graphics.cg.uni-saarland.de · 24 chapter 5. popular sampling patterns samples power spectrum...
TRANSCRIPT
![Page 1: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/1.jpg)
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
![Page 2: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/2.jpg)
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
![Page 3: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/3.jpg)
Recall: Monte Carlo Integration
3Realistic Image Synthesis SS2020
I =
Z
Df(x) dx
⇡Z
Df(x)S(x) dx
![Page 4: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/4.jpg)
Recall: Monte Carlo Integration
3Realistic Image Synthesis SS2020
I =
Z
Df(x) dx
⇡Z
Df(x)S(x) dx
![Page 5: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/5.jpg)
Recall: Monte Carlo Integration
3Realistic Image Synthesis SS2020
I =
Z
Df(x) dx
⇡Z
Df(x)S(x) dx
![Page 6: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/6.jpg)
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
![Page 7: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/7.jpg)
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
![Page 8: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/8.jpg)
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
![Page 9: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/9.jpg)
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
![Page 10: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/10.jpg)
Realistic Image Synthesis SS2020
Independent Random Sampling
4
for (int k = 0; k < num; k++){
samples(k).x = randf();samples(k).y = randf();
}
![Page 11: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/11.jpg)
Realistic Image Synthesis SS2020
Independent Random Sampling
4
for (int k = 0; k < num; k++){
samples(k).x = randf();samples(k).y = randf();
}
![Page 12: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/12.jpg)
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();
}
![Page 13: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/13.jpg)
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();
}
![Page 14: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/14.jpg)
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();
}
![Page 15: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/15.jpg)
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();
}
![Page 16: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/16.jpg)
Recall: Fourier Theory
5Realistic Image Synthesis SS2020
Power SpectrumInput Image
Image courtesy: Laurent Belcour
![Page 17: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/17.jpg)
Recall: Fourier Theory
5Realistic Image Synthesis SS2020
Power SpectrumInput Image
Image courtesy: Laurent Belcour
![Page 18: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/18.jpg)
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:
![Page 19: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/19.jpg)
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:
![Page 20: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/20.jpg)
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)
![Page 21: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/21.jpg)
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|)
![Page 22: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/22.jpg)
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|)
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
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N
NX
k=1
e�2⇡ ı (~!·~xk)
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![Page 23: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/23.jpg)
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
Independent Random Sampling
7Realistic Image Synthesis SS2020
Samples Power spectrum
1
N
NX
k=1
�(|~x� ~xk|)
~xx
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~!x
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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![Page 24: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/24.jpg)
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
Independent Random Sampling
7Realistic Image Synthesis SS2020
Samples Power spectrum
1
N
NX
k=1
�(|~x� ~xk|)
~xx
~xy ~!y
~!x
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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![Page 25: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/25.jpg)
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
Independent Random Sampling
7Realistic Image Synthesis SS2020
Samples Power spectrum
1
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NX
k=1
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~xx
~xy ~!y
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24 Chapter 5. Popular sampling patterns
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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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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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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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iththe
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sequenceis
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mersley
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patterns.He
advocatedthree
importantfeatures
foranidealradialpow
erspectrum;First,its
peakshould
beat
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alm
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ersp
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ence
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mer
sley
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enu
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roft
otal
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re5.
