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Issue No.06 - June (2012 vol.18)
pp: 879-889
Shi-Qing Xin , Nanyang Technological University, Singapore
Ying He , Nanyang Technological University, Singapore
Chi-Wing Fu , Nanyang Technological University, Singapore
ABSTRACT
Closed geodesics, or geodesic loops, are crucial to the study of differential topology and differential geometry. Although the existence and properties of closed geodesics on smooth surfaces have been widely studied in mathematics community, relatively little progress has been made on how to compute them on polygonal surfaces. Most existing algorithms simply consider the mesh as a graph and so the resultant loops are restricted only on mesh edges, which are far from the actual geodesics. This paper is the first to prove the existence and uniqueness of geodesic loop restricted on a closed face sequence; it contributes also with an efficient algorithm to iteratively evolve an initial closed path on a given mesh into an exact geodesic loop within finite steps. Our proposed algorithm takes only an O(k) space complexity and an O(mk) time complexity (experimentally), where m is the number of vertices in the region bounded by the initial loop and the resultant geodesic loop, and k is the average number of edges in the edge sequences that the evolving loop passes through. In contrast to the existing geodesic curvature flow methods which compute an approximate geodesic loop within a predefined threshold, our method is exact and can apply directly to triangular meshes without needing to solve any differential equation with a numerical solver; it can run at interactive speed, e.g., in the order of milliseconds, for a mesh with around 50K vertices, and hence, significantly outperforms existing algorithms. Actually, our algorithm could run at interactive speed even for larger meshes. Besides the complexity of the input mesh, the geometric shape could also affect the number of evolving steps, i.e., the performance. We motivate our algorithm with an interactive shape segmentation example shown later in the paper.
INDEX TERMS
Discrete geodesic, geodesic loop, triangular mesh.
CITATION
Shi-Qing Xin, Ying He, Chi-Wing Fu, "Efficiently Computing Exact Geodesic Loops within Finite Steps", IEEE Transactions on Visualization & Computer Graphics, vol.18, no. 6, pp. 879-889, June 2012, doi:10.1109/TVCG.2011.119
REFERENCES
[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.
[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.
[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.
[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 Parameterizations 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.
[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.
[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.
[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.
[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] W. Zeng, Y. He, J. Xia, X. Gu, and H. Qin, “${\rm C}^\infty$ 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 ${\rm s}{\ddot{a}}$ tzes Von Edelheit ${\ddot{u}}$ 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.
[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.
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