This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Metric-Driven RoSy Field Design and Remeshing
January/February 2010 (vol. 16 no. 1)
pp. 95-108
Yu-Kun Lai, Tsinghua University, Beijing
Miao Jin, University of Louisiana at Lafayette, LA
Xuexiang Xie, Nanyang Technological University, Singapore
Ying He, Nanyang Technological University, Singapore
Jonathan Palacios, Oregon State University, Corvallis
Eugene Zhang, Oregon State University, Corvallis
Shi-Min Hu, Tsinghua University, Beijing
Xianfeng Gu, State University of New York at Stony Brook, Stony Brook
Designing rotational symmetry fields on surfaces is an important task for a wide range of graphics applications. This work introduces a rigorous and practical approach for automatic N-RoSy field design on arbitrary surfaces with user-defined field topologies. The user has full control of the number, positions, and indexes of the singularities (as long as they are compatible with necessary global constraints), the turning numbers of the loops, and is able to edit the field interactively. We formulate N-RoSy field construction as designing a Riemannian metric such that the holonomy along any loop is compatible with the local symmetry of N-RoSy fields. We prove the compatibility condition using discrete parallel transport. The complexity of N-RoSy field design is caused by curvatures. In our work, we propose to simplify the Riemannian metric to make it flat almost everywhere. This approach greatly simplifies the process and improves the flexibility such that it can design N-RoSy fields with single singularity and mixed-RoSy fields. This approach can also be generalized to construct regular remeshing on surfaces. To demonstrate the effectiveness of our approach, we apply our design system to pen-and-ink sketching and geometry remeshing. Furthermore, based on our remeshing results with high global symmetry, we generate Celtic knots on surfaces directly.

[1] J. Palacios and E. Zhang, “Rotational Symmetry Field Design on Surfaces,” ACM Trans. Graphics, vol. 26, no. 3, p. 55, 2007.
[2] N. Ray, B. Vallet, W.-C. Li, and B. Levy, “N-Symmetry Direction Field Design,” ACM Trans. Graphics, vol. 27, no. 2, pp. 10:1-10:13, 2008.
[3] L. Kharevych, B. Springborn, and P. Schröder, “Discrete Conformal Mappings via Circle Patterns,” ACM Trans. Graphics, vol. 25, no. 2, pp. 412-438, 2006.
[4] M. Jin, J. Kim, F. Luo, and X. Gu, “Discrete Surface Ricci Flow: Theories and Applications,” Math. of Surfaces XII, pp. 209-232, Springer, 2007.
[5] E. Praun, A. Finkelstein, and H. Hoppe, “Lapped Textures,” Proc. ACM SIGGRAPH, pp. 465-470, Aug. 2000.
[6] G. Turk, “Texture Synthesis on Surfaces,” Proc. ACM SIGGRAPH, pp. 347-354, 2001.
[7] L.Y. Wei and M. Levoy, “Texture Synthesis over Arbitrary Manifold Surfaces,” Proc. ACM SIGGRAPH, pp. 355-360, 2001.
[8] J. Stam, “Flows on Surfaces of Arbitrary Topology,” Proc. ACM SIGGRAPH, vol. 22, no. 3, pp. 724-731, July 2003.
[9] J.J. van Wijk, “Image Based Flow Visualization,” Proc. ACM SIGGRAPH, vol. 21, no. 3, pp. 745-754, July 2002.
[10] J.J. van Wijk, “Image Based Flow Visualization for Curved Surfaces,” Proc. IEEE Visualization Conf., pp. 123-130, Oct. 2003.
[11] H. Theisel, “Designing 2d Vector Fields of Arbitrary Topology,” Proc. Eurographics Conf., vol. 21, pp. 595-604, 2002.
[12] E. Zhang, K. Mischaikow, and G. Turk, “Vector Field Design on Surfaces,” ACM Trans. Graphics, vol. 25, no. 4, pp. 1294-1326, 2006.
[13] G. Chen, K. Mischaikow, R.S. Laramee, P. Pilarczyk, and E. Zhang, “Vector Field Editing and Periodic Orbit Extraction Using Morse Decomposition,” IEEE Trans. Visualization and Computer Graphics, vol. 13, no. 4, pp. 769-785, July/Aug. 2007.
[14] E. Zhang, J. Hays, and G. Turk, “Interactive Tensor Field Design and Visualization on Surfaces,” IEEE Trans. Visualization and Computer Graphics, vol. 13, no. 1, pp. 94-107, Jan./Feb. 2007.
[15] M. Fisher, P. Schröder, M. Desbrun, and H. Hoppe, “Design of Tangent Vector Fields,” ACM Trans. Graphics, vol. 26, no. 3, p. 56, 2007.
[16] A. Hertzmann and D. Zorin, “Illustrating Smooth Surfaces,” Proc. ACM SIGGRAPH, pp. 517-526, Aug. 2000.
[17] 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.
[18] Y. Tong, P. Alliez, D. Cohen-Steiner, and M. Desbrun, “Designing Quadrangulations with Discrete Harmonic Forms,” Proc. Symp. Geometry Processing, pp. 201-210, 2006.
