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Monte Carlo Global Illumination

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The Global Illumination Compendium [Dutre 2001] contains transformations for a ... Phillip Dutre, Global Illumination Compendium, http://www.graphics.cornell. ... – PowerPoint PPT presentation

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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
  • 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
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
    about the half-way vector

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
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