On a Class of Gauss-Like Quadrature Rules

Abstract

We consider a problem that arises in the evaluation of computer graphics illumination models. In particular, there is a need to find a finite set of wavelengths at which the illumination model should be evaluated. The result of evaluating the illumniation model at these points is a sampled representation of spectral power density of light emanating from a point in the scene. These values are then used to determine the RGB coordinates of the light by evaluating three definite integrals, each with a common integrand (the SPD) and interval of integration by with distinct weight functions. We develop a method for selecting the sample wavelengths in an optimal manner. More abstractly, we examine the problem of numerically evaluating a set of m definite integrals taken with respect to distinct weight functions but related by a common integrand and interval of integration. It is shown that when m is greater than or equal to 3 it is not efficient to use a set of m Gauss reules because valuable information is wasted. We go on to extend the notions used in Gaussian quadrature to find an optimal set of shared abcissas that maximize precision in a well-defined sense. The classical Gauss rules come out when as the special case m=1 and some anslysis is given concerning the existence of these rules when m>1. In particular, we give conditions on the weight functions that are sufficient to guarantee that the shared abcissas are real, distinct, and lie in the interval of integration. Finally, we examine the comutational strategies for constructing these rules.