graphics.cg.uni-saarland.de · 24 Chapter 5. Popular sampling patterns Samples Power spectrum Radial mean Random Jitter Multi-jitter N-rooks Figure 5.6: Illustration of random and

ADVANCED SAMPLINGRandom Stratified N-Rooks

Multi-Jittered Quasi-Random Poisson-Disc

Random Stratified N-Rooks

Multi-Jittered Quasi-Random Poisson-Disc

Philipp Slusallek Karol Myszkowski Gurprit Singh

Realistic Image Synthesis SS2020

Part of Siggraph 2016 Course

2Realistic Image Synthesis SS2020

*Wojciech Jarosz Gurprit Singh Kartic Subr

Fourier Analysis of Numerical Integration in Monte Carlo Rendering

*First part of slides from Wojciech Jarosz

Recall: Monte Carlo Integration


I =

Z

Df(x) dx

⇡Z

Df(x)S(x) dx



I =

Z

Df(x) dx

⇡Z

Df(x)S(x) dx



I =

Z

Df(x) dx

⇡Z

Df(x)S(x) dx



I =

Z

Df(x) dx

⇡Z

Df(x)S(x) dx

I =

Z

Df(x) dx

⇡Z

Df(x)S(x) dx



S(x) =1

N

NX

k=1

�(x� xk)

xk

I =

Z

Df(x) dx

⇡Z

Df(x)S(x) dx

I =

Z

Df(x) dx

⇡Z

Df(x)S(x) dx



S(x) =1

N

NX

k=1

�(x� xk)

xk

I =

Z

Df(x) dx

⇡Z

Df(x)S(x) dx

I =

Z

Df(x) dx

⇡Z

Df(x)S(x) dx



S(x) =1

N

NX

k=1

�(x� xk)

How to generate the locations ?xk

xk

I =

Z

Df(x) dx

⇡Z

Df(x)S(x) dx

I =

Z

Df(x) dx

⇡Z

Df(x)S(x) dx


Independent Random Sampling

4

for (int k = 0; k < num; k++){

samples(k).x = randf();samples(k).y = randf();

}



4



}



4

✔Trivially extends to higher dimensions



}



4


✔Trivially progressive and memory-less



}



4



✘ Big gaps



}



4



✘ Big gaps

✘ Clumping



}

Recall: Fourier Theory


Power SpectrumInput Image

Image courtesy: Laurent Belcour

https://belcour.github.io/blog/course/2016/08/25/siggraph-course-part1.html

Recall: Fourier Theory


Power SpectrumInput Image

Image courtesy: Laurent Belcour

https://belcour.github.io/blog/course/2016/08/25/siggraph-course-part1.html

Recall: Fourier theory


f(!) =

Z

Df(x) e�2⇡ ı! x dx

f(~!) =

Z

Df(~x) e�2⇡ ı (~!·~x) d~x

Fourier transform:



f(!) =

Z


f(~!) =

Z

Df(~x) e�2⇡ ı (~!·~x) d~xFourier transform:



S(~!) =

Z

DS(~x) e�2⇡ ı (~!·~x) d~x

S(~!) =

Z

D

1

N

NX

k=1

�(|~x� ~xk|) e�2⇡ ı (~!·~x) d~x

S(~!) =1

N

NX

k=1

e�2⇡ ı (~!·~xk)

f(!) =

Z


f(~!) =

Z


Sampling function:

S(~!) =

Z

DS(~x) e�2⇡ ı (~!·~x) d~x

S(~!) =

Z

D

1

N

NX

k=1

�(|~x� ~xk|) e�2⇡ ı (~!·~x) d~x

S(~!) =1

N

NX

k=1

e�2⇡ ı (~!·~xk)

S(~!) =

Z

DS(~x) e�2⇡ ı (~!·~x) d~x

S(~!) =

Z

D

1

N

NX

k=1

�(|~x� ~xk|) e�2⇡ ı (~!·~x) d~x

S(~!) =1

N

NX

k=1

e�2⇡ ı (~!·~xk)

S(~!) =

Z

DS(~x) e�2⇡ ı (~!·~x) d~x

S(~!) =

Z

D

1

N

NX

k=1

�(|~x� ~xk|) e�2⇡ ı (~!·~x) d~x

S(~!) =1

N

NX

k=1

e�2⇡ ı (~!·~xk)



S(~!) =

Z

DS(~x) e�2⇡ ı (~!·~x) d~x

S(~!) =

Z

D

1

N

NX

k=1

�(|~x� ~xk|) e�2⇡ ı (~!·~x) d~x

S(~!) =1

N

NX

k=1

e�2⇡ ı (~!·~xk)

f(!) =

Z


f(~!) =

Z


Sampling function:

S(~!) =

Z

DS(~x) e�2⇡ ı (~!·~x) d~x

S(~!) =

Z

D

1

N

NX

k=1

�(|~x� ~xk|) e�2⇡ ı (~!·~x) d~x

S(~!) =1

N

NX

k=1

e�2⇡ ı (~!·~xk)

S(~!) =

Z

DS(~x) e�2⇡ ı (~!·~x) d~x

S(~!) =

Z

D

1

N

NX

k=1

�(|~x� ~xk|) e�2⇡ ı (~!·~x) d~x

S(~!) =1

N

NX

k=1

e�2⇡ ı (~!·~xk)

1

N

NX

k=1

�(|~x� ~xk|)



