Importance Driven Quasi-Random Walk Solution of the Rendering Equation L´ aszl´ o Szirmay-Kalos, Bal´ azs Cs´ ebfalvi, Werner Purgathofer Department of Control Engineering and Information Technology, Technical University of Budapest, Budapest, M˝ uegyetem rkp. 11, H-1111, HUNGARY szirmay@fsz.bme.hu Abstract: This paper presents a new method that combines quasi-Monte Carlo quadrature with impor- tance sampling to solve the general rendering equation efficiently. Since classical importance sampling has been proposed for Monte-Carlo integration, first an appropriate formulation is elaborated for de- terministic sample sets used in quasi-Monte Carlo methods. This formulation is based on integration by variable transformation. It is also shown that instead of multi-dimensional inversion methods, the variable transformation can be executed iteratively where each step works only with 2-dimensional mappings. Since the integrands of the Neumann expansion of the rendering equation is not available explicitely, some approximations are used, that are based on a particle-shooting step. Although the complete method works for the original geometry, in order to store the results of the initial particle- shooting, surfaces are decomposed into patches and the patches are interconnected by links. 1 Introduction The main goal of computer graphics is to calculate the image that could be seen by a camera in a vir- tual world. This requires the calculation of the radi- ance reaching the camera from a given direction, i.e. through a given pixel, taking into account the opti- cal properties of the surfaces and the lightsources in the virtual world. However, the radiance of an arbi- trary point is also a function of other point radiances, since these other points may illuminate the given point which may reflect a given portion of the illumination. This type of coupling is expressed by the rendering equation: (1) where is the hemisphere above point , is the radiance of the surface in point at direction , is the visibility function defining the point that is visible from point at direction , is the an- gle between the surface normal and direction , and is the bi-directional reflection/refraction function (figure 1). To simplify the notations, the integral operator of rendering equation is denoted by : (2) x h(x, - L(x, ) ϖ ϖ ϖ ’ Θ ’ ’ ϖ L(h(x, - ϖ ϖ ’ ’ ’ , ) ) ) Figure 1: Geometry of the rendering equation Thus the short form of the rendering equation is: (3) There are two families of techniques that can be used to solve such an integral equation: finite element methods and random walk methods. Un- like finite-element methods, random-walk techniques need not require the surfaces to be decomposed into small patches but can work with the original geome- try. This paper investigates the random-walk solution when the walk is not governed by random decisions but deterministic selection from appropriate uniform sequences. In equation 3 the unknown function appears on both sides. Substituting the right side’s by and assuming that is contractive, we get the follow- ing Neumann series: