graphics.cg.uni-saarland.de · 24 Chapter 5. Popular sampling patterns Samples Power spectrum...

ADVANCED SAMPLINGRandom Stratified N-Rooks

Multi-Jittered Quasi-Random Poisson-Disc

Random Stratified N-Rooks

Multi-Jittered Quasi-Random Poisson-Disc

Philipp Slusallek Karol Myszkowski Gurprit Singh

Realistic Image Synthesis SS2020

Part of Siggraph 2016 Course

2Realistic Image Synthesis SS2020

*Wojciech Jarosz Gurprit Singh Kartic Subr

Fourier Analysis of Numerical Integration in Monte Carlo Rendering

*First part of slides from Wojciech Jarosz

Recall: Monte Carlo Integration

Df(x) dx

Df(x)S(x) dx

Df(x) dx

Df(x)S(x) dx

Df(x) dx

Df(x)S(x) dx

Df(x) dx

Df(x)S(x) dx

Df(x) dx

Df(x)S(x) dx

S(x) =1

�(x� xk)

Df(x) dx

Df(x)S(x) dx

Df(x) dx

Df(x)S(x) dx

S(x) =1

�(x� xk)

Df(x) dx

Df(x)S(x) dx

Df(x) dx

Df(x)S(x) dx

S(x) =1

�(x� xk)

How to generate the locations ?xk

Df(x) dx

Df(x)S(x) dx

Df(x) dx

Df(x)S(x) dx

Independent Random Sampling

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

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

✔Trivially extends to higher dimensions

✔Trivially progressive and memory-less

✘ Big gaps

✘ Clumping

Recall: Fourier Theory

Power SpectrumInput Image

Image courtesy: Laurent Belcour

Recall: Fourier Theory

Power SpectrumInput Image

Image courtesy: Laurent Belcour

Recall: Fourier theory

f(!) =

Df(x) e�2⇡ ı! x dx

f(~!) =

Df(~x) e�2⇡ ı (~!·~x) d~x

Fourier transform:

f(!) =

f(~!) =

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

S(~!) =

DS(~x) e�2⇡ ı (~!·~x) d~x

S(~!) =

�(|~x� ~xk|) e�2⇡ ı (~!·~x) d~x

S(~!) =1

e�2⇡ ı (~!·~xk)

f(!) =

f(~!) =

Sampling function:

S(~!) =

DS(~x) e�2⇡ ı (~!·~x) d~x

S(~!) =

�(|~x� ~xk|) e�2⇡ ı (~!·~x) d~x

S(~!) =1

e�2⇡ ı (~!·~xk)

S(~!) =

DS(~x) e�2⇡ ı (~!·~x) d~x

S(~!) =

�(|~x� ~xk|) e�2⇡ ı (~!·~x) d~x

S(~!) =1

e�2⇡ ı (~!·~xk)

S(~!) =

DS(~x) e�2⇡ ı (~!·~x) d~x

S(~!) =

�(|~x� ~xk|) e�2⇡ ı (~!·~x) d~x

S(~!) =1

e�2⇡ ı (~!·~xk)

S(~!) =

DS(~x) e�2⇡ ı (~!·~x) d~x

S(~!) =

�(|~x� ~xk|) e�2⇡ ı (~!·~x) d~x

S(~!) =1

e�2⇡ ı (~!·~xk)

f(!) =

f(~!) =

Sampling function:

S(~!) =

DS(~x) e�2⇡ ı (~!·~x) d~x

S(~!) =

�(|~x� ~xk|) e�2⇡ ı (~!·~x) d~x

S(~!) =1

e�2⇡ ı (~!·~xk)

S(~!) =

DS(~x) e�2⇡ ı (~!·~x) d~x

S(~!) =

�(|~x� ~xk|) e�2⇡ ı (~!·~x) d~x

S(~!) =1

e�2⇡ ı (~!·~xk)

�(|~x� ~xk|)

S(~!) =

DS(~x) e�2⇡ ı (~!·~x) d~x

S(~!) =

�(|~x� ~xk|) e�2⇡ ı (~!·~x) d~x

S(~!) =1

e�2⇡ ı (~!·~xk)

f(!) =

f(~!) =

Sampling function:

S(~!) =

DS(~x) e�2⇡ ı (~!·~x) d~x

S(~!) =

�(|~x� ~xk|) e�2⇡ ı (~!·~x) d~x

S(~!) =1

e�2⇡ ı (~!·~xk)

S(~!) =

DS(~x) e�2⇡ ı (~!·~x) d~x

S(~!) =

�(|~x� ~xk|) e�2⇡ ı (~!·~x) d~x

S(~!) =1

e�2⇡ ı (~!·~xk)

