# Mathematical Models and Monte Carlo Algorithms for Physically Based Rendering

PhD dissertation, Eric Lafortune, February 1996

## Abstract

Algorithms for image synthesis render photo-realistic imagery, given the description of a scene. Physically based rendering specifically stresses the physical correctness of the algorithms and their results. The algorithms perform an accurate simulation of the behaviour of light, in order to faithfully render global illumination effects such as soft shadows, glossy reflections and indirect illumination. In this dissertation we investigate image-based Monte Carlo rendering algorithms. We pay special attention to their correctness, their versatility and their efficiency.

First of all, we discuss theoretical frameworks that describe the global illumination problem. These formal mathematical models are the first step to ensure the correctness of the eventual results. Moreover, they allow to apply standard numerical techniques to compute a solution. We give an overview of existing models, which are based on the rendering equation and the potential equation. We then introduce a model based on a new concept, called the global reflectance distribution function. It combines the ideas of radiance and potential into a single function, which is defined by a set of two integral equations. We later show that, while the three models are equivalent, a straightforward Monte Carlo approach to solve them leads to entirely different rendering algorithms.

We chose to apply Monte Carlo methods because of their versatility. First, we give an overview of Monte Carlo techniques in general. The variance of a technique provides a measure for the stochastic errors on its results. The basic strategy of Monte Carlo methods is to reduce the variance by averaging the results of large numbers of samples. Convergence is slow, however. Variance reduction techniques therefore try to improve the efficacy of the individual samples, by taking into account information about the integrand. We discuss techniques such as stratified sampling, importance sampling, the combining of estimators, control variates, Russian roulette and next event estimation. We stress their unbiasedness, which ensures that the estimators always converge to the exact solution. We specifically analyse aspects that are of importance for the global illumination problem and we present a few improvements to existing techniques.

Finally, we apply the Monte Carlo methods to the mathematical models of the global illumination problem. The different models give rise to entirely different algorithms. The rendering equation leads to the well-known path tracing algorithm, while the potential equation leads to the light tracing algorithm. We discuss their strengths and weaknesses. We study the application of the variance reduction techniques and present results from practical experiments. The global reflectance distribution function and its equations give rise to a new algorithm, which we have called bidirectional path tracing. This algorithm proves to have superior qualities for rendering typical indirectly illuminated interior scenes.