Separating Re?ection Functions

for Linear Radiosity

ALAIN FOURNIER

Department of Computer Science, University of British Columbia
Vancouver,BC, V6T 1Z4, Canada. [email protected]

Abstract: Classic radiosity assumes diffuse re?ectors in order to consider only pair-wise exchanges of light between elements. It has been previously shown that one can use the same system of equations with separable bi-directional re?ection distribution functions (BRDFs), that is BRDFs that can be put in the form of a product of two functions, one of the incident direction and one of the re?ected direction.

We s how here that this can be easily extended to BRDFs that can be approximated by sums of such terms. The classic technique of Singular Value Decomposition (SVD) can be used to compute those terms given an analytical or experimental BRDF.Weuse the example of the traditional Phong model for specularlike re?ection to extract a separable model, and show the results in term of closeness to ordinary Phong shading. Wealso show an example with experimental BRDF data. Further work will indicate whether the quality of linear radiosity images will be improved by this modi?cation.

Keywords: BRDF,separable BRDF,global illumination, form factors, singular value decomposition.

1. Introduction

There is no need to repeat here the principles involved in computing global illumination through the radiosity method (we will use this synecdoche, since it is now well accepted and understood). Wewill use here the notation and terminology of Cohen & Wallace [1] whenever applicable. We w ill include within the class linear pair-wise radiosity,or linear radiosity for short, the methods which use an equation of the form:

Bi = E i + Ri
N

j =1
S B j F ij (1)

Where Ri is only a function of the element i (the re?ectance r i in the classic case), and F ij are only geometric functions of the pairs ij,the form factors.Our goal in this paper is to show how one can extend considerably the class of re?ective behaviours which still lead to a linear radiosity solution.

2. Separable Models and Form Factors

The starting point for the radiosity equations is the rendering equation as given for example in [1]. The corresponding geometry is shown in Figure (1).

L(x?, w ?) = L e(x?, w ?) +

S
? fr (x?, w , w ?) L (x, w)G(x, x?)V (x, x?) dA (2)