Polyline-sourced Geodesic Voronoi Diagrams on Triangle Meshes
This paper studies the Voronoi diagrams on 2‐manifold meshes based on geodesic metric (a.k.a. geodesic Voronoi diagrams or GVDs), which have polyline generators. We show that our general setting leads to situations more complicated than conventional 2D Euclidean Voronoi diagrams as well as point‐sou...
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| Published in: | Computer graphics forum Vol. 33; no. 7; pp. 161 - 170 |
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| Main Authors: | , , , , |
| Format: | Journal Article |
| Language: | English |
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Oxford
Blackwell Publishing Ltd
01.10.2014
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| ISSN: | 0167-7055, 1467-8659 |
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| Abstract | This paper studies the Voronoi diagrams on 2‐manifold meshes based on geodesic metric (a.k.a. geodesic Voronoi diagrams or GVDs), which have polyline generators. We show that our general setting leads to situations more complicated than conventional 2D Euclidean Voronoi diagrams as well as point‐source based GVDs, since a typical bisector contains line segments, hyperbolic segments and parabolic segments. To tackle this challenge, we introduce a new concept, called local Voronoi diagram (LVD), which is a combination of additively weighted Voronoi diagram and line‐segment Voronoi diagram on a mesh triangle. We show that when restricting on a single mesh triangle, the GVD is a subset of the LVD and only two types of mesh triangles can contain GVD edges. Based on these results, we propose an efficient algorithm for constructing the GVD with polyline generators. Our algorithm runs in O(nNlogN) time and takes O(nN) space on an n‐face mesh with m generators, where N = max{m, n}. Computational results on real‐world models demonstrate the efficiency of our algorithm. |
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| AbstractList | This paper studies the Voronoi diagrams on 2‐manifold meshes based on geodesic metric (a.k.a. geodesic Voronoi diagrams or GVDs), which have polyline generators. We show that our general setting leads to situations more complicated than conventional 2D Euclidean Voronoi diagrams as well as point‐source based GVDs, since a typical bisector contains line segments, hyperbolic segments and parabolic segments. To tackle this challenge, we introduce a new concept, called local Voronoi diagram (LVD), which is a combination of additively weighted Voronoi diagram and line‐segment Voronoi diagram on a mesh triangle. We show that when restricting on a single mesh triangle, the GVD is a subset of the LVD and only two types of mesh triangles can contain GVD edges. Based on these results, we propose an efficient algorithm for constructing the GVD with polyline generators. Our algorithm runs in O(nNlogN) time and takes O(nN) space on an n‐face mesh with m generators, where N =
max
{m, n}. Computational results on real‐world models demonstrate the efficiency of our algorithm. This paper studies the Voronoi diagrams on 2‐manifold meshes based on geodesic metric (a.k.a. geodesic Voronoi diagrams or GVDs), which have polyline generators. We show that our general setting leads to situations more complicated than conventional 2D Euclidean Voronoi diagrams as well as point‐source based GVDs, since a typical bisector contains line segments, hyperbolic segments and parabolic segments. To tackle this challenge, we introduce a new concept, called local Voronoi diagram (LVD), which is a combination of additively weighted Voronoi diagram and line‐segment Voronoi diagram on a mesh triangle. We show that when restricting on a single mesh triangle, the GVD is a subset of the LVD and only two types of mesh triangles can contain GVD edges. Based on these results, we propose an efficient algorithm for constructing the GVD with polyline generators. Our algorithm runs in O(nNlogN) time and takes O(nN) space on an n‐face mesh with m generators, where N = max{m, n}. Computational results on real‐world models demonstrate the efficiency of our algorithm. |
| Author | Sun, Qian Xu, Chunxu Li, Jinyan Liu, Yong-Jin He, Ying |
| Author_xml | – sequence: 1 givenname: Chunxu surname: Xu fullname: Xu, Chunxu organization: TNList, Department of Computer Science and Technology, Tsinghua University, Beijing, China – sequence: 2 givenname: Yong-Jin surname: Liu fullname: Liu, Yong-Jin organization: TNList, Department of Computer Science and Technology, Tsinghua University, Beijing, China – sequence: 3 givenname: Qian surname: Sun fullname: Sun, Qian organization: School of Computer Engineering, Nanyang Technological University, Singapore – sequence: 4 givenname: Jinyan surname: Li fullname: Li, Jinyan organization: Advanced Analytics Institute, University of Technology, Australia – sequence: 5 givenname: Ying surname: He fullname: He, Ying organization: School of Computer Engineering, Nanyang Technological University, Singapore |
