13 Monte Carlo Integration
Before we introduce the Integrators that compute radiance along ray paths between lights and the camera, we will first lay some groundwork regarding the techniques they will use to compute solutions to the integral equations that describe light scattering. These integral equations generally do not have analytic solutions, so we must turn to numerical methods. Although standard numerical integration techniques like trapezoidal integration or Gaussian quadrature are very effective at solving low-dimensional smooth integrals, their rate of convergence for the higher dimensional and discontinuous integrals that are common in rendering is poor.
Monte Carlo numerical integration methods provide one solution to this problem. They use randomness to evaluate integrals with a convergence rate that is independent of the dimensionality of the integrand. In this chapter, we review important concepts from probability and lay the foundation for using Monte Carlo techniques to evaluate the key integrals in rendering.
Judicious use of randomness has revolutionized the field of algorithm design. Randomized algorithms fall broadly into two classes: Las Vegas and Monte Carlo. Las Vegas algorithms are those that use randomness but always give the same result in the end (e.g., choosing a random array entry as the pivot element in Quicksort). Monte Carlo algorithms, on the other hand, give different results depending on the particular random numbers used along the way but give the right answer on average. So, by averaging the results of several runs of a Monte Carlo algorithm (on the same input), it is possible to find a result that is statistically very likely to be close to the true answer. Motwani and Raghavan (1995) have written an excellent introduction to the field of randomized algorithms.
Monte Carlo integration is a method for using random sampling to estimate the values of integrals. One very useful property of Monte Carlo is that one only needs the ability to evaluate an integrand at arbitrary points in the domain in order to estimate the value of its integral . This property not only makes Monte Carlo easy to implement but also makes the technique applicable to a broad variety of integrands, including those containing discontinuities.
Many of the integrals that arise in rendering are difficult or impossible to evaluate directly. For example, to compute the amount of light reflected by a surface at a point according to Equation (5.9), we must integrate the product of the incident radiance and the BSDF over the unit sphere. How to do so is not immediately clear: the incident radiance function is almost never available in closed form due to the complex and difficult-to-predict effect of object visibility in realistic scenes.
Even if the incident radiance function were available in closed form, performing the integral analytically would still not be possible in general. Monte Carlo integration makes it possible to estimate the reflected radiance simply by choosing a set of directions over the sphere, computing the incident radiance along them, multiplying by the BSDF’s value for those directions, and applying a weighting term. Arbitrary BSDFs, light source descriptions, and scene geometry are easily handled; evaluation of each of these functions at arbitrary points is all that is required.
The main disadvantage of Monte Carlo is that if samples are used to estimate the integral, the algorithm converges to the correct result at a rate of . In other words, to cut the error in half, it is necessary to evaluate four times as many samples. In rendering, each sample generally requires that one or more rays be traced in the process of computing the value of the integrand, a computationally expensive cost to bear when using Monte Carlo for image synthesis. In images, artifacts from Monte Carlo sampling manifest themselves as noise—pixels are randomly too bright or too dark. Most of the current research in Monte Carlo for computer graphics is about reducing this error as much as possible while minimizing the number of additional samples that must be taken.