S(~!) =

Z

DS(~x) e�2⇡ ı (~!·~x) d~x

S(~!) =

Z

D

1

N

NX

k=1

�(|~x� ~xk|) e�2⇡ ı (~!·~x) d~x

S(~!) =1

N

NX

k=1

e�2⇡ ı (~!·~xk)

f(!) =

Z


f(~!) =

Z


Sampling function:

S(~!) =

Z

DS(~x) e�2⇡ ı (~!·~x) d~x

S(~!) =

Z

D

1

N

NX

k=1

�(|~x� ~xk|) e�2⇡ ı (~!·~x) d~x

S(~!) =1

N

NX

k=1

e�2⇡ ı (~!·~xk)

S(~!) =

Z

DS(~x) e�2⇡ ı (~!·~x) d~x

S(~!) =

Z

D

1

N

NX

k=1

�(|~x� ~xk|) e�2⇡ ı (~!·~x) d~x

S(~!) =1

N

NX

k=1

e�2⇡ ı (~!·~xk)

1

N

NX

k=1

�(|~x� ~xk|)

S(~!) =

Z

DS(~x) e�2⇡ ı (~!·~x) d~x

S(~!) =

Z

D

1

N

NX

k=1

�(|~x� ~xk|) e�2⇡ ı (~!·~x) d~x

S(~!) =1

N

NX

k=1

e�2⇡ ı (~!·~xk)E

2

4��1

N

NX

k=1

e�2⇡ ı (~!·~xk)

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5

24 Chapter 5. Popular sampling patterns

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Figure 5.6: Illustration of random and some stochastic grid-based sampling patterns with thecorresponding Fourier expected power spectra and the corresponding radial mean of their expectedpower spectra.

sequence is called the Hammersley sequence, which can create a even lower discrepancy point setfor arbitrary dimensions, but due to the first dimension being a regular sampling, knowledge of thenumber of total samples is necessary. Figure 5.7 illustrates the Hammersley point set with 16 and64 points in 2D. The corresponding sampling power spectra for Halton and Hammersley samples(first two components) are summarised in Figures 5.8.

5.3 Blue noise

Any sampling pattern with Blue noise characteristics is suppose to be well distributed within thespatial domain without containing any regular structures. The term Blue noise was coined byUlichney [47], who investivated a radially averaged power spectra of various sampling patterns. Headvocated three important features for an ideal radial power spectrum; First, its peak should be at



Samples Power spectrum

1

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5.3 Blue noise




Many sample set realizations Expected power spectrum

1

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k=1

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~xx

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k=1

e�2⇡ ı (~!·~xk)

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k=1

�(|~x� ~xk|)

~xx

~xy

Samples Power spectrum Radial meanSamples Power spectrum Radial meanSamples Power spectrum Radial meanSamples Power spectrum Radial mean

~!y

~!x

24C

hap

ter5

.Pop

ula

rsa

mp

ling

pa

ttern

s

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ples

Pow

ersp

ectru

mR

adia

lmea

n

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6:Ill

ustra

tion

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ndom

and

som

est

ocha

stic

grid

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edsa

mpl

ing

patte

rns

with

the

corr

espo

ndin

gFo

urie

rexp

ecte

dpo

wer

spec

traan

dth

eco

rres

pond

ing

radi

alm

ean

ofth

eire

xpec

ted

pow

ersp

ectra

.

sequ

ence

isca

lled

the

Ham

mer

sley

sequ

ence

,whi

chca

ncr

eate

aev

enlo

wer

disc

repa

ncy

poin

tset

fora

rbitr

ary

dim

ensi

ons,

butd

ueto

the

first

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ensi

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ing

are

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plin

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dge

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enu

mbe

roft

otal

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ples

isne

cess

ary.

Figu

re5.

7ill

ustra

tes

the

Ham

mer

sley

poin

tset

with

16an

d64

poin

tsin

2D.T

heco

rres

pond

ing

sam

plin

gpo

wer

spec

trafo

rHal

ton

and

Ham

mer

sley

sam

ples

(firs

ttw

oco

mpo

nent

s)ar

esu

mm

aris

edin

Figu

res

5.8.

5.3

Blue

nois

e

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sam

plin

gpa

ttern

with

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eno

ise

char

acte

ristic

sis

supp

ose

tobe

wel

ldis

tribu

ted

with

inth

esp

atia

ldom

ain

with

outc

onta

inin

gan

yre

gula

rst

ruct

ures

.Th

ete

rmB

lue

nois

ew

asco

ined

byU

lichn

ey[4

7],w

hoin

vest

ivat

eda

radi

ally

aver

aged

pow

ersp

ectra

ofva

rious

sam

plin

gpa

ttern

s.H

ead

voca

ted

thre

eim

porta

ntfe

atur

esfo

ran

idea

lrad

ialp

ower

spec

trum

;Firs

t,its

peak

shou

ldbe

at

24C

hap

ter5.Popula

rsam

pling

pa

tterns

Samples

Powerspectrum

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ean

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Figure5.6:

Illustrationof

randomand

some

stochasticgrid-based

sampling

patternsw

iththe

correspondingFourierexpected

powerspectra

andthe

correspondingradialm

eanoftheirexpected

powerspectra.

sequenceis

calledthe

Ham

mersley

sequence,which

cancreate

aeven

lowerdiscrepancy

pointsetforarbitrary

dimensions,butdue

tothe

firstdimension

beinga

regularsampling,know

ledgeofthe

numberoftotalsam

plesis

necessary.Figure5.7

illustratesthe

Ham

mersley

pointsetwith

16and

64points

in2D

.Thecorresponding

sampling

powerspectra

forHalton

andH

amm

ersleysam

ples(firsttw

ocom

ponents)aresum

marised

inFigures

5.8.

