Surface Mapping Using Consistent Pants Decomposition

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]  Hiromasa Suzuki,et al.  Three-dimensional geometric metamorphosis based on harmonic maps , 1998, The Visual Computer.

[2]  Peter Schröder,et al.  Consistent mesh parameterizations , 2001, SIGGRAPH.

[3]  Elwood S. Buffa,et al.  Graph Theory with Applications , 1977 .

[4]  Steven M. LaValle,et al.  Deterministic sampling methods for spheres and SO(3) , 2004, IEEE International Conference on Robotics and Automation, 2004. Proceedings. ICRA '04. 2004.

[5]  Szymon Rusinkiewicz,et al.  Modeling by example , 2004, ACM Trans. Graph..

[6]  Dinesh Manocha,et al.  Feature-based surface decomposition for correspondence and morphing between polyhedra , 1998, Proceedings Computer Animation '98 (Cat. No.98EX169).

[7]  Hong Qin,et al.  Globally Optimal Surface Mapping for Surfaces with Arbitrary Topology , 2008, IEEE Transactions on Visualization and Computer Graphics.

[8]  Jeff Erickson,et al.  Greedy optimal homotopy and homology generators , 2005, SODA '05.

[9]  Marc Alexa,et al.  Merging polyhedral shapes with scattered features , 1999, Proceedings Shape Modeling International '99. International Conference on Shape Modeling and Applications.

[10]  Hugues Hoppe,et al.  Consistent Spherical Parameterization , 2005, International Conference on Computational Science.

[11]  Francis Lazarus,et al.  Optimal pants decompositions and shortest homotopic cycles on an orientable surface , 2003, JACM.

[12]  Wayne E. Carlson,et al.  Shape transformation for polyhedral objects , 1992, SIGGRAPH.

[13]  Anne Verroust-Blondet,et al.  Three-dimensional metamorphosis: a survey , 1998, The Visual Computer.

[14]  Gauthier Lafruit,et al.  Adaptive 3D Content for Multi-Platform On-Line Games , 2007, CW 2007.

[15]  Alla Sheffer,et al.  Cross-parameterization and compatible remeshing of 3D models , 2004, ACM Trans. Graph..

[16]  Hugues Hoppe,et al.  Inter-surface mapping , 2004, ACM Trans. Graph..

[17]  Paul M. Thompson,et al.  Genus zero surface conformal mapping and its application to brain surface mapping , 2004, IEEE Transactions on Medical Imaging.

[18]  Tony DeRose,et al.  Multiresolution analysis of arbitrary meshes , 1995, SIGGRAPH.

[19]  David P. Dobkin,et al.  Multiresolution mesh morphing , 1999, SIGGRAPH.

[20]  Leila Schneps,et al.  On the Teichmüller tower of mapping class groups , 2000 .

[21]  S. Yau,et al.  Global conformal surface parameterization , 2003 .

[22]  D. Badouel.  An efficient ray-polygon intersection , 1990 .

[23]  Michael S. Floater,et al.  Mean value coordinates , 2003, Comput. Aided Geom. Des..

[24]  Hong Qin,et al.  Topology-driven surface mappings with robust feature alignment , 2005, VIS 05. IEEE Visualization, 2005..

[25]  Douglas DeCarlo,et al.  Topological Evolution of Surfaces , 1996, Graphics Interface.

[26]  Tamal K. Dey,et al.  On Computing Handle and Tunnel Loops , 2007, 2007 International Conference on Cyberworlds (CW'07).

[27]  Hong Qin,et al.  Harmonic volumetric mapping for solid modeling applications , 2007, Symposium on Solid and Physical Modeling.

[28]  Kai Hormann,et al.  Surface Parameterization: a Tutorial and Survey , 2005, Advances in Multiresolution for Geometric Modelling.

[29]  Hans-Christian Hege,et al.  Fast and intuitive generation of geometric shape transitions , 2000, The Visual Computer.