[19] F. Kälberer, M. Nieser, and K. Polthier, “Quadcover—Surface Parameterization Using Branched Coverings,” Computer Graphics Forum, vol. 26, no. 10, pp. 375-384, Sept. 2007.
[20] A. Girshick, V. Interrante, S. Haker, and T. Lemoine, “Line Direction Matters: An Argument for the Use of Principal Directions in 3D Line Drawings,” Proc. Int'l Symp. Non-Photorealistic Animation and Rendering (NPAR), pp. 43-52, 2000.
[21] M. Meyer, M. Desbrun, P. Schröder, and A.H. Barr, “Discrete Differential-Geometry Operators for Triangulated 2-Manifolds,” Proc. Workshop Visualization and Math. (VisMath), 2002.
[22] D. Cohen-Steiner and J. Morvan, “Restricted Delaunay Triangulations and Normal Cycle,” Proc. ACM SIGGRAPH Symp. Computational Geometry, pp. 312-321, 2003.
[23] S. Rusinkiewicz, “Estimating Curvatures and Their Derivatives on Triangle Meshes,” Proc. Int'l Symp. 3D Data Processing, Visualization, and Transmission (3DPVT), pp. 486-493, 2004.
[24] S. Zelinka and M. Garland, “Jump Map-Based Interactive Texture Synthesis,” ACM Trans. Graphics, vol. 23, no. 4, pp. 930-962, 2004.
[25] X. Ni, M. Garland, and J.C. Hart, “Fair Morse Functions for Extracting the Topological Structure of a Surface Mesh,” ACM Trans. Graphics, vol. 23, no. 3, pp. 613-622, 2004.
[26] M. Kaplan and E. Cohen, “Computer Generated Celtic Design,” Proc. 14th Eurographics Workshop Rendering Techniques, pp. 2-19, 2003.
[27] K. Zhou, X. Huang, X. Wang, Y. Tong, M. Desbrun, B. Guo, and H.-Y. Shum, “Mesh Quilting for Geometric Texture Synthesis,” ACM Trans. Graphics, vol. 25, no. 3, pp. 690-697, 2006.
[28] P. Alliez, D. Cohen-Steiner, O. Devillers, B. Lévy, and M. Desbrun, “Anisotropic Polygonal Remeshing,” ACM Trans. Graphics, vol. 22, no. 3, pp. 485-493, July 2003.
[29] M. Marinov and L. Kobbelt, “Direct Anisotropic Quad-Dominant Remeshing,” Proc. 12th Pacific Conf. Computer Graphics and Applications, pp. 207-216, 2004.
[30] S. Dong, S. Kircher, and M. Garland, “Harmonic Functions for Quadrilateral Remeshing of Arbitrary Manifolds,” Computer Aided Geometric Design (CAGD), no. 5, pp. 392-423, 2005.
[31] S. Dong, P.-T. Bremer, M. Garland, V. Pascucci, and J.C. Hart, “Spectral Surface Quadrangulation,” ACM Trans. Graphics, vol. 25, no. 3, pp. 1057-1066, 2006.
[32] R.S. Hamilton, “Three-Manifolds with Positive Ricci Curvature,” J. Differential Geometry, vol. 17, pp. 255-306, 1982.
[33] 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.
[34] F. Luo, “Combinatorial Yamabe Flow on Surfaces,” Comm. Contemporary Math., vol. 6, pp. 765-780, 2004.
[35] B. Springborn, P. Schröder, and U. Pinkall, “Conformal Equivalence of Triangle Meshes,” ACM Trans. Graphics, vol. 27, no. 3, p.77, 2008.
[36] K. Polthier and M. Schmies, “Straightest Geodesics on Polyhedral Surfaces,” Proc. Math. Visualization Conf., pp. 391-398, 1998.
[37] M. Tarini, K. Hormann, P. Cignoni, and C. Montani, “Polycube-Maps,” ACM Trans. Graphics, vol. 23, no. 3, pp. 853-860, 2004.
[38] H. Wang, Y. He, X. Li, X. Gu, and H. Qin, “Polycube Splines,” Proc. ACM SIGGRAPH Symp. Solid and Physical Modeling, pp. 241-251, 2007.
[39] U. Pinkall and K. Polthier, “Computing Discrete Minimal Surfaces and Their Conjugates,” Experimental Math., vol. 2, pp. 15-36, 1993.
[40] X.D. Gu and S.-T. Yau, “Global Conformal Surface Parameterization,” Proc. Symp. Geometry Processing, pp. 127-137, 2003.

Index Terms:
metric, rotational symmetry, design, surface, parameterization, remeshing.
Citation:
Yu-Kun Lai, Miao Jin, Xuexiang Xie, Ying He, Jonathan Palacios, Eugene Zhang, Shi-Min Hu, Xianfeng Gu, "Metric-Driven RoSy Field Design and Remeshing," IEEE Transactions on Visualization and Computer Graphics, vol. 16, no. 1, pp. 95-108, Jan.-Feb. 2010, doi:10.1109/TVCG.2009.59
Usage of this product signifies your acceptance of the Terms of Use.