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The Role of Functional Analysis in Global Illumination Program of Computer Graphics

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


Source: Arvo, Jim - Departments of Information and Computer Science & Electrical and Computer Engineering, University of California, Irvine


Collections: Computer Technologies and Information Sciences