5.3Blue

noise

Any

sampling

patternw

ithB

luenoise

characteristicsis

supposeto

bew

elldistributedw

ithinthe

spatialdomain

withoutcontaining

anyregular

structures.The

termB

luenoise

was

coinedby

Ulichney

[47],who

investivateda

radiallyaveraged

powerspectra

ofvarioussampling

patterns.He

advocatedthree

importantfeatures

foranidealradialpow

erspectrum;First,its

peakshould

beat

24C

hap

ter5

.Pop

ula

rsa

mp

ling

pa

ttern

s

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ples

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ersp

ectru

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re5.

6:Ill

ustra

tion

ofra

ndom

and

som

est

ocha

stic

grid

-bas

edsa

mpl

ing

patte

rns

with

the

corr

espo

ndin

gFo

urie

rexp

ecte

dpo

wer

spec

traan

dth

eco

rres

pond

ing

radi

alm

ean

ofth

eire

xpec

ted

pow

ersp

ectra

.

sequ

ence

isca

lled

the

Ham

mer

sley

sequ

ence

,whi

chca

ncr

eate

aev

enlo

wer

disc

repa

ncy

poin

tset

fora

rbitr

ary

dim

ensi

ons,

butd

ueto

the

first

dim

ensi

onbe

ing

are

gula

rsam

plin

g,kn

owle

dge

ofth

enu

mbe

roft

otal

sam

ples

isne

cess

ary.

Figu

re5.

7ill

ustra

tes

the

Ham

mer

sley

poin

tset

with

16an

d64

poin

tsin

2D.T

heco

rres

pond

ing

sam

plin

gpo

wer

spec

trafo

rHal

ton

and

Ham

mer

sley

sam

ples

(firs

ttw

oco

mpo

nent

s)ar

esu

mm

aris

edin

Figu

res

5.8.

5.3

Blue

nois

e

Any

sam

plin

gpa

ttern

with

Blu

eno

ise

char

acte

ristic

sis

supp

ose

tobe

wel

ldis

tribu

ted

with

inth

esp

atia

ldom

ain

with

outc

onta

inin

gan

yre

gula

rst

ruct

ures

.Th

ete

rmB

lue

nois

ew

asco

ined

byU

lichn

ey[4

7],w

hoin

vest

ivat

eda

radi

ally

aver

aged

pow

ersp

ectra

ofva

rious

sam

plin

gpa

ttern

s.H

ead

voca

ted

thre

eim

porta

ntfe

atur

esfo

ran

idea

lrad

ialp

ower

spec

trum

;Firs

t,its

peak

shou

ldbe

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Samples Expected power spectrum

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5.3 Blue noise


Radial meanDC Peak

1

N

NX

k=1

�(|~x� ~xk|) E

2

4��1

N

NX

k=1

e�2⇡ ı (~!·~xk)

��

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5



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5.3 Blue noise



Regular Samplingfor (uint i = 0; i < numX; i++)

for (uint j = 0; j < numY; j++){

samples(i,j).x = (i + 0.5)/numX;samples(i,j).y = (j + 0.5)/numY;

}

10





}

10





}

10

✔Extends to higher dimensions, but…





}

10


✘ Curse of dimensionality





}

10



✘ Aliasing





}

11


Jittered/Stratified Samplingfor (uint i = 0; i < numX; i++)


samples(i,j).x = (i + randf())/numX;samples(i,j).y = (j + randf())/numY;

}

12





}

12

✔Provably cannot increase variance





}

12







}

12








}

12




✘ Not progressive



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Jittered Sampling


Samples Expected power spectrum Radial mean





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5.3 Blue noise



Monte Carlo (16 random samples)


Monte Carlo (16 jittered samples)


Stratifying in Higher DimensionsStratification requires O(Nd) samples- e.g. pixel (2D) + lens (2D) + time (1D) = 5D



• splitting 2 times in 5D = 25 = 32 samples

• splitting 3 times in 5D = 35 = 243 samples!



• splitting 2 times in 5D = 25 = 32 samples

• splitting 3 times in 5D = 35 = 243 samples!

Inconvenient for large d- cannot select sample count with fine granularity


Uncorrelated Jitter [Cook et al. 84]


Compute stratified samples in sub-dimensionsUncorrelated Jitter [Cook et al. 84]


Compute stratified samples in sub-dimensions- 2D jittered (x,y) for pixel


18Image source: PBRTe2 [Pharr & Humphreys 2010]Realistic Image Synthesis SS2020


- 2D jittered (u,v) for lens





- 1D jittered (t) for time





- 1D jittered (t) for time

- combine dimensions in random order



Depth of Field (4D)


Reference Random Sampling Uncorrelated Jitter

Image source: PBRTe2 [Pharr & Humphreys 2010]

Stratify samples in each dimension separatelyUncorrelated Jitter ➔ Latin Hypercube


Stratify samples in each dimension separately- for 5D: 5 separate 1D jittered point sets

Uncorrelated Jitter ➔ Latin Hypercube

20

x1 x2 x3 x4x

y1 y2 y3 y4y

u1 u2 u3 u4u

v1 v2 v3 v4v

t1 t2 t3 t4t


Stratify samples in each dimension separately- for 5D: 5 separate 1D jittered point sets



20

x1 x2 x3 x4x

y1 y2 y3 y4y

u1 u2 u3 u4u

v1 v2 v3 v4v

t1 t2 t3 t4t


Stratify samples in each dimension separately - for 5D: 5 separate 1D jittered point sets



