Approximating the Location of Integrand Discontinuities for Penumbral Illumination with Linear Light Sources Marc J. Ouellette Eugene Fiume Department of Computer Science University of Toronto Toronto, ON M5S 3G4 Canada e-mail: vv1 elf @dgp.toronto.edu Abstract One of the benefits of shading with linear light sources is also one of its major challenges: generating soft shad- ows. The primary difficulty in this task is determining the discontinuities in the linear light source integrals that are caused by occluding objects. We demonstrate in this pa- per that the computed location of each discontinuity only needs to be moderately accurate, provided that the ex- pected value of this location is a continuous function of the actual value of the location. We introduce Random Seed Bisection (RSB), an algorithm that has this property. We use this algorithm to efficiently find the approximate location of a discontinuity, in order to partitionthe domain of integration into subintervals (panels) over which the integrand is naturally smooth, and approximate the inte- gral efficiently over each panel using low-degree numeri- cal quadratures. We demonstrate the effectiveness of this solution for shadowing problems with at most 1 disconti- nuity in the domain of integration. We also provide effi- cient heuristics that take advantage of the coherence in a scene to handle shadowing problems with at most 2 dis- continuities in the domain of integration. This work is a first step toward a comprehensive approach to efficiently solving numerical integration problems for extended light sources. R´ esum´ e L’un des b´ en´ efices de l’utilisation de sources lumineuses lin´ eaires repr´ esente aussi son plus grand d´ efi, soit le cal- cul des ombres progressives. Pour r´ esoudre ce probl` eme, on doit d´ eterminer les discontinuit´ es dans la fonction ` a int´ egrer qui sont dues ` a une visibilit´ e partielle de l’en- vironnement. Nous d´ emontrons qu’une approximation mod´ er´ ement pr´ ecise de ces discontinuit´ es est suffisante, ` a condition que la valeur calcul´ ee varie de fac ¸on con- tinue en fonction de la valeur actuelle. Nous pr´ esentons un algorithme avec cette caract´ eristique, soit le Ran- dom Seed Bisection (RSB). Nous nous servons de cet algorithme pour trouver d’une fac ¸on efficace l’endroit approximatif d’une discontinuit´ e, et pour ensuite sub- diviser le domaine de l’int´ egrale en sous-domaines o` u l’on peut ´ evaluer l’int´ egrale rapidement en se servant d’une m´ ethode d’int´ egration ` a base d’interpolation poly- nomiale de faible degr´ e. Nous d´ emontrons l’efficacit´ e de cette solution pour des probl` emes d’ombres o` u l’int´ egrale poss` ede une seule discontinuit´ e. Nous proposons aussi des heuristiques pour les probl` emes ` a 2 discontinuit´ es qui sont bas´ ees sur la coh´ erence inh´ erente d’une sc` ene. Ce travail constitue une premi` ere ´ etape d’une nouvelle fac ¸on de r´ esoudre les probl` emes d’ombres progressives dues aux sources lumineuses ´ etendues bas´ ee sur une approche num´ erique efficace. Key words: Numerical quadratures, integration, linear light sources, soft shadows, random seed bisection. 1 Introduction The process of determining the illumination provided by an extended light source can be separated into two phases. First, given a point to be shaded, we must determine the visibility of the source, that is, the domain of the source that is fully visible from the given point. Second, given the visible portion of the source, we must compute the re- flected light due to this portion. In environments where sources emit light uniformly and surfaces reflect light dif- fusely, the latter integration problem can either be solved analytically — for example if the resulting area to inte- grate is polygonal [9] — or quickly approximated using low degree quadratures. In such cases, the limiting factor in obtaining accurate penumbral shadows lies in the abil- ity to solve the visibility problem. In the general setting of extended light sources, three main techniques have been used to handle the visibil- ity of a source. The earliest techniques determined vis- ibility of the source by either approximating it by point light sources [1], or by point sampling the source itself [4, 15]. This is prone to aliasing if an insufficient num- ber of samples is used. Images of a higher quality can be achieved using algorithms that use shadow volumes and/or discontinuity meshing to determine the exact vis-