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The Quasi-Random Walk

  • Alexander Keller
Part of the Lecture Notes in Statistics book series (LNS, volume 127)

Abstract

We present the method of the quasi-random walk for the approximation of functionals of the solution of second kind Fredholm integral equations. This deterministic approach efficiently uses low discrepancy sequences for the quasi-Monte Carlo integration of the Neumann series. The fast procedure is illustrated in the setting of computer graphics, where it is applied to several aspects of the global illumination problem.

Keywords

Computer Graphic Importance Sampling Bidirectional Reflectance Distribution Function Neumann Series Global Illumination 
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References

  1. [CW93]
    M. Cohen and J. Wallace. Radiosity and Realistic Image Syn-thesis. Academic Press Professional, Cambridge, 1993.Google Scholar
  2. [GSCH93]
    S. Gortler, P. Schröder, M. Cohen, and P. Hanrahan. Wavelet Radiosity. In Computer Graphics (ACM SIGGRAPH Annual Conference Series), pages 221–230, 1993.Google Scholar
  3. [H1a62]
    E. Hlawka. Lösung von Integralgleichungen mittels zahlenthe-oretischer Methoden I. Sitzungsber., Abt. II,Österr. Akad. Wiss., Math.-Naturwiss. Kl., (171):103–123, 1962.zbMATHMathSciNetGoogle Scholar
  4. [HM72]
    E. Hlawka and R. Mück. Über eine Transformation von gleichverteilten Folgen II. Computing, (9):127–138, 1972.CrossRefzbMATHGoogle Scholar
  5. [HW64]
    J. Halton and G. Weller. Algorithm 247: Radical-inverse quasi-random point sequence. Comm. ACM, 7(12):701–702, 1964.CrossRefGoogle Scholar
  6. [Ke195]
    A. Keller. A Quasi-Monte Carlo Algorithm for the Global Illu-mination Problem in the Radiosity Setting. In H. Niederreiter and P. Shiue, editors, Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, volume 106, pages 239–251. Springer, 1995.CrossRefGoogle Scholar
  7. [Ke196a]
    A. Keller. Quasi-Monte Carlo Methods in Computer Graphics: The Global Illumination Problem. Lectures in App. Math., 32:455–469, 1996.Google Scholar
  8. [Ke196b]
    A. Keller. Quasi-Monte Carlo Radiosity. In X. Pueyo and P. Schröder, editors, Rendering Techniques ‘86 (Proc. 7th Eurographics Workshop on Rendering), pages 101–110. Springer, 1996.CrossRefGoogle Scholar
  9. [Ke196c]
    A. Keller. The fast Calculation of Form Factors using Low Discrepancy Sequences. In Proc. Spring Conference on Computer Graphics (SCCG ‘86), pages 195–204, Bratislava, Slovakia, 1996. Comenius University Press.Google Scholar
  10. [KMS94]
    A. Kersch, W. Morokoff, and A. Schuster. Radiative Heat Transfer with Quasi-Monte Carlo Methods. Transport Theory and Statistical Physics, 7(23):1001–1021, 1994.CrossRefGoogle Scholar
  11. [KW86]
    M. Kalos and P. Whitlock. Monte Carlo Methods, Volume I: Basics. J. Wiley & Sons, 1986.Google Scholar
  12. [Laf96]
    E. Lafortune. Mathematical Models and Monte Carlo Algo-rithms for Physically Based Rendering. PhD thesis, Katholieke Universitiet Leuven, Belgium, 1996.Google Scholar
  13. [MC95]
    W. Morokoff and R. Caflisch Quasi-Monte Carlo Integration. J. Comp. Physics, (122):218–230, 1995.CrossRefzbMATHMathSciNetGoogle Scholar
  14. [Mit92]
    D. Mitchell. Ray Tracing and Irregularities of Distribution. In Proc. 3rd Eurographics Workshop on Rendering, pages 61–69, Bristol, UK, 1992.Google Scholar
  15. [Mit96]
    D. Mitchell. Consequences of Stratified Sampling in Graphics. In Computer Graphics (ACM SIGGRAPH Annual Conference Series), pages 277–280, 1996.Google Scholar
  16. [Nie92]
    H. Niederreiter. Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Pennsylvania, 1992.CrossRefzbMATHGoogle Scholar
  17. [NNB+96]
    A. Neumann, L. Neumann, P. Bekaert, Y. Willem, and W. Purgathofer. Importance-Driven Stochastic Ray Radiosity. Rendering Techniques ‘86 (Proc. 7th Eurographics Workshop on Rendering), pages 111–122, 1996.Google Scholar
  18. [PTVF92]
    H. Press, S. Teukolsky, T. Vetterling, and B. Flannery. Numer- ical Recipes in C. Cambridge University Press, 1992.Google Scholar
  19. [Shi91]
    P. Shirley. Discrepancy as a Quality Measure for Sampling Dis-tributions. In Eurographics ‘81, pages 183–194, Amsterdam, North-Holland, 1991. Elsevier Science Publishers.Google Scholar
  20. [SM94]
    J. Spanier and E. Maize. Quasi-Random Methods for Esti-mating Integrals using relatively small Samples. SIAM Review, 36(1):18–44, March 1994.CrossRefzbMATHMathSciNetGoogle Scholar
  21. [SP87]
    P. Sarkar and M. Prasad. A comparative Study of Pseudo and Quasi Random Sequences for the Solution of Integral Equations. J. Comp. Physics, (68):66–88, March 1987.CrossRefzbMATHMathSciNetGoogle Scholar
  22. [Spa95]
    J. Spanier. Quasi-Monte Carlo Methods for Particle Transport Problems. In H. Niederreiter and P. Shiue, editors, Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, pages 121–148. Springer, 1995.CrossRefGoogle Scholar
  23. [VG95]
    E. Veach and L. Guibas. Optimally Combining Sampling Tech-niques for Monte Carlo Rendering. In Computer Graphics (ACM SIGGRAPH Annual Conference Series), pages 419–428, 1995.Google Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Alexander Keller
    • 1
  1. 1.FB InformatikUniversität KaiserslauternKaiserslauternGermany

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