This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Hexagonal Global Parameterization of Arbitrary Surfaces
June 2012 (vol. 18 no. 6)
pp. 865-878
Matthias Nieser, Freie Universität Berlin, Berlin
Jonathan Palacios, Oregon State University, Corvallis
Konrad Polthier, Freie Universität Berlin, Berlin
Eugene Zhang, Oregon State University, Corvallis
HASH(0x3a3e494)

[1] A. Schiftner, M. Höbinger, J. Wallner, and H. Pottmann, “Packing Circles and Spheres on Surfaces,” Proc. SIGGRAPH Asia, pp. 1-8, 2009.
[2] C.S. Kaplan and D.H. Salesin, “Islamic Star Patterns in Absolute Geometry,” ACM Trans. Graphics, vol. 23, no. 2, pp. 97-119, 2004.
[3] Z.J. Wood, P. Schröder, D. Breen, and M. Desbrun, “Semi-Regular Mesh Extraction from Volumes,” Proc. Conf. Visualization (VIS), pp. 275-282, 2000.
[4] N. Ray, W.C. Li, B. Lévy, A. Sheffer, and P. Alliez, “Periodic Global Parameterization,” ACM Trans. Graphics, vol. 25, no. 4, pp. 1460-1485, 2006.
[5] F. Kälberer, M. Nieser, and K. Polthier, “QuadCover - Surface Parameterization Using Branched Coverings,” Computer Graphics Forum, vol. 26, no. 3, pp. 375-384, 2007.
[6] D. Bommes, H. Zimmer, and L. Kobbelt, “Mixed-Integer Quadrangulation,” ACM Trans. Graphics, vol. 28, no. 3, pp. 1-10, 2009.
[7] J. Palacios and E. Zhang, “Rotational Symmetry Field Design on Surfaces,” Proc. SIGGRAPH, p. 55, 2007.
[8] M.S. Floater and K. Hormann, “Surface Parameterization: A Tutorial and Survey,” Advances in Multiresolution for Geometric Modelling, pp. 157-186, Springer Verlag, 2005.
[9] K. Hormann, K. Polthier, and A. Sheffer, “Mesh Parameterization: Theory and Practice,” Proc. SIGGRAPH Asia, no. 11, 2008.
[10] S. Haker, S. Angenent, A. Tannenbaum, R. Kikinis, G. Sapiro, and M. Halle, “Conformal Surface Parameterization for Texture Mapping,” IEEE Trans. Visualization and Computer Graphics, vol. 6, no. 2, pp. 181-189, Apr.-June 2000.
[11] X. Gu and S.-T. Yau, “Global Conformal Surface Parameterization,” Proc. ACM SIGGRAPH Symp. Geometry Processing (SGP), pp. 127-137, 2003.
[12] M. Jin, Y. Wang, S.-T. Yau, and X. Gu, “Optimal Global Conformal Surface Parameterization,” Proc. IEEE Visualization, pp. 267-274, 2004.
[13] L. Kharevych, B. Springborn, and P. Schröder, “Discrete Conformal Mappings Via Circle Patterns,” ACM Trans. Graphics, vol. 25, no. 2, pp. 412-438, 2006.
[14] M. Ben-Chen, C. Gotsman, and G. Bunin, “Conformal Flattening by Curvature Prescription and Metric Scaling,” Computer Graphics Forum, vol. 27, no. 2, pp. 449-458, 2008.
[15] B. Springborn, P. Schröder, and U. Pinkall, “Conformal Equivalence of Triangle Meshes,” ACM Trans. Graphics, vol. 27, pp. 77:1-77:11, Aug. 2008.
[16] S. Dong, S. Kircher, and M. Garland, “Harmonic Functions for Quadrilateral Remeshing of Arbitrary Manifolds,” Computer Aided Geometric Design, vol. 22, pp. 392-423, 2005.
[17] S. Dong, P.-T. Bremer, M. Garland, V. Pascucci, and J.C. Hart, “Spectral Surface Quadrangulation,” Proc. SIGGRAPH '06, pp. 1057-1066, 2006.
[18] Y. Tong, P. Alliez, D. Cohen-Steiner, and M. Desbrun, “Designing Quadrangulations with Discrete Harmonic Forms,” Proc. Fourth Eurographics Symp. Geometry Processing (SGP), 2006.
[19] E. Zhang, J. Hays, and G. Turk, “Interactive Tensor Field Design and Visualization on Surfaces,” IEEE Trans. Visualization Computer Graphics, vol. 13, no. 1, pp. 94-107, Jan./Feb. 2007.
