An Adaptive Gauss Method for Computing Irradiance Coeecients of Galerkin Radiosity Systems

1 Abstract Computing energy transfer between objects is the most expensive operation in radiosity systems. This energy transfer operation, known as the irradiance operator, is an integral that, in general, must be calculated numerically. We wish to increase the speed of this computation without severely compromising delity and perform a study of numerical integration techniques, Quadrature Methods. The results of our study show the strengths of Gauss Quadrature Rules and give us insights into greatly reducing the cost of the irradiance operator while maintaining accuracy. An adaptive method for choosing Gauss quadrature rules is presented, and our performance analysis of the new adaptive algorithm shows that it can be up to 10 times faster than previous methods. 2 Introduction The radiosity equation can be written as a simpliication of the rendering equation , introduced by Kajiya 8], by assuming that all surfaces are perfectly diiuse (Lambertian): B(x) = B e (x) + (x) Z dA x 0 G(x; x 0) ?1 B(x 0) where B(x) is the radiosity at point x, consisting of emitted and reeected ra-diosity, (x) is the reeectance, and G(x; x 0) = cossxcoss x 0 kx?x 0 k 2 V (x; x 0) characterizes the radiant coupling between points x and x 0. G accounts for relative surface orientation, distance, and visibility, V = 0 or V = 1, depending on whether x can or cannot see x 0. The integral is taken over the hemisphere about x and represents the amount of energy per unit area received from other surfaces, irradiance. Radiosity algorithms are usually based on nite element methods. It is assumed that all functions will be represented in the chosen orthonormal basis 1