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
3Realistic Image Synthesis SS2020
I =
Z
Df(x) dx
⇡Z
Df(x)S(x) dx
Recall: Monte Carlo Integration
3Realistic Image Synthesis SS2020
I =
Z
Df(x) dx
⇡Z
Df(x)S(x) dx
Recall: Monte Carlo Integration
3Realistic Image Synthesis SS2020
I =
Z
Df(x) dx
⇡Z
Df(x)S(x) dx
Recall: Monte Carlo Integration
3Realistic Image Synthesis SS2020
I =
Z
Df(x) dx
⇡Z
Df(x)S(x) dx
I =
Z
Df(x) dx
⇡Z
Df(x)S(x) dx
Recall: Monte Carlo Integration
3Realistic Image Synthesis SS2020
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
Recall: Monte Carlo Integration
3Realistic Image Synthesis SS2020
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
Recall: Monte Carlo Integration
3Realistic Image Synthesis SS2020
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
Realistic Image Synthesis SS2020
Independent Random Sampling
4
for (int k = 0; k < num; k++){
samples(k).x = randf();samples(k).y = randf();
}
Realistic Image Synthesis SS2020
Independent Random Sampling
4
for (int k = 0; k < num; k++){
samples(k).x = randf();samples(k).y = randf();
}
Realistic Image Synthesis SS2020
Independent Random Sampling
4
✔Trivially extends to higher dimensions
for (int k = 0; k < num; k++){
samples(k).x = randf();samples(k).y = randf();
}
Realistic Image Synthesis SS2020
Independent Random Sampling
4
✔Trivially extends to higher dimensions
✔Trivially progressive and memory-less
for (int k = 0; k < num; k++){
samples(k).x = randf();samples(k).y = randf();
}
Realistic Image Synthesis SS2020
Independent Random Sampling
4
✔Trivially extends to higher dimensions
✔Trivially progressive and memory-less
✘ Big gaps
for (int k = 0; k < num; k++){
samples(k).x = randf();samples(k).y = randf();
}
Realistic Image Synthesis SS2020
Independent Random Sampling
4
✔Trivially extends to higher dimensions
✔Trivially progressive and memory-less
✘ Big gaps
✘ Clumping
for (int k = 0; k < num; k++){
samples(k).x = randf();samples(k).y = randf();
}
Recall: Fourier Theory
5Realistic Image Synthesis SS2020
Power SpectrumInput Image
Image courtesy: Laurent Belcour
Recall: Fourier Theory
5Realistic Image Synthesis SS2020
Power SpectrumInput Image
Image courtesy: Laurent Belcour
Recall: Fourier theory
6Realistic Image Synthesis SS2020
f(!) =
Z
Df(x) e�2⇡ ı! x dx
f(~!) =
Z
Df(~x) e�2⇡ ı (~!·~x) d~x
Fourier transform:
Recall: Fourier theory
6Realistic Image Synthesis SS2020
f(!) =
Z
Df(x) e�2⇡ ı! x dx
f(~!) =
Z
Df(~x) e�2⇡ ı (~!·~x) d~xFourier transform:
Recall: Fourier theory
6Realistic Image Synthesis SS2020
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
Df(x) e�2⇡ ı! x dx
f(~!) =
Z
Df(~x) e�2⇡ ı (~!·~x) d~xFourier transform:
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)
Recall: Fourier theory
6Realistic Image Synthesis SS2020
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
Df(x) e�2⇡ ı! x dx
f(~!) =
Z
Df(~x) e�2⇡ ı (~!·~x) d~xFourier transform:
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|)
Recall: Fourier theory
6Realistic Image Synthesis SS2020
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
Df(x) e�2⇡ ı! x dx
f(~!) =
Z
Df(~x) e�2⇡ ı (~!·~x) d~xFourier transform:
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|)
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DS(~x) e�2⇡ ı (~!·~x) d~x
S(~!) =
Z
D
1
N
NX
k=1
�(|~x� ~xk|) e�2⇡ ı (~!·~x) d~x
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k=1
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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
Independent Random Sampling
7Realistic Image Synthesis SS2020
Samples Power spectrum
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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
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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
Independent Random Sampling
7Realistic Image Synthesis SS2020
Samples Power spectrum
1
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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
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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
Independent Random Sampling
7Realistic Image Synthesis SS2020
Samples Power spectrum
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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
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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
Independent Random Sampling
8Realistic Image Synthesis SS2020
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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
Independent Random Sampling
8Realistic Image Synthesis SS2020
Many sample set realizations Expected power spectrum
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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
Sam
ples
Pow
ersp
ectru
mR
adia
lmea
nRandom
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Figu
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
at
24 Chapter 5. Popular sampling patterns
Samples Power spectrum Radial mean
Ran
dom
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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
Independent Random Sampling
9Realistic Image Synthesis SS2020
Samples Expected power spectrum
1
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k=1
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Samples Power spectrum Radial mean
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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
24 Chapter 5. Popular sampling patterns
Samples Power spectrum Radial mean
Ran
dom
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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
Independent Random Sampling
9Realistic Image Synthesis SS2020
Samples Expected power spectrum
24 Chapter 5. Popular sampling patterns
Samples Power spectrum Radial mean
Ran
dom
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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
Radial mean
1
N
NX
k=1
�(|~x� ~xk|) E
2
4�����1
N
NX
k=1
e�2⇡ ı (~!·~xk)
�����
23
5
24 Chapter 5. Popular sampling patterns
Samples Power spectrum Radial mean
Ran
dom
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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
24 Chapter 5. Popular sampling patterns
Samples Power spectrum Radial mean
Ran
dom
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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
Independent Random Sampling
9Realistic Image Synthesis SS2020
Samples Expected power spectrum
24 Chapter 5. Popular sampling patterns
Samples Power spectrum Radial mean
Ran
dom
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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
Radial meanDC Peak
1
N
NX
k=1
�(|~x� ~xk|) E
2
4�����1
N
NX
k=1
e�2⇡ ı (~!·~xk)
�����
23
5
24 Chapter 5. Popular sampling patterns
Samples Power spectrum Radial mean
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dom
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ooks
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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
Realistic Image Synthesis SS2020
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
Realistic Image Synthesis SS2020
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
