1. Mesh Parameterizations Lizheng Lu Lulz_zju@yahoo.com.cn Oct. 19, 2005

2. Overview Introduction Planar Methods Non-Planar Methods Mean Value Methods Spherical Methods Summary

3. Motivation(1) Analysis on surfaces is usually performed in Eucli- dean plane, using appropriate (local) coordinates. ⇒ One has to assign to every surface point a parameter value in the plane The result of the analysis often depends on the choice of the parameterization.

4. Example: B-Spline Interpolation

5. Motivation(2) Q: What is a good parameterization ? A: One that preserve all the (basic) geometry length, angles, area,... ⇒ isometric parameterization but : possible only for developable surfaces e.g., there will always be distortion ! Try to keep the distortion as small as possible (change of length, area, angles,... )

6. Motivation(3): Applications Many operations, manipulations on/with surfaces require a parameterization as a preliminary step. e.g.: Texture mapping Surface fitting Hierarchical representations Mesh conversion Morphing & Deformation

7. Motivation: Applications of Parameterizations

10. Morphing

11. Problem Description For a triangulated set of data points find a parameteration with minimal distortion

12. Classifications Conformal mapping No distortion in angles Equiareal mapping No distortion in areas Isometric mappings No distortion, but usually impossible

13. Desirable Properties With minimal distortion So how to measure and minimize it? Guaranteeing one-to-one mapping Avoid overlapping, degenerating, flipping Most difficult and critical! Robustness Time and space efficiency Process meshes with genus, if possible

14. Previous Methods: Classifications Planar methods Early works Cube/Polycube methods Spherical methods Partition methods … Goal: Minimizing distortion for diverse meshes

15. Overview Introduction Planar Methods Non-Planar Methods Mean Value Methods Spherical Methods Summary

16. Theory Foundation Given: A planar 3-connected graph Boundary fixed to a convex shape in R 2 Result: Interior vertices form a planar triangular Each vertex is some convex combination of its neighbors

17. Main Challenge Measure of distortion Ratio of singular values (Hormann & Greiner 1998) Conformal (Levy, 2002) Dirichlet energy (Guskov, 2002) Mean value (Floater, 2003,2005 & Tao Ju,2005) Boundary fixing Choose the shape, e.g.. Circle, square, etc. Choose the distribution Seamless merging Partition and cutting

18. Main References M. S. Floater. Parameterization and smooth approximation of surface triangulations. CAGD, 1997, 14(3):231-250. Hormann, Greiner: MIPS: An efficient global parametrization method, in: Curve and Surface Design: Saint−Malo1999,153−162 M. S. Floater and M. Reimers. Meshless parameterization and surface reconstruction. CAGD, 2001, 18(2):77-92. M. S. Floater, Mean value coordinates, CAGD, 2003,20(1), 19-27. M. S. Floater, One-to-one piecewise linear mappings over triangulations, Math. Comp. 2003,72(242), 685-696. M. S. Floater and K. Hormann, Surface Parameterization: a Tutorial and Survey, in Advances in Multiresolution for Geometric Modelling, N. A. Dodgson, M. S. Floater, and M. A. Sabin (eds.), Springer-Verlag, Heidelberg, 2004, 157-186.

19. Linear Methods: Idea Fixing the boundary of the mesh onto an unit circle an unit square

20. Linear Methods: Idea For interior mesh points: ⇒ Forming a linear system.

21. Choices of the Weights Uniform: Chord length: Centripetal: Mean value:

22. Shortcomings Severe distortion Topology limiting Can't process non genus-zero meshes Introduce other artifacts Such as cutting seams

23. Non-linear Methods [Hormann et al. 1999] MIPS MIPS [Piponi et al. 2000] Seamless texture mapping of subdivision surfaces by model pelting and texture blending. SIGGRAPH [Sander et al. 2001] Texture mapping progressive meshed. SIGGRAPH [Zigelman et al. 2001] Texture mapping using surface flattening via multi-dimensional scaling. TVCG, 8(2), 198-207

24. Overview Introduction Planar Methods Non-Planar Methods Mean Value Methods Spherical Methods Summary

25. How to Obtain Good Parameterization Mesh independence? Very difficult Less distortion? Maybe, defining better measure function Possible method for minimizing distortion Choosing possible mapping domains! Sphere, Cube/polycub, Simplified domains...