�(|~x� ~xk|)

S(~!) =

DS(~x) e�2⇡ ı (~!·~x) d~x

S(~!) =

�(|~x� ~xk|) e�2⇡ ı (~!·~x) d~x

S(~!) =1

e�2⇡ ı (~!·~xk)E

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24 Chapter 5. Popular sampling patterns

Samples Power spectrum Radial meanR

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

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

5.3 Blue noise

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

Samples Power spectrum

�(|~x� ~xk|)

~xy ~!y

Samples Power spectrum Radial mean

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Many sample set realizations Expected power spectrum

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

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ter5.Popula

tterns

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Figure5.6:

Illustrationof

randomand

stochasticgrid-based

sampling

patternsw

iththe

correspondingFourierexpected

powerspectra

andthe

correspondingradialm

eanoftheirexpected

powerspectra.

sequenceis

calledthe

mersley

sequence,which

cancreate

lowerdiscrepancy

pointsetforarbitrary

dimensions,butdue

firstdimension

beinga

regularsampling,know

ledgeofthe

numberoftotalsam

plesis

necessary.Figure5.7

illustratesthe

mersley

pointsetwith

64points

.Thecorresponding

sampling

powerspectra

forHalton

ersleysam

ples(firsttw

ponents)aresum

marised

inFigures

5.3Blue

sampling

patternw

luenoise

characteristicsis

supposeto

elldistributedw

ithinthe

spatialdomain

withoutcontaining

anyregular

structures.The

luenoise

coinedby

Ulichney

[47],who

investivateda

radiallyaveraged

powerspectra

ofvarioussampling

patterns.He

advocatedthree

importantfeatures

foranidealradialpow

erspectrum;First,its

peakshould

nRandom

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Samples Expected power spectrum

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Radial meanDC Peak

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5.3 Blue noise

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

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

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

✔Extends to higher dimensions, but…

✘ Curse of dimensionality

✘ Aliasing

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

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

✔Provably cannot increase variance

✘ Not progressive

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Jittered Sampling

Samples Expected power spectrum Radial mean

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5.3 Blue noise

Monte Carlo (16 random samples)

Monte Carlo (16 jittered samples)

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

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

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

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

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

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

Uncorrelated Jitter [Cook et al. 84]

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

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

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

- 2D jittered (u,v) for lens

- 1D jittered (t) for time

- combine dimensions in random order

Depth of Field (4D)

Reference Random Sampling Uncorrelated Jitter

Image source: PBRTe2 [Pharr & Humphreys 2010]

Stratify samples in each dimension separatelyUncorrelated Jitter ➔ Latin Hypercube

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

Uncorrelated Jitter ➔ Latin Hypercube

x1 x2 x3 x4x

y1 y2 y3 y4y

u1 u2 u3 u4u

v1 v2 v3 v4v

t1 t2 t3 t4t

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

x1 x2 x3 x4x

y1 y2 y3 y4y

u1 u2 u3 u4u

v1 v2 v3 v4v

t1 t2 t3 t4t

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

x1 x2 x3 x4

y4 y2 y1 y3

u3 u4 u2 u1

v2 v1 v3 v4

t2 t1 t4 t3

Shuffle order

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

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

x1 x2 x3 x4

y4 y2 y1 y3

Latin Hypercube (N-Rooks) Sampling

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

[Shirley 91]

// 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,:));

24Initialize

25Shuffle rows

26Shuffle rows

27Shuffle columns

Evenly distributed in each individual dimension

Unevenly distributed in n-dimensions

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5.3 Blue noise

N-Rooks Sampling

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

– combine N-Rooks and Jittered stratification constraints

Multi-Jittered Sampling

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

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

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

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

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

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

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

Initialize

Shuffle x-coords

Shuffle y-coords

Multi-Jittered Sampling (Projections)

Evenly distributed in 2D!

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N-Rooks Sampling

Samples Radial meanExpected power spectrum

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5.3 Blue noise

Jittered Sampling

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

stochastic sampling.” ACM SIGGRAPH, 1985.

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

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

Random Dart Throwing

5.4 Interpreting and exploiting knowledge of the sampling spectra 27

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

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

5.4 Interpreting and exploiting knowledge of the sampling spectra

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

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

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

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

Poisson Disk Sampling

Blue-Noise Sampling (Relaxation-based)

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

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

from each other.

