Review: Basic MC Estimator

Transcription

1 Overview Earlier lectures Monte Carlo I: Integration Signal processing and sampling Today: Monte Carlo II Noise and variance Efficiency Stratified sampling Importance sampling

2 Review: Basic MC Estimator " NX # Z E 1 f(x i ) = f(x)dx X i 2 [0, 1) n N i=1

3 High-Dimensional Integration Complete set of samples: The curse of dimensionality N = n n n {z } d = n d Random sampling error: E V 1/2 1 p N Numerical integration error: E 1 n = 1 N 1/d In high dimensions, Monte Carlo requires fewer samples than quadrature-based numerical integration for the same error

4 A Tale of Two Functions f(x) =e 20(x 1 2 )2 f(x) =e 5000(x 1 2 )2

5 Monte Carlo Estimates, n= f(x) =e 20(x 1 2 )2 f(x) =e 5000(x 1 2 )2 Z Z f(x)dx = f(x)dx =

6 Random Sampling Introduces Noise Center Random 1 shadow ray

7 Less Noise with More Rays 1 shadow ray 16 shadow rays

8 Variance Definition V [Y ] E[(Y E[Y ]) 2 ] = E[Y 2 ] E[Y ] 2 Variance decreases linearly with sample size V " 1 N NX i=1 Y i # = 1 N 2 NX i=1 V [Y i ]= 1 N 2 NV[Y ]= 1 N V [Y ] V [ay ]=a 2 V [Y ]

9 Comparing Different Techniques Efficiency measure E ciency / 1 Variance Cost Comparing two sampling techniques A and B If A has twice the variance as B, then it takes twice as many samples from A to achieve the same variance as B If A has twice the cost of B, then it takes twice as much time reduce the variance using A compared to using B The product of variance and cost is a constant independent of the number of samples Recall: Variance goes as 1/N, time goes as C*N

10 (Live Demo of Variance)

11 High Variance From Narrow Function Estimate of integral: Z f(x)dx =

12 Stratified Sampling Allocate samples per region Estimate each region separately F N 1 N = N i = 1 F i New variance 1 V[ F ] = V[ F ] N i N N 2 i = 1 If the variance in each region is the same, then total variance goes as 1/N

13 Stratified Sampling Sample a polygon N V F V F E 2 j N 1.5 i= 1 1 [ ] V[ F ] = [ ] = N N If the variance in some regions are smaller, then the overall variance will be reduced

14 Stratified Samples, n= Variance Uniform Stratified 3.99E E-04 Z f(x)dx =

15 Jittered Sampling Add uniform random jitter to each sample

16 Jittered vs. Uniform Supersampling 4x4 Jittered Sampling 4x4 Uniform

17 Theory: Analysis of Jitter Non-uniform sampling n= s( x) = δ ( x x ) n n= x = nt + j n n jn Jittered sampling j( x) ~ j( x) " $ 1 x 1/ 2 = % $& 0 x > 1/ 2 J( ω) = sincω 1 2 2π n= 2 2 π n S( ω) = & 1 J( ω) ' + J( ω) δ ( ω ) T ( ) T 2 T 1 & 1 sinc 2 ω' δ ( ω ) = + T ( ) n=

18 Poisson Disk Sampling Dart throwing algorithm Less energy near origin

19 Distribution of Extrafoveal Cones Monkey eye cone distribution Fourier transform Yellot

20 Fourier Radial Power Spectrum 2 Jitter Power Frequency Poisson Disk Power Frequency Pilleboue et al. [2015]

21 Fourier Radial Power Spectrum 2 Jitter Power Frequency Poisson Disk Power Frequency Pilleboue et al. [2015]

22 Theoretical Convergence Rates in 2D Sampler Worst Case Best Case Jitter O(N 1.5 ) O(N 2 ) Poisson Disk O(N 1 ) O(N 1 ) Random O(N 1 ) O(N 1 )

23 Measuring Power at a Pixel Lens Aperture p 0 r 2 0 Film Plane p Pixel area

24 Measuring Power at a Pixel Lens Aperture p 0 r 2 0 Film Plane p Pixel area ZAlens L(p 0! p) E(p) = cos cos 0 p 0 p 2 da0

25 Measuring Power at a Pixel Lens Aperture p 0 r 2 0 Film Plane p Pixel area ZAlens L(p 0! p) E(p) = cos cos 0 p 0 p 2 da0 Z W = A pixel ZAlens L(p 0! p) cos cos 0 p 0 p 2 da0 da

26 Lens Exit Pupils In Focus CS348b Lecture 11 Out of Focus Pat Hanrahan / Matt Pharr, Spring 2017

27 How to Stratify? n strata in d dimensions: O(n d ) 8 strata in 4 dimensions: 4096 samples!

28 How to Stratify? n strata in d dimensions: O(n d ) 8 strata in 4 dimensions: 4096 samples! Solution: padding Generate stratified lower dimensional point sets, randomly associate pairs of samples (p0, p1, p2, p3)

