fourier analysis of numerical integration in monte …€¦ · freq. analysis of mc sampling: this...
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FOURIERANALYSISOFNUMERICALINTEGRATIONINMONTECARLORENDERING
KarticSubrGurprit SinghWojciech JaroszHeriotWattUniversity,Edinburgh DartmouthCollege DartmouthCollege
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Motivationforanalysis
• assess,compareexistingmethodsforMonteCarlorendering
• provideinsight,inspireimprovement
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[Subretal2014]
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Errorvscostplotsofrenderingmethods
method1
method2
method3
method4
[Subretal2014]
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Errorvscostplotsofrenderingmethods
method1
method2
method3
method4
[Subretal2014]
method4isbestmethod4isworst
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Errorvscostplotsofrenderingmethods
method1
method2
method3
method4
[Subretal2014]
method4isworstmethod4isbest
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Coursestructure
Preliminaries Sampling
Formaltreatment
30m
30m
20m
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OpenGL[Stachowiak 2010]
Raytracing[Whitted 1980]
Rendering=geometry+radiometry
cameraobscura
geometry/projectionfor pin-hole model known since 400BC
radiometrically accurate simulationis important for photorealism
[photocredit:videomaker.comJune2015]
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Rendering=geometry+radiometry
geometry/projectionfor pin-hole model known since 400BC
radiometrically accurate simulationis important for photorealism
[photocredit:videomaker.comJune2015]
OpenGL[Stachowiak 2010]
Raytracing[Whitted 1980]
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Radiometricfidelityimprovesphotorealism
PedroCampos
manuallypaintedphotograph
Colourbox.com
computergenerated
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Simulatingthephysicsoflightischallenging
lensesdefocus
materials
light,media
exposuretime
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Lighttransport
12
Image?
virtuallightemitter
virtualcamera
virtualscene:geometry+materials
exitant radiance
estimateincidentradianceatallpixelsonthevirtualsensor
Wm2 Sr
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Eachreflectionismodeledbyanintegration
13
radiance:
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14
radiance:
Eachreflectionismodeled byanintegration
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Eachreflectionismodeledbyanintegration
15
radiance:
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Recursiveintegrals
16
Image?
virtuallightemitter
virtualcamera
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Recursiveintegrals
17
Image?
virtuallightemitter
virtualcamera
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Lighttransport:recursiveintegralequation
18
radiance
integraloperatoremittedradiance
LightTransportOperators[Arvo 94]TheRenderingequation[Kajiya 86]
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L isasumofhigh-dimensionalintegrals
19
Onebounce Threebounces
radiance
integraloperatoremittedradiance
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Reconstructionandintegrationinrendering
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Reconstruction:estimateimagesamples
X
Y
X
Ygroundtruth(high-res)image reconstructon(low-res)pixelgrid
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Naïvemethod:sampleimageatgridlocations
X
Y
X
Yreconstructon(low-res)pixelgridgroundtruth(high-res)image
sampling copy
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Naïvemethod:whensamplingisincreased
X
Y
X
Ygroundtruth(high-res)image reconstructon(low-res)pixelgrid
aliasing
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Antialiasing:assuming`square’pixels
X
Y
X
Y
multi-sampling average
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Antialiasingiscostlyduetomulti-sampling
X
Y
X
Y
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Antialiasingusinggeneralreconstructionfilter
X
Y
X
Y
multi-sampling weightedaverage
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Rendering:Reconstructingintegrals
multi-samplingforreconstruction
deterministic
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Rendering:Reconstructingintegrals
multi-samplingforreconstruction
multi-pathsamplingforintegrationestimatepersampledpixel
path1
path2
path3
estimate(probabilisticforMonteCarlo)
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Function-spaceview:Samplinginpathspace
29
n-dimensionalpathspace
light
camera
lightpaths
eachsamplerepresentsapathandhasanassociatedradiancevalue
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Samplelocationsshowninpath-pixelspace
30
n-dimen
sionalpathspace
pixelsonsensor
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31
n-dimen
sionalpathspace
pixelsonsensor
path-spaceintegration(projection)
pixelsonsensor
reconstruction usingintegratedradiance
pixelvalue
(radiance)
Rendering=integration+reconstruction
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Frequencyanalysisoflightfields inrendering
n-dimen
sionalpathspace
pixelsonsensor
path-spaceintegration(projection)
pixelsonsensor
integratedradiance
pixelvalue
(radiance)
localvariation/anisotropy? useinregression/reconstruction
localvariationofintegrand reconstructionfilter
[Ramamoorthi etal.04][Durandetal.05][Soler etal.2009][Overbeck etal.2009][Eganetal.2009,2011][Ramamoorthi etal.2012]
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Freq.analysisofMCsampling:Thiscourse!
n-dimen
sionalpathspace
localvariation/anisotropy?
pixelsonsensor[Durand2011][Ramamoorthi etal.12][SubrandKautz 2013][Pilleboue etal.2015]
AssessingMSE,bias,varianceandconvergenceofMonteCarloestimatorsasafunctionoftheFourierspectrumofthesamplingfunction.
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Freq.analysisofMCsampling:Thiscourse!
n-dimen
sionalpathspace
[Durand2011][Ramamoorthi etal.12][SubrandKautz 2013][Pilleboue etal.2015]
AssessingMSE,bias,varianceandconvergenceofMonteCarloestimatorsasafunctionoftheFourierspectrumofthesamplingfunction.
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Freq.analysisofMCsampling:Thiscourse!
n-dimen
sionalpathspace
localvariation/anisotropy?
AssessingMSE,bias,varianceandconvergenceofMonteCarloestimatorsasafunctionoftheFourierspectrumofthesamplingfunction.
