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Monte Carlo Integration
Robert Lin
April 20, 2004
Outline
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Integration Applications
Random variables, probability, expected value, variance
Integration Approximation
Monte Carlo Integration
Variance Reduction (sampling methods)
Integration Applications
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Antialiasing
Integration Applications
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Soft Shadows
Integration Applications
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Indirect Lighting
Random Variables, Probability Density Function
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Continuous random variable x:
scalar or vector quantity that randomly takes on a value (-∞,+∞)
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Probability Density Function p associated with x (denoted x ~ p)
describes the distribution of x:
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Properties:
Random Variables, Probability Density Function
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Example:
Let ε be a random variable taking on values [0, 1) uniformly
Probability Density Function ε ~ q
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Probability that ε takes on a certain value [a, b] in [0, 1) is
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Expected Value
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The average value of a function f(x) with probability distribution
function (pdf) p(x) is called the expected value:
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The expected value of a 1D random variable can be calculated by
letting f(x) = x.
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Expected Value Properties:
1.
2.
Multidimensionality
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Random variables and expected values can be extended to multiple
dimensions easily
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Let S represent a multidimensional space with measure μ
Let x be a random variable with pdf p
Probability that x takes on a value in region in Si, a subset of S, is
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Multidimensionality
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Example:
Let α be a 2D random variable uniformly distributed on a disk of radius R
 p(α) = 1 / (πR2)
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Multidimensionality
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Example
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Given a unit square S = [0, 1] x [0, 1]
 Given pdf p(x, y) = 4xy
 The expected value of the x coordinate is found by setting f(x, y) = x:
Variance
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The variance of a random variable is defined as the expected
value of the square of the difference between x and E(x).
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Some algebra lets us convert this to the form:
Integration Problems
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Integrals for rendering can be difficult to evaluate
 Multi-dimensional integrals
 Non-continuous functions
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Highlights
Occluders
Integration Approximation
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How to evaluate integral of f(x)?
Integration Approximation
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Can approximate using another function g(x)
Integration Approximation
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Can approximate by taking the average value
Integration Approximation
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Estimate the average by taking N samples
Monte Carlo Integration
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Im = Monte Carlo estimate
N = number of samples
x1, x2, …, xN are uniformly distributed random numbers between
a and b
Monte Carlo Integration
Monte Carlo Integration
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We have the definition of expected value and how to estimate it.
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Since the expected value can be expressed as an integral, the
integral is also approximated by the sum.
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To simplify the integral, we can substitute g(x) = f(x)p(x).
Variance
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The variance describes how much the sampled values vary
from each other.
Variance proportional to 1/N
Variance
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Standard Deviation is just the square root of the variance
Standard Deviation proportional to 1 / sqrt(N)
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Need 4X samples to halve the error
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Variance
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Problem:
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Variance (noise) decreases slowly
 Using more samples only removes a small amount of noise
Variance Reduction
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There are several ways to reduce the variance
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Importance Sampling
 Stratified Sampling
 Quasi-random Sampling
 Metropolis Random Mutations
Importance Sampling
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Idea: use more samples in important regions of the function
If function is high in small areas, use more samples there
Importance Sampling
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Want g/p to have low variance
Choose a good function p similar to g:
Stratified Sampling
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Partition S into smaller domains Si
Evaluate integral as sum of integrals over Si
Example: jittering for pixel sampling
Often works much better than importance sampling in practice
Examples
Examples
Conclusion
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Monte Carlo Integration Pros
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Good to estimate integrals with many dimensions
 Good to estimate integrals with complex functions
 General integration method with many applications
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Monte Carlo Integration Cons
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Variance reduces slowly (error appears as noise)
 Reduce variance with importance sampling, stratified sampling, etc.
 Can use other methods (filtering) to remove noise
References
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Peter Shirley, R. Keith Morley. Realistic Ray Tracing, Natick, MA: A K
Peters, Ltd., 2003, pages 47-51, 145-154.
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Henrik Wann Jensen. Realistic Image Synthesis Using Photon Mapping,
Natick, MA: A K Peters, Ltd., 2001, pages 153-155.
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Pat Hanrahan. Monte Carlo Integration 1 (Lecture Notes):
http://graphics.stanford.edu/courses/cs348b-02/lectures/lecture6
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Thomas Funkhouser, Monte Carlo Integration For Image Synthesis:
http://www.cs.princeton.edu/courses/archive/fall02/cs526/lectures/montecarl
o.pdf
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Eric Veach. Robust Monte Carlo Methods for Light Transport Simulation.
Ph.D Thesis, Stanford University, Dec 1997.