Simulating Non-Lambertian Phenomena Involving Linearly-Varying Luminaires

Abstract

We present a new technique for exactly computing glossy reflections and transmissions of polygonal Lambertian luminaires with linearly-varying radiant exitance. To derive the underlying closed-form expressions, we introduce a rational generalization of irradiance tensors and an associated recurrence relation. The generalized tensors allow us to integrate a useful class of rational polynomials over regions of the sphere; this class of rational polynomials can simultaneously account for the linear variation of radiant exitance across a planar luminaire and simple forms of non-Lambertian scattering. Applications include the computation of irradiance at a point, view-dependent reflections from glossy surfaces, and transmissions through glossy surfaces, where the scattering is limited to Phong distributions and the incident illumination is due to linearlyvarying luminaires. In polyhedral environments, the resulting expressions can be exactly evaluated in quadratic time (in the Phong exponent) using dynamic programming or efficiently approximated in linear time using standard numerical quadrature. To illustrate the use of generalized irradiance tensors, we present a greatly simplified derivation of a previously published closed-form expression for the irradiance due to linearly-varying luminaires, and simulate Phong-like scattering effects from such emitters. The validity of our algorithm is demonstrated by comparison with Monte Carlo.