# Mathematical Models and Monte Carlo Algorithms for Physically Based Rendering

PhD dissertation, Eric Lafortune, February 1996

## Contents

### Chapter 1: Introduction

This chapter introduces the rendering problem and this work. First we briefly discuss the typical input and the output of a rendering algorithm. Then we present a cursory inspection of different types of physically-based rendering algorithms. We conclude the introduction with a survey of the objectives of this thesis and an overview of the contents of the text.

### Chapter 2: The Global Illumination Problem

This chapter introduces the physical concepts that are relevant to physically-based rendering and to the global illumination problem. The essential radiometric quantities are radiance and radiant flux. We will formalise the problem itself using alternative mathematical models. The classical model to describe the radiance function is the rendering equation. The potential function which has been introduced into computer graphics more recently is defined by the potential equation. This equation presents a dual view of the problem. We will then present a new concept, the global reflection distribution function, which combines the ideas of the radiance function and the potential function in a single function. We will show how it is defined by two equivalent integral equations.

### Chapter 3: Monte Carlo Methods

This chapter gives an overview of Monte Carlo methods in general. After an explanation of the basic principles the emphasis will lie on variance reduction techniques. We will present them in a coherent framework and stress the underlying ideas they have in common. We will note any elements that are of particular interest for application to the global illumination problem.

### Chapter 4: Monte Carlo Methods Applied to the Global Illumination Problem

This chapter applies the numerical techniques of Chapter 3 to the mathematical framework of Chapter 2. We will argue why we opt for an image-based approach and for Monte Carlo algorithms, given our objective of versatility. In this context we will show how the different models lead to different rendering algorithms. Most commonly known is the rendering equation as a basis for the path tracing algorithm. This algorithm gathers light starting from the viewpoint. Similarly the potential equation leads to the light tracing algorithm, which distributes light starting from the light sources. The specific strengths and weaknesses of these algorithms will be discussed. We will show how the global reflectance distribution function and its equations give rise to a new algorithm called bidirectional path tracing. This algorithm successfully combines the strengths of the previous approaches by simultaneously distributing and gathering light from the light sources and from the viewpoint respectively.

The variance reduction techniques discussed in the previous chapter can improve the basic rendering algorithms further. We will present a systematic analysis of these general optimisations in the case of these specific algorithms.

### Chapter 5: Test Results

This chapter presents some practical test results. We will apply the various algorithms and optimisations of Chapter 4 to a set of test scenes. The results should give an impression of their qualities in practice.

### Chapter 6: Summary and Conclusions

This chapter summarises the results of this work. We will draw some final conclusions and indicate possible future research directions.

### Appendix A: Camera Models

The appendix discusses the goals of physically based rendering and some more practical aspects of modeling a camera.