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Poisson Disk Sampling

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

The Van der Corput SequenceRadical Inverse Φb in base 2

Subsequent points “fall into biggest holes”

k Base 2 Φb

k Base 2 Φb1 1 .1 = 1/2

2 10 .01 = 1/4

k Base 2 Φb1 1 .1 = 1/2

2 10 .01 = 1/4

3 11 .11 = 3/4

k Base 2 Φb1 1 .1 = 1/2

2 10 .01 = 1/4

3 11 .11 = 3/4

4 100 .001 = 1/8

k Base 2 Φb1 1 .1 = 1/2

2 10 .01 = 1/4

3 11 .11 = 3/4

4 100 .001 = 1/8

5 101 .101 = 5/8

k Base 2 Φb1 1 .1 = 1/2

2 10 .01 = 1/4

3 11 .11 = 3/4

4 100 .001 = 1/8

5 101 .101 = 5/8

6 110 .011 = 3/8

k Base 2 Φb1 1 .1 = 1/2

2 10 .01 = 1/4

3 11 .11 = 3/4

4 100 .001 = 1/8

5 101 .101 = 5/8

6 110 .011 = 3/8

7 111 .111 = 7/8

k Base 2 Φb1 1 .1 = 1/2

2 10 .01 = 1/4

3 11 .11 = 3/4

4 100 .001 = 1/8

5 101 .101 = 5/8

6 110 .011 = 3/8

7 111 .111 = 7/8

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

Halton and Hammersley Points

Halton: Radical inverse with different base for each dimension:

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

- The bases should all be relatively prime.

- Incremental/progressive generation of samples

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

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

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

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

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

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

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

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

The Hammersley Sequence

1 sample in each “elementary interval”

Monte Carlo (16 random samples)

Monte Carlo (16 jittered samples)

Scrambled Low-Discrepancy Sampling

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

How can we predict error from these?

Samples’ Radial Spectrum

Integrand Radial SpectrumPo

Frequency

Part 2: Formal Treatment of MSE, Bias and Variance

Convergence rate for Random Samples

75Increasing Samples

Increasing Samples

O(N�1)

Increasing Samples

Convergence rate for Jittered Samples

O(N�1)

Increasing Samples

Convergence rate for Jittered Samples

O(N�1)

O(N�1.5)

Increasing Samples

Convergence rate Jittered vs Poisson Disk

O(N�1.5)

O(N�1)

Increasing Samples

O(N�1.5)

O(N�1)

Increasing Samples

O(N�1.5)

O(N�1)

Increasing Samples

O(N�1)

O(N�1.5)

Samples and function in Fourier Domain

Spatial Domain Fourier Domain

f(x)0 w-w

Convolution

Source: vdumoulin-github

Convolution

Source: vdumoulin-github

Sampling in Primal Domain is Convolution in Fourier Domain

f(x)S(x)

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

0 w-w0

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

0 w-w0

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

Aliasing in Reconstruction

Aliasing

Error in Monte Carlo Integration

Error in Integration

Aliasing (Reconstruction) vs. Error (Integration)

Error in IntegrationAliasing

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

Integration in the Fourier Domain

Integration is the DC term in the Fourier Domain

Df(x)dx

Spatial Domain:

Df(x)dx

Spatial Domain:

Fourier Domain:

Df(x)dx

Spatial Domain:

Fourier Domain:

Df(x)S(x)dx =

⌦f⇤(!)S(!)d!

Monte Carlo Estimator in Spatial Domain

Df(x)S(x)dx =

⌦f⇤(!)S(!)d!

S(x) =1

�(x� xk)

Df(x)S(x)dx =

⌦f⇤(!)S(!)d!

S(x) =1

�(x� xk)

Df(x)S(x)dx =

⌦f⇤(!)S(!)d!

S(x) =1

�(x� xk)

Df(x)S(x)dx =

⌦f⇤(!)S(!)d!

S(x) =1

�(x� xk)

Df(x)S(x)dx =

⌦f⇤(!)S(!)d!

S(x) =1

�(x� xk)

Monte Carlo Estimator in Fourier Domain

Df(x)S(x)dx =

⌦f⇤(!)S(!)d!

S(x) =1

�(x� xk)

Df(x)S(x)dx =

⌦f⇤(!)S(!)d!

S(x) =1

�(x� xk)

Df(x)S(x)dx =

⌦f⇤(!)S(!)d!

S(x) =1

�(x� xk)

S(!) =1

e�i2⇡!xk

How to Formulate Error in Fourier Domain ?

Fredo Durand [2011]

⌦f⇤(!)S(!)d!I = f(0)

How to Formulate Error in Fourier Domain ?