Realistic Image Synthesis SS2020
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
✔Extends to higher dimensions, but…
Realistic Image Synthesis SS2020
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
✔Extends to higher dimensions, but…
✘ Curse of dimensionality
Realistic Image Synthesis SS2020
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
✔Extends to higher dimensions, but…
✘ Curse of dimensionality
✘ Aliasing
Realistic Image Synthesis SS2020
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;
}
11
Realistic Image Synthesis SS2020
Jittered/Stratified Samplingfor (uint i = 0; i < numX; i++)
for (uint j = 0; j < numY; j++){
samples(i,j).x = (i + randf())/numX;samples(i,j).y = (j + randf())/numY;
}
12
Realistic Image Synthesis SS2020
Jittered/Stratified Samplingfor (uint i = 0; i < numX; i++)
for (uint j = 0; j < numY; j++){
samples(i,j).x = (i + randf())/numX;samples(i,j).y = (j + randf())/numY;
}
12
✔Provably cannot increase variance
Realistic Image Synthesis SS2020
Jittered/Stratified Samplingfor (uint i = 0; i < numX; i++)
for (uint j = 0; j < numY; j++){
samples(i,j).x = (i + randf())/numX;samples(i,j).y = (j + randf())/numY;
}
12
✔Provably cannot increase variance
✔Extends to higher dimensions, but…
Realistic Image Synthesis SS2020
Jittered/Stratified Samplingfor (uint i = 0; i < numX; i++)
for (uint j = 0; j < numY; j++){
samples(i,j).x = (i + randf())/numX;samples(i,j).y = (j + randf())/numY;
}
12
✔Provably cannot increase variance
✔Extends to higher dimensions, but…
✘ Curse of dimensionality
Realistic Image Synthesis SS2020
Jittered/Stratified Samplingfor (uint i = 0; i < numX; i++)
for (uint j = 0; j < numY; j++){
samples(i,j).x = (i + randf())/numX;samples(i,j).y = (j + randf())/numY;
}
12
✔Provably cannot increase variance
✔Extends to higher dimensions, but…
✘ Curse of dimensionality
✘ Not progressive
24 Chapter 5. Popular sampling patterns
Samples Power spectrum Radial mean
Ran
dom
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ooks
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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
Jittered Sampling
13Realistic Image Synthesis SS2020
Samples Expected power spectrum Radial mean
Independent Random Sampling
14Realistic Image Synthesis SS2020
24 Chapter 5. Popular sampling patterns
Samples Power spectrum Radial mean
Ran
dom
� � � � �
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ooks
� � � � �
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�
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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 Expected power spectrum Radial mean
Monte Carlo (16 random samples)
15Realistic Image Synthesis SS2020
Monte Carlo (16 jittered samples)
16Realistic Image Synthesis SS2020
Stratifying in Higher DimensionsStratification requires O(Nd) samples- e.g. pixel (2D) + lens (2D) + time (1D) = 5D
17Realistic Image Synthesis SS2020
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!
17Realistic Image Synthesis SS2020
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!
Inconvenient for large d- cannot select sample count with fine granularity
17Realistic Image Synthesis SS2020
Uncorrelated Jitter [Cook et al. 84]
18Realistic Image Synthesis SS2020
Compute stratified samples in sub-dimensionsUncorrelated Jitter [Cook et al. 84]
18Realistic Image Synthesis SS2020
Compute stratified samples in sub-dimensions- 2D jittered (x,y) for pixel
Uncorrelated Jitter [Cook et al. 84]
18Image source: PBRTe2 [Pharr & Humphreys 2010]Realistic Image Synthesis SS2020
Compute stratified samples in sub-dimensions- 2D jittered (x,y) for pixel
- 2D jittered (u,v) for lens
Uncorrelated Jitter [Cook et al. 84]
18Image source: PBRTe2 [Pharr & Humphreys 2010]Realistic Image Synthesis SS2020
Compute stratified samples in sub-dimensions- 2D jittered (x,y) for pixel
- 2D jittered (u,v) for lens
- 1D jittered (t) for time
Uncorrelated Jitter [Cook et al. 84]
18Image source: PBRTe2 [Pharr & Humphreys 2010]Realistic Image Synthesis SS2020
Compute stratified samples in sub-dimensions- 2D jittered (x,y) for pixel
- 2D jittered (u,v) for lens
- 1D jittered (t) for time
- combine dimensions in random order
Uncorrelated Jitter [Cook et al. 84]
18Image source: PBRTe2 [Pharr & Humphreys 2010]Realistic Image Synthesis SS2020
Depth of Field (4D)
19Realistic Image Synthesis SS2020
Reference Random Sampling Uncorrelated Jitter
Image source: PBRTe2 [Pharr & Humphreys 2010]
Stratify samples in each dimension separatelyUncorrelated Jitter ➔ Latin Hypercube
20Realistic Image Synthesis SS2020
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
Realistic Image Synthesis SS2020
Stratify samples in each dimension separately- for 5D: 5 separate 1D jittered point sets
- combine dimensions in random order
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
Realistic Image Synthesis SS2020
Stratify samples in each dimension separately - for 5D: 5 separate 1D jittered point sets
- combine dimensions in random order
Uncorrelated Jitter ➔ Latin Hypercube
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
Realistic Image Synthesis SS2020
Stratify samples in each dimension separately - for 2D: 2 separate 1D jittered point sets
- combine dimensions in random order
N-Rooks = 2D Latin Hypercube [Shirley 91]
22
x1 x2 x3 x4
y4 y2 y1 y3
x
y
Realistic Image Synthesis SS2020
Latin Hypercube (N-Rooks) Sampling
23Realistic Image Synthesis SS2020 Image source: Michael Maggs, CC BY-SA 2.5
[Shirley 91]
Realistic Image Synthesis SS2020
// 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,:));
Latin Hypercube (N-Rooks) Sampling
24
Realistic Image Synthesis SS2020
// 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,:));
Latin Hypercube (N-Rooks) Sampling
24Initialize
Realistic Image Synthesis SS2020
// 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,:));
Latin Hypercube (N-Rooks) Sampling
24
Realistic Image Synthesis SS2020
Latin Hypercube (N-Rooks) Sampling
25Shuffle rows
// 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,:));
Realistic Image Synthesis SS2020
Latin Hypercube (N-Rooks) Sampling
25Shuffle rows
// 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,:));
Realistic Image Synthesis SS2020
// 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,:));
Latin Hypercube (N-Rooks) Sampling
26Shuffle rows
Realistic Image Synthesis SS2020
// 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,:));
Latin Hypercube (N-Rooks) Sampling
26
Realistic Image Synthesis SS2020
Latin Hypercube (N-Rooks) Sampling
27Shuffle columns
// 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,:));
Realistic Image Synthesis SS2020
Latin Hypercube (N-Rooks) Sampling
27Shuffle columns
// 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,:));
Realistic Image Synthesis SS2020
Latin Hypercube (N-Rooks) Sampling
28
// 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,:));
Latin Hypercube (N-Rooks) Sampling
29Realistic Image Synthesis SS2020
Latin Hypercube (N-Rooks) Sampling
30Realistic Image Synthesis SS2020
Latin Hypercube (N-Rooks) Sampling
31Realistic Image Synthesis SS2020
Latin Hypercube (N-Rooks) Sampling
31Realistic Image Synthesis SS2020
Evenly distributed in each individual dimension
Latin Hypercube (N-Rooks) Sampling