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mer
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tsin
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rres
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plin
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rHal
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mer
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mm
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res
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eno
ise
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rst
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ete
rmB
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lichn
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ntfe
atur
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ran
idea
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ialp
ower
spec
trum
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ldbe
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24 Chapter 5. Popular sampling patterns
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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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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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24 Chapter 5. Popular sampling patterns
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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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9Realistic Image Synthesis SS2020
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24 Chapter 5. Popular sampling patterns
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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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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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24 Chapter 5. Popular sampling patterns
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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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9Realistic Image Synthesis SS2020
Samples Expected power spectrum
24 Chapter 5. Popular sampling patterns
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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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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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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
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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
![Page 33: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/33.jpg)
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…
![Page 34: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/34.jpg)
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
![Page 35: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/35.jpg)
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
![Page 36: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/36.jpg)
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
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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
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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
![Page 39: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/39.jpg)
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…
![Page 40: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/40.jpg)
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
![Page 41: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/41.jpg)
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
![Page 42: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/42.jpg)
24 Chapter 5. Popular sampling patterns
Samples Power spectrum Radial mean
Ran
dom
� � � � �
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r
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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
![Page 43: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/43.jpg)
Independent Random Sampling
14Realistic Image Synthesis SS2020
24 Chapter 5. Popular sampling patterns
Samples Power spectrum Radial mean
Ran
dom
� � � � �
������
�
�
�
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Jitte
r
� � � � �
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�
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Mul
ti-jit
ter
� � � � �
������
�
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N-r
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
Samples Expected power spectrum Radial mean
![Page 44: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/44.jpg)
Monte Carlo (16 random samples)
15Realistic Image Synthesis SS2020
![Page 45: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/45.jpg)
Monte Carlo (16 jittered samples)
16Realistic Image Synthesis SS2020
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Stratifying in Higher DimensionsStratification requires O(Nd) samples- e.g. pixel (2D) + lens (2D) + time (1D) = 5D
17Realistic Image Synthesis SS2020
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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
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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
![Page 49: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/49.jpg)
Uncorrelated Jitter [Cook et al. 84]
18Realistic Image Synthesis SS2020
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Compute stratified samples in sub-dimensionsUncorrelated Jitter [Cook et al. 84]
18Realistic Image Synthesis SS2020
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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
![Page 52: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/52.jpg)
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
![Page 53: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/53.jpg)
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
![Page 54: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/54.jpg)
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
![Page 55: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/55.jpg)
Depth of Field (4D)
19Realistic Image Synthesis SS2020
Reference Random Sampling Uncorrelated Jitter
Image source: PBRTe2 [Pharr & Humphreys 2010]
![Page 56: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/56.jpg)
Stratify samples in each dimension separatelyUncorrelated Jitter ➔ Latin Hypercube
20Realistic Image Synthesis SS2020
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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
![Page 58: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/58.jpg)
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
![Page 59: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/59.jpg)
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
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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
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Latin Hypercube (N-Rooks) Sampling
23Realistic Image Synthesis SS2020 Image source: Michael Maggs, CC BY-SA 2.5
[Shirley 91]
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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
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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
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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
![Page 65: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/65.jpg)
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,:));
![Page 66: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/66.jpg)
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,:));
![Page 67: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/67.jpg)
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
![Page 68: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/68.jpg)
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
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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,:));
![Page 70: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/70.jpg)
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,:));
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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,:));
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Latin Hypercube (N-Rooks) Sampling
29Realistic Image Synthesis SS2020
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Latin Hypercube (N-Rooks) Sampling
30Realistic Image Synthesis SS2020
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Latin Hypercube (N-Rooks) Sampling
31Realistic Image Synthesis SS2020
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Latin Hypercube (N-Rooks) Sampling
