# Efficient triangulation of Poisson-disk sampled point sets

- 339 Downloads
- 5 Citations

## Abstract

In this paper, we present a simple yet efficient algorithm for triangulating a 2D input domain containing a Poisson-disk sampled point set. The proposed algorithm combines a regular grid and a discrete clustering approach to speedup the triangulation. Moreover, our triangulation algorithm is flexible and performs well on more general point sets such as adaptive, non-maximal Poisson-disk sets. The experimental results demonstrate that our algorithm is robust for a wide range of input domains and achieves significant performance improvement compared to the current state-of-the-art approaches.

## Keywords

Triangulation Poisson-disk sampling Geometric algorithms## Notes

### Acknowledgments

This research was partially funded by National Natural Science Foundation of China (Nos. 61372168, 61172104, 61331018, and 61271431), the KAUST Visual Computing Center, and the National Science Foundation.

## References

- 1.Barber, C.B., Dobkin, D., Huhdanpaa, H.: The quickhull algorithm for convex hulls. ACM Trans. Math. Softw.
**22**(4), 469–483 (1996)CrossRefMATHMathSciNetGoogle Scholar - 2.Blelloch, G.E., Miller, G.L., Hardwick, J.C., Talmor, D.: Design and implementation of a practical parallel delaunay algorithm. Algorithmica
**24**(3–4), 243–269 (1999)CrossRefMATHMathSciNetGoogle Scholar - 3.CGAL, Computational Geometry Algorithms Library. http://www.cgal.org
- 4.Cheng, S.W., Dey, T.K., Levine, J.A.: A practical Delaunay meshing algorithm for a large class of domains. In: Proceedings of the 16th International Meshing Roundtable, pp. 477–494 (2007)Google Scholar
- 5.Cheng, S.W., Dey, T.K., Shewchuk, J.R.: Delaunay Mesh Generation. CRC Press, Boca Raton (2012)MATHGoogle Scholar
- 6.Chew, L.P.: Guaranteed-quality triangular meshes. Department of Computer Science Tech Report 89-983, Cornell University (1989)Google Scholar
- 7.Chrisochoides, N., Nave, D.: Parallel delaunay mesh generation kernel. Int. J. Numer. Methods Eng.
**58**(2), 161–176 (2003)CrossRefMATHGoogle Scholar - 8.Cohen-Steiner, D., Alliez, P., Desbrun, M.: Variational shape approximation. ACM Trans. Graph. (Proc. SIGGRAPH)
**23**(3), 905–914 (2004)CrossRefGoogle Scholar - 9.Cook, R.L.: Stochastic sampling in computer graphics. ACM Trans. Graph.
**5**(1), 69–78 (1986)Google Scholar - 10.Dunbar, D., Humphreys, G.: A spatial data structure for fast poisson-disk sample generation. ACM Trans. Graph. (Proc. SIGGRAPH)
**25**(3), 503–508 (2006)CrossRefGoogle Scholar - 11.Ebeida, M.S., Mitchell, S.A., Davidson, A.A., Patney, A., Knupp, P.M., Owens, J.D.: Efficient and good delaunay meshes from random points. Comput. Aided Des.
**43**(11), 1506–1515 (2011)CrossRefGoogle Scholar - 12.Ebeida, M.S., Mitchell, S.A., Patney, A., Davidson, A.A., Owens, J.D.: A simple algorithm for maximal poisson-disk sampling in high dimensions. Comput. Graph. Forum (Proc. EUROGRAPHICS)
**31**(2), 785–794 (2012)CrossRefGoogle Scholar - 13.Ebeida, M.S., Patney, A., Mitchell, S.A., Davidson, A., Knupp, P.M., Owens, J.D.: Efficient maximal poisson-disk sampling. ACM Trans. Graph. (Proc. SIGGRAPH)
**30**(4), 49:1–49:12 (2011)CrossRefGoogle Scholar - 14.Edelsbrunner, H.: Geometry and Topology for Mesh Generation. Cambridge University Press, Cambridge (2001)CrossRefMATHGoogle Scholar
- 15.Gamito, M.N., Maddock, S.C.: Accurate multidimensional poisson-disk sampling. ACM Trans. Graph.
**29**(1), 8:1–8:19 (2009)Google Scholar - 16.Hoff III, K.E., Keyser, J., Lin, M., Manocha, D., Culver, T.: Fast computation of generalized Voronoi diagrams using graphics hardware. In: Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH ’99, pp. 277–286 (1999)Google Scholar
- 17.Jones, T.R.: Efficient generation of poisson-disk sampling patterns. J. Graph. Tools
**11**(2), 27–36 (2006)Google Scholar - 18.Lagae, A., Dutré, P.: A comparison of methods for generating poisson disk distributions. Comput. Graph. Forum
**27**(1), 114–129 (2008)Google Scholar - 19.Lu, Y., Lien, J.-M., Ghosh, M., Amato, N.M.: Alpha-decomposition of polygons. Comput. Graph. (Proc. SMI)
**36**(5), 466–476 (2012)Google Scholar - 20.Qi, M., Cao, T.T., Tan, T.S.: Computing 2d constrained delaunay triangulation using the gpu. IEEE Trans. Vis. Comput. Graph.
**19**(5), 736–748 (2013)CrossRefGoogle Scholar - 21.Rong, G., Tan, T.S., Cao, T.T., et al.: Computing two-dimensional Delaunay triangulation using graphics hardware. In: Proceedings of the 2008 Symposium on Interactive 3D Graphics and Games, pp. 89–97. ACM (2008)Google Scholar
- 22.Schechter, H., Bridson, R.: Ghost sph for animating water. ACM Trans. Graph. (Proc. SIGGRAPH)
**31**(4), 61:1–61:8 (2012)CrossRefGoogle Scholar - 23.Shewchuk, J.R.: Triangle: engineering a 2d quality mesh generator and delaunay triangulator. In: Lin, M.C., Manocha, D. (eds.) Applied Computational Geometry: Towards Geometric Engineering. Lecture Notes in Computer Science, vol. 1148, pp. 203–222. Springer, Berlin (1996)CrossRefGoogle Scholar
- 24.Shewchuk, J.R.: Delaunay refinement algorithms for triangular mesh generation. Comput. Geom. Theory Appl.
**22**(1), 21–74 (2002)MATHMathSciNetGoogle Scholar - 25.Wei, L.Y.: Parallel poisson disk sampling. ACM Trans. Graph. (Proc. SIGGRAPH)
**27**(3), 20:1–20:9 (2008)Google Scholar - 26.Wei, L.Y.: Multi-class blue noise sampling. ACM Trans. Graph. (Proc. SIGGRAPH)
**29**(4), 79:1–79:8 (2010)Google Scholar - 27.White, K.B., Cline, D., Egbert, P.K.: Poisson disk point sets by hierarchical dart throwing. In: Proceedings of the IEEE Symposium on Interactive Ray Tracing, pp. 129–132 (2007)Google Scholar
- 28.Yan, D.M., Wonka, P.: Gap processing for adaptive maximal poisson-disk sampling. ACM Trans. Graph.
**32**(5), 148:1–148:15 (2013)Google Scholar