Many applications in graphics require “fair” sampling of unusual spaces, such as the space of all possible lines. For example, we might need to generate random edges within a pixel, or random sample points on a pixel that vary in density according to some density function. This chapter provides the machinery for such probability operations. These techniques will also prove useful for numerically evaluating complicated integrals using Monte Carlo integration, also covered in this chapter.
Although the words “integral” and “measure” often seem intimidating, they relate to some of the most intuitive concepts found in mathematics, and they should not be feared. For our very non-rigorous purposes, a measure is ...