Surface Parameterization: a Tutorial and Survey

This paper provides a tutorial and survey of methods for parameterizing surfaces with a view to applications in geometric modelling and computer graphics. We gather various concepts from differential geometry which are relevant to surface mapping and use them to understand the strengths and weaknesses of the many methods for parameterizing piecewise linear surfaces and their relationship to one another.

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

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

[3]  W. J. Gordon,et al.  B-SPLINE CURVES AND SURFACES , 1974 .

[4]  C. Chui,et al.  Approximation theory X : abstract and classical analysis , 2002 .

[5]  Bruno Lévy,et al.  Non-distorted texture mapping for sheared triangulated meshes , 1998, SIGGRAPH.

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

[7]  Craig Gotsman,et al.  High quality compatible triangulations , 2004, Engineering with Computers.

[8]  K. Hormann,et al.  Hierarchical Parametrization of Triangulated Surfaces , 2002 .

[9]  Laxmi Parida,et al.  Constraint-satisfying planar development of complex surfaces , 1993, Comput. Aided Des..

[10]  J. McCartney,et al.  The flattening of triangulated surfaces incorporating darts and gussets , 1999, Comput. Aided Des..

[11]  Erwin Kreyszig,et al.  Differential geometry , 1991 .

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

[13]  A. She.  SURFACE PARAMETERIZATION FOR MESHING BY TRIANGULATION FLATTENING , 2000 .

[14]  Jean-Marc Vézien,et al.  Piecewise surface flattening for non-distorted texture mapping , 1991, SIGGRAPH.

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

[16]  Nur Arad,et al.  Isometric Texture Mapping for Free‐form Surfaces , 1997, Comput. Graph. Forum.

[17]  W. Stuetzle,et al.  HIERARCHICAL COMPUTATION OF PL HARMONIC EMBEDDINGS , 1997 .

[18]  W. T. Tutte.  How to Draw a Graph , 1963 .

[19]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

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

[21]  Bruno Lévy,et al.  Hierarchical least squares conformal map , 2003, 11th Pacific Conference onComputer Graphics and Applications, 2003. Proceedings..

[22]  Eric L. Schwartz,et al.  Applications of computer graphics and image processing to 2D and 3D modeling of the functional architecture of visual cortex , 1988, IEEE Computer Graphics and Applications.

[23]  Adam Finkelstein,et al.  Lapped textures , 2000, SIGGRAPH.

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

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

[26]  A. Sheffer.  Spanning tree seams for reducing parameterization distortion of triangulated surfaces , 2002, Proceedings SMI. Shape Modeling International 2002.

[27]  John C. Hart,et al.  Seamster: inconspicuous low-distortion texture seam layout , 2002, IEEE Visualization, 2002. VIS 2002..

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

[29]  Eric A. Bier,et al.  Two-Part Texture Mappings , 1986, IEEE Computer Graphics and Applications.

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

[31]  E. Sturler,et al.  Surface Parameterization for Meshing by Triangulation Flattenin , 2000 .

[32]  Andrei Khodakovsky,et al.  Hybrid meshes: multiresolution using regular and irregular refinement , 2002, SCG '02.

[33]  Bruno Lévy,et al.  Constrained texture mapping for polygonal meshes , 2001, SIGGRAPH.

[34]  Andrei Khodakovsky,et al.  Multilevel Solvers for Unstructured Surface Meshes , 2005, SIAM J. Sci. Comput..

[35]  Michael S. Floater,et al.  Meshless Parameterization and B-Spline Surface Approximation , 2000, IMA Conference on the Mathematics of Surfaces.

[36]  Igor Guskov.  An anisotropic mesh parameterization scheme , 2004, Engineering with Computers.

[37]  Nira Dyn,et al.  Robust Spherical Parameterization of Triangular Meshes , 2004, Computing.

[38]  Anne Verroust-Blondet,et al.  Interactive texture mapping , 1993, SIGGRAPH.

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

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

[41]  Charlie C. L. Wang,et al.  Surface flattening based on energy model , 2002, Comput. Aided Des..

[42]  Peter Schröder,et al.  Normal meshes , 2000, SIGGRAPH.

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

[44]  Yunjin Lee,et al.  Mesh parameterization with a virtual boundary , 2002, Comput. Graph..

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

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

[47]  Nikola Guid,et al.  The Flattening of Arbitrary Surfaces by Approximation with Developable Stripes , 2000, Workshop on Geometric Modeling.

[48]  Vladimir A. Garanzha.  Maximum norm optimization of quasi-isometric mappings , 2002, Numer. Linear Algebra Appl..

