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Monte Carlo Global Illumination

Brandon Lloyd

COMP 238December 16, 2002

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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)( NO

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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)

dxxpxfxfE )()()]([

N

iixfN

xfE1

)(1

)]([

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Monte Carlo Method

• … but we are interested in the integral of an arbitrary function f.

N

i i

i

xp

xf

Nxf

1 )(

)(1)(

N

iii xf

Nxpxf

1

)(1

)()(

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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.

N

i i

i

xp

xf

Nxf

1 )(

)(1)(

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Importance Sampling

• Directional formulation of the rendering equation:

• We don’t know Li . We can sample according to: f, cos , or f cos

iiiioiro dLxfxL cos)(),(),(

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Importance Sampling

• Point formulation of the rendering equation:

• A bit more complicated. Usually just generate points on the surfaces.

)(coscos

),(),(),,(),( 2 xdAxx

xxVxxLxxxfxxLS

ir

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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.

x

dsspxP )()(

)(1 UPX

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Example: Diffuse BRDF

• Choose

iiiio dLxL cos)(),(

2

0 0

sincos)(2

iiiiii ddL

sincos),( p

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Example: Diffuse BRDF

• p is separable so we treat each dimension independently

• Invert by solving for u0 = P and u1 = P

22

1

cossincos

0

0

2

dP

dP

)2),(arccos(),( 21 uu

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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

)(1

),( iio LN

xL

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Tranforming the Distribution

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

ZN

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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

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Results

Test Scene

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BRDF sampling Area sampling

Path tracing (combined sampling)

Multiple Importancesampling

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Path tracing

Multiple ImportanceSampling

Multiple ImportanceSampling

Bias!

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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