Efficiently computing exact geodesic loops within finite steps

IEEE Transactions on Visualization and Computer Graphics

Volumn 18, Issue 6, 2012, Pages 879-889

  Xin, Shi Qing ^a   He, Ying ^a   Fu, Chi Wing ^a  

^a NANYANG TECHNOLOGICAL UNIVERSITY   (Singapore)

Author keywords

Discrete geodesic; geodesic loop; triangular mesh

Indexed keywords

AVERAGE NUMBERS; CLOSED GEODESICS; DIFFERENTIAL GEOMETRY; DIFFERENTIAL TOPOLOGY; DISCRETE GEODESIC; EFFICIENT ALGORITHM; EXISTENCE AND UNIQUENESS; FACE SEQUENCES; FINITE STEPS; GEODESIC CURVATURES; GEODESIC LOOP; GEOMETRIC SHAPE; NUMERICAL SOLVERS; POLYGONAL SURFACE; SHAPE SEGMENTATION; SMOOTH SURFACE; SPACE COMPLEXITY; TIME COMPLEXITY; TRIANGULAR MESHES;

ALGORITHMS; DIFFERENTIAL EQUATIONS; GEOMETRY; MESH GENERATION; NUMERICAL METHODS;

GEODESY;

EID: 84862777416     PISSN: 10772626     EISSN: None     Source Type: Journal    
DOI: 10.1109/TVCG.2011.119     Document Type: Article

References (38)

