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Monte Carlo Integration. Robert Lin April 20, 2004. Outline. Integration Applications Random variables, probability, expected value, variance Integration Approximation Monte Carlo Integration Variance Reduction (sampling methods). Integration Applications. Antialiasing.

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monte carlo integration

Monte Carlo Integration

Robert Lin

April 20, 2004

  • Integration Applications
  • Random variables, probability, expected value, variance
  • Integration Approximation
  • Monte Carlo Integration
  • Variance Reduction (sampling methods)
integration applications5
Integration Applications
  • Indirect Lighting
random variables probability density function
Random Variables, Probability Density Function
  • Continuous random variable x:

scalar or vector quantity that randomly takes on a value (-∞,+∞)

  • Probability Density Function p associated with x (denoted x ~ p) describes the distribution of x:
  • Properties:
random variables probability density function7
Random Variables, Probability Density Function
  • Example:

Let ε be a random variable taking on values [0, 1) uniformly

  • Probability Density Function ε~ q
  • Probability that ε takes on a certain value [a, b] in [0, 1) is
expected value
Expected Value
  • The average value of a function f(x) with probability distribution function (pdf) p(x) is called the expected value:
  • The expected value of a 1D random variable can be calculated by letting f(x) = x.
  • Expected Value Properties:



  • Random variables and expected values can be extended to multiple dimensions easily
  • 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
  • Example:
    • Let α be a 2D random variable uniformly distributed on a disk of radius R
    • p(α) = 1 / (πR2)
  • Example
    • 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:
  • The variance of a random variable is defined as the expected value of the square of the difference between x and E(x).
  • Some algebra lets us convert this to the form:
integration problems
Integration Problems
  • Integrals for rendering can be difficult to evaluate
    • Multi-dimensional integrals
    • Non-continuous functions
      • Highlights
      • Occluders
integration approximation
Integration Approximation
  • How to evaluate integral of f(x)?
integration approximation15
Integration Approximation
  • Can approximate using another function g(x)
integration approximation16
Integration Approximation
  • Can approximate by taking the average value
integration approximation17
Integration Approximation
  • Estimate the average by taking N samples
monte carlo integration18
Monte Carlo Integration
  • Im = Monte Carlo estimate
  • N = number of samples
  • x1, x2, …, xN are uniformly distributed random numbers between a and b
monte carlo integration20
Monte Carlo Integration
  • We have the definition of expected value and how to estimate it.
  • Since the expected value can be expressed as an integral, the integral is also approximated by the sum.
  • To simplify the integral, we can substitute g(x) = f(x)p(x).
  • The variance describes how much the sampled values vary from each other.
  • Variance proportional to 1/N
  • Standard Deviation is just the square root of the variance
  • Standard Deviation proportional to 1 / sqrt(N)
  • Need 4X samples to halve the error
  • Problem:
    • Variance (noise) decreases slowly
    • Using more samples only removes a small amount of noise
variance reduction
Variance Reduction
  • There are several ways to reduce the variance
    • Importance Sampling
    • Stratified Sampling
    • Quasi-random Sampling
    • Metropolis Random Mutations
importance sampling
Importance Sampling
  • Idea: use more samples in important regions of the function
  • If function is high in small areas, use more samples there
importance sampling26
Importance Sampling
  • Want g/p to have low variance
  • Choose a good function p similar to g:
stratified sampling
Stratified Sampling
  • 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
  • Monte Carlo Integration Pros
    • Good to estimate integrals with many dimensions
    • Good to estimate integrals with complex functions
    • General integration method with many applications
  • Monte Carlo Integration Cons
    • Variance reduces slowly (error appears as noise)
    • Reduce variance with importance sampling, stratified sampling, etc.
    • Can use other methods (filtering) to remove noise
  • Peter Shirley, R. Keith Morley. Realistic Ray Tracing, Natick, MA: A K Peters, Ltd., 2003, pages 47-51, 145-154.
  • Henrik Wann Jensen. Realistic Image Synthesis Using Photon Mapping, Natick, MA: A K Peters, Ltd., 2001, pages 153-155.
  • Pat Hanrahan. Monte Carlo Integration 1 (Lecture Notes):
  • Thomas Funkhouser, Monte Carlo Integration For Image Synthesis:
  • Eric Veach. Robust Monte Carlo Methods for Light Transport Simulation. Ph.D Thesis, Stanford University, Dec 1997.