Monte Carlo Global Illumination Brandon Lloyd COMP 238 December 16, 2002
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
Monte Carlo Method • The expected value of a function f according to a pdf p: • Can be approximated with a discrete number of samples xi~ p (converges as N)
Monte Carlo Method • … but we are interested in the integral of an arbitrary function f.
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.
Importance Sampling • Directional formulation of the rendering equation: • We don’t know Li .We can sample according to: f, cos , or f cos
Importance Sampling • Point formulation of the rendering equation: • A bit more complicated. Usually just generate points on the surfaces.
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.
Example: Diffuse BRDF • Choose
Example: Diffuse BRDF • p is separable so we treat each dimension independently • Invert by solving for u0= P and u1= P
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
Tranforming the Distribution • The distribution is created in a canonical space but we need to have it about the surface normal. Z N
Z H N 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
Results Test Scene
BRDF sampling Area sampling Path tracing (combined sampling) Multiple Importance sampling
Bias! Path tracing Multiple Importance Sampling Multiple Importance Sampling
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