31Realistic Image Synthesis SS2020
Evenly distributed in each individual dimension
Unevenly distributed in n-dimensions
24 Chapter 5. Popular sampling patterns
Samples Power spectrum Radial mean
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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
N-Rooks Sampling
32Realistic Image Synthesis SS2020
Samples Expected power spectrum Radial mean
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
33Realistic Image Synthesis SS2020
Multi-Jittered Sampling
34Realistic Image Synthesis SS2020
Realistic Image Synthesis SS2020
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
Multi-Jittered Sampling
36Realistic Image Synthesis SS2020
Initialize
Multi-Jittered Sampling
36Realistic Image Synthesis SS2020
Shuffle x-coords
Multi-Jittered Sampling
37Realistic Image Synthesis SS2020
Shuffle x-coords
Multi-Jittered Sampling
38Realistic Image Synthesis SS2020
Shuffle x-coords
Multi-Jittered Sampling
39Realistic Image Synthesis SS2020
Shuffle x-coords
Multi-Jittered Sampling
40Realistic Image Synthesis SS2020
Shuffle x-coords
Multi-Jittered Sampling
40Realistic Image Synthesis SS2020
Multi-Jittered Sampling
40Realistic Image Synthesis SS2020
Shuffle y-coords
Multi-Jittered Sampling
41Realistic Image Synthesis SS2020
Shuffle y-coords
Multi-Jittered Sampling
42Realistic Image Synthesis SS2020
Shuffle y-coords
Multi-Jittered Sampling
43Realistic Image Synthesis SS2020
Shuffle y-coords
Multi-Jittered Sampling
44Realistic Image Synthesis SS2020
Shuffle y-coords
Multi-Jittered Sampling (Projections)
45Realistic Image Synthesis SS2020
Multi-Jittered Sampling (Projections)
46Realistic Image Synthesis SS2020
Multi-Jittered Sampling (Projections)
47Realistic Image Synthesis SS2020
Multi-Jittered Sampling (Projections)
48Realistic Image Synthesis SS2020
Multi-Jittered Sampling (Projections)
48Realistic Image Synthesis SS2020
Evenly distributed in each individual dimension
Multi-Jittered Sampling (Projections)
48Realistic Image Synthesis SS2020
Evenly distributed in each individual dimension
Evenly distributed in 2D!
24 Chapter 5. Popular sampling patterns
Samples Power spectrum Radial mean
Ran
dom
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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
Multi-Jittered Sampling
49Realistic Image Synthesis SS2020
Samples Expected power spectrum Radial mean
24 Chapter 5. Popular sampling patterns
Samples Power spectrum Radial mean
Ran
dom
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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
N-Rooks Sampling
50Realistic Image Synthesis SS2020
Samples Radial meanExpected power spectrum
24 Chapter 5. Popular sampling patterns
Samples Power spectrum Radial mean
Ran
dom
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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
Jittered Sampling
51Realistic Image Synthesis SS2020
Samples Radial meanExpected power spectrum
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.
52Realistic Image Synthesis SS2020
Random Dart Throwing
53Realistic Image Synthesis SS2020
Random Dart Throwing
53Realistic Image Synthesis SS2020
Random Dart Throwing
53Realistic Image Synthesis SS2020
Random Dart Throwing
53Realistic Image Synthesis SS2020
Random Dart Throwing
54Realistic Image Synthesis SS2020
Random Dart Throwing
54Realistic Image Synthesis SS2020
Random Dart Throwing
55Realistic Image Synthesis SS2020
5.4 Interpreting and exploiting knowledge of the sampling spectra 27
Samples Power spectrum Radial mean
Pois
son
Dis
k
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VT
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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
56Realistic Image Synthesis SS2020
Samples Radial meanExpected power spectrum
Blue-Noise Sampling (Relaxation-based)
57Realistic Image Synthesis SS2020
Blue-Noise Sampling (Relaxation-based)1. Initialize sample positions (e.g. random)
57Realistic Image Synthesis SS2020
Blue-Noise Sampling (Relaxation-based)1. Initialize sample positions (e.g. random)2. Use an iterative relaxation to move samples away
from each other.
57Realistic Image Synthesis SS2020
5.4 Interpreting and exploiting knowledge of the sampling spectra 27
Samples Power spectrum Radial mean
Pois
son
Dis
k
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�
�
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CC
VT
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�
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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.
CCVT Sampling [Balzer et al. 2009]
58Realistic Image Synthesis SS2020
Samples Radial meanExpected power spectrum
5.4 Interpreting and exploiting knowledge of the sampling spectra 27
Samples Power spectrum Radial mean
Pois
son
Dis
k
� � � � �
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�
�
�
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CC
VT
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�
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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
59Realistic Image Synthesis SS2020
Samples Radial meanExpected power spectrum
Low-Discrepancy SamplingDeterministic sets of points specially crafted to be evenly distributed (have low discrepancy). Entire field of study called Quasi-Monte Carlo (QMC)
60Realistic Image Synthesis SS2020
The Van der Corput SequenceRadical Inverse Φb in base 2
Subsequent points “fall into biggest holes”
61Realistic Image Synthesis SS2020
k Base 2 Φb
The Van der Corput SequenceRadical Inverse Φb in base 2
Subsequent points “fall into biggest holes”
61Realistic Image Synthesis SS2020
k Base 2 Φb1 1 .1 = 1/2
The Van der Corput SequenceRadical Inverse Φb in base 2
Subsequent points “fall into biggest holes”
61Realistic Image Synthesis SS2020
k Base 2 Φb1 1 .1 = 1/2
2 10 .01 = 1/4
The Van der Corput SequenceRadical Inverse Φb in base 2
Subsequent points “fall into biggest holes”
61Realistic Image Synthesis SS2020
k Base 2 Φb1 1 .1 = 1/2
2 10 .01 = 1/4
3 11 .11 = 3/4
The Van der Corput SequenceRadical Inverse Φb in base 2
Subsequent points “fall into biggest holes”
61Realistic Image Synthesis SS2020
k Base 2 Φb1 1 .1 = 1/2
2 10 .01 = 1/4
3 11 .11 = 3/4
4 100 .001 = 1/8
The Van der Corput SequenceRadical Inverse Φb in base 2
Subsequent points “fall into biggest holes”
61Realistic Image Synthesis SS2020
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
The Van der Corput SequenceRadical Inverse Φb in base 2
Subsequent points “fall into biggest holes”
61Realistic Image Synthesis SS2020
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
The Van der Corput SequenceRadical Inverse Φb in base 2
Subsequent points “fall into biggest holes”
61Realistic Image Synthesis SS2020
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
The Van der Corput SequenceRadical Inverse Φb in base 2
Subsequent points “fall into biggest holes”
61Realistic Image Synthesis SS2020
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
62Realistic Image Synthesis SS2020
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))
Halton and Hammersley Points
62Realistic Image Synthesis SS2020
Halton: Radical inverse with different base for each dimension:
- The bases should all be relatively prime.