[Durand2011][Ramamoorthi etal.12][SubrandKautz 2013][Pilleboue etal.2015]
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Freq.analysisofMCsampling:Thiscourse!
n-dimen
sionalpathspace
AssessingMSE,bias,varianceandconvergenceofMonteCarloestimatorsasafunctionoftheFourierspectrumofthesamplingfunction.
[Durand2011][Ramamoorthi etal.12][SubrandKautz 2013][Pilleboue etal.2015]
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Freq.analysisofMCsampling:Thiscourse!
n-dimen
sionalpathspace
localvariation/anisotropy?
AssessingMSE,bias,varianceandconvergenceofMonteCarloestimatorsasafunctionoftheFourierspectrumofthesamplingfunction.
[Durand2011][Ramamoorthi etal.12][SubrandKautz 2013][Pilleboue etal.2015]
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Rendering=integration+reconstruction
Shinyball,outoffocusShinyballinmotion
…imagelocation multi-dimintegral
Domain:shuttertimex apertureareax 1st bouncex 2nd bounceIntegrand:radiance(Wm-2 Sr-1)
…
…
38
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Theproblemin1D
0
39
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thesamplingfunction
integrandsamplingfunction
sampledintegrand
multiply
40
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samplingfunctiondecidesintegrationquality
integrandsampledfunction
multiplysamplingfunction
41
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strategiestoimproveestimators1.modifyweights 2.modifylocations
eg.quadraturerules,importancesampling,jitteredsampling,etc.
42
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insightintoimpact:Fourierdomain1.modifyweights 2.modifylocations
analyse samplingfunctioninFourierdomain
43
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Fourieranalysis:originandintuition
• Eigenfunction ofthedifferentialoperator
• Turnsdifferentialequationsintoalgebraicequations
scaling
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Fourieranalysis:originandintuition
• Eigenfunction ofthedifferentialoperator
• Turnsdifferentialequationsintoalgebraicequations
• if
scaling
projection
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TheFourierdomain
Imagecredit:Wikipedia
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ThecontinuousFouriertransform
primal(space,time,etc.)
domain
Fourierdomain
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TheFouriertransform:`frequency’domain
projectionontosinandcos
frequencyfrequencydomain
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Asinglesample:
frequency
amplitude=1phase
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Fourierseries:replaceintegralwithsum
approximatingasquarewaveusing4sinusoids
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frequency
amplitude(samplingspectrum)
phase(samplingspectrum)
51
Fourierspectrumofthesamplingfunction
samplingfunction
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samplingfunction=sumofDiracdeltas
+
+
+
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IntheFourierdomain…
primal Fourier
DiracdeltaFouriertransform
Frequency
Real
Imaginary
Complexplane
amplitudephase
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Review:intheFourierdomain…
primal Fourier
DiracdeltaFouriertransform
Frequency
Real
Imaginary
Complexplane
Real
Imaginary
Complexplane
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amplitudespectrumisnotflat
=
+
+
+
primal Fourier
=
+
+
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Fouriertransform
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samplecontributionsatagivenfrequency
Real
Imaginary
Complexplane
5
1 2 3 4 5
Atagivenfrequency
3
2
41
samplingfunction
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thesamplingspectrumatagivenfrequencysamplingspectrum
Complexplane
53
2
41
centroid
givenfrequency
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thesamplingspectrumatagivenfrequencysamplingspectrumrealizations
expectedcentroid centroid variancegivenfrequency
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expectedsamplingspectrumandvariance
expectedamplitudeofsamplingspectrum varianceofsamplingspectrum
frequency
DC
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1.modifyweights
a.Distributioneg.importancesampling)
2.modifylocations
eg.quadrature rules
samplingfunctionintheFourierdomain
frequency
amplitude(samplingspectrum)
phase(samplingspectrum)
60
Abstractingsamplingstrategyusingspectra
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stochasticsampling&instancesofspectra
Sampler(Strategy1)
Fouriertransform
draw
realizationsofsamplingfunctions realizationsofsamplingspectra
61
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assessingestimatorsusingsamplingspectra
Sampler(Strategy1)
Sampler(Strategy2)
Instancesofsamplingfunctions Instancesofsamplingspectra
Whichstrategyisbetter?Metric?
62
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accuracy(bias)andprecision(variance)
estimatedvalue(bins)
freq
uency
reference
Estimator2
Estimator1
Estimator2isunbiasedbuthashighervariance
63
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Varianceandbias
Highvariance Highbias
predictasafunctionofsamplingstrategyand
integrand
64
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MonteCarlointegration:summaryanderror
• Error• MSE,bias,variance• convergencerate:errorasNisincreased
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Bird’s-eyeviewofanalysis
• RewriteMCestimatorintermsofsamplingfunction
where
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Bird’s-eyeviewofanalysis
• RewriteMCestimatorintermsofsamplingfunction
• Fouriertransformpreservesinnerproducts,so
where
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Bird’s-eyeviewofanalysis
• RewriteMCestimatorintermsofsamplingfunction
• Fouriertransformpreservesinnerproducts,so
• AnalyseMSEerror,biasandconvergenceintermsof
where
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Summary
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Summary
lighttransport&integration high-dimensionalsampling samplingfunction&spectrum
fS average
errorprediction
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Next
lighttransport&integration high-dimensionalsampling samplingfunction&spectrum
fS average
errorprediction
GurpritWojciech
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Localvariationisusefulforadaptivesampling
72
n-dimen
sionalpathspace
pixelsonsensor
path-spaceintegration(projection)
pixelsonsensor
integratedradiance
pixelvalue
(radiance)
localvariation/anisotropy? useinregression/reconstruction