** László Szirmay-Kalos, Péter Dornbach, Werner Purgathofer
Department of Control Engineering and Information Technology, Technical University of Budapest,
Budapest, Muegyetem rkp. 11, H-1111, HUNGARY
szirmay@fsz.bme.hu
**

**Abstract:**

The paper presents an analysis of the start-up bias problem of Metropolis sampling. The analysis is carried out both theoretically and using simulations. In order to allow theoretical treatment, a simplified model is established and the convergence of the Metropolis sampling is examined by Fourier analysis. It is concluded that the start-up bias can be quite significant if the integrand is relatively uniform, thus Metropolis sampling becomes really efficient only for difficult integrands, that is for difficult lighting situations. The theoretical results are then validated for two different integral formulations of the rendering equation. The first is based on bi-directional path tracing, and the second is on ray-bundle tracing.

**Keywords:**

Metropolis sampling, importance sampling, Monte-Carlo integration, rendering equation.