The role of functional analysis in global illumination.

In P. M. Hanrahan and Werner Purgathofer, editors, Rendering Techniques '95. Eurographics, Springer-Verlag, June 1995.

The problem of global illumination is virtually synonymous with solving the rendering equation. Although a great deal of research has been directed toward Monte Carlo and finite element methods for solving the rendering equation, little is known about the continuous equation beyond the existence and uniqueness of its solution. The continuous problem may be posed in terms of linear operators acting on infinite-dimensional function spaces. Such operators are fundamentally different from their finite-dimensional counterparts, and are properly studied using the methods of functional analysis. This paper summarizes some of the basic 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, several fundamental operators that arise in global illumination are shown to be nearly finite-dimensional in that they can be uniformly approximated by matrices.

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