Bijective maps from simplicial foliations

This paper presents a method for bijective parametrization of 2D and 3D objects over canonical domains. While a range of solutions for the two-dimensional case are well-known, our method guarantees bijectivity of mappings also for a large, combinatorially-defined class of tetrahedral meshes (shellable meshes). The key concept in our method is the piecewise-linear (PL) foliation, decomposing the mesh into one-dimensional submanifolds and reducing the mapping problem to parametrization of a lower-dimensional manifold (a foliation section). The maps resulting from these foliations are proved to be bijective and continuous, and shown to have provably bijective PL approximations. We describe exact, numerically robust evaluation methods and demonstrate our implementation's capabilities on a large variety of meshes.

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

[2]  Denis Zorin,et al.  Locally injective parametrization with arbitrary fixed boundaries , 2014, ACM Trans. Graph..

[3]  Yaron Lipman,et al.  Injective and bounded distortion mappings in 3D , 2013, ACM Trans. Graph..

[4]  Yaron Lipman,et al.  Bounded distortion mapping spaces for triangular meshes , 2012, ACM Trans. Graph..

[5]  Shing-Tung Yau,et al.  Conformal Spherical Parametrization for High Genus Surfaces , 2007, Commun. Inf. Syst..

[6]  Yotam I. Gingold,et al.  Bijective Deformations in Rn via Integral Curve Coordinates , 2014, ArXiv.

[7]  Marc Alexa,et al.  As-rigid-as-possible shape interpolation , 2000, SIGGRAPH.

[8]  Afra Zomorodian,et al.  Computational topology , 2010 .

[9]  Ying He,et al.  Direct-Product Volumetric Parameterization of Handlebodies via Harmonic Fields , 2010, 2010 Shape Modeling International Conference.

[10]  A. W. J. Stoddart,et al.  The shape of level surfaces of harmonic functions in three dimensions. , 1964 .

[11]  Scott Schaefer,et al.  Bijective parameterization with free boundaries , 2015, ACM Trans. Graph..

[12]  Dmitry Sokolov,et al.  Tracing cross-free polylines oriented by a N-symmetry direction field on triangulated surfaces , 2013, ArXiv.

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

[14]  Michael Kerber,et al.  ROBUST GEOMETRIC COMPUTATION , 2012 .

[15]  Bert Jüttler,et al.  Parameterization of Contractible Domains Using Sequences of Harmonic Maps , 2010, Curves and Surfaces.

[16]  Udo Pachner,et al.  P.L. Homeomorphic Manifolds are Equivalent by Elementary 5hellingst , 1991, Eur. J. Comb..

[17]  Yalin Wang,et al.  Volumetric Harmonic Map , 2003, Commun. Inf. Syst..

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

[19]  M. Fiedler.  A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory , 1975 .

[20]  J. W. Alexander.  On the Subdivision of 3-Space by a Polyhedron. , 1924, Proceedings of the National Academy of Sciences of the United States of America.

[21]  Michael S. Floater,et al.  One-to-one piecewise linear mappings over triangulations , 2003, Math. Comput..

[22]  Ying He,et al.  Interior structure transfer via harmonic 1-forms , 2013, Multimedia Tools and Applications.

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

[24]  O. P. Lossers.  Lösung der Aufgabe 741 , 1975 .

[25]  P. Mani,et al.  Shellable Decompositions of Cells and Spheres. , 1971 .

[26]  Aleksej F. Filippov,et al.  Differential Equations with Discontinuous Righthand Sides , 1988, Mathematics and Its Applications.

[27]  R. Laugesen.  Injectivity can fail for higher-dimensional harmonic extensions , 1996 .

[28]  Mary Ellen Rudin,et al.  An unshellable triangulation of a tetrahedron , 1958 .

[29]  Michael S. Floater,et al.  Convex combination maps over triangulations, tilings, and tetrahedral meshes , 2006, Adv. Comput. Math..

[30]  Chee-Keng Yap,et al.  Robust Geometric Computation , 2016, Encyclopedia of Algorithms.

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

[32]  Hans-Peter Seidel,et al.  Topological Construction and Visualization of Higher Order 3D Vector Fields , 2004, Comput. Graph. Forum.

