1. Monte Carlo Rendering Central theme is sampling:
Determine what happens at a set of discrete points, and extrapolate from there
Central algorithm is ray tracing
Rays from the eye or the light
Theoretical basis is probability theory
The study of random events

2. 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  :

3. Expected Value The expected value of a random variable is defined as:
The expected value of a function is defined as:

4. Variance and Standard Deviation
The variance of a random variable is defined as:
The standard deviation of a random variable is defined as the square root of its variance:

5. Sampling
A process samples according to the distribution p(x) if it randomly chooses a value for x such that:
Weak Law of Large Numbers: If xi are independent samples from p(x), then:

6. Algorithms for Sampling
Psuedo-random number generators give independent samples from the uniform distribution on [0,1), p(x)=1
Transformation method: take random samples from the uniform distribution and convert them to samples from another
Rejection sampling: sample from a different distribution, but reject some samples

7. Distribution Functions
Assume a density function f(y) defined on [a,b]
Define the probability distribution function, or cumulative distribution function, as:
Monotonically increasing function with F(a)=0 and F(b)=1

8. 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 distribution function F

9. 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

10. 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 the domain
Geometric interpretation: generate sample 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

11. Important Example Consider uniformly sampling a point on a sphere
Uniformly means that the probability that the point is in a region depends 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

12. Estimating Integrals Say we wish to estimate
Write h=gf, where f is something you choose
If we sample xi according to f, then:

13. Standard Deviation of the Estimate
Expected error in the estimate after n samples is measured by the standard deviation 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

14. Example: Form Factor Integrals
We wish to estimate
Define
Sample from f by sampling xi uniformly on Pi and yi uniformly on Pj
Estimate is

15. Basic Ray Tracing For each pixel in the image
Shoot a ray from the eye through the pixel, to determine what is seen through that pixel, and what its intensity is
Intensity takes contributions from:
Direct illumination (shadow rays, diffuse, Phong)
Reflected rays (recurse on reflected direction)
Transmitted rays (recurse of refraction direction)

16. Casting Rays Given a ray,
Determine the first surface hit by the ray (intersection with lowest t)
Algorithms exist for most representations of surfaces, including splines, fractals, height fields, CSG, implicit surfaces…
Hence, algorithms based on ray tracing can be very general with respect to the geometry

17. Distributed Ray Tracing
Cook, Porter, Carpenter 1984
Addresses the inability of ray tracing to capture:
non-ideal reflection/transmission
soft shadows
motion blur
depth of field
Basic idea: Cast more than one ray for each pixel, for each reflection, for each frame…
Rays are distributed, not the algorithm. Should probably be called distribution ray tracing.

18. Specific Cases
Sample several directions around the reflected direction to get non-ideal reflection, and specularities
Send multiple rays to area light sources to get soft shadows
Cast multiple rays per pixel, spread in time, to get motion blur
Cast multiple rays per pixel, through different lens paths, to get depth-of-field

19. What’s Good? What’s Bad?
Easy to implement - standard ray tracer plus simple sampling
Which L(D|S)*E paths does it get?
Which previous method could it be used with to good effect?