“Geometry-Aware Direction Field Processing” by Ray, Vallet, Alonso and Levy

Many algorithms in texture synthesis, nonphotorealistic rendering (hatching), or remeshing require to define the orientation of some features (texture, hatches, or edges) at each point of a surface. In early works, tangent vector (or tensor) fields were used to define the orientation of these features. Extrapolating and smoothing such fields is usually performed by minimizing an energy composed of a smoothness term and of a data fitting term. More recently, dedicated structures (N-RoSy and N-symmetry direction fields ) were introduced in order to unify the manipulation of these fields, and provide control over the field’s topology (singularities). On the one hand, controlling the topology makes it possible to have few singularities, even in the presence of high frequencies (fine details) in the surface geometry. On the other hand, the user has to explicitly specify all singularities, which can be a tedious task. It would be better to let them emerge naturally from the direction extrapolation and smoothing.

This article introduces an intermediate representation that still allows the intuitive design operations such as smoothing and directional constraints, but restates the objective function in a way that avoids the singularities yielded by smaller geometric details. The resulting design tool is intuitive, simple, and allows to create fields with simple topology, even in the presence of high geometric frequencies. The generated field can be used to steer global parameterization methods (e.g., QuadCover).

1. Alliez, P., Cohen-Steiner, D., Devillers, O., Levy, B., and Desbrun, M. 2003. Anisotropic polygonal remeshing. ACM Trans. Graph. Special issue for SIGGRAPH Conference, 485–493. 
2. Ben-Chen, M., Gotsman, C., and Bunin, G. 2008. Conformal flattening by curvature prescription and metric scaling. In Proceedings of the IEEE Eurographics. Eurographics.
3. Davis, T. A. and Hager, W. W. 2005. Row modifications of a sparse Cholesky factorization. J. Matrix Anal. Appl. 26, 3, 621–639. 
4. Desbrun, M., Kanzo, E., and Tong, Y. 2005a. Discrete differential forms for computational modeling. Siggraph’05 Course Notes on Discrete Differential Geometry. Chapter 7. 
5. Desbrun, M., Leok, M., and Marsden, J. 2005b. Discrete Poincaré lemma. Appl. Numer. Math. 53, 2-4, 231–248. 
6. Fisher, M., Schröder, P., Desbrun, M., and Hoppe, H. 2007. Design of tangent vector fields. In Proceedings of the SIGGRAPH’07 Conference. ACM, New York, 56. 
7. Gu, X., Gortler, S., and Hoppe, H. 2002. Geometry images. In Proceedings of the SIGGRAPH’02 Conference. 355–361. 
8. Hertzmann, A. and Zorin, D. 2000. Illustrating smooth surfaces. In Proceedings of the ACM SIGGRAPH Conference. 517–526. 
9. Kalberer, F., Nieser, M., and Polthier, K. 2007. Quadcover—Surface parameterization using branched coverings. Comput. Graph. Forum 26, 3, 375–384.
10. Kharevych, L., Springborn, B., and Schröder, P. 2006. Discrete conformal mappings via circle patterns. ACM Trans. Graph. 25, 2, 412–438. 
11. Lefebvre, S. and Hoppe, H. 2006. Appearance-space texture synthesis. In Proceedings of the SIGGRAPH’06 Conference. ACM Press, New York, 541–548. 
12. Levy, B. 2005. Numerical methods for digital geometry processing. In Proceedings of Israel Korea Bi-National Conference.
13. Li, W.-C., Ray, N., and Levy, B. 2006a. Automatic and interactive mesh to T-spline conversion. In Proceedings of EG/ACM Symposium on Geometry Processing. 
14. Li, W.-C., Vallet, B., Ray, N., and Levy, B. 2000b. Representing higher-order singularities in vector fields on piecewise linear surfaces. IEEE Trans. Visual. Comput. Graph. 
15. Palacios, J. and Zhang, E. 2007. Rotational symmetry field design on surfaces. In Proceedings of the SIGGRAPH’07 Conference. ACM, New York, 55. 
16. Pinkall, U. and Polthier, K. 1993. Computing discrete minimal surfaces and their conjugates. Experiment. Math. 2, 1, 15–36.
17. Praun, E., Finkelstein, A., and Hoppe, H. 2000. Lapped textures. In Proceedings of the SIGGRAPH’00 Conference, K. Akeley, Ed. 465–470. 
18. Praun, E., Hoppe, H., Webb, M., and Finkelstein, A. 2001. Real-Time hatching. In Proceedings of the SIGGRAPH’01 Conference, E. Fiume, Ed. 579–584. 
19. Ray, N., Li, W. C., Levy, B., Sheffer, A., and Alliez, P. 2006. Periodic global parameterization. ACM Trans. Graph. 25, 4, 1460–1485. 
20. Ray, N., Vallet, B., Li, W.-C., and Levy, B. 2008. N-symmetry direction fields on surfaces of arbitrary genus. ACM Trans. Graph. 
21. Springborn, B., Schröder, P., and Pinkall, U. 2008. Conformal equivalence of triangle meshes. ACM Trans. Graph. 27, 3, 1–11. 
22. Tong, Y., Alliez, P., Cohen-Steiner, D., and Desbrun, M. 2006. Designing quadrangulations with discrete harmonic forms. In Proceedings of the Symposium on Geometry Processing. Eurographics, 201–210. 
23. Turk, G. 2001. Texture synthesis on surfaces. In Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH’01). ACM Press, New York, 347–354. 
24. Zhang, E., Mischaikow, K., and Turk, G. 2006. Vector field design on surfaces. ACM Trans. Graph. 25, 4, 1294–1326.