[33]  Teseo Schneider,et al.  Smooth bijective maps between arbitrary planar polygons , 2015, Comput. Aided Geom. Des..

[34]  Chi-Wing Fu,et al.  Parameterization of Star-Shaped Volumes Using Green's Functions , 2010, GMP.

[35]  James F. Glazebrook,et al.  INTRODUCTION TO FOLIATIONS AND LIE GROUPOIDS (Cambridge Studies in Advanced Mathematics 91) , 2005 .

[36]  Baining Guo,et al.  Computing locally injective mappings by advanced MIPS , 2015, ACM Trans. Graph..

[37]  J. J. Gergen.  Note on the Green Function of a Star-Shaped Three Dimensional Region , 1931 .

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

[39]  Patrice Ossona de Mendez,et al.  Bipolar orientations Revisited , 1995, Discret. Appl. Math..

[40]  J. Geelen.  ON HOW TO DRAW A GRAPH , 2012 .

[41]  Edwin E. Moise,et al.  Affine structures in 3-manifolds, V, The triangulation theorem and Hauptvermutung , 1952 .

[42]  Ronen Basri,et al.  Controlling singular values with semidefinite programming , 2014, ACM Trans. Graph..

[43]  Simon A. King.  How to make a triangulation of S^3 polytopal , 2000 .

[44]  Alexander Russell,et al.  Computational topology: ambient isotopic approximation of 2-manifolds , 2003, Theor. Comput. Sci..

[45]  R. Pollack,et al.  Advances in Discrete and Computational Geometry , 1999 .

[46]  Roi Poranne,et al.  Seamless surface mappings , 2015, ACM Trans. Graph..

[47]  Karim A. Adiprasito,et al.  Derived subdivisions make every PL sphere polytopal , 2013 .

[48]  Pierre Alliez,et al.  Integer-grid maps for reliable quad meshing , 2013, ACM Trans. Graph..

[49]  Olga Sorkine-Hornung,et al.  Locally Injective Mappings , 2013 .

[50]  Ieke Moerdijk,et al.  Introduction to Foliations and Lie Groupoids: Prerequisites , 2003 .

[51]  Torbjrn Granlund,et al.  GNU MP 6.0 Multiple Precision Arithmetic Library , 2015 .

[52]  Yaron Lipman,et al.  Orbifold Tutte embeddings , 2015, ACM Trans. Graph..

[53]  Michel Pocchiola,et al.  Tutte's barycenter method applied to isotopies , 2003, Comput. Geom..

[54]  R. Groff.  Piecewise linear homeomorphisms for approximation of invertible maps. , 2003 .

[55]  John J. Gergen.  Mapping of a General Type of Three Dimensional Region on a Sphere , 1930 .

[56]  Jian-Jun Zhang,et al.  Stretch-Minimizing Volumetric Parameterization , 2015, Journal of Computer Science and Technology.

[57]  Jian-Ping Wu,et al.  An efficient and accurate method to compute the Fiedler vector based on Householder deflation and inverse power iteration , 2014, J. Comput. Appl. Math..

[58]  Dmitry Sokolov,et al.  Robust Polylines Tracing for N-Symmetry Direction Field on Triangulated Surfaces , 2013, ACM Trans. Graph..

[59]  Denis Zorin,et al.  Robust field-aligned global parametrization , 2014, ACM Trans. Graph..

[60]  Bruno Benedetti,et al.  Subdivisions, shellability, and collapsibility of products , 2012, Comb..

[61]  Günter M. Ziegler,et al.  Shelling Polyhedral 3-Balls and 4-Polytopes , 1998, Discret. Comput. Geom..

[62]  Yao Jin,et al.  Remeshing‐assisted Optimization for Locally Injective Mappings , 2014, Comput. Graph. Forum.

[63]  Richard E. Goodrick,et al.  Non-simplicially collapsible triangulations of In , 1968, Mathematical Proceedings of the Cambridge Philosophical Society.

[64]  Konstantin Mischaikow,et al.  Vector field design on surfaces , 2006, TOGS.

[65]  Robert Furch,et al.  Zur grundlegung der kombinatorischen topologie , 1924 .

[66]  Guojun Liao,et al.  A note on harmonic maps , 1996 .