Parallel Poisson disk sampling with spectrum analysis on surfaces

Abstract

The ability to place surface samples with Poisson disk distribution can benefit a variety of graphics applications. Such a distribution satisfies the blue noise property, i.e. lack of low frequency noise and structural bias in the Fourier power spectrum. While many techniques are available for sampling the plane, challenges remain for sampling arbitrary surfaces. In this paper, we present new methods for Poisson disk sampling with spectrum analysis on arbitrary manifold surfaces. Our first contribution is a parallel dart throwing algorithm that generates high-quality surface samples at interactive rates. It is flexible and can be extended to adaptive sampling given a user-specified radius field. Our second contribution is a new method for analyzing the spectral quality of surface samples. Using the spectral mesh basis derived from the discrete mesh Laplacian operator, we extend standard concepts in power spectrum analysis such as radial means and anisotropy to arbitrary manifold surfaces. This provides a way to directly evaluate the spectral distribution quality of surface samples without requiring mesh parameterization. Finally, we implement our Poisson disk sampling algorithm on the GPU, and demonstrate practical applications involving interactive sampling and texturing on arbitrary surfaces.

References

1. Alcantara, D. A., Sharf, A., Abbasinejad, F., Sengupta, S., Mitzenmacher, M., Owens, J. D., and Amenta, N. 2009. Real-time parallel hashing on the GPU. ACM Trans. Graph. 28, 5, 154:1--9.
2. Alexa, M. 2002. Recent advances in mesh morphing. Comput. Graph. Forum 21, 2, 173--196.
3. Alliez, P., Meyer, M., and Desbrun, M. 2002. Interactive geometry remeshing. ACM Trans. Graph. 21, 3, 347--354.
4. Balzer, M., Schlömer, T., and Deussen, O. 2009. Capacity-constrained point distributions: a variant of Lloyd's method. ACM Trans. Graph. 28, 3, 86:1--8.
5. Cheslack-Postava, E., Wang, R., Akerlund, O., and Pellacini, F. 2008. Fast, realistic lighting and material design using nonlinear cut approximation. ACM Trans. Graph. 27, 5, 128:1--10.
6. Cline, D., Jeschke, S., White, K., Razdan, A., and Wonka, P. 2009. Dart throwing on surfaces. Computer Graphics Forum 28, 4, 1217--1226.
7. Cook, R. L. 1986. Stochastic sampling in computer graphics. ACM Transactions on Graphics 5, 1, 51--72.
8. Desbrun, M., Meyer, M., Schröder, P., and Barr, A. H. 2002. Discrete differential-geometry operators for triangulated 2-manifolds. In Visualization and Mathematics III, Springer-Verlag, 35--57.
9. Dong, S., Bremer, P.-T., Garland, M., Pascucci, V., and Hart, J. C. 2006. Spectral surface quadrangulation. ACM Trans. Graph. 25, 3, 1057--1066.
10. Dunbar, D., and Humphreys, G. 2006. A spatial data structure for fast Poisson-disk sample generation. ACM Trans. Graph. 25, 3, 503--508.
11. Dyer, R., Zhang, H., Moeller, T., and Clements, A. 2007. An investigation of the spectral robustness of mesh Laplacians. Tech. Rep. TR-2007-17, School of Computing Science, Simon Fraser University.
12. Fu, Y., and Zhou, B. 2008. Direct sampling on surfaces for high quality remeshing. In SPM '08, 115--124.
13. Glanville, S. 2004. Texture bombing. GPU Gems.
14. Grossman, J. P., and Dally, W. J. 1998. Point sample rendering. In Rendering Techniques, 181--192.
15. Jensen, H. W., and Buhler, J. 2002. A rapid hierarchical rendering technique for translucent materials. ACM Trans. Graph. 21, 3, 576--581.
16. Jones, T. R. 2006. Efficient generation of Poisson-disk sampling patterns. Journal of Graphics Tools 11, 2, 27--36.