26. Main References(1) Spherical Domain Sheffer, A., Gotsman, C., Dyn, N. 2004. Robust Spherical Parameterization of Triangular Meshes. Computing, 72(1-2), 185 – 193. Praun, E., Hoppe, H. Spherical Parametrization and Remeshing. SIGGRAPH2003. Gotsman, C., Gu, X., Sheffer, A. Fundamentals of Spherical Parameterization for 3D Meshes. SIGGRAPH 2003. Alexa, M. Recent advances in mesh morphing. 2002. Computer Graphics Forum, 21(2), 173-196. Grimm, C. Simple manifolds for surface modeling and parametrization. Shape Modeling International 2002. Haker, S., Angenent, S., et al. Conformal surface parameterization for texture mapping. 2000. TVCG, 6(2), 181-189. Kobbelt, L.P., Vorsatz, J., Labisk, U., Seidel, J.-p.. A shrink-wrapping approach to remeshing polygonal surfaces. 1999. CGF. 18(3), 119-129. Kent, J., Carlson, W., Parent, R. 1992. Shape transformation for polyhedral objects. SIGGRAPH 1992, 47-54.

27. Main References(2) Cube/Polycube Domain Tarini, M., Hormann, K., Cigononi, P., Montani, C. PolyCube-Maps. SIGGRAPH 2004.

28. Main References(3) Simplified Domains Schreiner, J., Asirvatham, A, Praun, E., Hoppe, H. Inter-Surface Mapping. SIGGRAPH 2004. Khodakovsky, A., Litke, N., Schr ö der, P. Globally Smooth Parameterizations with Low Distortion. SIGGRAPH 2003. Gu, X., Gortler, J., Hoppe, H. Geometry images. SIGGRAPH 2002. Sorkine, O., Cohen-or, D., et al. Bounded-distortion piecewise mesh parametrization. 2002. IEEE Visualization, 355-362. Praun, E. Sweldens, W. Schr ö der, P. Consistent mesh parametrizations. SIGGRAPH 2001. Guskov, I., Vidimce, K., Sweldens, W., Schr ö der, P. Normal meshes. SIGGRAPH 2000. Lee, A., Dobkin, D., Sweldens, W., Schr ö der, P. Multiresolution mesh morphing. SIGGRAPH 1999. Hoppe, H. Progressive meshes. SIGGRAPH 1996, 99-108.

29. Example Spherical Methods

30. Example Polycube Methods

31. Example Simplification/Cutting Methods

32. Overview Introduction Planar Methods Non-Planar Methods Mean Value Methods Spherical Methods Summary

33. Mean Value Coordinates for Closed Triangular Mesh Tao Ju, Scott Schaefer, Joe Warren Rice University SIGGRAPH2005

34. Mean Value Methods References M. S. Floater. Mean value coordinates. CAGD, 14(3):19 – 27, 2003. M. S. Floater. Mean value coordinates in 3D. CAGD, 22(7):623 – 631, 2005. Tao, Ju Scott Schaefer, Joe Warren. Mean Value Coordinates for Closed Triangular Meshes. SIGGRAPH 2005.

35. Barycentric Coordinates Give, find weights such that with barycentric coordinates

36. Boundary Value Interpolation Given, compute such that Given values at, construct a function Good properties: Interpolates values at vertices Linear on boundary Smooth on interior

37. Continuous Barycentric Coordinates Discrete Continuous

38. Mean Value Interpolation Properties: Interpolates boundary Generates smooth function Reproduces linear functions

39. Relation to Discrete Coordinates MV coordinates ⇒ Closed-form solution of continuous interpolant for piecewise linear shapes Discrete Continuous

40. 3D Mean Value Coordinates Project surface onto sphere centered at v m = mean vector (integral of unit normal over spherical triangle) Stokes ’ Theorem:

41. Computing the Mean Vector Given spherical triangle, compute mean vector (integral of unit normal) Build wedge with face normals Apply Stokes ’ Theorem,

42. Interpolant Computation Compute mean vector: Calculate weights By Sum over all triangles

43. Implementation Considerations Special cases On boundary (co-planar) Numerical stability Small spherical triangles Large meshes Pseudo-code provided in paper