31Realistic Image Synthesis SS2020
Evenly distributed in each individual dimension
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Latin Hypercube (N-Rooks) Sampling
31Realistic Image Synthesis SS2020
Evenly distributed in each individual dimension
Unevenly distributed in n-dimensions
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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
32Realistic Image Synthesis SS2020
Samples Expected power spectrum Radial mean
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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
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Multi-Jittered Sampling
34Realistic Image Synthesis SS2020
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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
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Multi-Jittered Sampling
36Realistic Image Synthesis SS2020
Initialize
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Multi-Jittered Sampling
36Realistic Image Synthesis SS2020
Shuffle x-coords
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Multi-Jittered Sampling
37Realistic Image Synthesis SS2020
Shuffle x-coords
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Multi-Jittered Sampling
38Realistic Image Synthesis SS2020
Shuffle x-coords
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Multi-Jittered Sampling
39Realistic Image Synthesis SS2020
Shuffle x-coords
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Multi-Jittered Sampling
40Realistic Image Synthesis SS2020
Shuffle x-coords
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Multi-Jittered Sampling
40Realistic Image Synthesis SS2020
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Multi-Jittered Sampling
40Realistic Image Synthesis SS2020
Shuffle y-coords
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Multi-Jittered Sampling
41Realistic Image Synthesis SS2020
Shuffle y-coords
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Multi-Jittered Sampling
42Realistic Image Synthesis SS2020
Shuffle y-coords
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Multi-Jittered Sampling
43Realistic Image Synthesis SS2020
Shuffle y-coords
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Multi-Jittered Sampling
44Realistic Image Synthesis SS2020
Shuffle y-coords
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Multi-Jittered Sampling (Projections)
45Realistic Image Synthesis SS2020
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Multi-Jittered Sampling (Projections)
46Realistic Image Synthesis SS2020
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Multi-Jittered Sampling (Projections)
47Realistic Image Synthesis SS2020
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Multi-Jittered Sampling (Projections)
48Realistic Image Synthesis SS2020
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Multi-Jittered Sampling (Projections)
48Realistic Image Synthesis SS2020
Evenly distributed in each individual dimension
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Multi-Jittered Sampling (Projections)
48Realistic Image Synthesis SS2020
Evenly distributed in each individual dimension
Evenly distributed in 2D!
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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
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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
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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
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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
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Random Dart Throwing
53Realistic Image Synthesis SS2020
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Random Dart Throwing
53Realistic Image Synthesis SS2020
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Random Dart Throwing
53Realistic Image Synthesis SS2020
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Random Dart Throwing
53Realistic Image Synthesis SS2020
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Random Dart Throwing
54Realistic Image Synthesis SS2020
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Random Dart Throwing
54Realistic Image Synthesis SS2020
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Random Dart Throwing
55Realistic Image Synthesis SS2020
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5.4 Interpreting and exploiting knowledge of the sampling spectra 27
Samples Power spectrum Radial mean
Pois
son
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k
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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
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Blue-Noise Sampling (Relaxation-based)
57Realistic Image Synthesis SS2020
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Blue-Noise Sampling (Relaxation-based)1. Initialize sample positions (e.g. random)
57Realistic Image Synthesis SS2020
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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
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5.4 Interpreting and exploiting knowledge of the sampling spectra 27
Samples Power spectrum Radial mean
Pois
son
Dis
k
� � � � �
������
�
�
�
�����
CC
VT
� � � � �
������
�
�
�
�����
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
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5.4 Interpreting and exploiting knowledge of the sampling spectra 27
Samples Power spectrum Radial mean
Pois
son
Dis
k
� � � � �
������
�
�
�
�����
CC
VT
� � � � �
������
�
�
�
�����
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
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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
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The Van der Corput SequenceRadical Inverse Φb in base 2
Subsequent points “fall into biggest holes”
61Realistic Image Synthesis SS2020
k Base 2 Φb
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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
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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
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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
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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
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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
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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
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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
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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
...
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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
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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
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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
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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
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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
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The Hammersley Sequence
63Realistic Image Synthesis SS2020
1 sample in each “elementary interval”
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The Hammersley Sequence
64Realistic Image Synthesis SS2020
1 sample in each “elementary interval”
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The Hammersley Sequence
65Realistic Image Synthesis SS2020
1 sample in each “elementary interval”
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The Hammersley Sequence
66Realistic Image Synthesis SS2020
1 sample in each “elementary interval”
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The Hammersley Sequence
67Realistic Image Synthesis SS2020
1 sample in each “elementary interval”
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The Hammersley Sequence
68Realistic Image Synthesis SS2020
1 sample in each “elementary interval”
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Monte Carlo (16 random samples)
69Realistic Image Synthesis SS2020
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Monte Carlo (16 jittered samples)
70Realistic Image Synthesis SS2020
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Scrambled Low-Discrepancy Sampling
71Realistic Image Synthesis SS2020
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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
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How can we predict error from these?