21

x1 x2 x3 x4

y4 y2 y1 y3

u3 u4 u2 u1

v2 v1 v3 v4

t2 t1 t4 t3

x

y

u

v

t

Shuffle order


Stratify samples in each dimension separately - for 2D: 2 separate 1D jittered point sets


N-Rooks = 2D Latin Hypercube [Shirley 91]

22

x1 x2 x3 x4

y4 y2 y1 y3

x

y


Latin Hypercube (N-Rooks) Sampling

23Realistic Image Synthesis SS2020 Image source: Michael Maggs, CC BY-SA 2.5

[Shirley 91]

https://commons.wikimedia.org/w/index.php?curid=3318748


// initialize the diagonalfor (uint d = 0; d < numDimensions; d++)

for (uint i = 0; i < numS; i++)samples(d,i) = (i + randf())/numS;

// shuffle each dimension independentlyfor (uint d = 0; d < numDimensions; d++)

shuffle(samples(d,:));


24







24Initialize







24



25Shuffle rows







25Shuffle rows











26Shuffle rows







26



27Shuffle columns







27Shuffle columns







28













Evenly distributed in each individual dimension




Unevenly distributed in n-dimensions



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5.3 Blue noise


N-Rooks Sampling



Multi-Jittered SamplingKenneth Chiu, Peter Shirley, and Changyaw Wang. “Multi-jittered sampling.” In Graphics Gems IV, pp. 370–374. Academic Press, May 1994.

– combine N-Rooks and Jittered stratification constraints


Multi-Jittered Sampling



Multi-Jittered Sampling// initializefloat cellSize = 1.0 / (resX*resY);for (uint i = 0; i < resX; i++)

for (uint j = 0; j < resY; j++){

samples(i,j).x = i/resX + (j+randf()) / (resX*resY);samples(i,j).y = j/resY + (i+randf()) / (resX*resY);

}

// shuffle x coordinates within each column of cellsfor (uint i = 0; i < resX; i++)

for (uint j = resY-1; j >= 1; j--)swap(samples(i, j).x, samples(i, randi(0, j)).x);

// shuffle y coordinates within each row of cellsfor (unsigned j = 0; j < resY; j++)

for (unsigned i = resX-1; i >= 1; i--)swap(samples(i, j).y, samples(randi(0, i), j).y);

35



Initialize



Shuffle x-coords



Shuffle x-coords



Shuffle x-coords



Shuffle x-coords



Shuffle x-coords





Shuffle y-coords



Shuffle y-coords



Shuffle y-coords



Shuffle y-coords



Shuffle y-coords

Multi-Jittered Sampling (Projections)














Evenly distributed in 2D!



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N-Rooks Sampling


Samples Radial meanExpected power spectrum



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Jittered Sampling



Poisson-Disk/Blue-Noise SamplingEnforce a minimum distance between points Poisson-Disk Sampling: - Mark A. Z. Dippé and Erling Henry Wold. “Antialiasing through

stochastic sampling.” ACM SIGGRAPH, 1985.

- Robert L. Cook. “Stochastic sampling in computer graphics.” ACM Transactions on Graphics, 1986.

- Ares Lagae and Philip Dutré. “A comparison of methods for generating Poisson disk distributions.” Computer Graphics Forum, 2008.


Random Dart Throwing














5.4 Interpreting and exploiting knowledge of the sampling spectra 27


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Figure 5.9: Illustration of some well known blue noise samplers with the corresponding Fourierexpected power spectra and the corresponding radial mean of their expected power spectra.

5.3.3 Tiling-based methodsThere are some tile-based approaches that can be used to generate blue noise samples Tile-basedmethods overcome the computational complexity of dart-throwing and/or relaxation based ap-proaches in generating blue noise sampling patterns. In computer graphics community, twotile-based approaches are well known: First approach uses a set of precomputed tiles [10, 25], witheach tile composed of multiple samples, and later use these tiles, in a sophisticated way, to pave thesampling domain. Second approach employed tiles with one sample per tile [34, 33, 49] and usessome relaxation-based schemes, with look-up tables, to improve the over all quality of samples.Although many blue noise sample generation algorithms exist, none of them are easily extendableto higher dimensions (> 3).

5.4 Interpreting and exploiting knowledge of the sampling spectra

Recently [39], it has been shown that the low frequency region of the radial power spectrum (of agiven sampling pattern) plays a crucial role in deciding the overall variance convergence rates ofsampling patterns used for Monte Carlo integration. Since blue noise sampling patterns containsalmost no radial energy in the low frequency region, they are of great interest for future researchto obtain fast results in rendering problems. Surprisingly, Poisson Disk samples have shown theconvergence rate of O

�N�1� which is the same as given by purely random samples. This can

be explained by looking at the low frequency region in the radial power spectrum of PoissonDisk samples (Fig. 5.9) which is not zero. The importance of the shape of the radial mean powerspectrum in the low frequency region demands methods and algorithms that could eventually allowsample generation directly from a target Fourier spectrum.

5.4.1 Radially-averaged periodogramsFigures 5.6, 5.8 and 5.9 depict radially averaged periodograms of the various sampling strategiesdescribed in this chapter. These spectra reveal two important characteristics of estimators builtusing the corresponding sampling strategies.

Poisson Disk Sampling



Blue-Noise Sampling (Relaxation-based)


Blue-Noise Sampling (Relaxation-based)1. Initialize sample positions (e.g. random)


Blue-Noise Sampling (Relaxation-based)1. Initialize sample positions (e.g. random)2. Use an iterative relaxation to move samples away

from each other.




