1. Monte Carlo Integration COS 323 Acknowledgment: Tom Funkhouser

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

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

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

5. Estimating the average x1x1x1x1 f(x) xNxNxNxN E(f(x)) E(f(x)) Slide courtesy of Peter Shirley

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

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

8. Variance x1x1x1x1 xNxNxNxN E(f(x)) E(f(x))

9. Variance x1x1x1x1 xNxNxNxN Variance decreases as 1/N Error decreases as 1/sqrt(N)

10. Variance Problem: variance decreases with 1/NProblem: variance decreases with 1/N – Increasing # samples removes noise slowly x1x1x1x1 xNxNxNxN E(f(x)) E(f(x))

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

12. Stratified Sampling Estimate subdomains separatelyEstimate subdomains separately x1x1x1x1 xNxNxNxN E k (f(x)) E k (f(x)) Arvo

13. Stratified Sampling This is still unbiasedThis is still unbiased x1x1x1x1 xNxNxNxN E k (f(x)) E k (f(x))

14. Stratified Sampling Less overall variance if less variance in subdomainsLess overall variance if less variance in subdomains x1x1x1x1 xNxNxNxN E k (f(x)) E k (f(x))

15. Importance Sampling Put more samples where f(x) is biggerPut more samples where f(x) is bigger x1x1x1x1 xNxNxNxN E(f(x)) E(f(x))

16. Importance Sampling This is still unbiasedThis is still unbiased x1x1x1x1 xNxNxNxN E(f(x)) E(f(x)) for all N

17. Importance Sampling Zero variance if p(x) ~ f(x)Zero variance if p(x) ~ f(x) x1x1 xNxN E(f(x)) Less variance with better importance sampling

18. Generating Random Points Uniform distribution:Uniform distribution: – Use pseudorandom number generator Probability 0 1 

19. Pseudorandom Numbers Deterministic, but have statistical properties resembling true random numbersDeterministic, but have statistical properties resembling true random numbers Common approach: each successive pseudorandom number is function of previousCommon approach: each successive pseudorandom number is function of previous Linear congruential method:Linear congruential method: – Choose constants carefully, e.g. a = 1664525 b = 1013904223 c = 2 32 – 1

20. Pseudorandom Numbers To get floating-point numbers in [0..1), divide integer numbers by c + 1To 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 uTo 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:Uniform distribution: – Use pseudorandom number generator Probability 0 1 

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

23. Importance Sampling “Inversion method”“Inversion method” – Integrate f ( x ): Cumulative Distribution Function 11

24. Importance Sampling “Inversion method”“Inversion method” – Integrate f ( x ): Cumulative Distribution Function – Invert CDF, apply to uniform random variable 11

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

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

27. Generating Random Points “Rejection method”“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. AnimationAnimation Computer Graphics Pipeline RenderingRendering LightingandReflectanceLightingandReflectance ModelingModeling

32. Rendering Equation Surface Light [Kajiya 1986] Viewer

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

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

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

36. Monte Carlo Integration f(x) Shirley

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

38. Monte Carlo Global Illumination Rendering = integrationRendering = integration – Antialiasing – Soft shadows – Indirect illumination – Caustics

39. Monte Carlo Global Illumination Rendering = integrationRendering = integration – Antialiasing – Soft shadows – Indirect illumination – Caustics SurfaceEyePixel x x

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

41. Monte Carlo Global Illumination Rendering = integrationRendering = integration – Antialiasing – Soft shadows – Indirect illumination – Caustics Herf

42. Monte Carlo Global Illumination Rendering = integrationRendering = integration – Antialiasing – Soft shadows – Indirect illumination – Caustics Surface Eye Light x  Surface ’’’’

43. Monte Carlo Global Illumination Rendering = integrationRendering = integration – Antialiasing – Soft shadows – Indirect illumination – Caustics Debevec

44. Monte Carlo Global Illumination Rendering = integrationRendering = integration – Antialiasing – Soft shadows – Indirect illumination – Caustics Diffuse Surface Eye Light x  Specular Surface ’’’’

45. Monte Carlo Global Illumination Rendering = integrationRendering = integration – Antialiasing – Soft shadows – Indirect illumination – Caustics Jensen

46. Challenge Rendering integrals are difficult to evaluateRendering integrals are difficult to evaluate – Multiple dimensions – Discontinuities Partial occludersPartial occluders HighlightsHighlights CausticsCaustics Drettakis

47. Challenge Rendering integrals are difficult to evaluateRendering integrals are difficult to evaluate – Multiple dimensions – Discontinuities Partial occludersPartial occluders HighlightsHighlights CausticsCaustics Jensen

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

49. Monte Carlo Path Tracing 1000 paths/pixel Jensen

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

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

52. Reducing Variance Observation: some paths more important (carry more energy) than othersObservation: 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 biggerIdea: put more samples where f(x) is bigger

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

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