Integrable PolyVector fields

Abstract

We present a framework for designing curl-free tangent vector fields on discrete surfaces. Such vector fields are gradients of locally-defined scalar functions, and this property is beneficial for creating surface parameterizations, since the gradients of the parameterization coordinate functions are then exactly aligned with the designed fields. We introduce a novel definition for discrete curl between unordered sets of vectors (PolyVectors), and devise a curl-eliminating continuous optimization that is independent of the matchings between them. Our algorithm naturally places the singularities required to satisfy the user-provided alignment constraints, and our fields are the gradients of an inversion-free parameterization by design.

References

1. Alliez, P., Cohen-Steiner, D., Devillers, O., Lévy, B., and Desbrun, M. 2003. Anisotropic polygonal remeshing. ACM Trans. Graph. 22, 3, 485--493.
2. Azencot, O., Ben-Chen, M., Chazal, F., and Ovsjanikov, M. 2013. An operator approach to tangent vector field processing. Comput. Graph. Forum 32, 5, 73--82.
3. Bommes, D., Zimmer, H., and Kobbelt, L. 2009. Mixed-integer quadrangulation. ACM Trans. Graph. 28, 3, 77:1--77:10.
4. Bommes, D., Campen, M., Ebke, H.-C., Alliez, P., and Kobbelt, L. 2013. Integer-grid maps for reliable quad meshing. ACM Trans. Graph. 32, 4, 98:1--98:12.
5. Bommes, D., Lévy, B., Pietroni, N., Puppo, E., Silva, C., Tarini, M., and Zorin, D. 2013. Quad-mesh generation and processing: A survey. Computer Graphics Forum 32, 6, 51--76.
6. Botsch, M., Kobbelt, L., Pauly, M., Alliez, P., and Lévy, B. 2010. Polygon Mesh Processing. AK Peters.
7. Campen, M., and Kobbelt, L. 2014. Dual strip weaving: Interactive design of quad layouts using elastica strips. ACM Trans. Graph. 33, 6, 183:1--183:10.
8. Crane, K., Desbrun, M., and Schröder, P. 2010. Trivial connections on discrete surfaces. Comput. Graph. Forum 29, 5.
9. Diamanti, O., Vaxman, A., Panozzo, D., and Sorkine-Hornung, O. 2014. Designing N-PolyVector fields with complex polynomials. Computer Graphics Forum 33, 5, 1--11.
10. Dong, S., Bremer, P.-T., Garland, M., Pascucci, V., and Hart, J. C. 2006. Spectral surface quadrangulation. ACM Trans. Graph. 25, 3 (July), 1057--1066.
11. Ebke, H.-C., Bommes, D., Campen, M., and Kobbelt, L. 2013. QEx: Robust quad mesh extraction. ACM Trans. Graph. 32, 6, 168:1--168:10.
12. Ebke, H.-C., Campen, M., Bommes, D., and Kobbelt, L. 2014. Level-of-detail quad meshing. ACM Trans. Graph. 33, 6, 184:1--184:11.
13. Fisher, M., Schröder, P., Desbrun, M., and Hoppe, H. 2007. Design of tangent vector fields. ACM Trans. Graph. 26, 3.
14. Hertzmann, A., and Zorin, D. 2000. Illustrating smooth surfaces. In Proc. ACM SIGGRAPH, 517--526.
15. Kälberer, F., Nieser, M., and Polthier, K. 2007. QuadCover -- surface parameterization using branched coverings. Computer Graphics Forum 26, 3, 375--384.
16. Knöppel, F., Crane, K., Pinkall, U., and Schröder, P. 2013. Globally optimal direction fields. ACM Trans. Graph. 32, 4.
17. Kuzmin, A., Luisier, M., and Schenk, O. 2013. Fast methods for computing selected elements of the Green's function in massively parallel nanoelectronic device simulations. In Proc. Euro-Par, 533--544.
