Rotational symmetry field design on surfaces

ABSTRACT

Designing rotational symmetries on surfaces is a necessary task for a wide variety of graphics applications, such as surface parameterization and remeshing, painterly rendering and pen-and-ink sketching, and texture synthesis. In these applications, the topology of a rotational symmetry field such as singularities and separatrices can have a direct impact on the quality of the results. In this paper, we present a design system that provides control over the topology of rotational symmetry fields on surfaces.

As the foundation of our system, we provide comprehensive analysis for rotational symmetry fields on surfaces and present efficient algorithms to identify singularities and separatrices. We also describe design operations that allow a rotational symmetry field to be created and modified in an intuitive fashion by using the idea of basis fields and relaxation. In particular, we provide control over the topology of a rotational symmetry field by allowing the user to remove singularities from the field or to move them to more desirable locations.

At the core of our analysis and design implementations is the observations that N-way rotational symmetries can be described by symmetric N-th order tensors, which allows an efficient vector-based representation that not only supports coherent definitions of arithmetic operations on rotational symmetries but also enables many analysis and design operations for vector fields to be adapted to rotational symmetry fields.

To demonstrate the effectiveness of our approach, we apply our design system to pen-and-ink sketching and geometry remeshing.

References

1. Alliez, P., Cohen-Steiner, D., Devillers, O., Lévy, B., and Desbrun, M. 2003. Anisotropic polygonal remeshing. ACM Transactions on Graphics (SIGGRAPH 2003) 22, 3 (July), 485--493.
2. Chen, G., Mischaikow, K., Laramee, R. S., Pilarczyk, P., and Zhang, E. 2007. Vector field editing and periodic orbit extraction using morse decomposition. IEEE Transactions on Visualization and Computer Graphics 13, 4, 769--785.
3. Cohen-Steiner, D., and Morvan, J. 2003. Restricted delaunay triangulations and normal cycle. In 19th Annual ACM Symposium Computational Geometry.
4. Delmarcelle, T., and Hesselink, L. 1994. The topology of symmetric, second-order tensor fields. IEEE Computer Graphics and Applications, 140--147.
5. Dong, S., Kircher, S., and Garland, M. 2005. Harmonic functions for quadrilateral remeshing of arbitrary manifolds. Computer Aided Geometry Design. (to appear in upcoming Special Issue on Geometry Processing).
6. Dong, S., Bremer, P.-T., Garland, M., Pascucci, V., and Hart, J. C. 2006. Spectral surface quadrangulation. ACM Trans. Graph. 25, 3, 1057--1066.
7. Floater, M. S. 2003. Mean value coordinates. CAGD, 20, 19--27.
8. Girshick, A., Interrante, V., Haker, S., and Lemoine, T. 2000. Line direction matters: an argument for the use of principal directions in 3D line drawings. Proceedings of NPAR, 43--52.
9. Helman, J. L., and Hesselink, L. 1991. Visualizing vector field topology in fluid flows. IEEE Computer Graphics and Applications 11 (May), 36--46.
10. Hertzmann, A., and Zorin, D. 2000. Illustrating smooth surfaces. Computer Graphics Proceedings, Annual Conference Series (SIGGRAPH 2000) (Aug.), 517--526.
11. Marinov, M., and Kobbelt, L. 2004. Direct anisotropic quad-dominant remeshing. Computer Graphics and Applications, 12th Pacific Conference on (PG'04), 207--216.
12. Meyer, M., Desbrun, M., Schröder, P., and Barr, A. H. 2002. Discrete differential-geometry operators for triangulated 2-manifolds. VisMath.
13. Mischaikow, K., and Mrozek, M. 2002. Conley index. Hand-book of Dynamic Systems, North-Holland 2, 393--460.
14. Praun, E., Finkelstein, A., and Hoppe, H. 2000. Lapped textures. Computer Graphics Proceedings, Annual Conference Series (SIGGRAPH 2000) (Aug.), 465--470.
15. Ray, N., Li, W. C., Lévy, B., Sheffer, A., and Alliez, P. 2006. Periodic global parameterization. ACM Transactions on Graphics 25, 4, 1460--1485.
16. Ray, N., Vallet, B., Li, W.-C., and Levy, B. 2006. N-symmetry direction fields on surfaces of arbitrary genus. In Tech Report.
17. Rusinkiewicz, S. 2004. Estimating curvatures and their derivatives on triangle meshes. In 3DPVT '04: Proceedings of the 3D Data Processing, Visualization, and Transmission, 2nd International Symposium on (3DPVT'04), 486--493.
18. Scheuermann, G., Krger, H., Menzel, M., and Rockwood, A. P. 1998. Visualizing nonlinear vector field topology. IEEE Transactions on Visualization and Computer Graphics 4, 2, 109--116.
19. Stam, J. 2003. Flows on surfaces of arbitrary topology. ACM Transactions on Graphics (SIGGRAPH 2003) 22, 3 (July), 724--731.
20. Theisel, H. 2002. Designing 2d vector fields of arbitrary topology. In Computer Graphics Forum (Proceedings Eurographics 2002), vol. 21, 595--604.
21. Tong, Y., Lombeyda, S., Hirani, A., and Desbrun, M. 2003. Discrete multiscale vector field decomposition. ACM Transactions on Graphics (SIGGRAPH 2003) 22, 3 (July), 445--452.
22. Tong, Y., Alliez, P., Cohen-Steiner, D., and Desbrun, M. 2006. Designing quadrangulations with discrete harmonic forms. ACM/Eurographics Symposium on Geometry Processing, 201--210.
23. Tricoche, X. 2002. Vector and Tensor Field Topology Simplification, Tracking, and Visualization. PhD thesis, Universität Kaiserslautern.
24. Turk, G. 2001. Texture synthesis on surfaces. Computer Graphics Proceedings, Annual Conference Series (SIGGRAPH 2001), 347--354.
25. van Wijk, J. J. 2002. Image based flow visualization. ACM Transactions on Graphics (SIGGRAPH 2002) 21, 3 (July), 745--754.
26. van Wijk, J. J. 2003. Image based flow visualization for curved surfaces. In: G. Turk, J. van Wijk, R. Moorhead (eds.), Proceedings IEEE Visualization (Oct.), 123--130.
27. Wang, K., Weiwei, Tong, Y., Desbrun, M., and Schröder, P. 2006. Edge subdivision schemes and the construction of smooth vector fields. ACM Transactions on Graphics 25, 3, 1041--1048.
28. Wei, L. Y., and Levoy, M. 2001. Texture synthesis over arbitrary manifold surfaces. Computer Graphics Proceedings, of the Conference Series (SIGGRAPH 2001), 355--360.
29. Ying, L., Hertzmann, A., Biermann, H., and Zorin, D. 2001. Texture and shape synthesis on surfaces. Proc. 12th Eurographics Workshop on Rendering, 301--312.
30. Zhang, E., Mischaikow, K., and Turk, G. 2006. Vector field design on surfaces. ACM Transactions on Graphics 25, 4, 1294--1326.
31. Zhang, E., Hays, J., and Turk, G. 2007. Interactive tensor field design and visualization on surfaces. IEEE Transactions on Visualization and Computer Graphics 13, 1, 94--107.

Index Terms

• Computing methodologies

  • Computer graphics

    • Shape modeling

• Theory of computation

  • Randomness, geometry and discrete structures