1. 1 Monte Carlo Global Illumination Brandon Lloyd COMP 238 December 16, 2002

2. 2 Monte Carlo Method Advantages –Good for integrals of high dimension –All you need is point samples –Allows for arbitrary number of samples Disadvantages –Susceptible to noise (caused by high frequencies in the integrand) –Slow convergence where N is the number of samples

3. 3 Monte Carlo Method The expected value of a function f according to a pdf p: Can be approximated with a discrete number of samples x i ~ p (converges as N  )

4. 4 Monte Carlo Method … but we are interested in the integral of an arbitrary function f.

5. 5 Importance Sampling We can use any distribution p that is non-zero over the domain The distribution affects variance The more closely p matches f the less variance you will have. If p = f then you get the right answer with one sample! But that requires we know f.

6. 6 Importance Sampling Directional formulation of the rendering equation: We don’t know L i. We can sample according to: f, cos , or f cos 

7. 7 Importance Sampling Point formulation of the rendering equation: A bit more complicated. Usually just generate points on the surfaces.

8. 8 Generating Samples We can easily generate a uniform random variable U. Use the Inversion Method to transform U to X ~ p. –Create the CDF of p –Use the inverse of P to transform U.

9. 9 Example: Diffuse BRDF Choose

10. 10 Example: Diffuse BRDF p is separable so we treat each dimension independently Invert by solving for u 0 = P  and u 1 = P 

11. 11 Example: Diffuse BRDF Final Estimator The Global Illumination Compendium [Dutre 2001] contains transformations for a number of useful pdfs that arise in global illumination problems

12. 12 Tranforming the Distribution The distribution is created in a canonical space but we need to have it about the surface normal. Z N

13. 13 Tranforming the Distribution Obvious method. Create a coordinate frame by picking arbitrary S. T = ||NxS|| S=||TxN|| Can be done more cheaply [Hughes99] If the distribution is isotropic then reflect about the half-way vector Z H N

14. 14 Results Test Scene

15. 15 BRDF samplingArea sampling Path tracing (combined sampling) Multiple Importance sampling

16. 16 Path tracing Multiple Importance Sampling Multiple Importance Sampling Bias!

17. 17 References [Hughes99] John F. Hughes and Tomas Möller, “Building an Orthonormal Basis from a Unit Vector'' Journal of Graphics Tools, vol. 4, no. 4, pp. 33-35, 1999. [Dutre01] Phillip Dutre, Global Illumination Compendium, http://www.graphics.cornell.edu/~phil/GI/, 2001 http://www.graphics.cornell.edu/~phil/GI/