The Community for Technology Leaders
RSS Icon
Issue No.08 - Aug. (2013 vol.19)
pp: 1298-1306
Xin Feng , Dept. of Comput. Sci. & Eng., Michigan State Univ., East Lansing, MI, USA
Yiying Tong , Dept. of Comput. Sci. & Eng., Michigan State Univ., East Lansing, MI, USA
We present a method for computing “choking” loops-a set of surface loops that describe the narrowing of the volumes inside/outside of the surface and extend the notion of surface homology and homotopy loops. The intuition behind their definition is that a choking loop represents the region where an offset of the original surface would get pinched. Our generalized loops naturally include the usual 2g handles/tunnels computed based on the topology of the genus-g surface, but also include loops that identify chokepoints or bottlenecks, i.e., boundaries of small membranes separating the inside or outside volume of the surface into disconnected regions. Our definition is based on persistent homology theory, which gives a measure to topological structures, thus providing resilience to noise and a well-defined way to determine topological feature size. More precisely, the persistence computed here is based on the lower star filtration of the interior or exterior 3D domain with the distance field to the surface being the associated 3D Morse function.
Face, Generators, Topology, Inductors, Noise, Surface treatment, Noise measurement, object representations, Computer graphics, computational geometry and object modeling, curve, surface
Xin Feng, Yiying Tong, "Choking Loops on Surfaces", IEEE Transactions on Visualization & Computer Graphics, vol.19, no. 8, pp. 1298-1306, Aug. 2013, doi:10.1109/TVCG.2013.9
[1] D. Letscher and J. Fritts, "Image Segmentation Using Topological Persistence," Proc. 12th Int'l Conf. Computer Analysis of Images and Patterns (CAIP '07), citation.cfm?id=1770904.1770985 , pp. 587-595, 2007.
[2] Z. Wood, H. Hoppe, M. Desbrun, and P. Schröder, "Removing Excess Topology from Isosurfaces," ACM Trans. Graphics, vol. 23, pp. 190-208,, Apr. 2004.
[3] P.-T. Bremer, E.M. Bringa, M.A. Duchaineau, A.G. Gyulassy, D. Laney, A. Mascarenhas, and V. Pascucci, "Topological Feature Extraction and Tracking," J. Physics: Conf. Series, vol. 78, no. 1, , p. 012007, 2007.
[4] T.K. Dey, K. Li, J. Sun, and D. Cohen-Steiner, "Computing Geometry-Aware Handle and Tunnel Loops in 3D Models," ACM Trans. Graphics, vol. 27, pp. 45:1-45:9, , Aug. 2008.
[5] M. Desbrun, E. Kanso, and Y. Tong, "Discrete Differential Forms for Computational Modeling," Proc. ACM SIGGRAPH ASIA Courses, pp. 15:1-15:17, 1508059 , 2008.
[6] D. Boltcheva, D. Canino, S.M. Aceituno, J.-C. Léon, L. De Floriani, and F. Hétroy, "An Iterative Algorithm for Homology Computation on Simplicial Shapes," Comput.-Aided Design, vol. 43, no. 11, pp. 1457-1467, Nov. 2011.
[7] E. Zhang, K. Mischaikow, and G. Turk, "Feature-Based Surface Parameterization and Texture Mapping," ACM Trans. Graphics, vol. 24, no. 1, pp. 1-27, , Jan. 2005.
[8] S. Katz and A. Tal, "Hierarchical Mesh Decomposition Using Fuzzy Clustering and Cuts," ACM Trans. Graphics, vol. 22, no. 3, pp. 954-961,, July 2003.
[9] T.K. Dey, K. Li, and J. Sun, "Computing Handle And Tunnel Loops with Knot Linking," Computer-Aided Design, vol. 41, no. 10, pp. 730-738, piiS0010448509000153, Oct. 2009.
[10] J. Erickson and K. Whittlesey, "Greedy Optimal Homotopy and Homology Generators," Proc. 16th Ann. ACM-SIAM Symp. Discrete Algorithm, http://dl.acm.orgcitation.cfm?id=1070432.1070581 , pp. 1038-1046, 2005.
[11] C. Chen and D. Freedman, "Quantifying Homology Classes," Science, pp. 169-180,, 2008.
[12] T.K. Dey, A.N. Hirani, and B. Krishnamoorthy, "Optimal Homologous Cycles, Total Unimodularity, and Linear Programming," SIAM J. Computing, vol. 40, no. 4, pp. 1026-1044, 2011.
[13] E. Chambers, J. Erickson, and A. Nayyeri, "Minimum Cuts and Shortest Homologous Cycles," Proc. Symp. Computer Geometry, 2009.
