Mathematical Models and Monte Carlo Algorithms for Physically Based Rendering


Eric Lafortune, February 1996.

[Dissertation] [Abstract] [Overview] [Contents] [Images]

Table of contents

Chapter 1: Introduction ..... 3
1.1 Rendering ..... 3
1.2 The Input ..... 4
1.3 The Output ..... 6
1.4 Physically-Based Rendering Algorithms ..... 6
1.4.1 Object-Based and Image-Based Algorithms ..... 7
1.4.2 Deterministic and Monte Carlo Algorithms ..... 7
1.5 Objectives of this Thesis ..... 8
1.6 Overview ..... 9
Chapter 2: The Global Illumination Problem ..... 11
2.1 Concepts ..... 12
2.1.1 Radiance ..... 12
2.1.2 Radiant Flux ..... 14
2.1.3 The Bidirectional Reflectance Distribution Function ..... 16
2.2 The Rendering Equation ..... 18
2.3 The Potential Equation ..... 20
2.4 The Global Reflectance Distribution Function ..... 23
2.4.1 Definition ..... 23
2.4.2 Flux in Terms of the GRDF ..... 24
2.4.3 Equations Defining the GRDF ..... 24
2.5 Summary ..... 27
Chapter 3: Monte Carlo Methods ..... 29
3.1 Introduction ..... 29
3.2 Monte Carlo Integration ..... 31
3.3 Stratified Sampling ..... 34
3.4 Importance Sampling ..... 38
3.5 Combining Estimators ..... 42
3.6 Control Variates ..... 54
3.7 Monte Carlo Methods to Solve Integral Equations ..... 56
3.8 Russian roulette ..... 57
3.9 Next Event Estimation ..... 61
3.10 Summary ..... 62
Chapter 4: Monte Carlo Methods Applied to the Global Illumination Problem ..... 65
4.1 Monte Carlo Methods Applied to the Rendering Equation -- Path Tracing ..... 68
4.1.1 Basic Algorithm ..... 68
4.1.2 Stratified Sampling ..... 72
4.1.3 Importance Sampling ..... 73
4.1.4 Next Event Estimation ..... 74
4.1.5 Combining Estimators ..... 79
4.1.6 Control Variates ..... 80
4.1.7 Improved Importance Sampling and Control Variates ..... 81
4.2 Monte Carlo Methods Applied to the Potential Equation -- Light Tracing ..... 84
4.2.1 Basic Algorithm ..... 84
4.2.2 Importance Sampling ..... 87
4.2.3 Next Event Estimation ..... 87
4.3 Monte Carlo Methods Applied to the Integral Equations of the GRDF -- Bidirectional Path Tracing ..... 91
4.3.1 Basic Algorithm ..... 91
4.3.2 Next Event Estimation ..... 92
4.3.3 Importance Sampling ..... 100
4.3.4 Combining Estimators ..... 101
4.4 Summary ..... 102
Chapter 5: Test Results ..... 103
5.1 Implementation ..... 103
5.2 Test Scenes ..... 105
5.3 Stratified Sampling ..... 107
5.4 Importance Sampling ..... 109
5.5 Next Event Estimation ..... 110
5.6 Combining Estimators ..... 113
5.7 Control Variates ..... 114
5.8 Bidirectional Path Tracing ..... 116
5.9 Summary ..... 118
Chapter 6: Summary and Conclusions ..... 119
6.1 Summary ..... 119
6.2 Conclusions ..... 121
6.3 Future Work ..... 124
Appendix A: Camera Models ..... 127
List of Figures ..... 133
Bibliography ..... 135

This file is maintained by Eric Lafortune (eric@graphics.cornell.edu).