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CCVT Sampling [Balzer et al. 2009]





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Poisson Disk Sampling



Low-Discrepancy SamplingDeterministic sets of points specially crafted to be evenly distributed (have low discrepancy). Entire field of study called Quasi-Monte Carlo (QMC)


The Van der Corput SequenceRadical Inverse Φb in base 2

Subsequent points “fall into biggest holes”


k Base 2 Φb




k Base 2 Φb1 1 .1 = 1/2




k Base 2 Φb1 1 .1 = 1/2

2 10 .01 = 1/4




k Base 2 Φb1 1 .1 = 1/2

2 10 .01 = 1/4

3 11 .11 = 3/4




k Base 2 Φb1 1 .1 = 1/2

2 10 .01 = 1/4

3 11 .11 = 3/4

4 100 .001 = 1/8




k Base 2 Φb1 1 .1 = 1/2

2 10 .01 = 1/4

3 11 .11 = 3/4

4 100 .001 = 1/8

5 101 .101 = 5/8




k Base 2 Φb1 1 .1 = 1/2

2 10 .01 = 1/4

3 11 .11 = 3/4

4 100 .001 = 1/8

5 101 .101 = 5/8

6 110 .011 = 3/8




k Base 2 Φb1 1 .1 = 1/2

2 10 .01 = 1/4

3 11 .11 = 3/4

4 100 .001 = 1/8

5 101 .101 = 5/8

6 110 .011 = 3/8

7 111 .111 = 7/8




k Base 2 Φb1 1 .1 = 1/2

2 10 .01 = 1/4

3 11 .11 = 3/4

4 100 .001 = 1/8

5 101 .101 = 5/8

6 110 .011 = 3/8

7 111 .111 = 7/8

...

Halton: Radical inverse with different base for each dimension:~xk = (�2(k),�3(k),�5(k), . . . ,�pn(k))

Halton and Hammersley Points


Halton: Radical inverse with different base for each dimension:

- The bases should all be relatively prime.~xk = (�2(k),�3(k),�5(k), . . . ,�pn(k))




- The bases should all be relatively prime.

- Incremental/progressive generation of samples

~xk = (�2(k),�3(k),�5(k), . . . ,�pn(k))






Hammersley: Same as Halton, but first dimension is k/N:

~xk = (�2(k),�3(k),�5(k), . . . ,�pn(k))

~xk = (k/N,�2(k),�3(k),�5(k), . . . ,�pn(k))






Hammersley: Same as Halton, but first dimension is k/N:

- Not incremental, need to know sample count, N, in advance

~xk = (�2(k),�3(k),�5(k), . . . ,�pn(k))

~xk = (k/N,�2(k),�3(k),�5(k), . . . ,�pn(k))



The Hammersley Sequence


1 sample in each “elementary interval”
















Monte Carlo (16 random samples)


Monte Carlo (16 jittered samples)


Scrambled Low-Discrepancy Sampling


More info on QMC in RenderingS. Premoze, A. Keller, and M. Raab. Advanced (Quasi-) Monte Carlo Methods for Image Synthesis. In SIGGRAPH 2012 courses.


How can we predict error from these?



Samples’ Radial Spectrum

Integrand Radial SpectrumPo

wer

Frequency

Part 2: Formal Treatment of MSE, Bias and Variance


Convergence rate for Random Samples

75Increasing Samples

Varia

nce




Varia

nce



75

…

Increasing Samples

Varia

nce



75

…

Increasing Samples

Varia

nce



75

…

Increasing Samples

Varia

nce


76

…

Increasing Samples

Varia

nce



76

…

Increasing Samples

Varia

nce


O(N�1)


77

…

Increasing Samples

Varia

nce

Convergence rate for Jittered Samples

O(N�1)


77

…

Increasing Samples

Varia

nce

Convergence rate for Jittered Samples

O(N�1)

O(N�1.5)


78

…

Increasing Samples

Varia

nce

Convergence rate Jittered vs Poisson Disk

O(N�1.5)

O(N�1)


78

…

Increasing Samples

Varia

nce


O(N�1.5)

O(N�1)


78

…

Increasing Samples

Varia

nce


O(N�1.5)

O(N�1)


79

…

Increasing Samples

Varia

nce


O(N�1)

O(N�1.5)


Samples and function in Fourier Domain

80

Spatial Domain Fourier Domain



80




80


0 w-w

S(!)



80


f(x)0 w-w

S(!)



80


f(x)0 w-w

S(!)



80


f(x)

0 w-w

f(!)

0 w-w

S(!)

Convolution


Source: vdumoulin-github

https://github.com/vdumoulin/conv_arithmetic

Convolution


Source: vdumoulin-github

https://github.com/vdumoulin/conv_arithmetic


Sampling in Primal Domain is Convolution in Fourier Domain

82

f(x)S(x)



82

f(x)S(x)Fredo Durand [2011]



82

f(x)S(x)Fredo Durand [2011]



83

*

0 w-w0

f(x)S(x) f(!) ⌦ S(!)Fredo Durand [2011]



83

*

0 w-w0

f(x)S(x) f(!) ⌦ S(!)Fredo Durand [2011]


