This Article 
 Bibliographic References 
 Add to: 
Surface Mapping Using Consistent Pants Decomposition
July/August 2009 (vol. 15 no. 4)
pp. 558-571
Xin Li, Louisiana State University, Baton Rouge
Xianfeng Gu, Stony Brook University, Stony Brook
Hong Qin, Stony Brook University, Stony Brook
Surface mapping is fundamental to shape computing and various downstream applications. This paper develops a pants decomposition framework for computing maps between surfaces with arbitrary topologies. The framework first conducts pants decomposition on both surfaces to segment them into consistent sets of pants patches (a pants patch is intuitively defined as a genus-0 surface with three boundaries), then composes global mapping between two surfaces by using harmonic maps of corresponding patches. This framework has several key advantages over existing techniques. First, it is automatic. It can automatically construct mappings for surfaces with complicated topology, guaranteeing the one-to-one continuity. Second, it is general and powerful. It flexibly handles mapping computation between surfaces with different topologies. Third, it is flexible. Despite topology and geometry, it can also integrate semantics requirements from users. Through a simple and intuitive human-computer interaction mechanism, the user can flexibly control the mapping behavior by enforcing point/curve constraints. Compared with traditional user-guided, piecewise surface mapping techniques, our new method is less labor intensive, more intuitive, and requires no user's expertise in computing complicated surface maps between arbitrary shapes. We conduct various experiments to demonstrate its modeling potential and effectiveness.

[1] X. Li, Y. Bao, X. Guo, M. Jin, X. Gu, and H. Qin, “Globally Optimal Surface Mapping for Surfaces with Arbitrary Topology,” IEEE Trans. Visualization and Computer Graphics, vol. 14, no. 4, pp. 805-819, July/Aug. 2008.
[2] D. DeCarlo and J. Gallier, “Topological Evolution of Surfaces,” Proc. Graphics Interface, pp. 194-203, 1996.
[3] T. Kanai, H. Suzuki, and F. Kimura, “Three-Dimensional Geometric Metamorphosis Based on Harmonic Maps,” The Visual Computer, vol. 14, no. 4, pp. 166-176, 1998.
[4] A. Asirvatham, E. Praun, and H. Hoppe, “Consistent Spherical Parameterization,” Proc. Int'l Conf. Computational Science, vol. 2, pp.265-272, 2005.
[5] V. Kraevoy and A. Sheffer, “Cross-Parameterization and Compatible Remeshing of 3D Models,” ACM Trans. Graphics, vol. 23, no. 3, pp. 861-869, 2004.
[6] J. Schreiner, A. Asirvatham, E. Praun, and H. Hoppe, “Inter-Surface Mapping,” Proc. ACM SIGGRAPH '04, vol. 23, no. 3, pp.870-877, 2004.
[7] A. Gregory, A. State, M. Lin, D. Manocha, and M. Livingston, “Feature-Based Surface Decomposition for Correspondence and Morphing Between Polyhedra,” Proc. Computer Animation Conf., pp. 64-71, 1998.
[8] M.S. Floater and K. Hormann, “Surface Parameterization: A Tutorial and Survey,” Advances in Multiresolution for Geometric Modelling, series mathematics and visualization, pp. 157-186, 2005.
[9] A. Sheffer, E. Praun, and K. Rose, “Mesh Parameterization Methods and Their Applications,” Foundations and Trends in Computer Graphics and Vision, vol. 2, no. 2, pp. 105-171, 2006.
[10] J. Kent, W. Carlson, and R. Parent, “Shape Transformation for Polyhedral Objects,” Proc. ACM SIGGRAPH '92, pp. 47-54, 1992.
[11] M. Alexa, “Merging Polyhedral Shapes with Scattered Features,” Proc. Int'l Conf. Shape Modeling and Applications, pp. 202-210, 1999.
[12] M. Zöckler, D. Stalling, and H.-C. Hege, “Fast and Intuitive Generation of Geometric Shape Transitions,” Visual Computer, vol. 16, no. 5, pp. 241-253, 2000.