[20] J.L. Helman and L. Hesselink, “Visualizing Vector Field Topology in Fluid Flows,” IEEE Computer Graphics and Applications, vol. 11, no. 3, pp. 36-46, May 1991.
[21] T. Delmarcelle and L. Hesselink, “The Topology of Symmetric, Second-Order Tensor Fields,” IEEE Conf. Visualization, pp. 140-147, 1994.
[22] E. Praun, A. Finkelstein, and H. Hoppe, “Lapped Textures,” Proc. SIGGRAPH, pp. 465-470, 2000.
[23] G. Turk, “Texture Synthesis on Surfaces,” Proc. SIGGRAPH, 2001.
[24] L.Y. Wei and M. Levoy, “Texture Synthesis over Arbitrary Manifold Surfaces,” Proc. SIGGRAPH, pp. 355-360, 2001.
[25] J. Stam, “Flows on Surfaces of Arbitrary Topology,” Proc. SIGGRAPH, 2003.
[26] J.J. van Wijk, “Image Based Flow Visualization for Curved Surfaces,” Proc. IEEE Visualization, pp. 123-130, 2003.
[27] M. Fisher, P. Schröder, M. Desbrun, and H. Hoppe, “Design of Tangent Vector Fields,” Proc. SIGGRAPH, p. 56, 2007.
[28] H. Theisel, “Designing 2D Vector Fields of Arbitrary Topology,” Proc. Eurographics Conf., vol. 21, pp. 595-604, 2002.
[29] W.-C. Li, B. Vallet, N. Ray, and B. Lévy, “Representing Higher-Order Singularities in Vector Fields on Piecewise Linear Surfaces,” IEEE Trans. Visualization and Computer Graphics, vol. 12, no. 5, pp. 1315-1322, Sept./Oct. 2006.
[30] E. Zhang, K. Mischaikow, and G. Turk, “Vector Field Design on Surfaces,” ACM Trans. Graphics, vol. 25, no. 4, pp. 1294-1326, 2006.
[31] G. Chen, G. Esch, P. Wonka, P. Müller, and E. Zhang, “Interactive Procedural Street Modeling,” ACM Trans. Graphics, vol. 27, no. 3, pp. 1-10, 2008.
[32] A. Hertzmann and D. Zorin, “Illustrating Smooth Surfaces,” Proc. SIGGRAPH, pp. 517-526, 2000.
[33] N. Ray, B. Vallet, W.C. Li, and B. Lévy, “N-Symmetry Direction Field Design,” ACM Trans. Graphics, vol. 27, no. 2, pp. 1-13, 2008.
[34] J. Palacios and E. Zhang, “Interactive Visualization of Rotational Symmetry Fields on Surfaces,” IEEE Trans. Visualization and Computer Graphics, vol. 17, no. 7, pp. 947-955, July 2011.
[35] Y.-K. Lai, M. Jin, X. Xie, Y. He, J. Palacios, E. Zhang, S.-M. Hu, and X. Gu, “Metric-Driven Rosy Field Design and Remeshing,” IEEE Trans. Visualization and Computer Graphics, vol. 16, no. 1, pp. 95-108, Jan./Feb. 2010.
[36] M. Zhang, J. Huang, X. Liu, and H. Bao, “A Wave-Based Anisotropic Quadrangulation Method,” ACM Trans. Graphics, vol. 29, pp. 1-8, 2010.
[37] K. Crane, M. Desbrun, and P. Schröder, “Trivial Connections on Discrete Surfaces,” Computer Graphic Forum, vol. 29, no. 5, pp. 1525-1533, 2010.
[38] N. Ray, B. Vallet, L. Alonso, and B. Lévy, “Geometry Aware Direction Field Design,” ACM Trans. Graphics, vol. 29, no. 1, 2009.
[39] K. Xu, D. Cohen-Or, T. Ju, L. Liu, H. Zhang, S. Zhou, and Y. Xiong, “Feature-Aligned Shape Texturing,” ACM Trans. Graphics, vol. 28, no. 5, pp. 1-7, 2009.
[40] M. Meyer, M. Desbrun, P. Schröder, and A.H. Barr, “Discrete Differential Geometry Operators for Triangulated 2-Manifolds,” Visualization and Mathematics III, Springer, 2002.