Aliasing in Reconstruction

84

Hig

h Sa

mpl

ing

Rate

0 w-w

C



84

Hig

h Sa

mpl

ing

Rate

0 w-w

C



84

Hig

h Sa

mpl

ing

Rate

Low

Sam

plin

g Ra

te

0 w-w

C



84

Hig

h Sa

mpl

ing

Rate

Low

Sam

plin

g Ra

te

0 w-w

C



84

Hig

h Sa

mpl

ing

Rate

Low

Sam

plin

g Ra

te

0 w-w

C

C



84

Hig

h Sa

mpl

ing

Rate

Low

Sam

plin

g Ra

te

0 w-w

C

Aliasing

C



85

Hig

h Sa

mpl

ing

Rate

Low

Sam

plin

g Ra

te

0 w-w

C

C


Error in Monte Carlo Integration

86

Hig

h Sa

mpl

ing

Rate

Low

Sam

plin

g Ra

te

0 w-w

C

C



86

Hig

h Sa

mpl

ing

Rate

Low

Sam

plin

g Ra

te

0 w-w

C

C



86

Hig

h Sa

mpl

ing

Rate

Low

Sam

plin

g Ra

te

0 w-w

C

C



86

Error in Integration

Hig

h Sa

mpl

ing

Rate

Low

Sam

plin

g Ra

te

0 w-w

C

C


Aliasing (Reconstruction) vs. Error (Integration)

87

0 w-w

Error in IntegrationAliasing

C C



87

0 w-w

Fredo Durand [2011]Belcour et al. [2013]


C C



87

0 w-w

Fredo Durand [2011]Belcour et al. [2013]


C C


Integration in the Fourier Domain

88


Integration is the DC term in the Fourier Domain

89

I =

Z

Df(x)dx

Spatial Domain:



89

I =

Z

Df(x)dx

Spatial Domain:

Fourier Domain:



89

I =

Z

Df(x)dx

f(0)

Spatial Domain:

Fourier Domain:


µN =

Z

Df(x)S(x)dx =

Z

⌦f⇤(!)S(!)d!

Monte Carlo Estimator in Spatial Domain

90


µN =

Z

Df(x)S(x)dx =

Z

⌦f⇤(!)S(!)d!


91

S(x) =1

N

NX

k=1

�(x� xk)


µN =

Z

Df(x)S(x)dx =

Z

⌦f⇤(!)S(!)d!


91

S(x) =1

N

NX

k=1

�(x� xk)


µN =

Z

Df(x)S(x)dx =

Z

⌦f⇤(!)S(!)d!


91

S(x) =1

N

NX

k=1

�(x� xk)


µN =

Z

Df(x)S(x)dx =

Z

⌦f⇤(!)S(!)d!


91

S(x) =1

N

NX

k=1

�(x� xk)


µN =

Z

Df(x)S(x)dx =

Z

⌦f⇤(!)S(!)d!


91

S(x) =1

N

NX

k=1

�(x� xk)


Monte Carlo Estimator in Fourier Domain

92

µN =

Z

Df(x)S(x)dx =

Z

⌦f⇤(!)S(!)d!

S(x) =1

N

NX

k=1

�(x� xk)



92

µN =

Z

Df(x)S(x)dx =

Z

⌦f⇤(!)S(!)d!

S(x) =1

N

NX

k=1

�(x� xk)



93

µN =

Z

Df(x)S(x)dx =

Z

⌦f⇤(!)S(!)d!

S(x) =1

N

NX

k=1

�(x� xk)

S(!) =1

N

NX

k=1

e�i2⇡!xk


How to Formulate Error in Fourier Domain ?

94

Fredo Durand [2011]

µN =

Z

⌦f⇤(!)S(!)d!I = f(0)

0 w-w


C


How to Formulate Error in Fourier Domain ?

94

Fredo Durand [2011]

µN =

Z

⌦f⇤(!)S(!)d!I = f(0)

0 w-w


C


Error in Spatial Domain

95

I � µN =

Z

Df(x)dx�

Z

Df(x)S(x)dx

µN =

Z

⌦f⇤(!)S(!)d!I = f(0)



95

I � µN =

Z

Df(x)dx�

Z

Df(x)S(x)dx

µN =

Z

⌦f⇤(!)S(!)d!I = f(0)

True Integral



95

I � µN =

Z

Df(x)dx�

Z

Df(x)S(x)dx

Monte Carlo Estimator

µN =

Z

⌦f⇤(!)S(!)d!I = f(0)

True Integral



96

I � µN =

Z

Df(x)dx�

Z

Df(x)S(x)dx

µN =

Z

⌦f⇤(!)S(!)d!I = f(0)



96

I � µN =

Z

Df(x)dx�

Z

Df(x)S(x)dx

µN =

Z

⌦f⇤(!)S(!)d!I = f(0)


Error in Fourier Domain

97

I � µN = f(0)�Z

⌦f⇤(!)S(!)d!

Fredo Durand [2011]

µN =

Z

⌦f⇤(!)S(!)d!I = f(0)

I � µN =

Z

Df(x)dx�

Z

Df(x)S(x)dx


Error in Fourier Domain

98

I � µN = f(0)�Z

⌦f⇤(!)S(!)d!


0 w-w

Fredo Durand [2011]


99

Error = Bias2 +Variance


Properties of Error

• Bias: Expected value of the Error

• Variance:

100

hI � µN i

Var(I � µN )


Properties of Error


• Variance:

100

hI � µN i

Var(I � µN )


Properties of Error


• Variance:

100

hI � µN i

Var(I � µN )


Properties of Error


• Variance:

100

hI � µN i

Subr and Kautz [2013]

Var(I � µN )


101

Bias in the Monte Carlo Estimator


Bias in Fourier Domain

102

I � µN = f(0)�Z

⌦f⇤(!)S(!)d!Error:


Bias:


103

I � µN = f(0)�Z


hI � µN i = f(0)�⌧Z

⌦f⇤(!)S(!)d!