[14] G.F. Italiano, Y. Nussbaum, P. Sankowski, and C. Wulff-Nilsen, "Improved Algorithms for Min Cut and Max Flow in Undirected Planar Graphs," Proc. 43rd Ann. ACM Symp. Theory of Computing (STOC), pp. 313-322, 2011.
[15] S. Cabello, E. Colin de Verdière, and F. Lazarus, "Finding Shortest Non-Trivial Cycles in Directed Graphs on Surfaces," Proc. Ann. Symp. Computational Geometry, pp. 156-165, , 2010.
[16] E.C.d. Verdière and F. Lazarus, "Optimal System of Loops on an Orientable Surface," Proc. 43rd Symp. Foundations of Computer Science (FOCS '02), pp. 627-636, http://dl.acm.orgcitation.cfm? id=645413.652186 , 2002.
[17] F. Lazarus, M. Pocchiola, G. Vegter, and A. Verroust, "Computing a Canonical Polygonal Schema of an Orientable Triangulated Surface," Proc. 17th Ann. Symp. Computer Geometry (SCG '01), pp. 80-89,, 2001.
[18] 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, http://dx.doi. org/10.1109TVCG.2009.103, July 2010.
[19] S.-Q. Xin, Y. He, and C.-W. Fu, "Efficiently Computing Exact Geodesic Loops within Finite Steps," IEEE Trans. Visualization and Computer Graphics, vol. 18, no. 6, pp. 879-889, June 2012.
[20] F. Hétroy, "Constriction Computation Using Surface Curvature," Eurographics (Short Paper), pp. 1-4, http://hal.inria.frinria-00001135, 2005.
[21] F. Hétroy and D. Attali, "Detection of Constrictions on Closed Polyhedral Surfaces," Proc. Symp. Data Visualisation (VISSYM '03), pp. 67-74, http://dl.acm.orgcitation.cfm?id=769922.769929 , 2003.
[22] H. Edelsbrunner, D. Letscher, and A. Zomorodian, "Topological Persistence and Simplification," Proc. 41st Ann. Symp. Foundation of Computer Sciences, pp. 454-, http://dl.acm.orgcitation.cfm?id= 795666.796607 , 2000.
[23] A. Gyulassy, M. Duchaineau, V. Natarajan, V. Pascucci, E. Bringa, A. Higginbotham, and B. Hamann, "Topologically Clean Distance Fields," IEEE Trans. Visualization and Computer Graphics, vol. 13, no. 6, pp. 1432-1439, 70603 , Nov. 2007.
[24] L. Liu, E.W. Chambers, D. Letscher, and T. Ju, "A Simple and Robust Thinning Algorithm on Cell Complexes," Computer Graphics Forum, vol. 29, no. 7, pp. 2253-2260, 2010.
[25] L. Shapira, A. Shamir, and D. Cohen-Or, "Consistent Mesh Partitioning and Skeletonisation Using the Shape Diameter Function," Visual Computer, vol. 24, no. 4, pp. 249-259, , Mar. 2008.
[26] A. Golovinskiy and T. Funkhouser, "Consistent Segmentation of 3D Models," Computers and Graphics (Shape Modeling Int'l 09), vol. 33, no. 3, pp. 262-269, June 2009.
[27] F. Chazal, L.J. Guibas, S.Y. Oudot, and P. Skraba, "Analysis of Scalar Fields over Point Cloud Data," Proc. 20th Ann. ACM-SIAM Symp. Discrete Algorithms (SODA '09), pp. 1021-1030, http://dl. acm.orgcitation.cfm?id=1496770.1496881 , 2009.
[28] H. Edelsbrunner and J. Harer, Computational Topology: An Introduction, series Applied Math. Am. Math. Soc., http:// books. google.combooks?id=MDXa6gFRZuIC , 2010.
[29] H. Edelsbrunner and J. Harer, Computational Topology: An Introduction. Am. Math. Soc., 2010.
[30] H. Si, "Constrained Delaunay Tetrahedral Mesh Generation and Refinement," Finite Elements Analysis Design, vol. 46, pp. 33-46, http://dl.acm.orgcitation.cfm?id=1660153.1660230 , Jan. 2010.
[31] S. Mauch, "Closest Point Transform to a Triangle Surface," cpthtml3/, 2004.
[32] J.J. van Wijk and A.M. Cohen, "Visualization of Seifert Surfaces," IEEE Trans. Visualization and Computer Graphics, vol. 12, no. 4, pp. 485-496, http://dl.acm.orgcitation.cfm?id=1137246.1137503 , July 2006.
15 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool