Surface parameterization via aligning optimal local flattening

This paper presents a novel parameterization method for a non-closed triangular mesh. For every flattened 1-ring neighbors, we choose a local coordinate frame, and the local geometry structure is represented as local parametric coordinates. Then the global optimal parametric coordinates are attained by aligning all the local parametric planes while preserving the local structure as much as possible. The boundary conditions are not necessary in our method, thus no high distortion appears around the boundary, and distortion is uniformly distributed over parametric domain. In addition, our method can operate directly on mesh surface which has holes without any preprocessing of surface partition. Furthermore, linear constraints are allowed in the parameterization in a least squares sense.

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

[2]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

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

[4]  Pedro V. Sander,et al.  Multi-Chart Geometry Images , 2003, Symposium on Geometry Processing.

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

[6]  Alla Sheffer,et al.  Parameterization of Faceted Surfaces for Meshing using Angle-Based Flattening , 2001, Engineering with Computers.

[7]  Craig Gotsman,et al.  Discrete one-forms on meshes and applications to 3D mesh parameterization , 2006, Comput. Aided Geom. Des..

[8]  Hugues Hoppe,et al.  Spherical parametrization and remeshing , 2003, ACM Trans. Graph..

[9]  Andrew P. Witkin,et al.  Free-form shape design using triangulated surfaces , 1994, SIGGRAPH.

[10]  S T Roweis,et al.  Nonlinear dimensionality reduction by locally linear embedding. , 2000, Science.

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

[12]  Pedro V. Sander,et al.  Texture mapping progressive meshes , 2001, SIGGRAPH.

[13]  Mikhail Belkin,et al.  Laplacian Eigenmaps and Spectral Techniques for Embedding and Clustering , 2001, NIPS.

[14]  Heng Tao Shen,et al.  Principal Component Analysis , 2009, Encyclopedia of Biometrics.

[15]  张振跃,et al.  Principal Manifolds and Nonlinear Dimensionality Reduction via Tangent Space Alignment , 2004 .

[16]  Ron Kimmel,et al.  Texture Mapping Using Surface Flattening via Multidimensional Scaling , 2002, IEEE Trans. Vis. Comput. Graph..

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

[18]  Christian Rössl,et al.  Setting the boundary free: a composite approach to surface parameterization , 2005, SGP '05.

[19]  Michael S. Floater,et al.  Parametrization and smooth approximation of surface triangulations , 1997, Comput. Aided Geom. Des..

[20]  Alla Sheffer,et al.  Fundamentals of spherical parameterization for 3D meshes , 2003, ACM Trans. Graph..

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

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

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

[24]  Konstantin Mischaikow,et al.  Feature-based surface parameterization and texture mapping , 2005, TOGS.

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

[26]  J. Tenenbaum,et al.  A global geometric framework for nonlinear dimensionality reduction. , 2000, Science.

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

[28]  Hugues Hoppe,et al.  Spherical parametrization and remeshing , 2003, ACM Trans. Graph..

[29]  Bruno Lévy,et al.  ABF++: fast and robust angle based flattening , 2005, TOGS.