44. Applications Boundary Value Problems

45. Applications Solid Textures

46. Applications Surface Deformations Real-time! Control MeshSurfaceComp. WeightsDeformation 216 Triangles30k Triangles1.9 Sec.0.03 Sec. Initial mesh

47. Summary Integral formulation for closed surfaces Closed-form solution for triangle meshes Numerically stable evaluation Applications Boundary Value Interpolation Volumetric textures Surface Deformation

48. Overview Introduction Planar Methods Non-Planar Methods Mean Value Methods Spherical Methods Summary

49. Challenges on Spherical Domain Robustness Non-overlapping -->> Difficult and critical 1-to-1 spherical map -->> Required Less distortion Diverse meshes -->> Highly deformed Oversampling/downsamping -->>Irregular So, how to miminize it? Visually pleasing, regular, …

50. Previous Spherical Methods(1) [Kent et al. 92]Shape Transformation for Polyhedral Objects. SIGGRAPH. Projections Methods: Convex and Star-Shaped Objects Methods using model knowledge Physically-Based Simulation Simulating balloon inflation process Hybrid methods Unsolved problem, 1-to-1 map? …

51. Previous Spherical Methods(2) [Shapiro & Tal 98] Polygon Realization for Shape Transformation. The Visu. Comp. 8-9,429-444. Limitations Difficult to optimize, due to greedy nature Lack desirable mathematical properties So simple, inefficient to large mesh

52. Previous Spherical Methods(3) [Kobbelt et al. 99] A Shrink-wrapping Approach to Remeshing Polygonal Surfaces. CGF, 18(3),119-129. [Alexa 00] Merging Polyhedral Shapes with Scattered Features. The Visu. Comp., 16(1), 26-37. [Alexa 02] Recent Advances in Mesh Morphing. CGF, 21(2), 173-196. Heuristic iterative No guarantee to converge Sometimes invalid embedding [Alexa 02] Several heuristics to converge validness

53. Previous Spherical Methods(4) [Haker 00]Conformal Surface Parameterization for Texture Mapping. TVCG, 6(2),181-189. Conformal mapping Remove a single point, harmonic map: remain surface  an infinite plane Stereographic projection: plane  sphere Limitations: No guarantee to embedding despite bijective and conformal map

54. Previous Spherical Methods(5) [Grimm 02] Simple manifolds for surface modeling and parameterization. Shape Modeling International. Remark: A priori chart partitions constrain spherical parameterization

55. Previous Spherical Methods(6) [Sheffer 04] Robust Spherical Parameterization of Triangular Meshes. Computing, 72(1-2), 185 – 193. Angle-based method Constrained nonlinear system Valid embedding guaranteeing Limitations Highly no-linear optimization Lack efficient numerical computation procedure

56. Spherical Methods(1) Spherical Parameterization and Remeshing Emil Praun Hugues Hoppe University of Utah Microsoft Research SIGGRAPH2003

57. Main References GU, X., GORTLER, S., AND HOPPE, H. 2002. Geometry images. ACM SIGGRAPH 2002, pp. 355-361. SANDER, P., GORTLER, S., SNYDER, J., HOPPE, H. Signal-specialized parameterization. Eurographics Workshop on Rendering 2002, pp. 87-100. SANDER, P., SNYDER, J., GORTLER, S., AND HOPPE, H. 2001. Texture mapping progressive meshes. ACM SIGGRAPH 2001, pp. 409-416. PRAUN, E., SWELDENS, W., AND SCHR Ö DER, P. 2001. Consistent mesh parametrizations. ACM SIGGRAPH 2001, pp. 179-184. HAKER, S., ANGENENT, S., TANNENBAUM, S., KIKINIS, R., SAPIRO, G., AND HALLE, M. 2000. Conformal surface parametrization for texture mapping. IEEE TVCG, 6(2), pp. 181-189. LOSASSO, F., HOPPE, H., SCHAEFER, S., AND WARREN, J. 2003. Smooth geometry images. Submitted for publication. HOPPE, H. 1996. Progressive meshes. ACM SIGGRAPH 96, pp. 99-108. DAVIS, G. 1996. Wavelet image compression construction kit. http://www.geoffdavis.net/dartmouth/wavelet/wavelet.html. http://www.geoffdavis.net/dartmouth/wavelet/wavelet.html

