Advertisement

Enumerating Quasi-Monte Carlo Point Sequences in Elementary Intervals

  • Leonhard Grünschloß
  • Matthias Raab
  • Alexander Keller
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 23)

Abstract

Low discrepancy sequences, which are based on radical inversion, expose an intrinsic stratification. New algorithms are presented to efficiently enumerate the points of the Halton and (t, s)-sequences per stratum. This allows for consistent and adaptive integro-approximation as for example in image synthesis.

Keywords

Lookup Table Image Synthesis Radiance Function Elementary Interval Parallel Computing Environment 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Cormen, T., Leiserson, C., Rivest, R., Stein, C.: Introduction to Algorithms, Second Edition. MIT Press (2001)Google Scholar
  2. 2.
    van der Corput, J.: Verteilungsfunktionen. Proc. Ned. Akad. v. Wet. 38, 813–821 (1935)Google Scholar
  3. 3.
    Dammertz, H., Hanika, J., Keller, A., Lensch, H.: A hierarchical automatic stopping condition for Monte Carlo global illumination. In: Proc. of the WSCG 2009, pp. 159–164 (2009)Google Scholar
  4. 4.
    Faure, H.: Discrépance de suites associées à un système de numération (en dimension s). Acta Arith. 41(4), 337–351 (1982)Google Scholar
  5. 5.
    Faure, H.: Good permutations for extreme discrepancy. J. Number Theory 42, 47–56 (1992)Google Scholar
  6. 6.
    Grünschloß, L.: Motion blur. Master’s thesis, Ulm University (2008)Google Scholar
  7. 7.
    Halton, J.: On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numerische Mathematik 2, 84–90 (1960)Google Scholar
  8. 8.
    Joe, S., Kuo, F.: Remark on algorithm 659: Implementing Sobol’s quasirandom sequence generator. ACM Trans. Math. Softw. 29(1), 49–57 (2003)Google Scholar
  9. 9.
    Joe, S., Kuo, F.: Constructing Sobol’ sequences with better two-dimensional projections. SIAM Journal on Scientific Computing 30(5), 2635–2654 (2008)Google Scholar
  10. 10.
    Keller, A.: Strictly deterministic sampling methods in computer graphics. SIGGRAPH 2003 Course Notes, Course #44: Monte Carlo Ray Tracing (2003)Google Scholar
  11. 11.
    Keller, A.: Myths of computer graphics. In: D. Talay, H. Niederreiter (eds.) Monte Carlo and Quasi-Monte Carlo Methods, pp. 217–243. Springer (2004)Google Scholar
  12. 12.
    Kollig, T., Keller, A.: Efficient multidimensional sampling. Computer Graphics Forum 21(3), 557–563 (2002)Google Scholar
  13. 13.
    Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia (1992)Google Scholar
  14. 14.
    Niederreiter, H.: Error bounds for quasi-Monte Carlo integration with uniform point sets. J. Comput. Appl. Math. 150, 283–292 (2003)Google Scholar
  15. 15.
    Pharr, M., Humphreys, G.: Physically Based Rendering: From Theory to Implementation, 2nd edition. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2010)Google Scholar
  16. 16.
    Sobol’, I.: On the distribution of points in a cube and the approximate evaluation of integrals. Zh. vychisl. Mat. mat. Fiz. 7(4), 784–802 (1967)Google Scholar
  17. 17.
    Downloadable source code package for this article. http://gruenschloss.org/sample-enum/sample-enum-src.zip
  18. 18.
    Wächter, C.: Quasi-Monte Carlo light transport simulation by efficient ray tracing. Ph.D. thesis, Ulm University (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Leonhard Grünschloß
    • 1
  • Matthias Raab
    • 2
  • Alexander Keller
    • 2
  1. 1.Rendering Research, Weta DigitalWellingtonNew Zealand
  2. 2.NVIDIA ARC GmbHBerlinGermany

Personalised recommendations