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