# 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]

## Information

- Publication Type: Technical Report
- Workgroup(s)/Project(s): not specified
- Date: August 1998
- Number: TR-186-2-98-24
- Keywords: Hardy-Krause vari ation, quasi-Monte Carlo quadrature, Rendering equation

## 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.## Additional Files and Images

## Weblinks

No further information available.## 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/", }