Variance of integral approximation methods in ray tracing

Abstract

Ray tracing is a rendering technique in computer graphics which mimics how light bounces off objects. Creating images using this technique requires solving the rendering equation [Kaj86], which contains a complicated integral. Finding an analytical solutions is not viable for arbitrary scenes, hence approximation techniques are required. The goal of this thesis is to provide mathematical comparisons between these techniques, which can then be used to provide more insight into their strengths and weaknesses. We will start by providing a mathematical framework for ray tracing and path tracing, where we will demonstrate the need for stochastic approximation methods. We will then discuss commonly used approximation techniques, commonly referred to as estimators. Specifically, we will consider Monte Carlo Sampling, Stratification, Latin Hypercube Sampling and Russian Roulette. The variance of these estimators strongly correlates to the quality of the final image, which will contain more noise with increased variance. We will also compare these estimators to a general estimator, and finally provide some experimental results.