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

Integration Applications Soft Shadows

Integration Applications Indirect Lighting

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 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 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: 1. 2.

Multidimensionality 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 S i, a subset of S, is

Multidimensionality Example:  Let α be a 2D random variable uniformly distributed on a disk of radius R  p(α) = 1 / (πR 2 )

Multidimensionality 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:

Variance 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 Integrals for rendering can be difficult to evaluate  Multi-dimensional integrals  Non-continuous functions Highlights Occluders

Integration Approximation How to evaluate integral of f(x)?

Integration Approximation Can approximate using another function g(x)

Integration Approximation Can approximate by taking the average value

Integration Approximation Estimate the average by taking N samples

Monte Carlo Integration I m = Monte Carlo estimate N = number of samples x 1, x 2, …, x N are uniformly distributed random numbers between a and b

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).

Variance The variance describes how much the sampled values vary from each other. Variance proportional to 1/N

Variance Standard Deviation is just the square root of the variance Standard Deviation proportional to 1 / sqrt(N) Need 4X samples to halve the error

Variance Problem:  Variance (noise) decreases slowly  Using more samples only removes a small amount of noise

Variance Reduction There are several ways to reduce the variance  Importance Sampling  Stratified Sampling  Quasi-random Sampling  Metropolis Random Mutations

Importance Sampling Idea: use more samples in important regions of the function If function is high in small areas, use more samples there

Importance Sampling Want g/p to have low variance Choose a good function p similar to g:

Stratified Sampling Partition S into smaller domains S i Evaluate integral as sum of integrals over S i Example: jittering for pixel sampling Often works much better than importance sampling in practice

Examples

Conclusion 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

References Peter Shirley, R. Keith Morley. Realistic Ray Tracing, Natick, MA: A K Peters, Ltd., 2003, pages 47-51, Henrik Wann Jensen. Realistic Image Synthesis Using Photon Mapping, Natick, MA: A K Peters, Ltd., 2001, pages Pat Hanrahan. Monte Carlo Integration 1 (Lecture Notes): Thomas Funkhouser, Monte Carlo Integration For Image Synthesis: o.pdf Eric Veach. Robust Monte Carlo Methods for Light Transport Simulation. Ph.D Thesis, Stanford University, Dec 1997.