Analysis of the Quasi-Monte Carlo Integration of the Rendering Equation

László Szirmay-Kalos, Werner Purgathofer
Analysis of the Quasi-Monte Carlo Integration of the Rendering Equation
TR-186-2-98-24, August 1998 [paper]

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Abstract

Quasi-Monte Carlo integration is said to be better than Monte-Carlo integration since its error bound can be in the order of $O(N^{-(1-\epsilon)})$ instead of the $O(N^{-0.5})$ probabilistic bound of classical Monte-Carlo integration if the integrand has finite variation. However, since in computer graphics the integrand of the rendering equation is usually discontinous and thus has infinite variation, the superiority of quasi-Monte Carlo integration has not been theoretically justified. This paper examines the integration of discontinuous functions using both theoretical arguments and simulations and explains what kind of improvements can be expected from the quasi-Monte Carlo techniques in computer graphics.

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BibTeX

@techreport{Szir-1998-QMC,
  title =      " Analysis of the Quasi-Monte Carlo Integration of the
               Rendering Equation",
  author =     "L\'{a}szl\'{o} Szirmay-Kalos and Werner Purgathofer",
  year =       "1998",
  abstract =   "Quasi-Monte Carlo integration is said to be better than
               Monte-Carlo integration since its error bound can be in the
               order of $O(N^{-(1-\epsilon)})$ instead of the $O(N^{-0.5})$
               probabilistic bound of classical Monte-Carlo integration if
               the integrand has finite variation. However, since in
               computer graphics the integrand of the rendering equation is
               usually discontinous and thus has infinite variation, the
               superiority of quasi-Monte Carlo integration has not been
               theoretically justified. This paper examines the integration
               of discontinuous functions using both theoretical arguments
               and simulations and explains what kind of improvements can
               be expected from the quasi-Monte Carlo techniques in
               computer graphics.",
  month =      aug,
  number =     "TR-186-2-98-24",
  address =    "Favoritenstrasse 9-11/E193-02, A-1040 Vienna, Austria",
  institution = "Institute of Computer Graphics and Algorithms, Vienna
               University of Technology ",
  note =       "human contact: technical-report@cg.tuwien.ac.at",
  keywords =   "Hardy-Krause vari ation, quasi-Monte Carlo quadrature,
               Rendering equation",
  URL =        "https://www.cg.tuwien.ac.at/research/publications/1998/Szir-1998-QMC/",
}