A Mathematical Framework for Global Illumination Algorithms Ph. Dutré, E. Lafortune, Y. D. Willems {philipd, ericl, ydw}@cs.kuleuven.ac.be Department of Computing Science Katholieke Universiteit Leuven Celestijnenlaan 200A B-3001 Heverlee, Belgium 1 Abstract This paper describes a mathematical framework for rendering algorithms. Starting from the ren- dering equation and the potential equation, we will introduce the Global Reflection Distribution Function (GRDF). By using the GRDF, we are able to compute the behaviour of light in an envi- ronment, independent of the initial lighting or viewpoint conditions. This framework is able to describe most existing rendering algorithms. 2 Introduction The global illumination problem is formulated by the well known rendering equation [Kajiya86]. Different methods have been proposed to solve this equation: Monte Carlo Path Tracing, which is in fact an application of distributed ray tracing [Cook et al. 84, Shirley-Wang91, Shirley-Wang92]; various two-pass methods [Chen et al. 91, Sillion-Puech89, Wallace et al. 87], which combine a radiosity and a ray tracing pass; methods based on particle tracing [Pattanaik-Mudur92], which are related to solutions presented in recent heat transfer literature [Brewster92]. Algorithms which solve the global illumination problem can be subdivided into four different classes. A first group of methods is based upon gathering techniques: the illumination of a point or surface is computed by looking at its surroundings, and by taking into account possible contribu- tions towards the illumination of the surface. A second group of methods simulates the propaga- tion of light in an environment, starting from the light sources. Both these approaches can further be divided in deterministic and probabilistic algorithms. Gathering algorithms are described by the traditional rendering equation, but shooting algorithms are best described by the so called potential equation [Pattanaik-Mudur93, Pattanaik93]. 3 The rendering equation 3.1 Exitant and incident radiance Radiance is the basic quantity for describing light transport. It is expressed as power per unit sur- face area per unit solid angle. Exitant radiance (L → ) is the radiance leaving a surface point in a given direction of the hemisphere. Incident radiance (L ← ) is radiance arriving at a surface point from a direction belonging to the hemisphere. Equation 1 gives the realtionship between exitant and incident radiance (figure 1).