- Incremental/progressive generation of samples
~xk = (�2(k),�3(k),�5(k), . . . ,�pn(k))
Halton and Hammersley Points
62Realistic Image Synthesis SS2020
Halton: Radical inverse with different base for each dimension:
- The bases should all be relatively prime.
- Incremental/progressive generation of samples
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))
Halton and Hammersley Points
62Realistic Image Synthesis SS2020
Halton: Radical inverse with different base for each dimension:
- The bases should all be relatively prime.
- Incremental/progressive generation of samples
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))
Halton and Hammersley Points
62Realistic Image Synthesis SS2020
The Hammersley Sequence
63Realistic Image Synthesis SS2020
1 sample in each “elementary interval”
The Hammersley Sequence
64Realistic Image Synthesis SS2020
1 sample in each “elementary interval”
The Hammersley Sequence
65Realistic Image Synthesis SS2020
1 sample in each “elementary interval”
The Hammersley Sequence
66Realistic Image Synthesis SS2020
1 sample in each “elementary interval”
The Hammersley Sequence
67Realistic Image Synthesis SS2020
1 sample in each “elementary interval”
The Hammersley Sequence
68Realistic Image Synthesis SS2020
1 sample in each “elementary interval”
Monte Carlo (16 random samples)
69Realistic Image Synthesis SS2020
Monte Carlo (16 jittered samples)
70Realistic Image Synthesis SS2020
Scrambled Low-Discrepancy Sampling
71Realistic Image Synthesis SS2020
More info on QMC in RenderingS. Premoze, A. Keller, and M. Raab. Advanced (Quasi-) Monte Carlo Methods for Image Synthesis. In SIGGRAPH 2012 courses.
72Realistic Image Synthesis SS2020
How can we predict error from these?
73Realistic Image Synthesis SS2020
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Samples’ Radial Spectrum
Integrand Radial SpectrumPo
wer
Frequency
Part 2: Formal Treatment of MSE, Bias and Variance
Realistic Image Synthesis SS2020
Convergence rate for Random Samples
75Increasing Samples
Varia
nce
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Convergence rate for Random Samples
75Increasing Samples
Varia
nce
Realistic Image Synthesis SS2020
Convergence rate for Random Samples
75
…
Increasing Samples
Varia
nce
Realistic Image Synthesis SS2020
Convergence rate for Random Samples
75
…
Increasing Samples
Varia
nce
Realistic Image Synthesis SS2020
Convergence rate for Random Samples
75
…
Increasing Samples
Varia
nce
Realistic Image Synthesis SS2020
76
…
Increasing Samples
Varia
nce
Convergence rate for Random Samples
Realistic Image Synthesis SS2020
76
…
Increasing Samples
Varia
nce
Convergence rate for Random Samples
O(N�1)
Realistic Image Synthesis SS2020
77
…
Increasing Samples
Varia
nce
Convergence rate for Jittered Samples
O(N�1)
Realistic Image Synthesis SS2020
77
…
Increasing Samples
Varia
nce
Convergence rate for Jittered Samples
O(N�1)
O(N�1.5)
Realistic Image Synthesis SS2020
78
…
Increasing Samples
Varia
nce
Convergence rate Jittered vs Poisson Disk
O(N�1.5)
O(N�1)
Realistic Image Synthesis SS2020
78
…
Increasing Samples
Varia
nce
Convergence rate Jittered vs Poisson Disk
O(N�1.5)
O(N�1)
Realistic Image Synthesis SS2020
78
…
Increasing Samples
Varia
nce
Convergence rate Jittered vs Poisson Disk
O(N�1.5)
O(N�1)
Realistic Image Synthesis SS2020
79
…
Increasing Samples
Varia
nce
Convergence rate Jittered vs Poisson Disk
O(N�1)
O(N�1.5)
Realistic Image Synthesis SS2020
Samples and function in Fourier Domain
80
Spatial Domain Fourier Domain
Realistic Image Synthesis SS2020
Samples and function in Fourier Domain
80
Spatial Domain Fourier Domain
Realistic Image Synthesis SS2020
Samples and function in Fourier Domain
80
Spatial Domain Fourier Domain
0 w-w
S(!)
Realistic Image Synthesis SS2020
Samples and function in Fourier Domain
80
Spatial Domain Fourier Domain
f(x)0 w-w
S(!)
Realistic Image Synthesis SS2020
Samples and function in Fourier Domain
80
Spatial Domain Fourier Domain
f(x)0 w-w
S(!)
Realistic Image Synthesis SS2020
Samples and function in Fourier Domain
80
Spatial Domain Fourier Domain
f(x)
0 w-w
f(!)
0 w-w
S(!)
