Global parametrization by incremental flattening

Global parametrization of surfaces requires singularities (cones) to keep distortion minimal. We describe a method for finding cone locations and angles and an algorithm for global parametrization which aim to produce seamless parametrizations with low metric distortion. The idea of the method is to evolve the metric of the surface, starting with the original metric so that a growing fraction of the area of the surface is constrained to have zero Gaussian curvature; the curvature becomes gradually concentrated at a small set of vertices which become cones. We demonstrate that the resulting parametrizations have significantly lower metric distortion compared to previously proposed methods.

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

[2]  David P. Dobkin,et al.  MAPS: multiresolution adaptive parameterization of surfaces , 1998, SIGGRAPH.

[3]  Aaron Hertzmann,et al.  Illustrating smooth surfaces , 2000, SIGGRAPH.

[4]  Dani Lischinski,et al.  Bounded-distortion piecewise mesh parameterization , 2002, IEEE Visualization, 2002. VIS 2002..

[5]  Bruno Lévy,et al.  Least squares conformal maps for automatic texture atlas generation , 2002, ACM Trans. Graph..

[6]  Steven J. Gortler,et al.  Geometry images , 2002, SIGGRAPH.

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

[8]  Andrei Khodakovsky,et al.  Globally smooth parameterizations with low distortion , 2003, ACM Trans. Graph..

[9]  Yalin Wang,et al.  Optimal global conformal surface parameterization , 2004, IEEE Visualization 2004.

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

[11]  Leif Kobbelt,et al.  Automatic Generation of Structure Preserving Multiresolution Models , 2005, Comput. Graph. Forum.

[12]  Keenan Crane,et al.  Rectangular multi-chart geometry images , 2006, SGP '06.

[13]  Pierre Alliez,et al.  Designing quadrangulations with discrete harmonic forms , 2006, SGP '06.

[14]  Valerio Pascucci,et al.  Spectral surface quadrangulation , 2006, ACM Trans. Graph..

[15]  Peter Schröder,et al.  Discrete conformal mappings via circle patterns , 2005, TOGS.

[16]  Pierre Alliez,et al.  Periodic global parameterization , 2006, TOGS.

[17]  Eugene Zhang,et al.  Rotational symmetry field design on surfaces , 2007, ACM Trans. Graph..

[18]  Alla Sheffer,et al.  Mesh parameterization: theory and practice Video files associated with this course are available from the citation page , 2007, SIGGRAPH Courses.

[19]  Konrad Polthier,et al.  QuadCover ‐ Surface Parameterization using Branched Coverings , 2007, Comput. Graph. Forum.

[20]  Bruno Lévy,et al.  Mesh parameterization: theory and practice , 2007, SIGGRAPH Courses.

[21]  Peter Schröder,et al.  Conformal equivalence of triangle meshes , 2008, ACM Trans. Graph..

[22]  Guy Bunin,et al.  A continuum theory for unstructured mesh generation in two dimensions , 2006, Comput. Aided Geom. Des..

[23]  Ligang Liu,et al.  A Local/Global Approach to Mesh Parameterization , 2008, Comput. Graph. Forum.

[24]  Xianfeng Gu,et al.  Discrete Surface Ricci Flow , 2008, IEEE Transactions on Visualization and Computer Graphics.

[25]  Bruno Lévy,et al.  N-symmetry direction field design , 2008, TOGS.

[26]  Craig Gotsman,et al.  Conformal Flattening by Curvature Prescription and Metric Scaling , 2008, Comput. Graph. Forum.

[27]  YANQING CHEN,et al.  Algorithm 8 xx : CHOLMOD , supernodal sparse Cholesky factorization and update / downdate ∗ , 2006 .

[28]  Elaine Cohen,et al.  Localized Quadrilateral Coarsening , 2009, Comput. Graph. Forum.

[29]  D. Bommes,et al.  Mixed-integer quadrangulation , 2009, SIGGRAPH 2009.

[30]  Bruno Lévy,et al.  Geometry-aware direction field processing , 2009, TOGS.

[31]  Elaine Cohen,et al.  Semi‐regular Quadrilateral‐only Remeshing from Simplified Base Domains , 2009, Comput. Graph. Forum.

[32]  Peter Schröder,et al.  A simple geometric model for elastic deformations , 2010, ACM Trans. Graph..

[33]  Shi-Min Hu,et al.  Metric-Driven RoSy Field Design and Remeshing , 2010, IEEE Transactions on Visualization and Computer Graphics.

[34]  Keenan Crane,et al.  Trivial Connections on Discrete Surfaces , 2010, Comput. Graph. Forum.

[35]  Denis Zorin,et al.  Anisotropic quadrangulation , 2010, SPM '10.

[36]  Paolo Cignoni,et al.  Almost Isometric Mesh Parameterization through Abstract Domains , 2010, IEEE Transactions on Visualization and Computer Graphics.

[37]  Daniele Panozzo,et al.  Practical quad mesh simplification , 2010, Comput. Graph. Forum.

[38]  D. Zorin,et al.  Feature-aligned T-meshes , 2010, ACM Trans. Graph..