18. Lefebvre, S., and Hoppe, H. 2006. Appearance-space texture synthesis. ACM Trans. Graph. 25, 3, 541--548.
19. Li, Y., Bao, F., Zhang, E., Kobayashi, Y., and Wonka, P. 2011. Geometry synthesis on surfaces using field-guided shape grammars. IEEE Trans. Vis. Comput. Graph. 17, 2, 231--243.
20. Ling, R., Huang, J., Jüttler, B., Sun, F., Bao, H., and Wang, W. 2014. Spectral quadrangulation with feature curve alignment and element size control. ACM Trans. Graph. 34, 1.
21. Lipman, Y. 2012. Bounded distortion mapping spaces for triangular meshes. ACM Trans. Graph. 31, 4, 108:1--108:13.
22. Liu, Y., Xu, W., Wang, J., Zhu, L., Guo, B., Chen, F., and Wang, G. 2011. General planar quadrilateral mesh design using conjugate direction field. ACM Trans. Graph. 30, 6.
23. Marinov, M., and Kobbelt, L. 2004. Direct anisotropic quad-dominant remeshing. In Proc. Pacific Graphics, 207--216.
24. Myles, A., and Zorin, D. 2012. Global parametrization by incremental flattening. ACM Trans. Graph. 31, 4.
25. Myles, A., and Zorin, D. 2013. Controlled-distortion constrained global parametrization. ACM Trans. Graph. 32, 4.
26. Myles, A., Pietroni, N., and Zorin, D. 2014. Robust field-aligned global parametrization. ACM Trans. Graph. 33, 4.
27. Palacios, J., and Zhang, E. 2007. Rotational symmetry field design on surfaces. ACM Trans. Graph. 26, 3.
28. Panozzo, D., Lipman, Y., Puppo, E., and Zorin, D. 2012. Fields on symmetric surfaces. ACM Trans. Graph. 31, 4.
29. Panozzo, D., Puppo, E., Tarini, M., and Sorkine-Hornung, O. 2014. Frame fields: Anisotropic and non-orthogonal cross fields. ACM Trans. Graph. 33, 4, 134:1--134:11.
30. Polthier, K., and Preuss, E. 2003. Identifying vector field singularities using a discrete Hodge decomposition. In Visualization and Mathematics III, 113--134.
31. Ray, N., Li, W. C., Lévy, B., Sheffer, A., and Alliez, P. 2006. Periodic global parameterization. ACM Trans. Graph. 25, 4, 1460--1485.
32. Ray, N., Vallet, B., Li, W. C., and Lévy, B. 2008. N-symmetry direction field design. ACM Trans. Graph. 27, 2.
33. Ray, N., Vallet, B., Alonso, L., and Lévy, B. 2009. Geometry-aware direction field processing. ACM Trans. Graph. 29, 1, 1:1--1:11.
34. Schüller, C., Kavan, L., Panozzo, D., and Sorkine-Hornung, O. 2013. Locally injective mappings. Computer Graphics Forum 32, 5, 125--135.
35. Takayama, K., Panozzo, D., Sorkine-Hornung, A., and Sorkine-Hornung, O. 2013. Sketch-based generation and editing of quad meshes. ACM Trans. Graph. 32, 4, 97:1--97:8.
36. Weber, O., and Zorin, D. 2014. Locally injective parametrization with arbitrary fixed boundaries. ACM Trans. Graph. 33, 4 (July), 75:1--75:12.
37. Zhang, E., Mischaikow, K., and Turk, G. 2006. Vector field design on surfaces. ACM Trans. Graph. 25, 4, 1294--1326.
38. Zhang, M., Huang, J., Liu, X., and Bao, H. 2010. A wave-based anisotropic quadrangulation method. ACM Trans. Graph. 29, 4, 118:1--118:8.

Index Terms

• Computing methodologies

  • Computer graphics

    • Shape modeling

• Theory of computation

  • Randomness, geometry and discrete structures