Fredo Durand [2011]

⌦f⇤(!)S(!)d!I = f(0)

Error in Spatial Domain

I � µN =

Df(x)dx�

Df(x)S(x)dx

⌦f⇤(!)S(!)d!I = f(0)

I � µN =

Df(x)dx�

Df(x)S(x)dx

⌦f⇤(!)S(!)d!I = f(0)

True Integral

I � µN =

Df(x)dx�

Df(x)S(x)dx

Monte Carlo Estimator

⌦f⇤(!)S(!)d!I = f(0)

True Integral

I � µN =

Df(x)dx�

Df(x)S(x)dx

⌦f⇤(!)S(!)d!I = f(0)

I � µN =

Df(x)dx�

Df(x)S(x)dx

⌦f⇤(!)S(!)d!I = f(0)

Error in Fourier Domain

I � µN = f(0)�Z

⌦f⇤(!)S(!)d!

Fredo Durand [2011]

⌦f⇤(!)S(!)d!I = f(0)

I � µN =

Df(x)dx�

Df(x)S(x)dx

Error in Fourier Domain

I � µN = f(0)�Z

⌦f⇤(!)S(!)d!

Fredo Durand [2011]

Error = Bias2 +Variance

Properties of Error

• Bias: Expected value of the Error

• Variance:

hI � µN i

Var(I � µN )

Properties of Error

• Variance:

hI � µN i

Var(I � µN )

Properties of Error

• Variance:

hI � µN i

Var(I � µN )

Properties of Error

• Variance:

hI � µN i

Subr and Kautz [2013]

Var(I � µN )

Bias in the Monte Carlo Estimator

Bias in Fourier Domain

I � µN = f(0)�Z

⌦f⇤(!)S(!)d!Error:

I � µN = f(0)�Z

hI � µN i = f(0)�⌧Z

⌦f⇤(!)S(!)d!

I � µN = f(0)�Z

hI � µN i = f(0)�⌧Z

⌦f⇤(!)S(!)d!

hI � µN i = f(0)�⌧Z

⌦f⇤(!)S(!)d!

hI � µN i = f(0)�⌧Z

⌦f⇤(!)S(!)d!

hI � µN i = f(0)�⌧Z

⌦f⇤(!)S(!)d!

hI � µN i = f(0)�Z

⌦f⇤(!) hS(!)i d!

hI � µN i = f(0)�⌧Z

⌦f⇤(!)S(!)d!

hI � µN i = f(0)�Z

⌦f⇤(!) hS(!)i d!

hI � µN i = f(0)�Z

⌦f⇤(!) hS(!)i d!

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

hI � µN i = f(0)�Z

⌦f⇤(!) hS(!)i d!

To obtain an unbiased estimator:

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

hI � µN i = f(0)�Z

⌦f⇤(!) hS(!)i d!

How to obtain ?

hS(!)i = 0

Complex form in Amplitude and Phase

hS(!)i = |hS(!)i| e��(hS(!)i)

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

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

Phase change due to Random Shift

For a given frequency !

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

ImagMultiple realizations

… Multiple realizations

hS(!)i = 0

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

• 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.

Variance in the Fourier domain

I � µN = f(0)�Z

Var(I � µN ) = Var

✓f(0)�

⌦f⇤(!) S(!) d!

I � µN = f(0)�Z

✓f(0)�

⌦f⇤(!) S(!) d!

I � µN = f(0)�Z

✓f(0)�

⌦f⇤(!) S(!) d!

✓f(0)�

⌦f⇤(!) S(!) d!

✓f(0)�

⌦f⇤(!) S(!) d!

✓f(0)�

⌦f⇤(!) S(!) d!

✓f(0)�

⌦f⇤(!) S(!) d!

Var(µN ) = Var

⌦f⇤(!) S(!)d!

Var(µN ) = Var

⌦f⇤(!) S(!)d!

Var(µN ) = Var

⌦f⇤(!) S(!)d!

Var(µN ) = Var

⌦f⇤(!) S(!)d!

Var(µN ) =

⌦Pf (!)Var

⇣S(!)

Var(µN ) = Var

⌦f⇤(!) S(!)d!

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

Var(µN ) =

⌦Pf (!)Var

⇣S(!)

Var(µN ) =

⌦Pf (!)Var

⇣S(!)

Var(µN ) =

⌦Pf (!)Var

⇣S(!)

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

Var(µN ) =

⌦Pf (!)Var

⇣S(!)

Var(µN ) =

⌦Pf (!)Var

⇣S(!)

Var(µN ) =

⌦Pf (!)Var

⇣S(!)

For purely random samples:

Var(µN ) =

⌦Pf (!)Var

⇣S(!)

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

Var(µN ) =

⌦Pf (!)Var

⇣S(!)

