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

Szirmay-Kalos László and Werner Purgathofer
Department of Control Engineering and Information Technology, Technical University of Budapest,
Budapest, Muegyetem rkp. 11, H-1111, HUNGARY


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 discontinuous 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.


Rendering equation, quasi-Monte Carlo quadrature, Hardy-Krause variation.