1. Monte Carlo I Previous lecture Analytical illumination formula
This lecture
Numerical evaluation of illumination
Review random variables and probability
Monte Carlo integration
Sampling from distributions
Sampling from shapes
Variance and efficiency

2. Lighting and Soft Shadows
Challenges
Visibility and blockers
Varying light distribution
Complex source geometry
Source: Agrawala. Ramamoorthi, Heirich, Moll, 2000

3. Penumbras and Umbras
Should show the view of the light source from the floor at various points
Redraw this figure.
Showed how Lambert’s formulas could be extended to include blockers
Mentioned that Lambert considered regions with different curves as boundaries, including pieces of a sphere
Talked about the shape of the figure on the plane as being a convolution of the two shapes
The shape is a 4D hypercube-like structure
Boundaries introduce discontinuities in the function
Meshing involves breaking surfaces into mesh elements according to these boundaries

4. Monte Carlo Lighting 1 eye ray per pixel 1 shadow ray per eye ray
The 4 eye ray part of this is confusing. Just do 1 eye ray.
1 eye ray per pixel
1 shadow ray per eye ray
Fixed
Random

5. Monte Carlo Algorithms
Advantages
Easy to implement
Easy to think about (but be careful of statistical bias)
Robust when used with complex integrands and domains (shapes, lights, …)
Efficient for high dimensional integrals
Efficient solution method for a few selected points
Disadvantages
Noisy
Slow (many samples needed for convergence)

6. Random Variables is chosen by some random process
probability distribution (density) function

7. Discrete Probability Distributions
Discrete events Xi with probability pi
Cumulative PDF (distribution)
Construction of samples
To randomly select an event,
Select Xi if
Uniform random variable

8. Continuous Probability Distributions
Uniform
PDF
CDF
p(x) dx is the probability that x lies between x and x + dx.
P(x) : if we were to choose a random variable X, then Pr(X < x) should be P(x)
More generally, Pr(alpha ) …

9. Sampling Continuous Distributions
Cumulative probability distribution function
Construction of samples
Solve for X=P-1(U)
Must know:
1. The integral of p(x)
2. The inverse function P-1(x)

10. Example: Power Function
Assume
Trick
Write out the description of the trick. It is hard to explain this method clearly.

11. Sampling a Circle Went through the following argument.
Suppose we have a die with 6 faces. We justify that all faces are equally likely by saying that if we relabeled the faces, the probabilities would be unchanged.
The same argument applies to areas. If we translate an area, then the probability of that area should be unchanged. Translation basic leads to linearity.
We should write something like A = \int p(A) dA = …

12. Sampling a Circle
WRONG  Equi-Areal
RIGHT = Equi-Areal

13. Rejection Methods Algorithm Pick U1 and U2 Accept U1 if U2 < f(U1)
Wasteful?
This is written as if we are computing area. In the current flow, this should be converted to sampling according to a pdf.
Efficiency = Area / Area of rectangle

14. Sampling a Circle: Rejection
do {
X=1-2*U1
Y=1-2*U2
while( X2+ Y2 >1 )
May be used to pick random 2D directions
Circle techniques may also be applied to the sphere

15. Monte Carlo Integration
Definite integral
Expectation of f
Random variables
Estimator

16. Over Arbitrary Domains

17. Non-Uniform Distributions

18. Unbiased Estimator Properties Assume uniform probability
Skipped this derivation. Part that needs to be explained more clearly is E[Y_i] = I(f)
Assume uniform probability
distribution for now

19. Direct Lighting – Directional Sampling
Ray intersection
Sample uniformly by
Show an example of sampling according to solid angle immediately following this slide.

20. Direct Lighting – Area Sampling
Ray direction
Sample uniformly by

21. Examples 4 eye rays per pixel 1 shadow ray per eye ray Fixed Random
The 4 eye ray part of this is confusing. Just do 1 eye ray.
4 eye rays per pixel
1 shadow ray per eye ray
Fixed
Random

22. Examples 4 eye rays per pixel 16 shadow rays per eye ray Uniform grid
Stratified random

23. Examples 4 eye rays per pixel 64 shadow rays per eye ray Uniform grid
Stratified random

24. Examples 4 eye rays per pixel 100 shadow rays per eye ray Uniform grid
Stratified random

25. Examples 4 eye rays per pixel 16 shadow rays per eye ray
What is the cost of these two alternatives. Assume the cost is proportional to the number of intersection tests. Which is equal to the number of edges in the ray tree.
On the left, we have per eye ray for a total of
On the right, we have per eye ray for a total of 128.
Thus we expect the one on the right to cost more. Time the two and report the results.
Which should work better? The right example seems like it should always be better since there are more samples and there seems to be more information per sample. Yet, the left image looks better. How could this be? It is because the extra samples don’t give us much more infromation?
4 eye rays per pixel
16 shadow rays per eye ray
64 eye rays per pixel
1 shadow ray per eye ray

26. Variance
Definition
Properties
Variance decreases with sample size

27. Direct Lighting – Directional Sampling
Ray intersection
Sample uniformly by
Sample uniformly by

28. Sampling Projected Solid Angle
Generate cosine weighted distribution

29. Examples Projected solid angle 4 eye rays per pixel 100 shadow rays
Area
4 eye rays per pixel
100 shadow rays

30. Variance Reduction Efficiency measure Techniques Importance sampling
Sampling patterns: stratified, …
This could be clearer.
Fix the variance. Then the time need to achieve that variance will be n * cost per sample.

31. Sampling a Triangle

32. Sampling a Triangle Here u and v are not independent!
Conditional probability