. 361-370, 2003.
42. Y. Matsumoto, An Introduction to Morse Theory, Am. Math. Soc., 2002.
43. G. Reeb, "Sur Les Points Singuliers D'une Forme de Pfaff Complètement Intégrable ou D'une Fonction Numérique," Comptes Rendus de L'Académie ses Séances, vol. 222, pp. 847-849, 1946.
44. H. Edelsbrunner, Geometry and Topology for Mesh Generation. Cambridge Univ. Press, 2001.
45. R. Diestel, Graph Theory, Graduate Texts in Mathematics, third ed., vol. 173, Springer-Verlag, 2005.
46. E.P. Mü cke, "Shapes and Implementations in Three-Dimensional Geometry," PhD dissertation, Dept. of Computer Science, Univ. of Illinois, Urbana-Champaign, Illinois, 1993.
47. The Digital Michelangelo Project. http://graphics.stanford.edu/projects/ mich/, 2012.