CS348B Lecture 9Pat Hanrahan, Spring 2005 Overview Earlier lecture Statistical sampling and Monte...

CS348B Lecture 9 Pat Hanrahan, Spring 2005

Overview

Earlier lecture Statistical sampling and Monte Carlo integration

Last lecture Signal processing view of sampling

Today Variance reduction Importance sampling Stratified sampling Multidimensional sampling patterns Discrepancy and Quasi-Monte Carlo

Latter Path tracing for interreflection Density estimation


Cameras

( , , ) ( ) ( ) cosT A

R L x t P x S t dA d dt

Source: Cook, Porter, Carpenter, 1984Source: Mitchell, 1991

Depth of FieldMotion Blur


Variance

1 shadow ray per eye ray 16 shadow rays per eye ray


Variance

Definition

Variance decreases with sample size

2

2 2

[ ] [( [ ]) ]

[ ] [ ]

V Y E Y E Y

E 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


Variance Reduction

Efficiency measure

If one technique has twice the variance as another technique, then it takes twice as many samples to achieve the same variance

If one technique has twice the cost of another technique with the same variance, then it takes twice as much time to achieve the same variance

Techniques to increase efficiency

Importance sampling

Stratified sampling

1Efficiency

Variance Cost


Biasing

Previously used a uniform probability distribution

Can use another probability distribution

But must change the estimator

~ ( )iX p x

( )

( )i

ii

f XY

p X


Unbiased Estimate

Probability

Estimator

~ ( )iX p x

( )

( )i

ii

f XY

p X

( )[ ]

( )

( )( )

( )

( )

ii

i

i

i

f XE Y E

p X

f Xp x dx

p X

f x dx

I


Importance Sampling

( )( )

[ ]

1( )

[ ]

1

f xp x dx dx

E f

f x dxE f

( )( )

[ ]

f xp x

E f

Sample according to f


Importance Sampling

Variance

2 2[ ] [ ] [ ]V f E f E f 2

2

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 xp x

E f

( )( )

( )

f xf x

p x

Sample according to f

2[ ] 0V f

Zero variance!


Example

method Sampling

function

variance Samples needed for standard error of 0.008

importance

(6-x)/16 56.8N-1 887,500

importance

1/4 21.3N-1 332,812

importance

(x+2)/16 6.4N-1 98,432

importance

x/8 0 1

stratified 1/4 21.3N-3 70

4

0

8xdxI Peter Shirley – Realistic Ray Tracing


Examples

Projected solid angle

4 eye rays per pixel100 shadow rays

Area

4 eye rays per pixel100 shadow rays


Irradiance

Generate cosine weighted distribution

2

( )cosi i i i

H

E L d

( ) cosp d d


Cosine Weighted Distribution

2

0

22

0

2

0

))(cos21(2)sin()cos()cos(

2

dddH

)()()sin()cos(

),()sin()cos(

),(

pppddddp

2

1)( p

21 U

12 U

22

1)( 0

0

0

0

dP

)sin()cos(2)( p

)(sin)(sin)sin()cos(2)( 02

00

20

0

0

dP

)(sin 22 U )arcsin( 2U


Sampling a Circle

1

2

2 U

r U

Equi-Areal


Shirley’s Mapping

1

2

14

r U

U

U


Stratified Sampling

Stratified sampling is like jittered sampling

Allocate samples per region

New variance

Thus, if the variance in regions is less than the overall variance, there will be a reduction in resulting variance

For example: An edge through a pixel

21

1[ ] [ ]

N

N ii

V F V FN

2 1.51

1 [ ][ ] [ ]

NE

N ji

V FV F V F

N N

1

1 N

N ii

F FN


Mitchell 91

Uniform random Spectrally optimized


Discrepancy

x

y

( , )( , )

( , ) number of samples in

n x yx y xy

N

A xy

n x y A

,max ( , )Nx y

D x y


Theorem on Total Variation

Theorem:

Proof: Integrate by parts

1

1( ) ( ) ( )

N

i Ni

f X f x dx V f DN

( ) ( )1ix x x

x N

1

0

( )( ) [ 1]

( )( )

( ) ( )( ) ( )

( )( )

i

N N

x xf x dx

Nx

f x dxxf x f x

f x dx x dxx x

f xD dx V f D

x


Quasi-Monte Carlo Patterns

Radical inverse (digit reverse)

of integer i in integer base b

Hammersley points

Halton points (sequential)

2 1 0

0 1 2( ) 0.i

b i

i d d d d

i d d d d

1 1 .1 1/22 10 .01 1/4

3 11 .11 3/44 100 .001 3/85 101 .101 5/8

2( )i

2 3 5( / , ( ), ( ), ( ), )i N i i i 1log

( )d

N

ND O

N

2 3 5( ( ), ( ), ( ), )i i i log

( )d

N

ND O

N


Hammersly Points

2 3 5( / , ( ), ( ), ( ), )i N i i i


Edge Discrepancy

ax by c

Note: SGI IR Multisampling extension: 8x8 subpixel grid; 1,2,4,8 samples


Low-Discrepancy Patterns

Process 16 points 256 points 1600 points

Zaremba 0.0504 0.00478 0.00111

Jittered 0.0538 0.00595 0.00146

Poisson-Disk

0.0613 0.00767 0.00241

N-Rooks 0.0637 0.0123 0.00488

Random 0.0924 0.0224 0.00866

Discrepancy of random edges, From Mitchell (1992)

Random sampling converges as N-1/2

Zaremba converges faster and has lower discrepancy

Zaremba has a relatively poor blue noise spectraJittered and Poisson-Disk recommended


High-dimensional Sampling

Numerical quadrature

For a given error …

Random sampling

For a given variance …

1

1 1~

dE

n N

1/ 21/ 2

1~ ~E V

N

Monte Carlo requires fewer samplesfor the same error in high dimensional spaces


Block Design

a

bc

d

a

bc

d

a

b

c

d

a

b

c

d

Alphabet of size n

Each symbol appears exactly once ineach row and column

Rows and columns are stratified

Latin Square


Block Design

a

a

a

a

N-Rook Pattern

Incomplete block design

Replaced n2 samples with n samples

Permutations:

Generalizations: N-queens, 2D projection

1 2( ( ), ( ), ( ))di i i

( {1,2,3,4}, {4,2,3,1})x y


Space-time Patterns

Distribute samples in time

Complete in space

Samples in space should have blue-noise spectrum

Incomplete in time

Decorrelate space and time

Nearby samples in space should differ greatly in time

15

4

10

1

3

13

6

8

0

11

5

14

12

2

9

7

Pan-diagonal Magic Square

15

4

10

1

3

136

8

0 11

5

14 12

2

9

7

Cook Pattern


Path Tracing

4 eye rays per pixel16 shadow rays per eye ray

64 eye rays per pixel1 shadow ray per eye ray

Complete Incomplete


Views of Integration

1. Signal processing

Sampling and reconstruction, aliasing and antialiasing

Blue noise good

2. Statistical sampling (Monte Carlo)

Sampling like polling

Variance

High dimensional sampling: 1/N1/2

3. Quasi Monte Carlo

Discrepancy

Asymptotic efficiency in high dimensions

4. Numerical

Quadrature/Integration rules

Smooth functions

CS348B Lecture 9Pat Hanrahan, Spring 2005 Overview Earlier lecture Statistical sampling and Monte...

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