Technical Report/TR-186-2-98-18 (4 August 1998) Global Ray-bundle Tracing L´ aszl´ o Szirmay-Kalos Department of Control Engineering and Information Technology, Technical University of Budapest Budapest, M˝ uegyetem rkp. 11, H-1111, HUNGARY szirmay@fsz.bme.hu Abstract The paper presents a single-pass, view-dependent method to solve the general rendering equation, using a com- bined finite element and random walk approach. Applying finite element techniques, the surfaces are decomposed into planar patches on which the radiance is assumed to be combined from finite number of unknown directional radiance functions by predefined positional basis functions. The directional radiance functions are then computed by random walk or by stochastic iteration using bundles of parallel rays. To compute the radiance transfer in a single direction, several global visibility methods are considered, including the global versions of the painter’s, z-buffer, Weiler-Atherton’s and planar graph based algorithms. The method requires no preprocessing except for handling point lightsources, for which a first-shot technique is proposed. The proposed method is particularly efficient for scenes including not very specular materials illuminated by large area lightsources or sky-light. In order to increase the speed for difficult lighting situations, walks can be selected according to their importance. The importance can be explored adaptively by the Metropolis and VEGAS sampling techniques. Keywords: Rendering equation, global radiance, Monte-Carlo and quasi-Monte Carlo integration, Importance sampling, Metropolis method, z-buffer. 1. Introduction The fundamental task of computer graphics is to solve a Fredholm type integral equation describing the light trans- port. This equation is called the rendering equation and has the following form: (1) where and are the radiance and emission of the surface in point at direction , is the directional sphere, is the visibility function defining the point that is visible from point at direction , is the bi-directional reflection/refraction function, is the angle between the surface normal and direction , and if and zero otherwise (figure 1). Since the rendering equation contains the unknown radi- x h(x, - L(x, ) ϖ ϖ ϖ ’ θ ’ ’ ϖ L(h(x, - ϖ ϖ ’ ’ ’ , ) ) ) Figure 1: Geometry of the rendering equation ance function both inside and outside the integral, in order to express the solution, this coupling should be resolved. Gen- erally, two methods can be applied for this: finite element methods or random walk methods. Finite element methods project the problem into a finite function base and approximate the solution here. The pro- c The Institute of Computer Graphics, Vienna University of Technology.