Var(µN ) =

⌦Pf (!) hPS(!)i d!

Fredo Durand [2011]

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

Var(µN ) =

⌦Pf (!)Var

⇣S(!)

Var(µN ) =

⌦Pf (!) hPS(!)i d!

Fredo Durand [2011]

hS(!)i = 0

Variance using Homogenized Samples

Homogenizing any sampling pattern makeshS(!)i = 0

Homogenizing any sampling pattern makes

Pilleboue et al. [2015]

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

Var(µN ) =

⌦Pf (!) hPS(!)i d!

hS(!)i = 0

Var(µN ) =

⌦Pf (!) hPS(!)i d!

Var(µN ) =

⌦Pf (!) hPS(!)i d!

Variance in terms of n-dimensional Power Spectra

Var(µN ) =

⌦Pf (!) hPS(!)i d!

Variance in terms of n-dimensional Power Spectra

Var(µN ) =

⌦Pf (!) hPS(!)i d!

Variance in the Polar Coordinates

Var(µN ) =

⌦Pf (!) hPS(!)i d!

In polar coordinates:

Var(µN ) =

⌦Pf (!) hPS(!)i d!

V ar[µN ] = M(Sd�1)

Sd�1

Pf (⇢n) hPS(⇢n)i dn d⇢

Var(µN ) =

⌦Pf (!) hPS(!)i d!

Sd�1

Var(µN ) =

⌦Pf (!) hPS(!)i d!

Sd�1

Variance for Isotropic Power Spectra

Sd�1

For isotropic power spectra:

Sd�1

0Pf (⇢) hPS(⇢)i d⇢

Sd�1

0Pf (⇢) hPS(⇢)i d⇢

Sd�1

0Pf (⇢) hPS(⇢)i d⇢

Sd�1

Variance in terms of 1-dimensional Power Spectra

0Pf (⇢) hPS(⇢)i d⇢

Variance in terms of 1-dimensional Power Spectra

0Pf (⇢) hPS(⇢)i d⇢

Variance: Integral over Product of Power Spectra

0Pf (⇢) hPS(⇢)i d⇢

For given number of Samples

Sampling Radial Power Spectrum

Integrand Radial Power Spectrum

0Pf (⇢) hPS(⇢)i d⇢

Spatial Distribution vs Radial Mean Power Spectra

135� � � � �

��

� � � � �

��

Samplers Worst Case Best Case

Random

Jitter

Poisson Disk

For 2-dimensions

Random

Jitter

Poisson Disk

For 2-dimensions

O(N�1)

Random

Jitter

Poisson Disk

For 2-dimensions

O(N�1) O(N�1)

Random

Jitter

Poisson Disk

O(N�1.5)

For 2-dimensions

O(N�1) O(N�1)

Random

Jitter

Poisson Disk

O(N�1.5) O(N�2)

For 2-dimensions

O(N�1) O(N�1)

Random

Jitter

Poisson Disk

O(N�1.5) O(N�2)

For 2-dimensions

O(N�1)

O(N�1) O(N�1)

Random

Jitter

Poisson Disk

O(N�1)

O(N�1.5) O(N�2)

For 2-dimensions

O(N�1)

O(N�1) O(N�1)

Random

Jitter

Poisson Disk

O(N�1)

O(N�1.5) O(N�2)

For 2-dimensions

O(N�1)

O(N�1) O(N�1)

O(N�1.5)

Random

Jitter

Poisson Disk

O(N�1)

O(N�1.5) O(N�2)

For 2-dimensions

O(N�1)

O(N�1) O(N�1)

O(N�1.5) O(N�3)

For 2-dimensions

Random

Jitter

Poisson Disk

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)

Low Frequency Region

138� � � � �

��

� � � � �

��

138� � � � �

��

� � � � �

��

Zoom-in

� � � � �

��

� � � � �

��

Variance for Low Sample Count

Zoom-in

� � � � �

��

� � � � �

��

Variance for Low Sample Count

Zoom-in

� � � � �

��

� � � � �

��

Variance for Increasing Sample Count

Zoom-in

� � � � �

��

� � � � �

��

O(N�1)

O(N�2)

Experimental Verification

Convergence rate

Increasing Samples

nce Convergence rate

Disk Function as Worst Case

Gaussian as Best Case

Ambient Occlusion Examples

Random vs Jittered

96 Secondary Rays

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

CCVT vs. Poisson Disk

96 Secondary Rays

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

Convergence rates

Jittered vs Poisson Disk

Variance

What are the benefits of this analysis ?

• 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

graphics.cg.uni-saarland.de · 24 Chapter 5. Popular sampling patterns Samples Power spectrum...

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