73Realistic Image Synthesis SS2020
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Realistic Image Synthesis SS2020
Samples’ Radial Spectrum
Integrand Radial SpectrumPo
wer
Frequency
Part 2: Formal Treatment of MSE, Bias and Variance
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Realistic Image Synthesis SS2020
Convergence rate for Random Samples
75Increasing Samples
Varia
nce
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Realistic Image Synthesis SS2020
Convergence rate for Random Samples
75Increasing Samples
Varia
nce
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Realistic Image Synthesis SS2020
Convergence rate for Random Samples
75
…
Increasing Samples
Varia
nce
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Realistic Image Synthesis SS2020
Convergence rate for Random Samples
75
…
Increasing Samples
Varia
nce
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Realistic Image Synthesis SS2020
Convergence rate for Random Samples
75
…
Increasing Samples
Varia
nce
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Realistic Image Synthesis SS2020
76
…
Increasing Samples
Varia
nce
Convergence rate for Random Samples
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Realistic Image Synthesis SS2020
76
…
Increasing Samples
Varia
nce
Convergence rate for Random Samples
O(N�1)
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Realistic Image Synthesis SS2020
77
…
Increasing Samples
Varia
nce
Convergence rate for Jittered Samples
O(N�1)
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Realistic Image Synthesis SS2020
77
…
Increasing Samples
Varia
nce
Convergence rate for Jittered Samples
O(N�1)
O(N�1.5)
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Realistic Image Synthesis SS2020
78
…
Increasing Samples
Varia
nce
Convergence rate Jittered vs Poisson Disk
O(N�1.5)
O(N�1)
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Realistic Image Synthesis SS2020
78
…
Increasing Samples
Varia
nce
Convergence rate Jittered vs Poisson Disk
O(N�1.5)
O(N�1)
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Realistic Image Synthesis SS2020
78
…
Increasing Samples
Varia
nce
Convergence rate Jittered vs Poisson Disk
O(N�1.5)
O(N�1)
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Realistic Image Synthesis SS2020
79
…
Increasing Samples
Varia
nce
Convergence rate Jittered vs Poisson Disk
O(N�1)
O(N�1.5)
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Realistic Image Synthesis SS2020
Samples and function in Fourier Domain
80
Spatial Domain Fourier Domain
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Realistic Image Synthesis SS2020
Samples and function in Fourier Domain
80
Spatial Domain Fourier Domain
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Realistic Image Synthesis SS2020
Samples and function in Fourier Domain
80
Spatial Domain Fourier Domain
0 w-w
S(!)
![Page 159: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/159.jpg)
Realistic Image Synthesis SS2020
Samples and function in Fourier Domain
80
Spatial Domain Fourier Domain
f(x)0 w-w
S(!)
![Page 160: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/160.jpg)
Realistic Image Synthesis SS2020
Samples and function in Fourier Domain
80
Spatial Domain Fourier Domain
f(x)0 w-w
S(!)
![Page 161: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/161.jpg)
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(!)
