The Visual Computer

, Volume 33, Issue 6–8, pp 833–843 | Cite as

Forced Random Sampling: fast generation of importance-guided blue-noise samples

  • Daniel CornelEmail author
  • Robert F. Tobler
  • Hiroyuki Sakai
  • Christian Luksch
  • Michael Wimmer
Original Article


In computer graphics, stochastic sampling is frequently used to efficiently approximate complex functions and integrals. The error of approximation can be reduced by distributing samples according to an importance function, but cannot be eliminated completely. To avoid visible artifacts, sample distributions are sought to be random, but spatially uniform, which is called blue-noise sampling. The generation of unbiased, importance-guided blue-noise samples is expensive and not feasible for real-time applications. Sampling algorithms for these applications focus on runtime performance at the cost of having weak blue-noise properties. Blue-noise distributions have also been proposed for digital halftoning in the form of precomputed dither matrices. Ordered dithering with such matrices allows to distribute dots with blue-noise properties according to a grayscale image. By the nature of ordered dithering, this process can be parallelized easily. We introduce a novel sampling method called forced random sampling that is based on forced random dithering, a variant of ordered dithering with blue noise. By shifting the main computational effort into the generation of a precomputed dither matrix, our sampling method runs efficiently on GPUs and allows real-time importance sampling with blue noise for a finite number of samples. We demonstrate the quality of our method in two different rendering applications.


Blue noise Importance sampling Ordered dithering 



The competence center VRVis is funded by BMVIT, BMWFW and City of Vienna (ZIT) within the scope of COMET Competence Centers for Excellent Technologies. The program COMET is managed by FFG. Hiroyuki Sakai is partly supported by the Austrian Science Fund (FWF), project no. P 27974. We thank Christoph Weinzierl-Heigl for providing us access to his implementation of reflective shadow mapping.

Supplementary material

371_2017_1392_MOESM1_ESM.rar (16.8 mb)
Supplementary material 1 (rar 17210 KB)