58. Scope Assumed meshes Geneus-0 Manifold Closed, can handle open ones also -->> Topology equal to a sphere!!

59. [Gu et al. ’ 02] Motivation: Geometry Images [Gu et al. ’ 02] Completely regular sampling Completely regular sampling Geometry image 257 x 257; 12 bits/channel Geometry image 257 x 257; 12 bits/channel 3D Geometry 3D Geometry

60. Motivation: Geometry Images Geometry Images [Gu et al. ’ 02] No connectivity to store Render without memory gather operations No vertex indices No texture coordinates Regularity allows use of image processing tools

61. Spherical Parametrization Genus-0 models: no a priori cuts Genus-0 models: no a priori cuts Geometry image 257 x 257; 12 bits/channel Geometry image 257 x 257; 12 bits/channel

62. Contribution Genus-0  no constraining cuts Less distortion in map; Better compression New applications: Morphing GPU splines DSP

63. Overview Original Spherical parametrization Geometry image Remesh

64. Process overview

65. Outline Spherical parametrization Spherical remeshing Results & applications

66. Spherical Parametrization Mesh M Sphere S [Kent et al. ’92] [Haker et al. 2000] [Alexa 2002] [Grimm 2002] [Sheffer et al. 2003] [Gotsman et al. 2003] Goals: Robustness Good sampling [Hoppe 1996] [Sander et al. 2001] [Hormann et al. 1999] [Hormann et al. 1999] [Sander et al. 2002]  Coarse-to-fine  Stretch metric

67. Coarse-to-Fine Algorithm Convert to progressive mesh [Hoppe 96] Convert to progressive mesh [Hoppe 96] Parametrize coarse-to-fine Maintain embedding & minimize stretch

68. Before V-split: No degenerate/flipped   1-ring kernel  Apply V-split: No flips if V inside kernel V Coarse-to-Fine Algorithm

69. Before V-split: No degenerate/flipped   1-ring kernel  Apply V-split: No flips if V inside kernel Optimize stretch: No degenerate  (they have  stretch) V Coarse-to-Fine Algorithm

70. Traditional Conformal Metric Preserve angles but “ area compression ” Bad for sampling using regular grids

71. [Sander et al. 2001] [Sander et al. 2002] Penalizes undersampling Better samples the surface Stretch Metric

72. Regularized Stretch Stretch alone is unstable Add small fraction of inverse stretch without with

73. Outline Spherical parametrization Spherical remeshing Results & applications

74. Domains and Their Sphere Maps Tetrahedron Octahedron Cube

75. Domain Unfoldings

76. Boundary Constraints

77. Spherical Image Topology

80. Outline Spherical parametrization Spherical remeshing Results & applications

81. Example Results

82. Results

84. David Model courtesy of Stanford University

85. Timing Results Model# FacesInitialOptimized Cow23,2167 min.65 sec. David60,0008 min.80 sec. Bunny69,63010 min.1.5 min. Horse96,94815 min.2.5 min. Gargoyle200,00023 min.4 min. Tyrannosaurus200,00025 min.4 min. Pentium IV 3GHz, optimized code

86. Rendering interpret domain render tessellation

87. Level-of-Detail Control n=1 n=2 n=4 n=8 n=16 n=32 n=64

88. Align meshes & Interpolate Geometry Images

89. Geometry Compression Image wavelets Boundary extension rules Spherical topology Infinite C 1 lattice* Globally smooth parametrization* *(except edge midpoints)

90. Compression Results 12 KB3 KB1.5 KB

91. Compression Results

92. Smooth Geometry Images 33x33 geometry image C 1 surface GPU 3.17 ms [Losasso et al. 2003] Ordinary Uniform Bicubic B-spline

93. Conclusions Spherical parametrization Guaranteed one-to-one New construction for geometry images Specialized to genus-0 No a priori cuts  better performance New boundary extension rules Effective compression, DSP, GPU splines, …

94. Future Work Explore DSP on unfolded octahedron 4 singular points at image edge midpoints Fine-to-coarse integrated metric tensors Faster parametrization; signal-specialized map Direct D  S  M optimization Consistent inter-model parametrization