![Page 162: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/162.jpg)
Convolution
81Realistic Image Synthesis SS2020
Source: vdumoulin-github
![Page 163: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/163.jpg)
Convolution
81Realistic Image Synthesis SS2020
Source: vdumoulin-github
![Page 164: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/164.jpg)
Realistic Image Synthesis SS2020
Sampling in Primal Domain is Convolution in Fourier Domain
82
f(x)S(x)
![Page 165: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/165.jpg)
Realistic Image Synthesis SS2020
Sampling in Primal Domain is Convolution in Fourier Domain
82
f(x)S(x)Fredo Durand [2011]
![Page 166: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/166.jpg)
Realistic Image Synthesis SS2020
Sampling in Primal Domain is Convolution in Fourier Domain
82
f(x)S(x)Fredo Durand [2011]
![Page 167: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/167.jpg)
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]
![Page 168: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/168.jpg)
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]
![Page 169: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/169.jpg)
Realistic Image Synthesis SS2020
Aliasing in Reconstruction
84
Hig
h Sa
mpl
ing
Rate
0 w-w
C
![Page 170: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/170.jpg)
Realistic Image Synthesis SS2020
Aliasing in Reconstruction
84
Hig
h Sa
mpl
ing
Rate
0 w-w
C
![Page 171: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/171.jpg)
Realistic Image Synthesis SS2020
Aliasing in Reconstruction
84
Hig
h Sa
mpl
ing
Rate
Low
Sam
plin
g Ra
te
0 w-w
C
![Page 172: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/172.jpg)
Realistic Image Synthesis SS2020
Aliasing in Reconstruction
84
Hig
h Sa
mpl
ing
Rate
Low
Sam
plin
g Ra
te
0 w-w
C
![Page 173: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/173.jpg)
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
![Page 174: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/174.jpg)
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
![Page 175: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/175.jpg)
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
![Page 176: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/176.jpg)
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
![Page 177: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/177.jpg)
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
![Page 178: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/178.jpg)
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
![Page 179: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/179.jpg)
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
![Page 180: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/180.jpg)
Realistic Image Synthesis SS2020
Aliasing (Reconstruction) vs. Error (Integration)
87
0 w-w
Error in IntegrationAliasing
C C
![Page 181: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/181.jpg)
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
![Page 182: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/182.jpg)
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
![Page 183: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/183.jpg)
Realistic Image Synthesis SS2020
Integration in the Fourier Domain
88
![Page 184: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/184.jpg)
Realistic Image Synthesis SS2020
Integration is the DC term in the Fourier Domain
89
I =
Z
Df(x)dx
Spatial Domain:
![Page 185: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/185.jpg)
Realistic Image Synthesis SS2020
Integration is the DC term in the Fourier Domain
89
I =
Z
Df(x)dx
Spatial Domain:
Fourier Domain:
![Page 186: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/186.jpg)
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:
![Page 187: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/187.jpg)
Realistic Image Synthesis SS2020
µN =
Z
Df(x)S(x)dx =
Z
⌦f⇤(!)S(!)d!
Monte Carlo Estimator in Spatial Domain
90
![Page 188: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/188.jpg)
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)
![Page 189: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/189.jpg)
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)
![Page 190: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/190.jpg)
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)
![Page 191: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/191.jpg)
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)
![Page 192: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/192.jpg)
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)
![Page 193: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/193.jpg)
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)
![Page 194: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/194.jpg)
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)
![Page 195: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/195.jpg)
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
![Page 196: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/196.jpg)
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
![Page 197: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/197.jpg)
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
![Page 198: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/198.jpg)
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)
![Page 199: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/199.jpg)
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
![Page 200: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/200.jpg)
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
![Page 201: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/201.jpg)
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)
![Page 202: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/202.jpg)
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)
![Page 203: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/203.jpg)
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
![Page 204: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/204.jpg)
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]
![Page 205: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/205.jpg)
Realistic Image Synthesis SS2020
99
Error = Bias2 +Variance
![Page 206: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/206.jpg)
Realistic Image Synthesis SS2020
Properties of Error
• Bias: Expected value of the Error
• Variance:
100
hI � µN i
Var(I � µN )
![Page 207: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/207.jpg)
Realistic Image Synthesis SS2020
Properties of Error
• Bias: Expected value of the Error
• Variance:
100
hI � µN i
Var(I � µN )
![Page 208: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/208.jpg)
Realistic Image Synthesis SS2020
Properties of Error
• Bias: Expected value of the Error
• Variance:
100
hI � µN i
Var(I � µN )
![Page 209: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/209.jpg)
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 )
![Page 210: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/210.jpg)
Realistic Image Synthesis SS2020
101
Bias in the Monte Carlo Estimator
![Page 211: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/211.jpg)
Realistic Image Synthesis SS2020
Bias in Fourier Domain
102
I � µN = f(0)�Z
⌦f⇤(!)S(!)d!Error:
![Page 212: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/212.jpg)
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!
�
![Page 213: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/213.jpg)
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!
�
![Page 214: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/214.jpg)
Realistic Image Synthesis SS2020
Bias:
Bias in Fourier Domain
103
hI � µN i = f(0)�⌧Z
⌦f⇤(!)S(!)d!