Convolution
81Realistic Image Synthesis SS2020
Source: vdumoulin-github
Convolution
81Realistic Image Synthesis SS2020
Source: vdumoulin-github
Realistic Image Synthesis SS2020
Sampling in Primal Domain is Convolution in Fourier Domain
82
f(x)S(x)
Realistic Image Synthesis SS2020
Sampling in Primal Domain is Convolution in Fourier Domain
82
f(x)S(x)Fredo Durand [2011]
Realistic Image Synthesis SS2020
Sampling in Primal Domain is Convolution in Fourier Domain
82
f(x)S(x)Fredo Durand [2011]
Realistic Image Synthesis SS2020
Sampling in Primal Domain is Convolution in Fourier Domain
83
*
0 w-w0
f(x)S(x) f(!) ⌦ S(!)Fredo Durand [2011]
Realistic Image Synthesis SS2020
Sampling in Primal Domain is Convolution in Fourier Domain
83
*
0 w-w0
f(x)S(x) f(!) ⌦ S(!)Fredo Durand [2011]
Realistic Image Synthesis SS2020
Aliasing in Reconstruction
84
Hig
h Sa
mpl
ing
Rate
0 w-w
C
Realistic Image Synthesis SS2020
Aliasing in Reconstruction
84
Hig
h Sa
mpl
ing
Rate
0 w-w
C
Realistic Image Synthesis SS2020
Aliasing in Reconstruction
84
Hig
h Sa
mpl
ing
Rate
Low
Sam
plin
g Ra
te
0 w-w
C
Realistic Image Synthesis SS2020
Aliasing in Reconstruction
84
Hig
h Sa
mpl
ing
Rate
Low
Sam
plin
g Ra
te
0 w-w
C
Realistic Image Synthesis SS2020
Aliasing in Reconstruction
84
Hig
h Sa
mpl
ing
Rate
Low
Sam
plin
g Ra
te
0 w-w
C
C
Realistic Image Synthesis SS2020
Aliasing in Reconstruction
84
Hig
h Sa
mpl
ing
Rate
Low
Sam
plin
g Ra
te
0 w-w
C
Aliasing
C
Realistic Image Synthesis SS2020
Aliasing in Reconstruction
85
Hig
h Sa
mpl
ing
Rate
Low
Sam
plin
g Ra
te
0 w-w
C
C
Realistic Image Synthesis SS2020
Error in Monte Carlo Integration
86
Hig
h Sa
mpl
ing
Rate
Low
Sam
plin
g Ra
te
0 w-w
C
C
Realistic Image Synthesis SS2020
Error in Monte Carlo Integration
86
Hig
h Sa
mpl
ing
Rate
Low
Sam
plin
g Ra
te
0 w-w
C
C
Realistic Image Synthesis SS2020
Error in Monte Carlo Integration
86
Hig
h Sa
mpl
ing
Rate
Low
Sam
plin
g Ra
te
0 w-w
C
C
Realistic Image Synthesis SS2020
Error in Monte Carlo Integration
86
Error in Integration
Hig
h Sa
mpl
ing
Rate
Low
Sam
plin
g Ra
te
0 w-w
C
C
Realistic Image Synthesis SS2020
Aliasing (Reconstruction) vs. Error (Integration)
87
0 w-w
Error in IntegrationAliasing
C C
Realistic Image Synthesis SS2020
Aliasing (Reconstruction) vs. Error (Integration)
87
0 w-w
Fredo Durand [2011]Belcour et al. [2013]
Error in IntegrationAliasing
C C
Realistic Image Synthesis SS2020
Aliasing (Reconstruction) vs. Error (Integration)
87
0 w-w
Fredo Durand [2011]Belcour et al. [2013]
Error in IntegrationAliasing
C C
Realistic Image Synthesis SS2020
Integration in the Fourier Domain
88
Realistic Image Synthesis SS2020
Integration is the DC term in the Fourier Domain
89
I =
Z
Df(x)dx
Spatial Domain:
Realistic Image Synthesis SS2020
Integration is the DC term in the Fourier Domain
89
I =
Z
Df(x)dx
Spatial Domain:
Fourier Domain:
Realistic Image Synthesis SS2020
Integration is the DC term in the Fourier Domain
89
I =
Z
Df(x)dx
f(0)
Spatial Domain:
Fourier Domain:
Realistic Image Synthesis SS2020
µN =
Z
Df(x)S(x)dx =
Z
⌦f⇤(!)S(!)d!
Monte Carlo Estimator in Spatial Domain
90
Realistic Image Synthesis SS2020
µN =
Z
Df(x)S(x)dx =
Z
⌦f⇤(!)S(!)d!
Monte Carlo Estimator in Spatial Domain
91
S(x) =1
N
NX
k=1
�(x� xk)
Realistic Image Synthesis SS2020
µN =
Z
Df(x)S(x)dx =
Z
⌦f⇤(!)S(!)d!
Monte Carlo Estimator in Spatial Domain
91
S(x) =1
N
NX
k=1
�(x� xk)
Realistic Image Synthesis SS2020
µN =
Z
Df(x)S(x)dx =
Z
⌦f⇤(!)S(!)d!
Monte Carlo Estimator in Spatial Domain
91
S(x) =1
N
NX
k=1
�(x� xk)
Realistic Image Synthesis SS2020
µN =
Z
Df(x)S(x)dx =
Z
⌦f⇤(!)S(!)d!
Monte Carlo Estimator in Spatial Domain
91
S(x) =1
N
NX
k=1
�(x� xk)
Realistic Image Synthesis SS2020
µN =
Z
Df(x)S(x)dx =
Z
⌦f⇤(!)S(!)d!
Monte Carlo Estimator in Spatial Domain
91
S(x) =1
N
NX
k=1
�(x� xk)
Realistic Image Synthesis SS2020
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)
Realistic Image Synthesis SS2020
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)
Realistic Image Synthesis SS2020
Monte Carlo Estimator in Fourier Domain
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
Realistic Image Synthesis SS2020
How to Formulate Error in Fourier Domain ?
94
Fredo Durand [2011]
µN =
Z
⌦f⇤(!)S(!)d!I = f(0)
0 w-w
Error in Integration
C
Realistic Image Synthesis SS2020
How to Formulate Error in Fourier Domain ?
94
Fredo Durand [2011]
µN =
Z
⌦f⇤(!)S(!)d!I = f(0)
0 w-w
Error in Integration
C
Realistic Image Synthesis SS2020
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)
Realistic Image Synthesis SS2020
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)
True Integral
Realistic Image Synthesis SS2020
Error in Spatial Domain
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
Realistic Image Synthesis SS2020
Error in Spatial Domain
96
I � µN =
Z
Df(x)dx�
Z
Df(x)S(x)dx
µN =
Z
⌦f⇤(!)S(!)d!I = f(0)
Realistic Image Synthesis SS2020
Error in Spatial Domain
96
I � µN =
Z
Df(x)dx�
Z
Df(x)S(x)dx
µN =
Z
⌦f⇤(!)S(!)d!I = f(0)
Realistic Image Synthesis SS2020
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
Realistic Image Synthesis SS2020
Error in Fourier Domain
98
I � µN = f(0)�Z
⌦f⇤(!)S(!)d!
Error in Integration
0 w-w
Fredo Durand [2011]
Realistic Image Synthesis SS2020
99
Error = Bias2 +Variance
Realistic Image Synthesis SS2020
Properties of Error
• Bias: Expected value of the Error
• Variance:
100
hI � µN i
Var(I � µN )
Realistic Image Synthesis SS2020
Properties of Error
• Bias: Expected value of the Error
• Variance:
100
hI � µN i
Var(I � µN )
Realistic Image Synthesis SS2020
Properties of Error
• Bias: Expected value of the Error
• Variance:
100
hI � µN i
Var(I � µN )
Realistic Image Synthesis SS2020
Properties of Error
• Bias: Expected value of the Error
• Variance:
100
hI � µN i
Subr and Kautz [2013]
Var(I � µN )
Realistic Image Synthesis SS2020
101
Bias in the Monte Carlo Estimator
Realistic Image Synthesis SS2020
Bias in Fourier Domain
102
I � µN = f(0)�Z
⌦f⇤(!)S(!)d!Error:
Realistic Image Synthesis SS2020
Bias:
Bias in Fourier Domain
103
I � µN = f(0)�Z
⌦f⇤(!)S(!)d!Error:
hI � µN i = f(0)�⌧Z
⌦f⇤(!)S(!)d!
