 
Summary: Linear Operators and Integral Equations
in Global Illumination \Lambda
James Arvo
Program of Computer Graphics
Cornell University
Abstract
These notes introduce the basic concepts of integral equations and their application in
global illumination. Much of the discussion is expressed in the language of linear operators
to simplify the notation and to emphasize the algebraic properties of the integral equations.
We start by reviewing some facts about linear operators and examining some of the operators
that occur in global illumination. Six general methods of solving operator and integral
equations are then discussed: the Neumann series, successive approximations, the Nystršom
method, collocation, least squares, and the Galerkin method. Finally, we look at some of
the steps involved in applying these techniques in the context of global illumination.
1 Introduction
The transfer of energy by radiation has a character fundamentally different from the pro
cesses of conduction and convection. One reason for this difference is that the radiant energy
passing through a point in space cannot be completely described by a single scalar quan
tity, even for monochromatic light. In contrast, the process of heat transfer via conduction
can be quite adequately modeled using a scalar field. Another difference is the nonlocal