�
![Page 215: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/215.jpg)
Realistic Image Synthesis SS2020
Bias in Fourier Domain
104
hI � µN i = f(0)�⌧Z
⌦f⇤(!)S(!)d!
�
![Page 216: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/216.jpg)
Realistic Image Synthesis SS2020
Bias in Fourier Domain
104
hI � µN i = f(0)�⌧Z
⌦f⇤(!)S(!)d!
�
![Page 217: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/217.jpg)
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!
�
![Page 218: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/218.jpg)
Realistic Image Synthesis SS2020
Bias in Fourier Domain
106
Subr and Kautz [2013]
hI � µN i = f(0)�Z
⌦f⇤(!) hS(!)i d!
![Page 219: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/219.jpg)
Realistic Image Synthesis SS2020
Bias in Fourier Domain
106
Subr and Kautz [2013]
hI � µN i = f(0)�Z
⌦f⇤(!) hS(!)i d!
![Page 220: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/220.jpg)
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!
![Page 221: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/221.jpg)
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!
![Page 222: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/222.jpg)
Realistic Image Synthesis SS2020
How to obtain ?
107
hS(!)i = 0
![Page 223: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/223.jpg)
Realistic Image Synthesis SS2020
108
Complex form in Amplitude and Phase
hS(!)i = |hS(!)i| e��(hS(!)i)
![Page 224: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/224.jpg)
Realistic Image Synthesis SS2020
108
Complex form in Amplitude and Phase
hS(!)i = |hS(!)i| e��(hS(!)i)Amplitude
![Page 225: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/225.jpg)
Realistic Image Synthesis SS2020
108
Complex form in Amplitude and Phase
hS(!)i = |hS(!)i| e��(hS(!)i)Amplitude Phase
![Page 226: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/226.jpg)
Realistic Image Synthesis SS2020
Phase change due to Random Shift
109
Real
Imag
S(!)
For a given frequency !
![Page 227: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/227.jpg)
Realistic Image Synthesis SS2020
110
Real
Imag
Phase change due to Random Shift
S(!)
For a given frequency !
![Page 228: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/228.jpg)
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 !
![Page 229: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/229.jpg)
Realistic Image Synthesis SS2020
111
Real
Imag
Phase change due to Random Shift
S(!)
For a given frequency !
![Page 230: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/230.jpg)
Realistic Image Synthesis SS2020
112
Real
ImagMultiple realizations
Phase change due to Random Shift
For a given frequency !
![Page 231: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/231.jpg)
Realistic Image Synthesis SS2020
112
Real
ImagMultiple realizations
Phase change due to Random Shift
For a given frequency !
![Page 232: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/232.jpg)
Realistic Image Synthesis SS2020
112
Real
Imag
…
… Multiple realizations
Phase change due to Random Shift
For a given frequency !
![Page 233: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/233.jpg)
Realistic Image Synthesis SS2020
112
Real
Imag
…
… Multiple realizations
Phase change due to Random Shift
hS(!)i = 0
For a given frequency !
![Page 234: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/234.jpg)
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 !
![Page 235: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/235.jpg)
Realistic Image Synthesis SS2020
113
Error = Bias2 +Variance
![Page 236: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/236.jpg)
Realistic Image Synthesis SS2020
113
Error = Bias2 +Variance
![Page 237: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/237.jpg)
Realistic Image Synthesis SS2020
113
Error = Bias2 +Variance
• Homogenization allows representation of error only in terms of variance
![Page 238: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/238.jpg)
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.
![Page 239: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/239.jpg)
Realistic Image Synthesis SS2020
114
Variance in the Fourier domain
![Page 240: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/240.jpg)
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!
◆
![Page 241: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/241.jpg)
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!
◆
![Page 242: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/242.jpg)
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!
◆
![Page 243: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/243.jpg)
Realistic Image Synthesis SS2020
Variance in the Fourier domain
115
Var(I � µN ) = Var
✓f(0)�
Z
⌦f⇤(!) S(!) d!
