 
Summary: The Role of Functional Analysis in Global Illumination
James Arvo
Program of Computer Graphics
Cornell University
Ithaca, NY 14853
Abstract: The problem of global illumination is virtually synonymouswith solv
ing the rendering equation. Although a great deal of research has been directed to
ward Monte Carlo and finite element methods for solving the rendering equation,
little is knownabout the continuous equation beyond the existence anduniqueness
of its solution. The continuous problem may be posed in terms of linear operators
acting on infinitedimensional function spaces. Such operators are fundamentally
different from their finitedimensional counterparts, and are properly studied us
ing the methods of functional analysis. This paper summarizes some of the ba
sic concepts of functional analysis and shows how these concepts may be applied
to a linear operator formulation of the rendering equation. In particular, operator
norms are obtained from thermodynamic principles, and a number of common
function spaces are shown to be closed under global illumination. Finally, sev
eral fundamental operators that arise in global illumination are shown to be nearly
finitedimensional in that they can be uniformly approximated by matrices.
1 Introduction