�


Bias:


103

I � µN = f(0)�Z


hI � µN i = f(0)�⌧Z

⌦f⇤(!)S(!)d!

�


Bias:


103

hI � µN i = f(0)�⌧Z

⌦f⇤(!)S(!)d!

�



104

hI � µN i = f(0)�⌧Z

⌦f⇤(!)S(!)d!

�



104

hI � µN i = f(0)�⌧Z

⌦f⇤(!)S(!)d!

�



105


hI � µN i = f(0)�Z

⌦f⇤(!) hS(!)i d!

hI � µN i = f(0)�⌧Z

⌦f⇤(!)S(!)d!

�



106


hI � µN i = f(0)�Z

⌦f⇤(!) hS(!)i d!



106


hI � µN i = f(0)�Z

⌦f⇤(!) hS(!)i d!



106

To obtain an unbiased estimator: Subr and Kautz [2013]

hI � µN i = f(0)�Z

⌦f⇤(!) hS(!)i d!



106

To obtain an unbiased estimator:

hS(!)i = 0for frequencies other than zero


hI � µN i = f(0)�Z

⌦f⇤(!) hS(!)i d!


How to obtain ?

107

hS(!)i = 0


108

Complex form in Amplitude and Phase

hS(!)i = |hS(!)i| e��(hS(!)i)


108


hS(!)i = |hS(!)i| e��(hS(!)i)Amplitude


108


hS(!)i = |hS(!)i| e��(hS(!)i)Amplitude Phase


Phase change due to Random Shift

109

Real

Imag

S(!)

For a given frequency !


110

Real

Imag


S(!)



110

Real

Imag


S(!)

Pauly et al. [2000]Ramamoorthi et al. [2012]



111

Real

Imag


S(!)



112

Real

ImagMultiple realizations




112

Real

ImagMultiple realizations




112

Real

Imag

…

… Multiple realizations




112

Real

Imag

…



hS(!)i = 0



112

Real

Imag

…



hS(!)i = 0 8! 6= 0



113



113



113


• Homogenization allows representation of error only in terms of variance


113


• Homogenization allows representation of error only in terms of variance

• We can take any sampling pattern and homogenize it to make the Monte Carlo estimator unbiased.


114

Variance in the Fourier domain



115

I � µN = f(0)�Z


Var(I � µN ) = Var

✓f(0)�

Z

⌦f⇤(!) S(!) d!

◆



115

I � µN = f(0)�Z



✓f(0)�

Z

⌦f⇤(!) S(!) d!

◆



115

I � µN = f(0)�Z



✓f(0)�

Z

⌦f⇤(!) S(!) d!

◆



115


✓f(0)�

Z

⌦f⇤(!) S(!) d!

◆



116


✓f(0)�

Z

⌦f⇤(!) S(!) d!

◆



116


✓f(0)�

Z

⌦f⇤(!) S(!) d!

◆



117


✓f(0)�

Z

⌦f⇤(!) S(!) d!

◆

Var(µN ) = Var

✓Z

⌦f⇤(!) S(!)d!

◆



117

Var(µN ) = Var

✓Z

⌦f⇤(!) S(!)d!

◆



118

Var(µN ) = Var

✓Z

⌦f⇤(!) S(!)d!

◆



118

Var(µN ) = Var

✓Z

⌦f⇤(!) S(!)d!

◆



119

Var(µN ) =

Z

⌦Pf (!)Var

⇣S(!)

⌘d!

Var(µN ) = Var

✓Z

⌦f⇤(!) S(!)d!

◆

where,Pf (!) = |f⇤(!)|2 Power Spectrum



119

Var(µN ) =

Z

⌦Pf (!)Var

⇣S(!)

⌘d!



120

Var(µN ) =

Z

⌦Pf (!)Var

⇣S(!)

⌘d!




120

Var(µN ) =

Z

⌦Pf (!)Var

⇣S(!)

⌘d!


This is a general form, both for homogenised as well as non-homogenised sampling patterns



121

Var(µN ) =

Z

⌦Pf (!)Var

⇣S(!)

⌘d!



121

Var(µN ) =

Z

⌦Pf (!)Var

⇣S(!)

⌘d!


122

Var(µN ) =

Z

⌦Pf (!)Var

⇣S(!)

⌘d!



122

For purely random samples:

Var(µN ) =

Z

⌦Pf (!)Var

⇣S(!)

⌘d!



122


where,PS(!) = |S(!)|2

Var(µN ) =

Z

⌦Pf (!)Var

⇣S(!)

⌘d!

Var(µN ) =

Z

⌦Pf (!) hPS(!)i d!

Fredo Durand [2011]



122


where,PS(!) = |S(!)|2

Var(µN ) =

Z

⌦Pf (!)Var

⇣S(!)

⌘d!

Var(µN ) =

Z

⌦Pf (!) hPS(!)i d!

Fredo Durand [2011]

hS(!)i = 0



123

Variance using Homogenized Samples

Homogenizing any sampling pattern makeshS(!)i = 0


123


Homogenizing any sampling pattern makes

Pilleboue et al. [2015]

where,PS(!) = |S(!)|2

Var(µN ) =

Z

⌦Pf (!) hPS(!)i d!

hS(!)i = 0


124


Var(µN ) =

Z

⌦Pf (!) hPS(!)i d!


124


Var(µN ) =

Z

⌦Pf (!) hPS(!)i d!


