02/10/03
© 2003 University of Wisconsin
Last Time
•Participating Media
•Assignment 2
–A solution program now exists, so you can preview what yoursolution should look like
–It uses OpenGL with smooth shading, so you would have toimplement Gourand shading in your ray-tracer to see the same thing
–It is OK to use OpenGL to generate the image in this project (butyou’ll get better results if you “gather to a pixel” in a ray-tracer)
02/10/03
© 2003 University of Wisconsin
Today
•Monte-Carlo introduction
•Probability overview
•Sampling methods
02/10/03
© 2003 University of Wisconsin
Monte Carlo Rendering
•Central theme is sampling:
–Determine what happens at a set of discrete points, or for a set oflight paths, and extrapolate from there
•Central algorithm is ray tracing
–Rays from the eye, the lights or other surfaces
•Theoretical basis is probability theory
–The study of random events
02/10/03
© 2003 University of Wisconsin
Distributed Ray Tracing(Cook, Porter, Carpenter 1984)
•Arguably the simplest Monte-Carlo method
•Addresses the inability of ray tracing to capture:
–non-ideal reflection/transmission
–soft shadows
–motion blur
–depth of field
•Basic idea: Use multiple sample rays to build the image foreach pixel
•Rays are distributed, not the algorithm (should probably becalled distribution ray tracing)
02/10/03
© 2003 University of Wisconsin
Specific Cases
•Cast multiple rays per pixel, through different lens paths, toget depth-of-field
•Sample several directions around the reflected direction toget non-ideal reflection
•Send multiple rays to area light sources to get soft shadows
•Cast multiple rays per pixel, spread in time, to get motionblur
•Must be careful not to get correlated samples, which wouldbe particularly apparent in an image
–What should you do for lights?
02/10/03
© 2003 University of Wisconsin
Depth-of-Field
•Regardless of where they pass through the lens, all the rays from a pointon the focal plane end up on the image
•Rays from a non-focal plane point converge somewhere else, resultingin a circle on the image plane
•Note that the aperture also allows a sampling of directions through aworld point
Focal plane
Lens
Aperture
Image plane
Out of Focal plane givescircle of confusion
02/10/03
© 2003 University of Wisconsin
Distributed Depth-of-Field
•Sample multiple rays through the aperture from the image point
•If there are objects outside the focal plane, different rays will hitdifferent points, and contribute different colors
•Also cast rays from within area of pixel
•Average all rays to get resulting pixel color
Lens
Aperture
Image plane
02/10/03
© 2003 University of Wisconsin
Distributed Directional Reflectance
•Cast multiple rays to determine what is reflected in a surface
•Rays should be sampled according to the BRDF
•Average contribution to get pixel color
•Why is this a poor strategy for very diffuse surfaces?
02/10/03
© 2003 University of Wisconsin
Better Approaches
•Distributed ray tracing would need to cast extremely denseray-trees to get good diffuse effects
•Other approaches improve upon this:
–Don’t cast complete trees, just single paths
–Observe that the same point may be hit many times, but distributedray-tracing does not ever re-use partial ray-trees
•But first, some math…
02/10/03
© 2003 University of Wisconsin
Random Variables
•A random variable, x
–Takes values from some domain,
–Has an associated probability density function, p(x)
•A probability density function, p(x)
–Is positive over the domain,
–“Integrates” to 1 over :
02/10/03
© 2003 University of Wisconsin
Expected Value
•The expected value of a random variable, x, is defined as:
•The expected value of a function, f(x), is defined as:
•The sample mean, for samples xi is defined as:
02/10/03
© 2003 University of Wisconsin
Variance and Standard Deviation
•The variance of a random variable is defined as:
•The standard deviation of a random variable is defined asthe square root of its variance:
•The sample variance is:
02/10/03
© 2003 University of Wisconsin
Sampling
•A process samples according to the distribution p(x) if it randomlychooses a value for x such that:
•Weak Law of Large Numbers: If xi are independent samples from p(x),then in the limit of infinite samples, the sample mean is equal to theexpected value:
02/10/03
© 2003 University of Wisconsin
Algorithms for Sampling
•Psuedo-random number generators give independentsamples from the uniform distribution on [0,1), p(x)=1
•Transformation method: take random samples from theuniform distribution and convert them to samples fromanother
•Rejection sampling: sample from a different distribution,but reject some samples
•Importance sampling (to compute an expectation): Samplefrom an easy distribution, but weight the samples
02/10/03
© 2003 University of Wisconsin
Distribution Functions
•Assume a density function f(y) defined on [a,b]
•Define the probability distribution function, or cumulativedistribution function, as:
•Monotonically increasing function with F(a)=0 and F(b)=1
02/10/03
© 2003 University of Wisconsin
Transformation Method for 1D
•Generate zi uniform on [0,1)
•Compute
•Then the xi are distributed according to f(x)
•To apply, must be able to compute and invert distributionfunction F
02/10/03
© 2003 University of Wisconsin
Multi-Dimensional Transformation
•Assume density function f(x,y) defined on [a,b][c,d]
•Distribution function:
•Sample xi according to:
•Sample yi according to:
02/10/03
© 2003 University of Wisconsin
Rejection Sampling
•Say we wish to sample xi according to f(x)
•Find a function g(x) such that g(x)>f(x) for all x in thedomain
•Geometric interpretation: generate sample uniform in thearea under g, and accept if also under f
•Transform a weighted uniform sample according to xi=G-1(z), and generate yi uniform on [0,g(xi)]
•Keep the sample if yi<f(x), otherwise reject
02/10/03
© 2003 University of Wisconsin
Important Example
•Consider uniformly sampling a point on a sphere
–Uniformly means that the probability that the point is in a regiondepends only on the area of the region, not its location on the sphere
•Generate points inside a cube [-1,1]x[-1,1]x[-1,1]
•Reject if the point lies outside the sphere
•Push accepted point onto surface
•Fraction of pts accepted: /6
•Bad strategy in higher dimensions
02/10/03
© 2003 University of Wisconsin
Estimating Integrals
•Say we wish to estimate
•Write h=gf, where f is something you choose
•If we sample xi according to f, then:
02/10/03
© 2003 University of Wisconsin
Standard Deviation of the Estimate
•Expected error in the estimate after n samples is measured by the standarddeviation of the estimate:
•Note that error goes down with
•This technique is called importance sampling
•f should be as close as possible to g
•Same principle for higher dimensional integrals
02/10/03
© 2003 University of Wisconsin
Example: Form Factor Integrals
•We wish to estimate
•Sample from f by sampling xi uniformly on Pi and yiuniformly on Pj
•Define
•Estimate is
