fourier analysis of numerical integration in monte …€¦ · freq. analysis of mc sampling: this...

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