95. Spherical Methods(2) Fundamentals of Spherical Parameterization for 3D Meshes Craig Gotsman 1, Xianfeng Gu 2, Alla Sheffer 1 1.Technion – Israel Inst. of Tech. 2.Harvard University. SIGGRAPH2003

96. Main References GU, X., AND YAU, S.-T. 2002. Computing Conformal Structures of Surfaces. Communications in Information and Systems, 2,2, 121-146. LEVY, B., PETITJEAN, S., RAY, N., AND MAILLOT, J. 2002. Least Squares Conformal Maps for Automatic Texture Atlas Generation.TOG,21,3,362-671 ALEXA, M. 2000. Merging Polyhedral Shapes with Scattered Features. The Visual Computer 16, 1, 26-37. HAKER, S., ANGENENT, S., TANNENBAUM, A., et al. 2000. Conformal Surface Parameterization for Texture Mapping. IEEE TVCG, 6, 2, 1-9. LOVASZ, L., AND SCHRIJVER, A. 1999. On the Nullspace of a Colin de Verdiere Matrix. Annales de l'Institute Fourier 49, 1017-1026. COLEMAN, T.F., LI, Y. 1996. An Interior Trust Region Approach for Nonlinear Minimization Subject to Bounds. SIAM J. on Optimi.,6, 418-445. On a New Graph Invariant and a Criterion for Planarity. In Graph Structure Theory. 1993. (N. Robertson, P. Seymour,Eds.) Contemporary Mathematics, AMS, 137-147. TUTTE. W.T. 1963. How to Draw a Graph. Proc. London Math. Soc. 13, 3, 743-768.

97. Scope Assumed meshes Geneus-0 Manifold Closed -->> Topology equal to a sphere!!

98. Main Idea Overview Nonlinear extension of the linear theory barycentric coordinates 2D: General Normalized Laplacian operator 3D: Laplace-Beltrami operator [Gu&Yau02] Spectral Graph Theory CdV(Colin de Verdiere) number CdV eigenvalue, eigenvector CdV nullspace

99. Spectral Graph Theory: Basic Theorem Given: Planar 3-connected graph in Result: Each vetex is some convex combination of its neighbors, projected the on sphere Valid embedding

100. Barycentric Coordinates Planar Case Interior edge e=(i,j), assign weight, such that All other entries (i,j), let Embed boundary vertex to a closed convex region Solve linear systems Laplace equation

101. Barycentric Coordinates Spherical Case Define Laplace operator But, restrict to be symmetric

102. Barycentric Coordinates Extension [Gu & Yau 02] Laplace-Beltrami operator [Gu & Yau 02] Inspired by class differential geometry Nonlinear system Bijective embedding a continuous Riemann surface on the sphere But, how is the discrete case, e.g., mesh?

103. Spectral Graph Theory: CdV Number Given n-vretex graph G=  M(G) is a symmetric matrix Spectrum of M CdV number Maximal integer such that

104. Spectral Graph Theory: Nullspace Embedding [Lovasz & Schrijver 99] Supposed CdV eigenvalue=0 w.l.o.g CdV eigenvectors be coordinates vectors Result G describes the edges of a convex polyhedron in R 3 containing the origin

105. Spherical Nullspace Embeddings System 4n unknowns

106. Geometric Interpretation of the Embeddings

107. System Analysis Properties Quadratic non-linear Solution non-unique Degenerate always Solving the system fsolve procedure of MATLAB ---- a subspace trust region procedure [Coleman & Li 1996]

108. Example

109. Conclusion Non-linear  Large mesh Degenerate  Robustness Solution with degree of freedom How to control it? Generate to higher genus? Need further improved …

110. Summary of Spherical Methods Good Properties Equivalent to sphere  Most meshes Less distortion  Remedy metric Though, difficult to control! Needn ’ t prior cutting/partition  Mesh independence Better for application  Morphing

111. Summary of Spherical Methods Future Works Generalize to non genus-0 meshes? Associate with partition? Associate with better distortion metric? Consistent spherical parameterizations among several models (feature correspondence) Implement acceleration?

112. Q&A Thank You!