Monte Carlo Global Illumination

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Monte Carlo Global Illumination

• Brandon Lloyd
• COMP 238
• December 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

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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??)

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Monte Carlo Method

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

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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.

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Importance Sampling

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

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Importance Sampling

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

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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.

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Example Diffuse BRDF

• Choose

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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?

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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

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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.

Z
N
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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
  about the half-way vector

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