[13] F. Lazarus and A. Verroust, “Three-Dimensional Metamorphosis: A Survey,” The Visual Computer, vol. 14, nos. 8/9, pp. 373-389, 1998.
[14] A. Lee, D. Dobkin, W. Sweldens, and P. Schröder, “Multiresolution Mesh Morphing,” Proc. ACM SIGGRAPH '99, pp. 343-350, 1999.
[15] E. Praun, W. Sweldens, and P. Schröder, “Consistent Mesh Parameterizations,” Proc. ACM SIGGRAPH '01, pp. 179-184, 2001.
[16] C. Carner, M. Jin, GuX., and H. Qin, “Topology-Driven Surface Mappings with Robust Feature Alignment,” Proc. IEEE Visualization Conf., pp. 543-550, 2005.
[17] X. Gu and S.-T. Yau, “Global Conformal Surface Parameterization,” Proc. Symp. Geometry Processing, pp. 127-137, 2003.
[18] J. Erickson and K. Whittlesey, “Greedy Optimal Homotopy and Homology Generators,” Proc. ACM-SIAM Symp. Discrete Algorithms, pp.1038-1046, 2005.
[19] T.K. Dey, K. Li, and J. Sun, “On Computing Handle and Tunnel Loops,” Proc. Int'l Conf. Cyberworlds, pp. 357-366, 2007.
[20] A. Hatcher, P. Lochak, and L. Schneps, “On the Teichmüller Tower of Mapping Class Groups,” J. die Reine und Angewandte Math., vol. 521, pp. 1-24, 2000.
[21] E. Verdière and F. Lazarus, “Optimal Pants Decompositions and Shortest Homotopic Cycles on an Orientable Surface,” J. ACM, vol. 54, no. 4, p. 18, 2007.
[22] M. Eck, T. DeRose, T. Duchamp, H. Hoppe, M. Lounsbery, and W. Stuetzle, “Multiresolution Analysis of Arbitrary Meshes,” Proc. ACM SIGGRAPH '95, pp. 173-182, 1995.
[23] M.S. Floater, “Mean Value Coordinates.” Computer Aided Geometric Design, vol. 20, no. 1, pp. 19-27, 2003.
[24] J. Bondy and U. Murty, Graph Theory with Applications. North Holland, 1982.
[25] A. Yershova and S. Lavalle, “Deterministic Sampling Methods for Spheres and SO(3),” Proc. IEEE Int'l Conf. Robotics and Automation, pp. 3974-3980, 2004.
[26] D. Badouel, An Efficient Ry-Polygon Intersection, pp. 390-393. Academic Press Professional, Inc., 1990.
[27] T. Cormen, C. Leiserson, R. Rivest, and C. Stein, Introduction to Algorithms. The MIT Press, 2001.
[28] X. Gu, Y. Wang, T. Chan, P. Thompson, and S.T. Yau, “Genus Zero Surface Conformal Mapping and Its Application to Brain Surface Mapping,” IEEE Trans. Medical Imaging, vol. 23, no. 8, pp. 949-958, 2004.
[29] T. Funkhouser, M. Kazhdan, P. Shilane, P. Min, W. Kiefer, A. Tal, S. Rusinkiewicz, and D. Dobkin, “Modeling by Example,” ACM Trans. Graphics, vol. 23, no. 3, pp. 652-663, 2004.
[30] X. Li, X. Guo, H. Wang, Y. He, X. Gu, and H. Qin, “Harmonic Volumetric Mapping for Solid Modeling Applications,” Proc. ACM Symp. Solid and Physical Modeling, pp. 109-120, 2007.

Index Terms:
Mathematics of computing, computer graphics, computational geometry and object modeling, geometric algorithms, languages, and systems.
Xin Li, Xianfeng Gu, Hong Qin, "Surface Mapping Using Consistent Pants Decomposition," IEEE Transactions on Visualization and Computer Graphics, vol. 15, no. 4, pp. 558-571, July-Aug. 2009, doi:10.1109/TVCG.2008.200
Usage of this product signifies your acceptance of the Terms of Use.