◆
![Page 244: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/244.jpg)
Realistic Image Synthesis SS2020
Variance in the Fourier domain
116
Var(I � µN ) = Var
✓f(0)�
Z
⌦f⇤(!) S(!) d!
◆
![Page 245: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/245.jpg)
Realistic Image Synthesis SS2020
Variance in the Fourier domain
116
Var(I � µN ) = Var
✓f(0)�
Z
⌦f⇤(!) S(!) d!
◆
![Page 246: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/246.jpg)
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!
◆
![Page 247: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/247.jpg)
Realistic Image Synthesis SS2020
Variance in the Fourier domain
117
Var(µN ) = Var
✓Z
⌦f⇤(!) S(!)d!
◆
![Page 248: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/248.jpg)
Realistic Image Synthesis SS2020
Variance in the Fourier domain
118
Var(µN ) = Var
✓Z
⌦f⇤(!) S(!)d!
◆
![Page 249: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/249.jpg)
Realistic Image Synthesis SS2020
Variance in the Fourier domain
118
Var(µN ) = Var
✓Z
⌦f⇤(!) S(!)d!
◆
![Page 250: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/250.jpg)
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
![Page 251: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/251.jpg)
Realistic Image Synthesis SS2020
Variance in the Fourier domain
119
Var(µN ) =
Z
⌦Pf (!)Var
⇣S(!)
⌘d!
![Page 252: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/252.jpg)
Realistic Image Synthesis SS2020
Variance in the Fourier domain
120
Var(µN ) =
Z
⌦Pf (!)Var
⇣S(!)
⌘d!
Subr and Kautz [2013]
![Page 253: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/253.jpg)
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
![Page 254: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/254.jpg)
Realistic Image Synthesis SS2020
Variance in the Fourier domain
121
Var(µN ) =
Z
⌦Pf (!)Var
⇣S(!)
⌘d!
![Page 255: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/255.jpg)
Realistic Image Synthesis SS2020
Variance in the Fourier domain
121
Var(µN ) =
Z
⌦Pf (!)Var
⇣S(!)
⌘d!
![Page 256: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/256.jpg)
Realistic Image Synthesis SS2020
122
Var(µN ) =
Z
⌦Pf (!)Var
⇣S(!)
⌘d!
Variance in the Fourier domain
![Page 257: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/257.jpg)
Realistic Image Synthesis SS2020
122
For purely random samples:
Var(µN ) =
Z
⌦Pf (!)Var
⇣S(!)
⌘d!
Variance in the Fourier domain
![Page 258: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/258.jpg)
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
![Page 259: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/259.jpg)
Realistic Image Synthesis SS2020
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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
![Page 260: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/260.jpg)
Realistic Image Synthesis SS2020
123
Variance using Homogenized Samples
Homogenizing any sampling pattern makeshS(!)i = 0
![Page 261: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/261.jpg)
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
![Page 262: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/262.jpg)
Realistic Image Synthesis SS2020
124
Variance using Homogenized Samples
Var(µN ) =
Z
⌦Pf (!) hPS(!)i d!
![Page 263: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/263.jpg)
Realistic Image Synthesis SS2020
124
Variance using Homogenized Samples
Var(µN ) =
Z
⌦Pf (!) hPS(!)i d!
![Page 264: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/264.jpg)
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
![Page 265: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/265.jpg)
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
![Page 266: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/266.jpg)
Realistic Image Synthesis SS2020
126
Variance in the Polar Coordinates
Var(µN ) =
Z
⌦Pf (!) hPS(!)i d!
![Page 267: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/267.jpg)
Realistic Image Synthesis SS2020
126
Variance in the Polar Coordinates
In polar coordinates:
Var(µN ) =
Z
⌦Pf (!) hPS(!)i d!
![Page 268: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/268.jpg)
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!
![Page 269: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/269.jpg)
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!
