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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Cormen, T., Leiserson, C., Rivest, R., Stein, C.: Introduction to Algorithms, Second Edition. MIT Press (2001)
van der Corput, J.: Verteilungsfunktionen. Proc. Ned. Akad. v. Wet. 38, 813–821 (1935)
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)
Faure, H.: Discrépance de suites associées à un système de numération (en dimension s). Acta Arith. 41(4), 337–351 (1982)
Faure, H.: Good permutations for extreme discrepancy. J. Number Theory 42, 47–56 (1992)
Grünschloß, L.: Motion blur. Master’s thesis, Ulm University (2008)
Halton, J.: On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numerische Mathematik 2, 84–90 (1960)
Joe, S., Kuo, F.: Remark on algorithm 659: Implementing Sobol’s quasirandom sequence generator. ACM Trans. Math. Softw. 29(1), 49–57 (2003)
Joe, S., Kuo, F.: Constructing Sobol’ sequences with better two-dimensional projections. SIAM Journal on Scientific Computing 30(5), 2635–2654 (2008)
Keller, A.: Strictly deterministic sampling methods in computer graphics. SIGGRAPH 2003 Course Notes, Course #44: Monte Carlo Ray Tracing (2003)
Keller, A.: Myths of computer graphics. In: D. Talay, H. Niederreiter (eds.) Monte Carlo and Quasi-Monte Carlo Methods, pp. 217–243. Springer (2004)
Kollig, T., Keller, A.: Efficient multidimensional sampling. Computer Graphics Forum 21(3), 557–563 (2002)
Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia (1992)
Niederreiter, H.: Error bounds for quasi-Monte Carlo integration with uniform point sets. J. Comput. Appl. Math. 150, 283–292 (2003)
Pharr, M., Humphreys, G.: Physically Based Rendering: From Theory to Implementation, 2nd edition. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2010)
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)
Downloadable source code package for this article. http://gruenschloss.org/sample-enum/sample-enum-src.zip
Wächter, C.: Quasi-Monte Carlo light transport simulation by efficient ray tracing. Ph.D. thesis, Ulm University (2008)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Grünschloß, L., Raab, M., Keller, A. (2012). Enumerating Quasi-Monte Carlo Point Sequences in Elementary Intervals. In: Plaskota, L., Woźniakowski, H. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2010. Springer Proceedings in Mathematics & Statistics, vol 23. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27440-4_21
Download citation
DOI: https://doi.org/10.1007/978-3-642-27440-4_21
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-27439-8
Online ISBN: 978-3-642-27440-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)