Monte Carlo Integration

1. Monte Carlo Integration COS 323 Acknowledgment: Tom Funkhouser

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

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

4. 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. 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)) x1 xN

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 0 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 = 1664525b = 1013904223c = 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 0 W

22. Importance Sampling • Specific probability distribution: • Function inversion • Rejection

23. Importance Sampling • “Inversion method” • Integrate f(x): Cumulative Distribution Function 1

24. Importance Sampling • “Inversion method” • Integrate f(x): Cumulative Distribution Function • Invert CDF, apply to uniform random variable 1

25. Importance Sampling • Specific probability distribution: • Function inversion • Rejection

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 Equationsfor Fun and Profit

30. or, Ugly Equations, Pretty Pictures

31. Computer Graphics Pipeline Modeling Animation Rendering Lighting and Reflectance

32. Viewer Rendering Equation Surface Light [Kajiya 1986]

33. Rendering Equation • This is an integral equation • Hard to solve! • Can’t solve thisin closed form • Simulate complexphenomena Heinrich

34. Rendering Equation • This is an integral equation • Hard to solve! • Can’t solve thisin closed form • Simulate complexphenomena 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. Eye Pixel x Surface Monte Carlo Global Illumination • Rendering = integration • Antialiasing • Soft shadows • Indirect illumination • Caustics

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 SpecularSurface • 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 benoisy unless lots ofpaths simulated • 40 paths per pixel: Lawrence

50. Monte Carlo Path Tracing • Drawback: can benoisy unless lots ofpaths simulated • 1200 paths per pixel: Lawrence