�
Realistic Image Synthesis SS2020
Bias:
Bias in Fourier Domain
103
I � µN = f(0)�Z
⌦f⇤(!)S(!)d!Error:
hI � µN i = f(0)�⌧Z
⌦f⇤(!)S(!)d!
�
Realistic Image Synthesis SS2020
Bias:
Bias in Fourier Domain
103
hI � µN i = f(0)�⌧Z
⌦f⇤(!)S(!)d!
�
Realistic Image Synthesis SS2020
Bias in Fourier Domain
104
hI � µN i = f(0)�⌧Z
⌦f⇤(!)S(!)d!
�
Realistic Image Synthesis SS2020
Bias in Fourier Domain
104
hI � µN i = f(0)�⌧Z
⌦f⇤(!)S(!)d!
�
Realistic Image Synthesis SS2020
Bias in Fourier Domain
105
Subr and Kautz [2013]
hI � µN i = f(0)�Z
⌦f⇤(!) hS(!)i d!
hI � µN i = f(0)�⌧Z
⌦f⇤(!)S(!)d!
�
Realistic Image Synthesis SS2020
Bias in Fourier Domain
106
Subr and Kautz [2013]
hI � µN i = f(0)�Z
⌦f⇤(!) hS(!)i d!
Realistic Image Synthesis SS2020
Bias in Fourier Domain
106
Subr and Kautz [2013]
hI � µN i = f(0)�Z
⌦f⇤(!) hS(!)i d!
Realistic Image Synthesis SS2020
Bias in Fourier Domain
106
To obtain an unbiased estimator: Subr and Kautz [2013]
hI � µN i = f(0)�Z
⌦f⇤(!) hS(!)i d!
Realistic Image Synthesis SS2020
Bias in Fourier Domain
106
To obtain an unbiased estimator:
hS(!)i = 0for frequencies other than zero
Subr and Kautz [2013]
hI � µN i = f(0)�Z
⌦f⇤(!) hS(!)i d!
Realistic Image Synthesis SS2020
How to obtain ?
107
hS(!)i = 0
Realistic Image Synthesis SS2020
108
Complex form in Amplitude and Phase
hS(!)i = |hS(!)i| e��(hS(!)i)
Realistic Image Synthesis SS2020
108
Complex form in Amplitude and Phase
hS(!)i = |hS(!)i| e��(hS(!)i)Amplitude
Realistic Image Synthesis SS2020
108
Complex form in Amplitude and Phase
hS(!)i = |hS(!)i| e��(hS(!)i)Amplitude Phase
Realistic Image Synthesis SS2020
Phase change due to Random Shift
109
Real
Imag
S(!)
For a given frequency !
Realistic Image Synthesis SS2020
110
Real
Imag
Phase change due to Random Shift
S(!)
For a given frequency !
Realistic Image Synthesis SS2020
110
Real
Imag
Phase change due to Random Shift
S(!)
Pauly et al. [2000]Ramamoorthi et al. [2012]
For a given frequency !
Realistic Image Synthesis SS2020
111
Real
Imag
Phase change due to Random Shift
S(!)
For a given frequency !
Realistic Image Synthesis SS2020
112
Real
ImagMultiple realizations
Phase change due to Random Shift
For a given frequency !
Realistic Image Synthesis SS2020
112
Real
ImagMultiple realizations
Phase change due to Random Shift
For a given frequency !
Realistic Image Synthesis SS2020
112
Real
Imag
…
… Multiple realizations
Phase change due to Random Shift
For a given frequency !
Realistic Image Synthesis SS2020
112
Real
Imag
…
… Multiple realizations
Phase change due to Random Shift
hS(!)i = 0
For a given frequency !
Realistic Image Synthesis SS2020
112
Real
Imag
…
… Multiple realizations
Phase change due to Random Shift
hS(!)i = 0 8! 6= 0
For a given frequency !
Realistic Image Synthesis SS2020
113
Error = Bias2 +Variance
Realistic Image Synthesis SS2020
113
Error = Bias2 +Variance
Realistic Image Synthesis SS2020
113
Error = Bias2 +Variance
• Homogenization allows representation of error only in terms of variance
Realistic Image Synthesis SS2020
113
Error = Bias2 +Variance
• 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.
Realistic Image Synthesis SS2020
114
Variance in the Fourier domain
Realistic Image Synthesis SS2020
Variance in the Fourier domain
115
I � µN = f(0)�Z
⌦f⇤(!)S(!)d!Error:
Var(I � µN ) = Var
✓f(0)�
Z
⌦f⇤(!) S(!) d!
◆
Realistic Image Synthesis SS2020
Variance in the Fourier domain
115
I � µN = f(0)�Z
⌦f⇤(!)S(!)d!Error:
Var(I � µN ) = Var
✓f(0)�
Z
⌦f⇤(!) S(!) d!
◆
Realistic Image Synthesis SS2020
Variance in the Fourier domain
115
I � µN = f(0)�Z
⌦f⇤(!)S(!)d!Error:
Var(I � µN ) = Var
✓f(0)�
Z
⌦f⇤(!) S(!) d!
◆
Realistic Image Synthesis SS2020
Variance in the Fourier domain
115
Var(I � µN ) = Var
✓f(0)�
Z
⌦f⇤(!) S(!) d!
◆
Realistic Image Synthesis SS2020
Variance in the Fourier domain
116
Var(I � µN ) = Var
✓f(0)�
Z
⌦f⇤(!) S(!) d!
◆
Realistic Image Synthesis SS2020
Variance in the Fourier domain
116
Var(I � µN ) = Var
✓f(0)�
Z
⌦f⇤(!) S(!) d!
◆
Realistic Image Synthesis SS2020
Variance in the Fourier domain
117
Var(I � µN ) = Var
✓f(0)�
Z
⌦f⇤(!) S(!) d!
◆
Var(µN ) = Var
✓Z
⌦f⇤(!) S(!)d!
◆
Realistic Image Synthesis SS2020
Variance in the Fourier domain
117
Var(µN ) = Var
✓Z
⌦f⇤(!) S(!)d!
◆
Realistic Image Synthesis SS2020
Variance in the Fourier domain
118
Var(µN ) = Var
✓Z
⌦f⇤(!) S(!)d!
◆
Realistic Image Synthesis SS2020
Variance in the Fourier domain
118
Var(µN ) = Var
✓Z
⌦f⇤(!) S(!)d!
◆
Realistic Image Synthesis SS2020
Variance in the Fourier domain
119
Var(µN ) =
Z
⌦Pf (!)Var
⇣S(!)
⌘d!
