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## Monte Carlo Global Illumination

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Title: Monte Carlo Global Illumination

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

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

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 xi p (converges as N??)

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

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
Importance Sampling
• Directional formulation of the rendering
equation
• We dont know Li . We can sample according to
f, cos ?, or f cos ?

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

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
Example Diffuse BRDF
• Choose

10
Example Diffuse BRDF
• p is separable so we treat each dimension
independently
• Invert by solving for u0 P? and u1 P?

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

Z
N
13
Tranforming the Distribution
• Obvious method. Create a coordinate frame by
picking arbitrary S.
• T NxS STxN
• Can be done more cheaply Hughes99
• If the distribution is isotropic then reflect

14
Results
Test Scene
15
BRDF sampling
Area sampling
Path tracing (combined sampling)
Multiple Importance sampling
16
Bias!
Path tracing
Multiple Importance Sampling
Multiple Importance Sampling
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