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CS348b Lecture 9 Pat Hanrahan, Spring 2016

Overview

Earlier lectures

￭Monte Carlo I: Integration

￭ Signal processing and sampling

Today: Monte Carlo II

￭Noise and variance reduction

￭ Importance sampling

￭ Stratified sampling

￭Multidimensional sampling patterns

Undersampling: Aliasing

CS348b Lecture 9 Pat Hanrahan, Spring 2016

⊗ =

Band-limited signal

Aliasing occurs if the sampling rate is less than 1/2 the maximum frequency in the signal

Note that high frequencies can appear as low frequencies

Frequency Space

Zone plate:

Sampling a “Zone Plate”

CS348b Lecture 9 Pat Hanrahan, Spring 2016

2 2sin x y+

Left rings: signal Right rings: aliases Middle rings: sum of the signal and the aliases

Jittered Sampling

CS348b Lecture 9 Pat Hanrahan, Spring 2016

Add uniform random jitter to each sample

Jittered vs. Uniform Supersampling

CS348b Lecture 9 Pat Hanrahan, Spring 2016

4x4 Jittered Sampling 4x4 Uniform

Regular pattern of aliases replaced by noise

Theory: Analysis of Jitter

CS348b Lecture 9 Pat Hanrahan, Spring 2016

( ) ( )n

nn

s x x xδ=∞

=−∞

= −∑

n nx nT j= +

~ ( )

1 1/ 2( )

0 1/ 2

( ) sinc

nj j x

xj x

x

J ω ω

" ≤$= %

>$&=

2 22

2

1 2 2( ) 1 ( ) ( ) ( )

1 1 sinc ( )

n

n

nS J JT T T

T

π πω ω ω δ ω

ω δ ω

=−∞

=−∞

& '= − + −( )

& '= − +( )

∑

Non-uniform sampling Jittered sampling

Poisson Disk Sampling

CS348b Lecture 9 Pat Hanrahan, Spring 2016

Dart throwing algorithm Less energy near origin

Distribution of Extrafoveal Cones

CS348b Lecture 9 Pat Hanrahan, Spring 2016

Monkey eye cone distribution Fourier transform

Yellot

CS348b Lecture 9 Pat Hanrahan, Spring 2016

Intuition: Uniform Sampling

Uniform sampling

￭ The spectrum of uniformly spaced samples is also a set of uniformly spaced spikes

￭Multiplying the signal by the sampling pattern in the spatial domain, corresponds to placing a copy of the spectrum at each spike in the frequency domain

￭ Aliases are coherent, and very noticeable

CS348b Lecture 9 Pat Hanrahan, Spring 2016

Intuition: Non-uniform Sampling

Non-uniform sampling

￭ Samples at non-uniform locations have a different spectrum; a single spike at the center plus noise

￭ Sampling a signal in this way converts aliases into broadband noise

￭Noise is incoherent, and much less objectionable

Random Sampling Introduces Noise

CS348b Lecture 9 Pat Hanrahan, Spring 2016

1 shadow ray

Center Random

Less Noise with More Rays

CS348b Lecture 9 Pat Hanrahan, Spring 2016

16 shadow rays1 shadow ray

CS348b Lecture 9 Pat Hanrahan, Spring 2016

Variance

Definition

Variance decreases linearly with sample size

2

2 2

[ ] [( [ ]) ][ ] [ ]

V Y E Y E YE Y E Y

≡ −

= −

2 21 1

1 1 1 1[ ] [ ] [ ] [ ]N N

i ii i

V Y V Y NV Y V YN N N N= =

= = =∑ ∑

2[ ] [ ]V aY a V Y=

Why is Area Better than Solid Angle?

CS348b Lecture 9 Pat Hanrahan, Spring 2016

Solid Angle

100 shadow rays

Area

100 shadow rays

Uniform Random Sampling Works

CS348b Lecture 9 Pat Hanrahan, Spring 2016

1

1 11

1 01

1 01

0

1[ ] [ ]

1 1[ ] [ ( )]

1 ( ) ( )

1 ( )

( )

N

N ii

N N

i ii i

N

i

N

i

E F E YN

E Y E f XN N

f x p x dxN

f x dxN

f x dx

=

= =

=

=

=

= =

=

=

=

∑

∑ ∑

∑∫

∑∫

∫

[ ] ( )NE F I f=

Assume uniform probability distribution for now

[ ] [ ]i ii i

E Y E Y=∑ ∑[ ] [ ]E aY aE Y=

Properties

CS348b Lecture 9 Pat Hanrahan, Spring 2016

“Biased” Sampling is “Unbiased”

Probability

Estimator

Proof

~ ( )iX p x

( )( )

ii

i

f XYp X

=

∫

∫

=

=

"#

$%&

'=

dxxf

dxxpxpxf

xpxf

EYE i

)(

)()()()()(][

Importance Sampling

CS348b Lecture 9 Pat Hanrahan, Spring 2016

( )( )[ ]f x

p xE f

=!

( )( )( )f x

f xp x

=!!

Sample according to f

Importance Sampling

CS348b Lecture 9 Pat Hanrahan, Spring 2016

( )( )[ ]f x

p xE f

=!

( )( )( )f x

f xp x

=!!