  1. 1.
    Abe, Y.: Digital halftoning with optimized dither array. 2001 IEEE Int. Symp. Circuits Syst 2, 517–520 (2001)Google Scholar
  2. 2.
    Ahmed, A.G.M., Perrier, H., Coeurjolly, D., Ostromoukhov, V., Guo, J., Yan, D.M., Huang, H., Deussen, O.: Low-discrepancy blue noise sampling. ACM Trans. Graph. (Proc. SIGGRAPH Asia 2016) 13(6), 247:1–247:13 (2016)Google Scholar
  3. 3.
    Balzer, M., Schlömer, T., Deussen, O.: Capacity-constrained point distributions: a variant of Lloyd’s method. ACM Trans. Graph. (Proc. SIGGRAPH 2009) 28(3), 86:1–86:8 (2009)Google Scholar
  4. 4.
    Bayer, B.E.: An optimum method for two-level rendition of continuous-tone pictures. IEEE International Conference on Communication, Conference Record pp. (26–11)–(26–15) (1973)Google Scholar
  5. 5.
    Bowers, J., Wang, R., Wei, L.Y., Maletz, D.: Parallel poisson disk sampling with spectrum analysis on surfaces. ACM Trans. Graph. (SIGGRAPH Asia 2010 Papers) 29(6), 166:1–166:10 (2010)Google Scholar
  6. 6.
    Clarberg, P., Akenine-Möller, T.: Practical product importance sampling for direct illumination. Comput. Graph. Forum (Proc. Eurograph. 2008) 27(2), 681–690 (2008)CrossRefGoogle Scholar
  7. 7.
    Clarberg, P., Jarosz, W., Akenine-Möller, T., Jensen, H.W.: Wavelet importance sampling: efficiently evaluating products of complex functions. ACM Trans. Graph. (Proc. SIGGRAPH 2005) 24(3), 1166–1175 (2005)CrossRefGoogle Scholar
  8. 8.
    Cline, D., Egbert, P.K., Talbot, J.F., Cardon, D.L.: Two stage importance sampling for direct lighting. In: Proceedings of the 17th eurographics conference on rendering techniques, pp. 103–113. Eurographics Association (2006)Google Scholar
  9. 9.
    Cook, R.L.: Stochastic sampling in computer graphics. ACM Trans. Graph. 5(1), 51–72 (1986)CrossRefGoogle Scholar
  10. 10.
    Georgiev, I., Fajardo, M.: Blue-noise Dithered Sampling. In: ACM SIGGRAPH 2016 Talks, pp. 35:1–35:1. ACM (2016)Google Scholar
  11. 11.
    Gjøl, M., Svendsen, M.: High fidelity, low complexity–the rendering of INSIDE. Game Developers Conference Europe 2016 (2016)Google Scholar
  12. 12.
    de Goes, F., Breeden, K., Ostromoukhov, V., Desbrun, M.: Blue noise through optimal transport. ACM Trans. Graph. (Proc. SIGGRAPH Asia 2012) 31(6), 171:1–171:11 (2012)Google Scholar
  13. 13.
    Hiller, S., Deussen, O., Keller, A.: Tiled blue noise samples. In: Proceedings of the vision modeling and visualization conference 2001, pp. 265–272. Aka GmbH (2001)Google Scholar
  14. 14.
    Huang, H.d., Chen, Y., Tong, X., Wang, W.c.: Incremental wavelet importance sampling for direct illumination. In: Proceedings of the 2007 ACM symposium on virtual reality software and technology, pp. 149–152. ACM (2007)Google Scholar
  15. 15.
    Kajiya, J.T.: The rendering equation. In: Proceedings of the 13th annual conference on computer graphics and interactive techniques, pp. 143–150. ACM (1986)Google Scholar
  16. 16.
    Kopf, J., Cohen-Or, D., Deussen, O., Lischinski, D.: Recursive Wang tiles for real-time blue noise. ACM Trans. Graph. (Proc. SIGGRAPH 2006) 25(3), 509–518 (2006)CrossRefGoogle Scholar
  17. 17.
    Lagae, A., Dutré, P.: A procedural object distribution function. ACM Trans. Graph. 24(4), 1442–1461 (2005)CrossRefGoogle Scholar
  18. 18.
    Lagae, A., Dutré, P.: An alternative for Wang tiles: colored edges versus colored corners. ACM Trans. Graph. 25(4), 1442–1459 (2006)CrossRefGoogle Scholar
  19. 19.
    Lagae, A., Dutré, P.: A comparison of methods for generating Poisson disk distributions. Comput. Graph. Forum 27(1), 114–129 (2008)CrossRefGoogle Scholar
  20. 20.
    McCool, M., Fiume, E.: Hierarchical Poisson disk sampling distributions. In: Proceedings of the conference on graphics interface ’92, pp. 94–105. Morgan Kaufmann Publ. Inc. (1992)Google Scholar
  21. 21.
    Mitsa, T., Parker, K.J.: Digital halftoning technique using a blue-noise mask. J. Opt. Soc. Am. A 9(11), 1920–1929 (1992)CrossRefGoogle Scholar
  22. 22.
    Newbern, J.L., Bove Jr., V.M.: Generation of blue noise arrays by genetic algorithm. Proc. SPIE Hum. Vis. Electron. Imaging II 3016, 441–450 (1997)CrossRefGoogle Scholar
  23. 23.
    Ostromoukhov, V.: Sampling with polyominoes. ACM Trans. Graph. (Proc. SIGGRAPH 2007) 26(3), 1–6 (2007)Google Scholar
  24. 24.
    Ostromoukhov, V., Donohue, C., Jodoin, P.M.: Fast hierarchical importance sampling with blue noise properties. ACM Trans. Graph. (Proc. SIGGRAPH 2004) 23(3), 488–495 (2004)CrossRefGoogle Scholar
  25. 25.
    Purgathofer, W., Tobler, R.F., Geiler, M.: Forced random dithering: improved threshold matrices for ordered dithering. Proc. 1st IEEE Int. Conf. Image Process. 2, 1032–1035 (1994)CrossRefGoogle Scholar
  26. 26.
    Schmaltz, C., Gwosdek, P., Bruhn, A., Weickert, J.: Electrostatic halftoning. Comput. Graph. Forum 29(8), 2313–2327 (2010)CrossRefGoogle Scholar
  27. 27.
    Shade, J., Cohen, M.F., Mitchell, D.P.: Tiling layered depth images. Technical report 02-12-07, University of Washington, Department of Computer Science and Engineering (2002)Google Scholar
  28. 28.
    Ulichney, R.: Digital halftoning. MIT Press, Cambridge (1987)Google Scholar
  29. 29.
    Ulichney, R.: Dithering with blue noise. Proc. IEEE 76(1), 56–79 (1988)CrossRefGoogle Scholar
  30. 30.
    Ulichney, R.: The void-and-cluster method for dither array generation. Proc. of SPIE, Hum. Vis. Vis. Process. Digit. Disp IV 1913, 332–343 (1993)CrossRefGoogle Scholar
  31. 31.
    Wachtel, F., Pilleboue, A., Coeurjolly, D., Breeden, K., Singh, G., Cathelin, G., de Goes, F., Desbrun, M., Ostromoukhov, V.: Fast tile-based adaptive sampling with user-specified Fourier spectra. ACM Trans. Graph. (Proc. SIGGRAPH 2014) 33(4), 56:1–56:11 (2014)Google Scholar
  32. 32.
    Wei, L.Y.: Parallel Poisson disk sampling. ACM Trans. Graph. (Proc. SIGGRAPH 2008) 27(3), 20:1–20:9 (2008)MathSciNetGoogle Scholar
  33. 33.
    Wei, L.Y.: Public SVN repository of Li-Yi Wei, Revision 13 (2011). (accessed February 19, 2017)
  34. 34.
    Wei, L.Y., Wang, R.: Differential domain analysis for non-uniform sampling. ACM Trans. Graph. (Proc. SIGGRAPH 2011) 30(4), 50:1–50:10 (2011)Google Scholar
  35. 35.
    Weinzierl-Heigl, C.: Efficient VAL-based real-time global illumination. In: Proceedings of the 17th Central European Seminar on Computer Graphics (2013)Google Scholar
  36. 36.
    Xiang, Y., Xin, S.Q., Sun, Q., He, Y.: Parallel and accurate Poisson disk sampling on arbitrary surfaces. In: SIGGRAPH Asia 2011 Sketches, pp. 18:1–18:2. ACM (2011)Google Scholar
  37. 37.
    Yellott Jr., J.I.: Spectral consequences of photoreceptor sampling in the rhesus retina. Science 221, 382–385 (1983)CrossRefGoogle Scholar
  38. 38.
    Yuksel, C.: Sample elimination for generating Poisson disk sample sets. Comput. Graph. Forum (Proc. Eurograph. 2015) 34(2), 25–32 (2015)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.VRVis Research CenterViennaAustria
  2. 2.TU WienViennaAustria

Personalised recommendations