1. H. Lee, Y. Tong, and M. Desbrun, "Geodesics-Based One-to-One Parameterization of 3D Triangle Meshes," IEEE Multimedia, vol. 12, no. 1, pp. 27-33, Jan. 2005. (Pubitemid 40351836)
2. G. Peyré and L.D. Cohen, "Geodesic Remeshing Using Front Propagation," Int'l J. Computer Vision, vol. 69, no. 1, pp. 145-156, 2006. (Pubitemid 43787408)
3. Q.-X. Huang, B. Adams, M. Wicke, and L.J. Guibas, "Non-Rigid Registration under Isometric Deformations," SGP '08: Proc. Sixth Symp. Geometry Processing, pp. 1449-1457, 2008.
4. P.V. Sander, Z.J. Wood, S.J. Gortler, J. Snyder, and H. Hoppe, "Multi-Chart Geometry Images," SGP '03: Proc. Eurographics/ACM SIGGRAPH Symp. Geometry Processing, pp. 146-155, 2003.
5. K. Zhou, J. Synder, B. Guo, and H.-Y. Shum, "Iso-Charts: Stretch-driven Mesh Parameterization Using Spectral Analysis," SGP '04: Proc. Eurographics/ACM SIGGRAPH Symp. Geometry Processing, pp. 45-54, 2004.
6. J. Zhang, C. Wu, J. Cai, J. Zheng, and X.-C. Tai, "Mesh Snapping: Robust Interactive Mesh Cutting Using Fast Geodesic Curvature Flow," Computer Graphics Forum, vol. 29, no. 2, pp. 517-526, 2010.
7. M. Sharir and A. Schorr, "On Shortest Paths in Polyhedral Spaces," SIAM J. Computing, vol. 15, no. 1, pp. 193-215, 1986.
8. J.S.B. Mitchell, D.M. Mount, and C.H. Papadimitriou, "The Discrete Geodesic Problem," SIAM J. Computing, vol. 16, no. 4, pp. 647-668, 1987.
9. S.-Q. Xin and G.-J. Wang, "Improving Chen and Han's Algorithm on the Discrete Geodesic Problem," ACM Trans. Graphics, vol. 28, no. 4, pp. 1-8, 2009.
10. R. Bott, "On the Iteration of Closed Geodesics and the Sturm Intersection Theory," Comm. Pure and Applied Math., vol. 9, no. 2, pp. 171-206, 1956.
11. M. Freedman, J. Hass, and P. Scott, "Closed Geodesics on Surfaces," Bull. London Math. Soc., vol. 14, pp. 385-391, 1982.
12. M. Jin, F. Luo, S.-T. Yau, and X. Gu, "Computing Geodesic Spectra of Surfaces," Proc. ACM Symp. Solid and Physical Modeling, pp. 387-393, 2007. (Pubitemid 47582506)
13. F. Hétroy and D. Attali, "From a Closed Piecewise Geodesic to a Constriction on a Closed Triangulated Surface," PG '03: Proc. 11th Pacific Conf. Computer Graphics and Applications, p. 394, 2003.
14. D. Reniers and A. Telea, "Part-Type Segmentation of Articulated Voxel-Shapes Using the Junction Rule," Proc. Pacific Graphics, 2008.
15. M. Dixon, N. Jacobs, and R. Pless, "Finding Minimal Parameter-izations of Cylindrical Image Manifolds," CVPRW '06: Proc. Conf. Computer Vision and Pattern Recognition Workshop, p. 192, 2006.
16. T.K. Dey and S. Guha, "Transforming Curves on Surfaces," J. Computer and System Sciences, vol. 58, pp. 297-325, 1999.
17. X. Yin, M. Jin, and X. Gu, "Computing Shortest Cycles Using Universal Covering Space," The Visual Computer, vol. 23, no. 12, pp. 999-1004, 2007. (Pubitemid 350178422)
18. L. Fan, L. Liu, and K. Liu, "Paint Mesh Cutting," Computer Graphics Forum, vol. 30, no. 2, pp. 603-612, 2011.
19. C. Wu and X. Tai, "A Level Set Formulation of Geodesic Curvature Flow on Simplicial Surfaces," IEEE Trans. Visualization and Computer Graphics, vol. 16, no. 4, pp. 647-662, July/Aug. 2010.
20. J. Chen and Y. Han, "Shortest Paths on a Polyhedron," SCG '90: Proc. Sixth Ann. Symp. Computational Geometry, pp. 360-369, 1990. (Pubitemid 20728511)
21. V. Surazhsky, T. Surazhsky, D. Kirsanov, S.J. Gortler, and H. Hoppe, "Fast Exact and Approximate Geodesics on Meshes," ACM Trans. Graphics, vol. 24, no. 3, pp. 553-560, 2005.
22. K. Polthier and M. Schmies, "Straightest Geodesics on Polyhedral Surfaces," Mathematical Visualization, p. 391, Springer Verlag, 1998.
23. K. Polthier and M. Schmies, "Geodesic Flow on Polyhedral Surfaces," Proc. Eurographics-IEEE Symp. Scientific Visualization, pp. 179-188, 1999.
24. S.-Q. Xin and G.-J. Wang, "Efficiently Determining a Locally Exact Shortest Path on Polyhedral Surfaces," Computer-Aided Design, vol. 39, no. 12, pp. 1081-1090, 2007. (Pubitemid 350166328)
25. R. Kimmel and J.A. Sethian, "Computing Geodesic Paths on Manifolds," Proc. Nat'l Academy of Sciences of USA, vol. 95, pp. 8431-8435, 1998. (Pubitemid 28350504)
26. S. Har-Peled, "Approximate Shortest paths and Geodesic Diameter on a Convex Polytope in Three Dimensions," Discrete and Computational Geometry, vol. 21, no. 2, pp. 217-231, 1999.
27. P.K. Agarwal, S. Har-Peled, and M. Karia, "Computing Approximate Shortest Paths on Convex Polytopes," Proc. Symp. Computational Geometry, pp. 270-279, 2000.
28. A. Spira and R. Kimmel, "Geodesic Curvature Flow on Parametric Surfaces," Proc. Curve and Surface Design, pp. 365-373, 2002.
29. P. Diaz-Gutierrez, D. Eppstein, and M. Gopi, "Curvature Aware Fundamental Cycles," Computer Graphics Forum, vol. 28, no. 7, pp. 2015-2024, 2009.
30. W. Zeng, M. Jin, F. Luo, and X. Gu, "Canonical Homotopy Class Representative Using Hyperbolic Structure," Proc. IEEE Int'l Conf. Shape Modeling and Applications, 2009.
31. 1 Smooth Freeform Surfaces over Hyperbolic Domains," SPM '09: Proc. SIAM/ACM Joint Conf. Geometric and Physical Modeling, pp. 367-372, 2009.
32. J. Zhang, J. Zheng, and J. Cai, "Interactive Mesh Cutting Using Constrained Random Walks," IEEE Trans. Visualization and Computer Graphics, vol. 17, no. 3, pp. 357-367, Mar. 2011.
33. Y. Zheng and C.-L. Tai, "Mesh Decomposition with Cross-Boundary Brushes," Computer Graphics Forum, vol. 29, no. 2, pp. 527-535, 2010.
34. A. Golovinskiy and T. Funkhouser, "Randomized Cuts for 3D Mesh Analysis," Proc. ACM SIGGRAPH ASIA Papers, vol. 27, Dec. 2008.
35. L.J. Guibas and J. Hershberger, "Optimal Shortest Path Queries in a Simple Polygon," SCG '87: Proc. Third Ann. Symp. Computational Geometry, pp. 50-63, 1987.
36. S. Kakutani, "Ein Beweis des sa€tzes Von Edelheit u€ber Konvexe Mengen," Proc. Imp. Acad. Tokyo 13, pp. 93-94, 1937.
37. M. Jin, F. Luo, and X.D. Gu, "Computing General Geometric Structures on Surfaces Using Ricci Flow," Computer-Aided Design, vol. 39, no. 8, pp. 663-675, 2007. (Pubitemid 47016476)
38. E.C. de Verdière and F. Lazarus, "Optimal System of Loops on an Orientable Surface," Proc. 43rd Symp. Foundations of Computer Science, pp. 627-636, 2002.