Var(µN ) = Var
✓Z
⌦f⇤(!) S(!)d!
◆
where,Pf (!) = |f⇤(!)|2 Power Spectrum
Realistic Image Synthesis SS2020
Variance in the Fourier domain
119
Var(µN ) =
Z
⌦Pf (!)Var
⇣S(!)
⌘d!
Realistic Image Synthesis SS2020
Variance in the Fourier domain
120
Var(µN ) =
Z
⌦Pf (!)Var
⇣S(!)
⌘d!
Subr and Kautz [2013]
Realistic Image Synthesis SS2020
Variance in the Fourier domain
120
Var(µN ) =
Z
⌦Pf (!)Var
⇣S(!)
⌘d!
Subr and Kautz [2013]
This is a general form, both for homogenised as well as non-homogenised sampling patterns
Realistic Image Synthesis SS2020
Variance in the Fourier domain
121
Var(µN ) =
Z
⌦Pf (!)Var
⇣S(!)
⌘d!
Realistic Image Synthesis SS2020
Variance in the Fourier domain
121
Var(µN ) =
Z
⌦Pf (!)Var
⇣S(!)
⌘d!
Realistic Image Synthesis SS2020
122
Var(µN ) =
Z
⌦Pf (!)Var
⇣S(!)
⌘d!
Variance in the Fourier domain
Realistic Image Synthesis SS2020
122
For purely random samples:
Var(µN ) =
Z
⌦Pf (!)Var
⇣S(!)
⌘d!
Variance in the Fourier domain
Realistic Image Synthesis SS2020
122
For purely random samples:
where,PS(!) = |S(!)|2
Var(µN ) =
Z
⌦Pf (!)Var
⇣S(!)
⌘d!
Var(µN ) =
Z
⌦Pf (!) hPS(!)i d!
Fredo Durand [2011]
Variance in the Fourier domain
Realistic Image Synthesis SS2020
122
For purely random samples:
where,PS(!) = |S(!)|2
Var(µN ) =
Z
⌦Pf (!)Var
⇣S(!)
⌘d!
Var(µN ) =
Z
⌦Pf (!) hPS(!)i d!
Fredo Durand [2011]
hS(!)i = 0
Variance in the Fourier domain
Realistic Image Synthesis SS2020
123
Variance using Homogenized Samples
Homogenizing any sampling pattern makeshS(!)i = 0
Realistic Image Synthesis SS2020
123
Variance using Homogenized Samples
Homogenizing any sampling pattern makes
Pilleboue et al. [2015]
where,PS(!) = |S(!)|2
Var(µN ) =
Z
⌦Pf (!) hPS(!)i d!
hS(!)i = 0
Realistic Image Synthesis SS2020
124
Variance using Homogenized Samples
Var(µN ) =
Z
⌦Pf (!) hPS(!)i d!
Realistic Image Synthesis SS2020
124
Variance using Homogenized Samples
Var(µN ) =
Z
⌦Pf (!) hPS(!)i d!
Realistic Image Synthesis SS2020
125
Variance in terms of n-dimensional Power Spectra
Var(µN ) =
Z
⌦Pf (!) hPS(!)i d!
CC
VTPo
isso
n D
isk
Realistic Image Synthesis SS2020
125
Variance in terms of n-dimensional Power Spectra
Var(µN ) =
Z
⌦Pf (!) hPS(!)i d!
CC
VTPo
isso
n D
isk
Realistic Image Synthesis SS2020
126
Variance in the Polar Coordinates
Var(µN ) =
Z
⌦Pf (!) hPS(!)i d!
Realistic Image Synthesis SS2020
126
Variance in the Polar Coordinates
In polar coordinates:
Var(µN ) =
Z
⌦Pf (!) hPS(!)i d!
Realistic Image Synthesis SS2020
126
Variance in the Polar Coordinates
In polar coordinates:
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!
Realistic Image Synthesis SS2020
127
Variance in the Polar Coordinates
V ar[µN ] = M(Sd�1)
Z 1
0
Z
Sd�1
Pf (⇢n) hPS(⇢n)i dn d⇢
In polar coordinates:
Var(µN ) =
Z
⌦Pf (!) hPS(!)i d!
Realistic Image Synthesis SS2020
127
Variance in the Polar Coordinates
V ar[µN ] = M(Sd�1)
Z 1
0
Z
Sd�1
Pf (⇢n) hPS(⇢n)i dn d⇢
Realistic Image Synthesis SS2020
127
Variance in the Polar Coordinates
V ar[µN ] = M(Sd�1)
Z 1
0
Z
Sd�1
Pf (⇢n) hPS(⇢n)i dn d⇢
Realistic Image Synthesis SS2020
128
Variance for Isotropic Power Spectra
V ar[µN ] = M(Sd�1)
Z 1
0
Z
Sd�1
Pf (⇢n) hPS(⇢n)i dn d⇢
Realistic Image Synthesis SS2020
128
Variance for Isotropic Power Spectra
For isotropic power spectra:
V ar[µN ] = M(Sd�1)
Z 1
0
Z
Sd�1
Pf (⇢n) hPS(⇢n)i dn d⇢
Realistic Image Synthesis SS2020
128
Variance for Isotropic Power Spectra
For isotropic power spectra:
V ar[µN ] = M(Sd�1)
Z 1
0
Z
Sd�1
Pf (⇢n) hPS(⇢n)i dn d⇢
Realistic Image Synthesis SS2020
128
Variance for Isotropic Power Spectra
For isotropic power spectra:
V ar[µN ] = M(Sd�1)
Z 1
0
Z
Sd�1
Pf (⇢n) hPS(⇢n)i dn d⇢
Realistic Image Synthesis SS2020
128
Variance for Isotropic Power Spectra
For isotropic power spectra:
V ar[µN ] = M(Sd�1)
Z 1
0Pf (⇢) hPS(⇢)i d⇢
V ar[µN ] = M(Sd�1)
Z 1
0
Z
Sd�1
Pf (⇢n) hPS(⇢n)i dn d⇢
Realistic Image Synthesis SS2020
128
Variance for Isotropic Power Spectra
For isotropic power spectra:
V ar[µN ] = M(Sd�1)
Z 1
0Pf (⇢) hPS(⇢)i d⇢
V ar[µN ] = M(Sd�1)
Z 1
0
Z
Sd�1
Pf (⇢n) hPS(⇢n)i dn d⇢
Realistic Image Synthesis SS2020
129
Variance for Isotropic Power Spectra
V ar[µN ] = M(Sd�1)
Z 1
0Pf (⇢) hPS(⇢)i d⇢
For isotropic power spectra:
V ar[µN ] = M(Sd�1)
Z 1
0
Z
Sd�1
Pf (⇢n) hPS(⇢n)i dn d⇢
Realistic Image Synthesis SS2020
130
Variance in terms of 1-dimensional Power Spectra
V ar[µN ] = M(Sd�1)
Z 1
0Pf (⇢) hPS(⇢)i d⇢
CC
VTPo
isso
n D
isk
Realistic Image Synthesis SS2020
130
Variance in terms of 1-dimensional Power Spectra
V ar[µN ] = M(Sd�1)
Z 1
0Pf (⇢) hPS(⇢)i d⇢
CC
VTPo
isso
n D
isk
Realistic Image Synthesis SS2020
Variance: Integral over Product of Power Spectra
V ar[µN ] = M(Sd�1)
Z 1
0Pf (⇢) hPS(⇢)i d⇢
131
Realistic Image Synthesis SS2020
Variance: Integral over Product of Power Spectra
V ar[µN ] = M(Sd�1)
Z 1
0Pf (⇢) hPS(⇢)i d⇢
For given number of Samples
Sampling Radial Power Spectrum
