Realistic Image Synthesis Using Photon Mapping
A practical guide to photon mapping, this book provides both the theory and the insight necessary to implement photon mapping and simulate all types of direct and indirect illumination efficiently.
TABLE OF CONTENTS Realistic Image Synthesis Using Photon Mapping
Appendix A: Basic Monte Carlo Integration
In rendering and in particular global illumination, we often encounter multidimensional integration problems of functions (light fields) with many discontinuities. Since these integrals cannot be evaluated efficiently using standard quadrature rules, it is better to use another class of techniques based on Monte Carlo integration.
A.1 The Sample Mean Method
Monte Carlo integration uses random sampling of the function of interest to examine its properties. Given a function, f( x), that we wish to integrate over a one-dimensional domain from a to b:
| (A.1) |  |
An intuitive way to evaluate this integral is by computing the mean value of f( x) over the interval a to b, and then multiply this mean by the length of the interval b - a. For this purpose we can average the values of f( x) at N locations ? 1, ? 2, ..., ? N, where ? 1, ..., N are uniformly distributed random numbers between a and b. This gives:
| (A.2) |  |
Here I m is the Monte Carlo estimate of the integral. As we increase the number of samples, N, this estimate becomes more accurate and in the limit we find that:
| (A.3) |  |
How fast does the estimator I m converge towards the correct result I? To answer this question we can compute the variance ? 2 of our estimate I m:
| (A.4) |  |
Since the standard deviation
Copyright A K Peters, Ltd. 2001 under license agreement with Books24x7