![Page 270: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/270.jpg)
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⇢
![Page 271: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/271.jpg)
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⇢
![Page 272: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/272.jpg)
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⇢
![Page 273: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/273.jpg)
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⇢
![Page 274: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/274.jpg)
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⇢
![Page 275: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/275.jpg)
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⇢
![Page 276: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/276.jpg)
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⇢
![Page 277: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/277.jpg)
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⇢
![Page 278: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/278.jpg)
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⇢
![Page 279: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/279.jpg)
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
![Page 280: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/280.jpg)
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
![Page 281: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/281.jpg)
Realistic Image Synthesis SS2020
Variance: Integral over Product of Power Spectra
V ar[µN ] = M(Sd�1)
Z 1
0Pf (⇢) hPS(⇢)i d⇢
131
![Page 282: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/282.jpg)
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
![Page 283: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/283.jpg)
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
![Page 284: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/284.jpg)
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
![Page 285: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/285.jpg)
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
![Page 286: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/286.jpg)
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
![Page 287: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/287.jpg)
Realistic Image Synthesis SS2020
Spatial Distribution vs Radial Mean Power Spectra
135� � � � �
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Realistic Image Synthesis SS2020
136
Samplers Worst Case Best Case
Random
Jitter
Poisson Disk
CCVT
Pilleboue et al. [2015]
For 2-dimensions
![Page 289: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/289.jpg)
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)
![Page 290: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/290.jpg)
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)
![Page 291: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/291.jpg)
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)
![Page 292: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/292.jpg)
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)
![Page 293: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/293.jpg)
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)
![Page 294: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/294.jpg)
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)
![Page 295: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/295.jpg)
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)
![Page 296: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/296.jpg)
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)
![Page 297: 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](https://reader036.vdocuments.net/reader036/viewer/2022070904/5f72475c78897a4b8230a315/html5/thumbnails/297.jpg)
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)
Jitte
rPo
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Realistic Image Synthesis SS2020
Low Frequency Region
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Realistic Image Synthesis SS2020
Low Frequency Region
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Realistic Image Synthesis SS2020
Low Frequency Region
138
Zoom-in
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Realistic Image Synthesis SS2020
Variance for Low Sample Count
139
Zoom-in
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Realistic Image Synthesis SS2020
Variance for Low Sample Count
139
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Realistic Image Synthesis SS2020
Variance for Increasing Sample Count
140
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Realistic Image Synthesis SS2020
Experimental Verification
141
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Realistic Image Synthesis SS2020
Convergence rate
142Increasing Samples
Varia
nce
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Realistic Image Synthesis SS2020
Convergence rate
142Increasing Samples
Varia
nce
…
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Realistic Image Synthesis SS2020
143
…
Increasing Samples
Varia
nce Convergence rate
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Realistic Image Synthesis SS2020
Disk Function as Worst Case
144
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Realistic Image Synthesis SS2020
Disk Function as Worst Case
144
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Realistic Image Synthesis SS2020
Disk Function as Worst Case
144
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Realistic Image Synthesis SS2020
Disk Function as Worst Case
145
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Realistic Image Synthesis SS2020
Gaussian as Best Case
146
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Realistic Image Synthesis SS2020
Gaussian as Best Case
146
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Realistic Image Synthesis SS2020
Ambient Occlusion Examples
147
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Realistic Image Synthesis SS2020
Random vs Jittered
148
96 Secondary Rays
MSE: 8.56 x 10e-4MSE: 4.74 x 10e-3
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Realistic Image Synthesis SS2020
CCVT vs. Poisson Disk
149
96 Secondary Rays
MSE: 4.24 x 10e-4 MSE: 6.95 x 10e-4
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Realistic Image Synthesis SS2020
Convergence rates
150
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Realistic Image Synthesis SS2020
Convergence rates
150
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Realistic Image Synthesis SS2020
Jittered vs Poisson Disk
151
Variance
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Realistic Image Synthesis SS2020
What are the benefits of this analysis ?
152
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Realistic Image Synthesis SS2020
What are the benefits of this analysis ?
152
• For offline rendering, analysis tells which samplers would converge faster.
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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