Spherical Parameterization Balancing Angle and Area Distortions

This work presents a novel framework for spherical mesh parameterization. An efficient angle-preserving spherical parameterization algorithm is introduced, which is based on dynamic Yamabe flow and the conformal welding method with solid theoretic foundation. An area-preserving spherical parameterization is also discussed, which is based on discrete optimal mass transport theory. Furthermore, a spherical parameterization algorithm, which is based on the polar decomposition method, balancing angle distortion and area distortion is presented. The algorithms are tested on 3D geometric data and the experiments demonstrate the efficiency and efficacy of the proposed methods.

[1]  Donald E. Marshall,et al.  Convergence of a Variant of the Zipper Algorithm for Conformal Mapping , 2007, SIAM J. Numer. Anal..

[2]  Christian Rössl,et al.  Curvilinear Spherical Parameterization , 2006, IEEE International Conference on Shape Modeling and Applications 2006 (SMI'06).

[3]  Pierre Alliez,et al.  An Optimal Transport Approach to Robust Reconstruction and Simplification of 2d Shapes , 2022 .

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

[5]  Ayellet Tal,et al.  Polyhedron realization for shape transformation , 1998, The Visual Computer.

[6]  D. Chand,et al.  On Convex Polyhedra , 1970 .

[7]  Lei Zhu,et al.  Area-Preserving Mappings for the Visualization of Medical Structures , 2003, MICCAI.

[8]  R. McCann.  A Convexity Principle for Interacting Gases , 1997 .

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

[10]  M. Ben-Chen,et al.  Can Mean-Curvature Flow Be Made Non-Singular? , 2012, 1203.6819.

[11]  Mathieu Desbrun,et al.  Blue noise through optimal transport , 2012, ACM Trans. Graph..

[12]  Lei Zhu,et al.  Optimal Mass Transport for Registration and Warping , 2004, International Journal of Computer Vision.

[13]  Guillermo Sapiro,et al.  Conformal Surface Parameterization for Texture Mapping , 1999 .

[14]  Xin Zhao,et al.  Area-Preservation Mapping using Optimal Mass Transport , 2013, IEEE Transactions on Visualization and Computer Graphics.

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

[16]  Allen R. Tannenbaum,et al.  Texture Mapping via Optimal Mass Transport , 2010, IEEE Transactions on Visualization and Computer Graphics.

[17]  Marc Alexa,et al.  Recent Advances in Mesh Morphing , 2002, Comput. Graph. Forum.

[18]  Gallagher Pryor,et al.  3D nonrigid registration via optimal mass transport on the GPU , 2009, Medical Image Anal..

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

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

[21]  Michael M. Kazhdan,et al.  Can Mean‐Curvature Flow be Modified to be Non‐singular? , 2012, Comput. Graph. Forum.

[22]  Y. Brenier.  Polar Factorization and Monotone Rearrangement of Vector-Valued Functions , 1991 .

[23]  Quentin Mérigot,et al.  A Multiscale Approach to Optimal Transport , 2011, Comput. Graph. Forum.

[24]  Alla Sheffer,et al.  Practical spherical embedding of manifold triangle meshes , 2005, International Conference on Shape Modeling and Applications 2005 (SMI' 05).

[25]  Hans-Peter Seidel,et al.  A Shrink Wrapping Approach to Remeshing Polygonal Surfaces , 1999, Comput. Graph. Forum.

[26]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[27]  U. Pinkall,et al.  Discrete conformal maps and ideal hyperbolic polyhedra , 2010, 1005.2698.

[28]  Bruce Fischl,et al.  FreeSurfer , 2012, NeuroImage.

[29]  Reinhard Klein,et al.  An Adaptable Surface Parameterization Method , 2003, IMR.

[30]  Olivier Lablée,et al.  Spectral Theory in Riemannian Geometry , 2015 .

[31]  L. Kantorovich.  On a Problem of Monge , 2006 .

[32]  Keenan Crane,et al.  Robust fairing via conformal curvature flow , 2013, ACM Trans. Graph..

[33]  Xianfeng Gu,et al.  A discrete uniformization theorem for polyhedral surfaces , 2013, Journal of Differential Geometry.

[34]  Neil A. Dodgson,et al.  Advances in Multiresolution for Geometric Modelling , 2005 .

[35]  Wei Zeng,et al.  Area Preserving Brain Mapping , 2013, 2013 IEEE Conference on Computer Vision and Pattern Recognition.

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

[37]  K. Hormann,et al.  MIPS: An Efficient Global Parametrization Method , 2000 .

[38]  Leonidas J. Guibas,et al.  Earth mover's distances on discrete surfaces , 2014, ACM Trans. Graph..

[39]  Wei Zeng,et al.  Ricci Flow for Shape Analysis and Surface Registration: Theories, Algorithms and Applications , 2013 .

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

[41]  S. Yau,et al.  Variational Principles for Minkowski Type Problems, Discrete Optimal Transport, and Discrete Monge-Ampere Equations , 2013, 1302.5472.

[42]  Franz Aurenhammer,et al.  Power Diagrams: Properties, Algorithms and Applications , 1987, SIAM J. Comput..

[43]  DesbrunMathieu,et al.  Blue noise through optimal transport , 2012 .

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