Sampling

Many applications in graphics require “fair” sampling of unusual spaces, such as the space of all possible lines. This chapter provides the machinery for such probability operations and provides the techniques will also prove useful for numerically evaluating complicated integrals using Monte Carlo integration. Many graphics algorithms use probability to construct random samples to solve integration and averaging problems. The discussion of random variables and their expected values extends naturally to multidimensional spaces. Most graphics problems will be in such higher-dimensional spaces. The variance of a sum of random variables is the sum of the variances if the variables are independent. This summation property of variance is one of the reasons it is frequently used in analysis of probabilistic models. A popular method for quadrature is to replace the random points in Monte Carlo integration with quasi-random points. Quasi-random points can improve performance in many integration applications.