# Monte Carlo Evaluation of Scattering Functions for Computer Graphics

Matt Pharr,
*Ph.D. dissertation*, Stanford University, June 2005.

## Abstract

We present a mathematical framework for solving rendering problems in
computer graphics that is based on scattering as its basic theoretical
foundation, rather than the usual approach of simulating light transport
and equilibrium. This framework can lead to more efficient solution methods
than those based on previous methods as well as a number of new theoretical
tools for solving rendering problems. We demonstrate the applicability of
this approach to accurately and efficiently rendering subsurface scattering
from geometric objects.
We first introduce a non-linear integral scattering equation that describes
scattering from complex objects directly in terms of the composition of
their lower-level scattering properties. This equation was first derived
to solve scattering problems in astrophysics and has gone on to
revolutionize approaches to transport problems in a number of fields. We
derive this equation in a sufficiently general setting to be able to apply
it to a variety of problems in graphics, which typically has problems with
higher-dimensionality, more complexity, and less regularity than those in
other fields. Methods to solve this equation have a divide-and-conquer
flavor to them, in contrast to previous iterative methods based on the
equation of transfer (i.e. the rendering equation). We apply Monte Carlo
techniques to solve this scattering equation efficiently; to our knowledge,
this is the first application of Monte Carlo to solving it in any field.

We next introduce Preisendorfer's *Interaction Principle*, which
subsumes both scattering and light transport based approaches to transfer
problems. It leads to a derivation of a set of adding equations
that describe scattering from multiple objects in terms of how they scatter
light individually. We show how Monte Carlo techniques can be applied to
solve these adding equations and apply them to the problem of rendering
subsurface scattering.

## Dissertation

500 kB PDF