17. Karni, Z., and Gotsman, C. 2000. Spectral compression of mesh geometry. In SIGGRAPH '00, 279--286.
18. Kopf, J., Cohen-Or, D., Deussen, O., and Lischinski, D. 2006. Recursive Wang tiles for real-time blue noise. ACM Trans. Graph. 25, 3, 509--518.
19. Lagae, A., and Dutré, P. 2005. A procedural object distribution function. ACM Trans. Graph. 24, 4, 1442--1461.
20. Lagae, A., and Dutré, P. 2008. A comparison of methods for generating Poisson disk distributions. Computer Graphics Forum 27, 1, 114--129.
21. Levy, B. 2006. Laplace-Beltrami eigenfunctions towards an algorithm that "understands" geometry. In Proc. of SMI 2006, 13.
22. Li, H., Lo, K.-Y., Leung, M.-K., and Fu, C.-W. 2008. Dual Poisson-disk tiling: An efficient method for distributing features on arbitrary surfaces. IEEE TVCG 14, 5, 982--998.
23. Li, H., Wei, L.-Y., Sander, P., and Fu, C.-W. 2010. Anisotropic blue noise sampling. ACM Trans. Graph. 29, 5, to appear.
24. Lloyd, S. 1982. Least squares quantization in pcm. IEEE Transactions on Information Theory 28, 2, 129--137.
25. McCool, M., and Fiume, E. 1992. Hierarchical Poisson disk sampling distributions. In Graphics Interface '92, 94--105.
26. Meier, B. J. 1996. Painterly rendering for animation. In SIGGRAPH '96, 477--484.
27. Mitchell, D. P. 1987. Generating antialiased images at low sampling densities. In SIGGRAPH '87, 65--72.
28. Nehab, D., and Shilane, P. 2004. Stratified point sampling of 3D models. In Proc. of PBG, 49--56.
29. Ostromoukhov, V., Donohue, C., and Jodoin, P.-M. 2004. Fast hierarchical importance sampling with blue noise properties. ACM Trans. Graph. 23, 3, 488--495.
30. Ostromoukhov, V. 2007. Sampling with polyominoes. ACM Trans. Graph. 26, 3, 78:1--6.
31. Pastor, O. M., Freudenberg, B., and Strothotte, T. 2003. Real-time animated stippling. IEEE CG&A 23, 4, 62--68.
32. Praun, E., Hoppe, H., and Finkelstein, A. 1999. Robust mesh watermarking. In SIGGRAPH '99, 49--56.
33. Qu, L., and Meyer, G. W. 2006. Perceptually driven interactive geometry remeshing. In I3D '06, 199--206.
34. Ritschel, T., Engelhardt, T., Grosch, T., Seidel, H.-P., Kautz, J., and Dachsbacher, C. 2009. Micro-rendering for scalable, parallel final gathering. ACM Trans. Graph. 28, 5, 132:1--8.
35. Sethian, J. A. 1995. A fast marching level set method for monotonically advancing fronts. In Proc. Nat. Acad. Sci, 1591--1595.
36. Surazhsky, V., Surazhsky, T., Kirsanov, D., Gortler, S. J., and Hoppe, H. 2005. Fast exact and approximate geodesics on meshes. ACM Trans. Graph. 24, 3, 553--560.
37. Turk, G. 1992. Re-tiling polygonal surfaces. In SIGGRAPH '92, 55--64.
38. Turk, G. 2001. Texture synthesis on surfaces. In SIGGRAPH '01, 347--354.
39. Ulichney, R. 1987. Digital halftoning. MIT Press, Cambridge, MA.
40. Vallet, B., and Lévy, B. 2008. Spectral geometry processing with manifold harmonics. Comput. Graph. Forum 27, 2, 251--260.
41. Vanderhaeghe, D., Barla, P., Thollot, J., and Sillion, F. X. 2007. Dynamic point distribution for stroke-based rendering. In EGSR'07, 139--146.
42. Wei, L.-Y., and Levoy, M. 2001. Texture synthesis over arbitrary manifold surfaces. In SIGGRAPH '01, 355--360.
43. Wei, L.-Y. 2008. Parallel Poisson disk sampling. ACM Trans. Graph. 27, 3, 20:1--9.
44. White, K., Cline, D., and Egbert, P. 2007. Poisson disk point sets by hierarchical dart throwing. In Symposium on Interactive Ray Tracing, 129--132.
45. Yu, Q., Neyret, F., Bruneton, E., and Holzschuch, N. 2009. Scalable real-time animation of rivers. Comput. Graph. Forum 28, 2, 239--248.