On numerical solutions to one-dimensional integration problems with applications to linear light sources

Abstract

Many key problems in computer graphics require the computation of integrals. Due to the nature of the integrand and of the domain of integration, these integrals seldom can be computed analytically. As a result, numerical techniques are used to find approximate solutions to these problems. While the numerical analysis literature offers many integration techniques, the choice of which method to use for specific computer graphic problems is a difficult one. This choice must be driven by the numerical efficiency of the method, and ultimately, by its visual impact on the computed image. In this paper, we begin to address these issues by methodically analyzing deterministic and stochastic numerical techniques and their application to the type of one-dimensional problems that occur in computer graphics, especially in the context of linear light source integration. In addition to traditional methods such as Gauss-Legendre quadratures, we also examine Voronoi diagram-based sampling, jittered quadratures, random offset quadratures, weighted Monte Carlo, and a newly introduced method of compounding known as a difficulty driven compound quadrature .We compare the effectiveness of these methods using a three-pronged approach. First, we compare the frequency domain characteristics of all the methods using periodograms. Next, applying ideas found in the numerical analysis literature, we examine the numerical and visual performance profiles of these methods for seven different one-parameter problem families. We then present results from the application of the methods for the example of linear light sources. Finally, we summarize the relative effectiveness of the methods surveyed, showing the potential power of difficulty-driven compound quadratures.