29 (Live Demo of Stratification)

30 Biased Sampling is Unbiased Probability X i ~ p( x) Estimator Y i = f ( X ) i p( X ) i Proof E[ Y i ] = E ' % & f ( x) p( x) $ " # f ( x) = p( x) dx p( x) = f ( x) dx

31 Importance Sampling Sample according to f p! ( x) = f ( x) E[ f ] f! ( x) = f ( x) p! ( x)

32 Importance Sampling Sample according to f p! ( x) = f ( x) E[ f ] f! ( x) = f ( x) p! ( x) Variance 2 2 V[ f ] = E[ f ] E [ f ]

33 Importance Sampling Sample according to f p! ( x) = f ( x) E[ f ] f ( x) E[ f!! " 2 ] = # $ p! ( x) dx % p! ( x) & 2 f ( x) f! ( x) = p! ( x) Variance 2 2 V[ f ] = E[ f ] E [ f ]! f ( x) " f ( x) = # $ dx % f ( x) / E[ f ]& E[ f ] = = E E[ f ] f ( x) dx 2 [ f ] 2

34 Importance Sampling Sample according to f p! ( x) = f ( x) E[ f ] f ( x) E[ f!! " 2 ] = # $ p! ( x) dx % p! ( x) & 2 f ( x) f! ( x) = p! ( x) Variance 2 2 V[ f ] = E[ f ] E [ f ]! f ( x) " f ( x) = # $ dx % f ( x) / E[ f ]& E[ f ] = = E E[ f ] f ( x) dx 2 [ f ] 2 Zero variance! V[ f! 2 ] = 0

35 Importance Sampling Sample according to f p! ( x) = f ( x) E[ f ] f ( x) E[ f!! " 2 ] = # $ p! ( x) dx % p! ( x) & 2 f ( x) f! ( x) = p! ( x) Variance 2 2 V[ f ] = E[ f ] E [ f ]! f ( x) " f ( x) = # $ dx % f ( x) / E[ f ]& E[ f ] = = E E[ f ] f ( x) dx 2 [ f ] 2 Zero variance! V[ f! 2 ] = 0 Gotcha?

36 Importance Sampling Sample distribution f(x) =e 5000(x 1 2 )2

37 Importance Sampling Sample distribution f(x) =e 5000(x 1 2 )2

38 Importance Sampling 8 6 MC Estimates, n= Sample distribution Z f(x)dx = f(x) =e 5000(x 1 2 )2

39 Importance Sampling Sample distribution Variance Uniform Stratified Importance 3.99E E E f(x) =e 5000(x 1 2 )2

40 Importance Sampling Sample distribution? f(x) =e 5000(x 1 2 )2

41 Importance Sampling Sample distribution? f(x) =e 5000(x 1 2 )2

42 Importance Sampling: Area Solid Angle Area 100 shadow rays 100 shadow rays

43 Ambient Occlusion Landis, McGaugh, ILM Z V (!) cos d!

44 Importance Sampling Uniform hemisphere sampling: f(!) =L i (!) cos p(!) = 1 2 q q q ( 1, 2 )! ( cos(2 2), sin(2 2), )

45 Importance Sampling Uniform hemisphere sampling: f(!) =L i (!) cos p(!) = 1 2 Z NX NX NX f(!)d! 1 N i f(! i ) p(! i ) = 1 N i L i (! i ) cos o 1/2 = 2 N i L i (! i ) cos

46 Importance Sampling Cosine-weighted hemisphere sampling: f(!) =L i (!) cos p(!) = cos Z NX NX NX f(! i ) L i (! i ) cos i = N L i (! i ) f(!)d! 1 N i p(! i ) = 1 N i cos i / i

47 Ambient Occlusion Z V (!) cos d!

48 Reference: 4096 samples

49 Uniform, 32 samples Variance

50 Cosine, 32 samples Variance

51 Uniform, 72 samples Variance

52 Sampling a Circle Equi-Areal θ r = = 2πU U 2 1

53 Shirley s Mapping: Better Strata r = U 1 θ = π U 2 4 U 1

54 Views of Sampling 1. Numerical integration Quadrature/Integration rules Efficient for smooth functions Curse of dimensionality 2. Statistical sampling (Monte Carlo integration) Unbiased estimate of integral High dimensional sampling: 1/N 1/2 3. Signal processing Sampling and reconstruction Aliasing and antialiasing Blue noise good 4. Quasi Monte Carlo Bound error using discrepancy Asymptotic efficiency in high dimensions