[41] E. Zhang, H. Yeh, Z. Lin, and R.S. Laramee, “Asymmetric Tensor Analysis for Flow Visualization,” IEEE Trans. Visualization and Computer Graphics, vol. 15, no. 1, pp. 106-122, Jan./Feb. 2009.
[42] J.J. Koenderink and A.J. van Doorn, “Surface Shape and Curvature Scales,” Image and Vision Computing, vol. 10, pp. 557-565, Oct. 1992.
[43] A.N. Hirani, “Discrete Exterior Calculus,” PhD dissertation, 2003.
[44] M. Eck, T. DeRose, T. Duchamp, H. Hoppe, M. Lounsbery, and W. Stuetzle, “Multiresolution Analysis of Arbitrary Meshes,” Proc. SIGGRAPH, pp. 173-182, 1995.
[45] K. Polthier and E. Preuss, “Identifying Vector Field Singularities Using a Discrete Hodge Decomposition,” Visualization and Mathematics III, pp. 113-134, Springer Verlag 2003.
[46] J. Erickson and K. Whittlesey, “Greedy Optimal Homotopy and Homology Generators,” Proc. ACM-SIAM Symp. Discrete Algorithms, pp. 1038-1046, 2005.
[47] F. Kälberer, M. Nieser, and K. Polthier, “Stripe Parameterization of Tubular Surfaces,” Topological Methods in Data Analysis and Visualization: Theory, Algorithms and Applications, Springer Verlag, 2009.
[48] L.-Y. Wei, S. Lefebvre, V. Kwatra, and G. Turk, “State of the Art in Example-Based Texture Synthesis,” Proc. Eurographics, 2009.
[49] Y. Liu, W.-C. Lin, and J. Hays, “Near-Regular Texture Analysis and Manipulation,” Proc. SIGGRAPH, pp. 368-376, 2004.
[50] P. Alliez, G. Ucelli, C. Gotsman, and M. Attene, “Recent Advances in Remeshing of Surfaces,” Shape Analysis and Structuring, Springer, 2005.
[51] X. Gu, S.J. Gortler, and H. Hoppe, “Geometry Images,” ACM Trans. Graphics, vol. 21, no. 3, pp. 355-361, 2002.
[52] P. Alliez, M. Meyer, and M. Desbrun, “Interactive Geometry Remeshing,” Proc. SIGGRAPH, pp. 347-354, 2002.
[53] P. Alliez, E.C. de Verdiere, O. Devillers, and M. Isenburg, “Isotropic Surface Remeshing,” Proc. Shape Modeling Int'l (SMI), p. 49, 2003.
[54] A. Khodakovsky, N. Litke, and P. Schröder, “Globally Smooth Parameterizations with Low Distortion,” Proc. SIGGRAPH '03, 2003.
[55] H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, and W. Stuetzle, “Mesh Optimization,” Proc. SIGGRAPH, pp. 19-26, 1993.
[56] P. Lindstrom and G. Turk, “Image-Driven Simplification,” ACM Trans. Graphics, vol. 19, no. 3, pp. 204-241, 2000.
[57] V. Surazhsky and C. Gotsman, “Explicit Surface Remeshing,” Proc. ACM SIGGRAPH Symp. Geometry Processing (SGP), 2003.
[58] G. Turk, “Re-Tiling Polygonal Surfaces,” Proc. SIGGRAPH, 1992.
[59] D.-M. Yan, B. Lévy, Y. Liu, F. Sun, and W. Wang, “Isotropic Remeshing with Fast and Exact Computation of Restricted Voronoi Diagram,” Proc. Symp. Geometry Processing (SGP), pp. 1445-1454, 2009.
[60] E. Akleman, J. Chen, Q. Xing, and J.L. Gross, “Cyclic Plain-Weaving on Polygonal Mesh Surfaces with Graph Rotation Systems,” ACM Trans. Graphics, vol. 28, no. 3, pp. 1-8, 2009.

Index Terms:
Surface parameterization, rotational symmetry, hexagonal global parameterization, triangular remeshing, pattern synthesis on surfaces, texture synthesis, geometry synthesis, regular patterns.
Citation:
Matthias Nieser, Jonathan Palacios, Konrad Polthier, Eugene Zhang, "Hexagonal Global Parameterization of Arbitrary Surfaces," IEEE Transactions on Visualization and Computer Graphics, vol. 18, no. 6, pp. 865-878, June 2012, doi:10.1109/TVCG.2011.118
Usage of this product signifies your acceptance of the Terms of Use.