Sample according to f

2 2[ ] [ ] [ ]V f E f E f= −

Variance

Importance Sampling

CS348b Lecture 9 Pat Hanrahan, Spring 2016

22

2

2

( )[ ] ( )( )

( ) ( )( ) / [ ] [ ]

[ ] ( )

[ ]

f xE f p x dx

p x

f x f xdx

f x E f E f

E f f x dx

E f

! "= # $

% &

! "= # $

% &

=

=

∫

∫

∫

! !!

( )( )[ ]f x

p xE f

=!

( )( )( )f x

f xp x

=!!

Sample according to f

2 2[ ] [ ] [ ]V f E f E f= −

Variance

Importance Sampling

CS348b Lecture 9 Pat Hanrahan, Spring 2016

22

2

2

( )[ ] ( )( )

( ) ( )( ) / [ ] [ ]

[ ] ( )

[ ]

f xE f p x dx

p x

f x f xdx

f x E f E f

E f f x dx

E f

! "= # $

% &

! "= # $

% &

=

=

∫

∫

∫

! !!

( )( )[ ]f x

p xE f

=!

( )( )( )f x

f xp x

=!!

Sample according to f

2[ ] 0V f =!

Zero variance!

2 2[ ] [ ] [ ]V f E f E f= −

Variance

Importance Sampling

CS348b Lecture 9 Pat Hanrahan, Spring 2016

22

2

2

( )[ ] ( )( )

( ) ( )( ) / [ ] [ ]

[ ] ( )

[ ]

f xE f p x dx

p x

f x f xdx

f x E f E f

E f f x dx

E f

! "= # $

% &

! "= # $

% &

=

=

∫

∫

∫

! !!

( )( )[ ]f x

p xE f

=!

( )( )( )f x

f xp x

=!!

Sample according to f

2[ ] 0V f =!

Zero variance!

2 2[ ] [ ] [ ]V f E f E f= −

Variance

Gotcha?

Importance Sampling: Area

CS348b Lecture 9 Pat Hanrahan, Spring 2016

Solid Angle

100 shadow rays

Area

100 shadow rays

CS348b Lecture 9 Pat Hanrahan, Spring 2016

Comparing Different Techniques

Efficiency measure

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

E�ciency / 1

Variance · Cost

CS348b Lecture 9 Pat Hanrahan, Spring 2016

Jittered = Stratified Sampling

Allocate samples per region

Estimate each region separately

New variance

If the variance in each region is the same, then total variance goes as

21

1[ ] [ ]N

N ii

V F V FN =

= ∑

1

1 N

N ii

F FN =

= ∑

1/N

CS348b Lecture 9 Pat Hanrahan, Spring 2016

Stratified Sampling

Sample a polygon

If the variance in some regions are smaller, then the overall variance will be reduced

In this example, the variance in the regions inside and outside the polygon is 0. The variance in the regions along the edge are greater, but there are fewer edge regions

2 1.51

1 [ ][ ] [ ]N

EN j

i

V FV F V FN N=

= =∑

Sampling a Circle

CS348b Lecture 9 Pat Hanrahan, Spring 2016

1

2

2 U

r U

θ π=

=

Equi-Areal

Shirley’s Mapping: Better Strata

CS348b Lecture 9 Pat Hanrahan, Spring 2016

1

2

14

r UUU

πθ

=

=

Block Design

CS348b Lecture 9 Pat Hanrahan, Spring 2016

a

bcd

abc

d

ab

cd

a

bc

d

Alphabet of size n

Each symbol appears exactly once in each row and column

Rows and columns are stratified

Latin Square

Incomplete Block Design

CS348b Lecture 9 Pat Hanrahan, Spring 2016

aa

aa

N-Rook Pattern

Replace n2 samples with n samples

Permutations:

Generalizations: N-queens

1 2( ( ), ( ), ( ))di i iπ π π!

( {1,2,3,4}, {4,2,3,1})x yπ π= =

Path Tracing

CS348b Lecture 9 Pat Hanrahan, Spring 2016

4 eye rays per pixel 16 shadow rays per eye ray

64 eye rays per pixel 1 shadow ray per eye ray

Complete Incomplete

CS348b Lecture 9 Pat Hanrahan, Spring 2016

Space-Time Patterns

Distribute samples in time

￭ Complete in space

￭ Incomplete in time

￭ Decorrelate space and time

￭Nearby samples in space should differ greatly in time

154101

3136

8

011

514

12

297

Pan-diagonal Magic Square

154

10

1

31368

0 115

14 122

97

Cook Pattern

CS348b Lecture 9 Pat Hanrahan, Spring 2016

High-dimensional Sampling

Complete set of samples

Random sampling

Error (variance) …

Numerical integration

Error …

In high dimensional space, for the same error, Monte Carlo integration requires fewer samples than numerical integration

11 1~

dE

n N=

1/ 21/ 21~ ~E VN

1. Numerical integration

￭ Quadrature/Integration rules

￭ Efficient for smooth functions

2. Statistical sampling (Monte Carlo integration)

￭ Unbiased estimate of integral

￭ Variance reduction techniques

￭ High dimensional sampling:

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

CS348b Lecture 9 Pat Hanrahan, Spring 2016

Views of Sampling

1/N1/2