# Mathematical Models and Monte Carlo Algorithms for Physically Based
Rendering

PhD dissertation, Eric Lafortune, February 1996
[Dissertation]
[Abstract]
[Overview]
[Contents]
[Images]

## Contents

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.
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.
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.
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.

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.
This chapter summarises the results of this work. We will draw some final
conclusions and indicate possible future research directions.
The appendix discusses the goals of physically based rendering and some more
practical aspects of modeling a camera.

Copyright © 1996-2016 Eric Lafortune.