[49]  Marie-Paule Cani,et al.  Hierarchical pattern mapping , 2002, ACM Trans. Graph..

[50]  Günther Greiner,et al.  Remeshing triangulated surfaces with optimal parameterizations , 2001, Comput. Aided Des..

[51]  Ewald Quak,et al.  Tutorials on Multiresolution in Geometric Modelling, Summer School Lecture Notes , 2002 .

[52]  Carl Friedrich Gauss.  Disquisitiones generales circa superficies curvas , 1981 .

[53]  Tamás Várady,et al.  Parameterizing complex triangular meshes , 2003 .

[54]  Alla Sheffer.  Skinning 3D meshes , 2003, Graph. Model..

[55]  K. Hormann,et al.  Triangulating Point Clouds with Spherical Topology , 2002 .

[56]  L. Schumaker,et al.  Surface Fitting and Multiresolution Methods , 1997 .

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

[58]  Mark Meyer,et al.  Generalized Barycentric Coordinates on Irregular Polygons , 2002, J. Graphics, GPU, & Game Tools.

[59]  C. Chui,et al.  Parameterization of Manifold Triangulations , 2002 .

[60]  M. Floater.  Mean value coordinates , 2003, Computer Aided Geometric Design.

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

[62]  Kai Hormann,et al.  Parameterization of Triangulations and Unorganized Points , 2002, Tutorials on Multiresolution in Geometric Modelling.

[63]  Nikos A. Aspragathos,et al.  On using planar developments to perform texture mapping on arbitrarily curved surfaces , 2000, Comput. Graph..

[64]  Eric L. Schwartz,et al.  A Numerical Solution to the Generalized Mapmaker's Problem: Flattening Nonconvex Polyhedral Surfaces , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[65]  Craig Gotsman,et al.  Explicit Surface Remeshing , 2003, Symposium on Geometry Processing.

[66]  Günther Greiner,et al.  Variational Design and Fairing of Spline Surfaces , 1994, Comput. Graph. Forum.

[67]  Michael S. Floater,et al.  Parametric Tilings and Scattered Data Approximation , 1998, Int. J. Shape Model..

[68]  Christian Rössl,et al.  Variations on Angle Based Flattening , 2005, Advances in Multiresolution for Geometric Modelling.

[69]  Alla Sheffer,et al.  Matchmaker: constructing constrained texture maps , 2003, ACM Trans. Graph..

[70]  T. Shimada,et al.  Approximate transformation of an arbitrary curved surface into a plane using dynamic programming , 1991, Comput. Aided Des..

[71]  Scott Schaefer,et al.  Smooth Geometry Images , 2003, Symposium on Geometry Processing.

[72]  Hans-Peter Seidel,et al.  Parameterizing Meshes with Arbitrary Topology , 1998 .

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

[74]  Gabriel Taubin,et al.  Space‐Optimized Texture Maps , 2002 .

[75]  John C. Hart,et al.  Meshed atlases for real-time procedural solid texturing , 2002, TOGS.

[76]  Martin Reimers,et al.  Meshless parameterization and surface reconstruction , 2001, Comput. Aided Geom. Des..

[77]  Michael S. Floater.  Convex Combination Maps , 2001 .

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

[79]  Joe D. Warren,et al.  Barycentric coordinates for convex polytopes , 1996, Adv. Comput. Math..

[80]  Mark Meyer,et al.  Intrinsic Parameterizations of Surface Meshes , 2002, Comput. Graph. Forum.

[81]  Alla Sheffer,et al.  Preconditioners for Indefinite Linear Systems Arising in Surface Parameterization , 2001, International Meshing Roundtable Conference.

[82]  Weiyin Ma,et al.  Parameterization of randomly measured points for least squares fitting of B-spline curves and surfaces , 1995, Comput. Aided Des..

[83]  Gustav Junge.  Besonderheiten der griechischen Mathematik. Dritter Teil. , 1926 .

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

[85]  W. Klingenberg.  A course in differential geometry , 1978 .

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

[87]  Ulrich Pinkall,et al.  Computing Discrete Minimal Surfaces and Their Conjugates , 1993, Exp. Math..

[88]  Alain Fournier,et al.  Conformal Texture Mapping , 1987, Eurographics.

[89]  Bruno Lévy.  Dual domain extrapolation , 2003, ACM Trans. Graph..

[90]  Hong Lin,et al.  Optimal Texture Mapping , 1988, Eurographics.

[91]  Alla Sheffer,et al.  Smoothing an overlay grid to minimize linear distortion in texture mapping , 2002, TOGS.