N-symmetry direction field design

Abstract

Many algorithms in computer graphics and geometry processing use two orthogonal smooth direction fields (unit tangent vector fields) defined over a surface. For instance, these direction fields are used in texture synthesis, in geometry processing or in nonphotorealistic rendering to distribute and orient elements on the surface. Such direction fields can be designed in fundamentally different ways, according to the symmetry requested: inverting a direction or swapping two directions might be allowed or not.

Despite the advances realized in the last few years in the domain of geometry processing, a unified formalism is still lacking for the mathematical object that characterizes these generalized direction fields. As a consequence, existing direction field design algorithms are limited to using nonoptimum local relaxation procedures.

In this article, we formalize N-symmetry direction fields, a generalization of classical direction fields. We give a new definition of their singularities to explain how they relate to the topology of the surface. Specifically, we provide an accessible demonstration of the Poincaré-Hopf theorem in the case of N-symmetry direction fields on 2-manifolds. Based on this theorem, we explain how to control the topology of N-symmetry direction fields on meshes. We demonstrate the validity and robustness of this formalism by deriving a highly efficient algorithm to design a smooth field interpolating user-defined singularities and directions.

References

1. Alliez, P., Cohen-Steiner, D., Devillers, O., Levy, B., and Desbrun, M. 2003. Anisotropic polygonal remeshing. ACM Trans. Graph. (SIGGRAPH'03) 22, 3, 485--493.
2. Dichler, J.-M., Maritaud, K., Levy, B., and Chazanfarpour, D. 2002. Texture particles. Comput. Graph. For. 21, 3.
3. Fisher, M., Schroder, P., Desbrun, M., and Hoppe, H. 2007. Design of tangent vector fields. ACM Trans. Graph. (SIGGRAPH'07).
4. Gortler, S., Grzeszczuk, R., Szeliski, R., and Cohen, M.-F. 1996. The lumigraph. In Proceedings of ACM SIGGRAPH. 43--54.
5. Gu, X. and Yau, S.-T. 2003. Global conformal surface parameterization. In Eurographics/ACM SIGGRAPH Symposium on Geometry Processing. 127--137.
6. Hertzmann, A. and Zorin, D. 2000. Illustrating smooth surfaces. In Proceedings of ACM SIGGRAPH. 517--526.
7. Levy, B. 2005. Numerical methods for digital geometry processing. In Israel Korea Bi-National Conference.
8. Li, W.-C., Vallet, B., Ray, N., and Levy, B. 2006. Representing higher-order singularities in vector fields on piecewise linear surfaces. IEEE Trans. Visualiz. Comput. Graph.
9. Marinov, M. and Kobbelt, L. 2004. Direct anisotropic quad-dominant remeshing. In Proceedings of Pacific Graphics. 207--216.
10. Ni, X., Garland, M., and Hart, J. C. 2004. Fair morse functions for extracting the topological structure of a surface mesh. ACM Trans. Graph. (SIGGRAPH'04) 23, 3, 613--622.
11. Ohtake, Y., Horikawa, M., and Belyaev, A. 2001. Adaptive smoothing tangential direction fields on polygonal surfaces. In Proceedings of Pacific Graphics. 189.
12. Palacios, J. and Zhang, E. 2007. Rotational symetry field design on surfaces. ACM Trans. Graph. (SIGGRAPH'07).
13. Polthier, K. and Preuss, E. 2002. Identifying vector fields singularities using a discrete hodge decomposition. In Visualization and Mathematics III, H. Hege and K. Polthier, Eds. Springer Verlag, 113--134.
14. Praun, E., Finkelstein, A., and Hoppe, H. 2000. Lapped textures. In Proceedings of ACM SIGGRAPH. 465--470.
15. Ray, N., Li, W. C., Levy, B., Sheffer, A., and Alliez, P. 2006. Periodic global parameterization. ACM Trans. Graph. 25, 4, 1460--1485.
16. Tong, Y., Lombeyda, S., Hirani, A., and Desbrun, M. 2003. Discrete multiscale vector field decomposition. ACM Trans. Graph. (SIGGRAPH'03) 22, 3, 445--452.
17. Tricoche, X., Scheuermann, G., and Hagen, H. 2003. Topology simplification of symmetric, second-order 2D tensor fields. In Hierarchical and Geometrical Methods in Scientific Visualization, Farin, G., Hamann, B., and Haqen, H., Eds. Springer.
18. Turk, G. 2001. Texture synthesis on surfaces. In Proceedings of ACM SIGGRAPH. 347--354.
19. Wang, K., Weiwei, Tong, Y., Desbrun, M., and Schrder, P. 2006. Edge subdivision schemes and the construction of smooth vector fields. ACM Trans. Graph. (SIGGRAPH'06).
20. Wei, L.-Y. and Levoy, M. 2001. Texture synthesis over arbitrary manifold surfaces. In Proceedings of ACM SIGGRAPH. 355--360.
21. Zelinka, S. and Garland, M. 2004. Jump map-based interactive texture synthesis. ACM Trans. Graph. 23, 4, 930--962.
22. Zhang, E., Hays, J., and Turk, G. 2005. Interactive design and visualization of tensor fields on surfaces. Tech. rep., Oregon State University.
23. Zhang, E., Mischaikow, K., and Turk, G. 2006. Vector field design on surfaces. ACM Trans. Graph. 25, 4, 1294--1326.

Index Terms

• Applied computing

  • Physical sciences and engineering

    • Engineering

• Computing methodologies

  • Computer graphics

    • Shape modeling

• Mathematics of computing

  • Discrete mathematics

    • Graph theory

  • Mathematical analysis

    • Mathematical optimization

• Theory of computation

  • Design and analysis of algorithms

    • Mathematical optimization

  • Randomness, geometry and discrete structures

    • Computational geometry