Monte Carlo and Quasi-Monte Carlo Methods 2010 pp 399-408 | Cite as
Enumerating Quasi-Monte Carlo Point Sequences in Elementary Intervals
Conference paper
First Online:
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.Cormen, T., Leiserson, C., Rivest, R., Stein, C.: Introduction to Algorithms, Second Edition. MIT Press (2001)Google Scholar
- 2.van der Corput, J.: Verteilungsfunktionen. Proc. Ned. Akad. v. Wet. 38, 813–821 (1935)Google Scholar
- 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.Faure, H.: Discrépance de suites associées à un système de numération (en dimension s). Acta Arith. 41(4), 337–351 (1982)Google Scholar
- 5.Faure, H.: Good permutations for extreme discrepancy. J. Number Theory 42, 47–56 (1992)Google Scholar
- 6.Grünschloß, L.: Motion blur. Master’s thesis, Ulm University (2008)Google Scholar
- 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.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.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.Keller, A.: Strictly deterministic sampling methods in computer graphics. SIGGRAPH 2003 Course Notes, Course #44: Monte Carlo Ray Tracing (2003)Google Scholar
- 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.Kollig, T., Keller, A.: Efficient multidimensional sampling. Computer Graphics Forum 21(3), 557–563 (2002)Google Scholar
- 13.Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia (1992)Google Scholar
- 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.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.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.Downloadable source code package for this article. http://gruenschloss.org/sample-enum/sample-enum-src.zip
- 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