Integrand Radial Power Spectrum
131
Realistic Image Synthesis SS2020
Variance: Integral over Product of Power Spectra
V ar[µN ] = M(Sd�1)
Z 1
0Pf (⇢) hPS(⇢)i d⇢
For given number of Samples
Sampling Radial Power Spectrum
Integrand Radial Power Spectrum
132
Realistic Image Synthesis SS2020
Variance: Integral over Product of Power Spectra
V ar[µN ] = M(Sd�1)
Z 1
0Pf (⇢) hPS(⇢)i d⇢
For given number of Samples
Sampling Radial Power Spectrum
Integrand Radial Power Spectrum
133
Realistic Image Synthesis SS2020
Variance: Integral over Product of Power Spectra
V ar[µN ] = M(Sd�1)
Z 1
0Pf (⇢) hPS(⇢)i d⇢
For given number of Samples
Sampling Radial Power Spectrum
Integrand Radial Power Spectrum
134
Realistic Image Synthesis SS2020
Variance: Integral over Product of Power Spectra
V ar[µN ] = M(Sd�1)
Z 1
0Pf (⇢) hPS(⇢)i d⇢
For given number of Samples
Sampling Radial Power Spectrum
Integrand Radial Power Spectrum
134
Realistic Image Synthesis SS2020
Spatial Distribution vs Radial Mean Power Spectra
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Samplers Worst Case Best Case
Random
Jitter
Poisson Disk
CCVT
Pilleboue et al. [2015]
For 2-dimensions
Realistic Image Synthesis SS2020
136
Samplers Worst Case Best Case
Random
Jitter
Poisson Disk
CCVT
Pilleboue et al. [2015]
For 2-dimensions
O(N�1)
Realistic Image Synthesis SS2020
136
Samplers Worst Case Best Case
Random
Jitter
Poisson Disk
CCVT
Pilleboue et al. [2015]
For 2-dimensions
O(N�1) O(N�1)
Realistic Image Synthesis SS2020
136
Samplers Worst Case Best Case
Random
Jitter
Poisson Disk
CCVT
O(N�1.5)
Pilleboue et al. [2015]
For 2-dimensions
O(N�1) O(N�1)
Realistic Image Synthesis SS2020
136
Samplers Worst Case Best Case
Random
Jitter
Poisson Disk
CCVT
O(N�1.5) O(N�2)
Pilleboue et al. [2015]
For 2-dimensions
O(N�1) O(N�1)
Realistic Image Synthesis SS2020
136
Samplers Worst Case Best Case
Random
Jitter
Poisson Disk
CCVT
O(N�1.5) O(N�2)
Pilleboue et al. [2015]
For 2-dimensions
O(N�1)
O(N�1) O(N�1)
Realistic Image Synthesis SS2020
136
Samplers Worst Case Best Case
Random
Jitter
Poisson Disk
CCVT
O(N�1)
O(N�1.5) O(N�2)
Pilleboue et al. [2015]
For 2-dimensions
O(N�1)
O(N�1) O(N�1)
Realistic Image Synthesis SS2020
136
Samplers Worst Case Best Case
Random
Jitter
Poisson Disk
CCVT
O(N�1)
O(N�1.5) O(N�2)
Pilleboue et al. [2015]
For 2-dimensions
O(N�1)
O(N�1) O(N�1)
O(N�1.5)
Realistic Image Synthesis SS2020
136
Samplers Worst Case Best Case
Random
Jitter
Poisson Disk
CCVT
O(N�1)
O(N�1.5) O(N�2)
Pilleboue et al. [2015]
For 2-dimensions
O(N�1)
O(N�1) O(N�1)
O(N�1.5) O(N�3)
Realistic Image Synthesis SS2020
137
Pilleboue et al. [2015]
For 2-dimensions
Samplers Worst Case Best Case
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)
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Realistic Image Synthesis SS2020
Low Frequency Region
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Low Frequency Region
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Low Frequency Region
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Variance for Low Sample Count
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Variance for Low Sample Count
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Variance for Increasing Sample Count
140
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O(N�2)
Realistic Image Synthesis SS2020
Experimental Verification
141
Realistic Image Synthesis SS2020
Convergence rate
142Increasing Samples
Varia
nce
…
Realistic Image Synthesis SS2020
Convergence rate
142Increasing Samples
Varia
nce
…
Realistic Image Synthesis SS2020
143
…
Increasing Samples
Varia
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Realistic Image Synthesis SS2020
Disk Function as Worst Case
144
Realistic Image Synthesis SS2020
Disk Function as Worst Case
144
Realistic Image Synthesis SS2020
Disk Function as Worst Case
144
Realistic Image Synthesis SS2020
Disk Function as Worst Case
145
Realistic Image Synthesis SS2020
Gaussian as Best Case
146
Realistic Image Synthesis SS2020
Gaussian as Best Case
146
Realistic Image Synthesis SS2020
Ambient Occlusion Examples
147
Realistic Image Synthesis SS2020
Random vs Jittered
148
96 Secondary Rays
MSE: 8.56 x 10e-4MSE: 4.74 x 10e-3
Realistic Image Synthesis SS2020
CCVT vs. Poisson Disk
149
96 Secondary Rays
MSE: 4.24 x 10e-4 MSE: 6.95 x 10e-4
Realistic Image Synthesis SS2020
Convergence rates
150
Realistic Image Synthesis SS2020
Convergence rates
150
Realistic Image Synthesis SS2020
Jittered vs Poisson Disk
151
Variance
Realistic Image Synthesis SS2020
What are the benefits of this analysis ?
152
Realistic Image Synthesis SS2020
What are the benefits of this analysis ?
152
• For offline rendering, analysis tells which samplers would converge faster.
Realistic Image Synthesis SS2020
What are the benefits of this analysis ?
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