125

Variance in terms of n-dimensional Power Spectra

Var(µN ) =

Z

⌦Pf (!) hPS(!)i d!

CC

VTPo

isso

n D

isk


125

Variance in terms of n-dimensional Power Spectra

Var(µN ) =

Z

⌦Pf (!) hPS(!)i d!

CC

VTPo

isso

n D

isk


126

Variance in the Polar Coordinates

Var(µN ) =

Z

⌦Pf (!) hPS(!)i d!


126


In polar coordinates:

Var(µN ) =

Z

⌦Pf (!) hPS(!)i d!


126



V ar[µN ] = M(Sd�1)

Z 1

0

Z

Sd�1

Pf (⇢n) hPS(⇢n)i dn d⇢

Var(µN ) =

Z

⌦Pf (!) hPS(!)i d!


127



Z 1

0

Z

Sd�1



Var(µN ) =

Z

⌦Pf (!) hPS(!)i d!


127



Z 1

0

Z

Sd�1



127



Z 1

0

Z

Sd�1



128

Variance for Isotropic Power Spectra


Z 1

0

Z

Sd�1



128


For isotropic power spectra:


Z 1

0

Z

Sd�1



128




Z 1

0

Z

Sd�1



128




Z 1

0

Z

Sd�1



128




Z 1

0Pf (⇢) hPS(⇢)i d⇢


Z 1

0

Z

Sd�1



128




Z 1

0Pf (⇢) hPS(⇢)i d⇢


Z 1

0

Z

Sd�1



129



Z 1

0Pf (⇢) hPS(⇢)i d⇢



Z 1

0

Z

Sd�1



130

Variance in terms of 1-dimensional Power Spectra


Z 1

0Pf (⇢) hPS(⇢)i d⇢

CC

VTPo

isso

n D

isk


130

Variance in terms of 1-dimensional Power Spectra


Z 1

0Pf (⇢) hPS(⇢)i d⇢

CC

VTPo

isso

n D

isk


Variance: Integral over Product of Power Spectra


Z 1

0Pf (⇢) hPS(⇢)i d⇢

131




Z 1

0Pf (⇢) hPS(⇢)i d⇢

For given number of Samples

Sampling Radial Power Spectrum

Integrand Radial Power Spectrum

131




Z 1

0Pf (⇢) hPS(⇢)i d⇢




132




Z 1

0Pf (⇢) hPS(⇢)i d⇢




133




Z 1

0Pf (⇢) hPS(⇢)i d⇢




134




Z 1

0Pf (⇢) hPS(⇢)i d⇢




134


Spatial Distribution vs Radial Mean Power Spectra

135� � � � �

��

�

�

�

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Jitte

rPo

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136

Samplers Worst Case Best Case

Random

Jitter

Poisson Disk

CCVT


For 2-dimensions


136


Random

Jitter

Poisson Disk

CCVT


For 2-dimensions

O(N�1)


136


Random

Jitter

Poisson Disk

CCVT


For 2-dimensions

O(N�1) O(N�1)


136


Random

Jitter

Poisson Disk

CCVT

O(N�1.5)


For 2-dimensions

O(N�1) O(N�1)


136


Random

Jitter

Poisson Disk

CCVT

O(N�1.5) O(N�2)


For 2-dimensions

O(N�1) O(N�1)


136


Random

Jitter

Poisson Disk

CCVT

O(N�1.5) O(N�2)


For 2-dimensions

O(N�1)

O(N�1) O(N�1)


136


Random

Jitter

Poisson Disk

CCVT

O(N�1)

O(N�1.5) O(N�2)


For 2-dimensions

O(N�1)

O(N�1) O(N�1)


136


Random

Jitter

Poisson Disk

CCVT

O(N�1)

O(N�1.5) O(N�2)


For 2-dimensions

O(N�1)

O(N�1) O(N�1)

O(N�1.5)


136


Random

Jitter

Poisson Disk

CCVT

O(N�1)

O(N�1.5) O(N�2)


For 2-dimensions

O(N�1)

O(N�1) O(N�1)

O(N�1.5) O(N�3)


137


For 2-dimensions


Random

Jitter

Poisson Disk

CCVT

O(N�1)

O(N�1.5) O(N�2)

O(N�1)

O(N�1) O(N�1)

O(N�1.5) O(N�3)

Jitte

rPo

isso

n D

isk


Low Frequency Region

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138

Zoom-in

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Variance for Low Sample Count

139

Zoom-in

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Variance for Low Sample Count

139

Zoom-in

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Variance for Increasing Sample Count

140

Zoom-in

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O(N�1)

O(N�2)


Experimental Verification

141


Convergence rate


Varia

nce

…


Convergence rate


Varia

nce

…


143

…

Increasing Samples

Varia

nce Convergence rate


Disk Function as Worst Case

144



144



144



145


Gaussian as Best Case

146


Gaussian as Best Case

146


Ambient Occlusion Examples

147


Random vs Jittered

148

96 Secondary Rays

MSE: 8.56 x 10e-4MSE: 4.74 x 10e-3


CCVT vs. Poisson Disk

149

96 Secondary Rays

MSE: 4.24 x 10e-4 MSE: 6.95 x 10e-4


Convergence rates

150


Convergence rates

150


Jittered vs Poisson Disk

151

Variance


What are the benefits of this analysis ?

152



152

• For offline rendering, analysis tells which samplers would converge faster.



152

• For offline rendering, analysis tells which samplers would converge faster.

• For real time rendering, blue noise samples are more effective in reducing variance for a given number of samples

Documents

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