On the Mathematical Formulation of Radiance

TL;DR

This paper redefines radiance using differential forms and exterior calculus, addressing historical limitations and providing a rigorous foundation for modern light transport theory.

Contribution

It introduces a novel formulation of radiance as a differential form, enabling rigorous derivation of key light transport concepts without heuristics.

Findings

Radiance modeled as a differential form using exterior calculus.

Rigorous derivation of the cosine term and area formulation.

Addresses limitations of classical radiance formulations.

Abstract

Radiance is widely regarded as the principal quantity in light transport theory. Yet, the concept of radiance in use today has remained mostly unchanged since Lambert's work in the 18th century. His formulation of the measurement of light intensity is based on classical differentials, and is known to suffer from several theoretical and practical limitations. After tracing the historic development of radiance and its shortcomings, we provide a modern formulation of light intensity measurements that models radiance as a differential form. We demonstrate the utility of this use of exterior calculus for questions in light transport theory by rigorously deriving the cosine term and the area formulation, without the need for postulates or heuristic arguments. The formulation of radiance as a differential form introduced in this paper hence provides the first step towards a modern theory of light transport.