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| Cites_doi | 10.1145/77635.77639 10.1002/9780470317013 10.1016/0021-9991(85)90140-8 10.1016/j.cad.2011.08.027 10.1145/2508363.2508379 10.1145/1073204.1073228 10.1016/j.cad.2012.11.005 10.1145/2516971.2516977 10.1073/pnas.95.15.8431 10.1137/0216045 10.1145/73393.73407 10.1007/BF01840357 10.1016/S0010-4485(98)00052-9 10.1016/S0925-7721(02)00077-9 10.1007/s00371-007-0136-5 10.1109/70.678452 10.1145/1559755.1559758 10.1109/CGI.1997.601311 10.1145/98524.98601 10.1145/2159616.2159622 10.1016/0010-4485(95)99797-C 10.1109/TPAMI.2010.221 10.1111/j.1467-8659.2012.03058.x 10.1016/j.ipl.2012.12.010 |
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| Copyright | 2014 The Author(s) Computer Graphics Forum © 2014 The Eurographics Association and John Wiley & Sons Ltd. Published by John Wiley & Sons Ltd. 2014 The Eurographics Association and John Wiley & Sons Ltd. |
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| References_xml | – reference: Xin S.-Q., Ying X., He Y.: Efficiently computing geodesic offsets on triangle meshes by the extended xin-wang algorithm. Comput. Aided Des. 43, 11 (2011), 1468-1476. – reference: Na H.-S., Lee C.-N., Cheong O.: Voronoi diagrams on the sphere. Computational Geometry 23, 2 (2002), 183-194. – reference: Liu Y.-J., Tang K.: The complexity of geodesic Voronoi diagrams on triangulated 2-manifold surfaces. Information Processing Letters 113, 4 (2013), 132-136. – reference: Fortune S.: A sweepline algorithm for Voronoi diagrams. Algorithmica 2, 1-4 (1987), 153-174. – reference: Surazhsky V., Surazhsky T., Kirsanov D., Gortler S.J., Hoppe H.: Fast exact and approximate geodesics on meshes. In ACM Transactions on Graphics (TOG) (2005), vol. 24, pp. 553-560. – reference: Augenbaum J.M., Peskin C.S.: On the construction of the Voronoi mesh on a sphere. Journal of Computational Physics 59, 2 (1985), 177-192. – reference: Ying X., Wang X., He Y.: Saddle vertex graph (SVG): a novel solution to the discrete geodesic problem. ACM Transactions on Graphics (SIGGRAPH ASIA 2013) 32, 6 (2013), 170:1-170:12. – reference: Kim D.-S., Hwang I.-K., Park B.-J.: Representing the Voronoi diagram of a simple polygon using rational quadratic Bézier curves. Computer-Aided Design 27, 8 (1995), 605-614. – reference: Mitchell J.S., Mount D.M., Papadimitriou C.H.: The discrete geodesic problem. SIAM Journal on Computing 16, 4 (1987), 647-668. – reference: Kunze R., Wolter F.-E., Rausch T.: Geodesic Voronoi diagrams on parametric surfaces. In Computer Graphics International (1997), vol. 97, pp. 230-237. – reference: Edelsbrunner H., Mücke E.P.: Simulation of simplicity: A technique to cope with degenerate cases in geometric algorithms. ACM Trans. Graph. 9, 1 (Jan. 1990), 66-104. – reference: Kim D.-S., Lee S.-W., Shin H.: A cocktail algorithm for planar Bézier curve intersections. Computer-Aided Design 30, 13 (1998), 1047-1051. – reference: Liu Y.-J., Chen Z., Tang K.: Construction of isocontours, bisectors, and Voronoi diagrams on triangulated surfaces. IEEE Transactions on Pattern Analysis and Machine Intelligence 33, 8 (2011), 1502-1517. – reference: Liu Y.-J.: Exact geodesic metric in 2-manifold triangle meshes using edge-based data structures. Computer-Aided Design 45, 3 (2013), 695-704. – reference: Okabe A., Boots B., Sugihara K., Chiu S.N.: Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley, 2000. – reference: Kimmel R., Sethian J.: Computing geodesic paths on manifolds. Proceedings of National Academy of Sciences 95 (1998), 8431-8435. – reference: Crane K., Weischedel C., Wardetzky M.: Geodesics in heat: A new approach to computing distance based on heat flow. ACM Transactions on Graphics 32, 5 (2013), 152. – reference: Liu Y., Wang W., Levy B., Sun F., Yan D.-M., Lu L., Yang C.: On centroidal Voronoi tessellation - energy smoothness and fast computation. ACM Transactions on Graphics 28, 4 (2009), Article No. 101. – reference: Kimmel R., Kiryati N., Bruckstein A.M.: Multivalued distance maps for motion planning on surfaces with moving obstacles. IEEE Transactions on Robotics and Automation 14, 3 (1998), 427-436. – reference: Lu L., Lévy B., Wang W.: Centroidal Voronoi tessellation of line segments and graphs. Comp. Graph. Forum 31, 2pt4 (May 2012), 775-784. – reference: Liu Y.-J., Zhou Q.-Y., Hu S.-M.: Handling degenerate cases in exact geodesic computation on triangle meshes. The Visual Computer 23, 9-11 (2007), 661-668. – reference: Onishi K., Takayama N.: Construction of Voronoi diagram on the upper half-plane. 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| Snippet | This paper studies the Voronoi diagrams on 2‐manifold meshes based on geodesic metric (a.k.a. geodesic Voronoi diagrams or GVDs), which have polyline... This paper studies the Voronoi diagrams on 2-manifold meshes based on geodesic metric (a.k.a. geodesic Voronoi diagrams or GVDs), which have polyline... |
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| SubjectTerms | Algorithms Analysis and object representations Categories and Subject Descriptors (according to ACM CCS) Computer graphics I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling-Curve I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Curve, surface, solid, and object representations solid Studies surface Topological manifolds |
| Title | Polyline-sourced Geodesic Voronoi Diagrams on Triangle Meshes |
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