1. Monte Carlo Integration
COS 323
Acknowledgment: Tom Funkhouser

2. Slide courtesy of Peter Shirley
Integration in 1D
f(x)
x=1
Slide courtesy of Peter Shirley

3. Slide courtesy of Peter Shirley
We can approximate
g(x)
f(x)
x=1
Slide courtesy of Peter Shirley

4. Slide courtesy of Peter Shirley
Or we can average
f(x)
E(f(x))
x=1
Slide courtesy of Peter Shirley

5. Estimating the average
f(x)
E(f(x)) x1 xN
Slide courtesy of Peter Shirley

6. Slide courtesy of Peter Shirley
Other Domains
f(x)
< f >ab
x=a
x=b
Slide courtesy of Peter Shirley

7. Benefits of Monte Carlo
No “exponential explosion” in required number of samples with increase in dimension
Resistance to badly-behaved functions

8. Variance
E(f(x)) x1 xN

9. Variance Variance decreases as 1/N Error decreases as 1/sqrt(N)
E(f(x)) x1 xN

10. Variance Problem: variance decreases with 1/N
Increasing # samples removes noise slowly
E(f(x)) x1 xN

11. Variance Reduction Techniques
Problem: variance decreases with 1/N
Increasing # samples removes noise slowly
Variance reduction:
Stratified sampling
Importance sampling

12. Stratified Sampling
Estimate subdomains separately
Arvo
Ek(f(x)) x1 xN

13. Stratified Sampling
This is still unbiased
Ek(f(x)) x1 xN

14. Stratified Sampling
Less overall variance if less variance in subdomains
Ek(f(x)) x1 xN

15. Importance Sampling Put more samples where f(x) is bigger E(f(x)) x1xN

16. Importance Sampling
This is still unbiased
E(f(x))
for all N x1 xN

17. Importance Sampling Zero variance if p(x) ~ f(x) E(f(x))
Less variance with better
importance sampling x1 xN

18. Generating Random Points
Uniform distribution:
Use pseudorandom number generator 1
Probability W

19. Pseudorandom Numbers
Deterministic, but have statistical properties resembling true random numbers
Common approach: each successive pseudorandom number is function of previous
Linear congruential method:
Choose constants carefully, e.g. a = b = c = 232 – 1

20. Pseudorandom Numbers
To get floating-point numbers in [0..1), divide integer numbers by c + 1
To get integers in range [u..v], divide by (c+1)/(v–u+1), truncate, and add u
Better statistics than using modulo (v–u+1)
Only works if u and v small compared to c

21. Generating Random Points
Uniform distribution:
Use pseudorandom number generator 1
Probability W

22. Importance Sampling Specific probability distribution:
Function inversion
Rejection
Split into 2 slides: one for theta, one for phi.

23. Importance Sampling “Inversion method”
Integrate f(x): Cumulative Distribution Function 1
Split into 2 slides: one for theta, one for phi.

24. Importance Sampling “Inversion method”
Integrate f(x): Cumulative Distribution Function
Invert CDF, apply to uniform random variable 1
Split into 2 slides: one for theta, one for phi.

25. Importance Sampling Specific probability distribution:
Function inversion
Rejection
Split into 2 slides: one for theta, one for phi.

26. Generating Random Points
“Rejection method”
Generate random (x,y) pairs, y between 0 and max(f(x))

27. Generating Random Points
“Rejection method”
Generate random (x,y) pairs, y between 0 and max(f(x))
Keep only samples where y < f(x)

28. Monte Carlo in Computer Graphics

29. or, Solving Integral Equations for Fun and Profit

30. or, Ugly Equations, Pretty Pictures

31. Computer Graphics Pipeline
Modeling
Animation
Rendering
Lighting
and
Reflectance

32. Rendering Equation
Surface
Light
Viewer
[Kajiya 1986]

33. Rendering Equation This is an integral equation Hard to solve!
Can’t solve this in closed form
Simulate complex phenomena
Heinrich

34. Rendering Equation This is an integral equation Hard to solve!
Can’t solve this in closed form
Simulate complex phenomena
Jensen

35. Monte Carlo Integration
f(x)
Shirley

36. Monte Carlo Path Tracing
Estimate integral for each pixel by random sampling

37. Monte Carlo Global Illumination
Rendering = integration
Antialiasing
Soft shadows
Indirect illumination
Caustics

38. Monte Carlo Global Illumination
Rendering = integration
Antialiasing
Soft shadows
Indirect illumination
Caustics
Surface
Eye
Pixel x

39. Monte Carlo Global Illumination
Rendering = integration
Antialiasing
Soft shadows
Indirect illumination
Caustics
Light x’
Eye x
Surface

40. Monte Carlo Global Illumination
Rendering = integration
Antialiasing
Soft shadows
Indirect illumination
Caustics
Herf

41. Monte Carlo Global Illumination
Rendering = integration
Antialiasing
Soft shadows
Indirect illumination
Caustics
Surface
Light
Eye w w’ x
Surface

42. Monte Carlo Global Illumination
Rendering = integration
Antialiasing
Soft shadows
Indirect illumination
Caustics
Debevec

43. Monte Carlo Global Illumination
Specular Surface
Rendering = integration
Antialiasing
Soft shadows
Indirect illumination
Caustics
Light
Eye w w’ x
Diffuse Surface

44. Monte Carlo Global Illumination
Rendering = integration
Antialiasing
Soft shadows
Indirect illumination
Caustics
Jensen

45. Challenge Rendering integrals are difficult to evaluate
Multiple dimensions
Discontinuities
Partial occluders
Highlights
Caustics
Drettakis

46. Challenge Rendering integrals are difficult to evaluate
Multiple dimensions
Discontinuities
Partial occluders
Highlights
Caustics
Jensen

47. Monte Carlo Path Tracing
Big diffuse light source, 20 minutes
Jensen

48. Monte Carlo Path Tracing
1000 paths/pixel
Jensen

49. Monte Carlo Path Tracing
Drawback: can be noisy unless lots of paths simulated
40 paths per pixel:
Lawrence

50. Monte Carlo Path Tracing
Drawback: can be noisy unless lots of paths simulated
1200 paths per pixel:
Lawrence

51. Reducing Variance
Observation: some paths more important (carry more energy) than others
For example, shiny surfaces reflect more light in the ideal “mirror” direction
Idea: put more samples where f(x) is bigger

52. Importance Sampling
Idea: put more samples where f(x) is bigger

53. Effect of Importance Sampling
Less noise at a given number of samples
Equivalently, need to simulate fewer paths for some desired limit of noise
Uniform random sampling
Importance sampling