Monte Carlo Integration COS 323 Acknowledgment: Tom Funkhouser.

Monte Carlo Integration Monte Carlo Integration

COS 323COS 323

Acknowledgment: Tom FunkhouserAcknowledgment: Tom Funkhouser

Integration in 1DIntegration in 1D

x=1x=1

f(x)f(x)

?)(1

0 dxxf ?)(1

0 dxxf

Slide courtesy of Slide courtesy of Peter ShirleyPeter Shirley

We can approximateWe can approximate

x=1x=1

f(x)f(x) g(x)g(x)

1

0

1

0

)()( dxxgdxxf 1

0

1

0

)()( dxxgdxxf


Or we can averageOr we can average

x=1x=1

f(x)f(x) E(f(x))E(f(x))

))(()(1

0

xfEdxxf ))(()(1

0

xfEdxxf


Estimating the averageEstimating the average

xx11

f(x)f(x)

xxNN

N

iixfN

dxxf1

1

0

)(1

)(

N

iixfN

dxxf1

1

0

)(1

)(

E(f(x))E(f(x))


Other DomainsOther Domains

x=bx=b

f(x)f(x) < f >< f >abab

x=ax=a

N

ii

b

a

xfN

abdxxf

1

)()(

N

ii

b

a

xfN

abdxxf

1

)()(


Benefits of Monte CarloBenefits of Monte Carlo

• No “exponential explosion” in required No “exponential explosion” in required number of samples with increase in number of samples with increase in dimensiondimension

• Resistance to badly-behaved functionsResistance to badly-behaved functions

VarianceVariance

xx11 xxNN

E(f(x))E(f(x))

2

1

))](()([1

)( xfExfN

xfVarN

ii

2

1

))](()([1

)( xfExfN

xfVarN

ii

VarianceVariance

xx11 xxNN

E(f(x))E(f(x))

)(1))(( xfVar

NxfEVar )(1))(( xfVar

NxfEVar

Variance decreases as 1/NVariance decreases as 1/NError decreases as 1/sqrt(N)Error decreases as 1/sqrt(N)

VarianceVariance

• Problem: variance decreases with 1/NProblem: variance decreases with 1/N– Increasing # samples removes noise slowlyIncreasing # samples removes noise slowly

xx11 xxNN

E(f(x))E(f(x))

Variance Reduction TechniquesVariance Reduction Techniques

• Problem: variance decreases with 1/NProblem: variance decreases with 1/N– Increasing # samples removes noise slowlyIncreasing # samples removes noise slowly

• Variance reduction:Variance reduction:– Stratified samplingStratified sampling

– Importance samplingImportance sampling

Stratified SamplingStratified Sampling

• Estimate subdomains separatelyEstimate subdomains separately

xx11 xxNN

EEkk(f(x))(f(x))

ArvoArvo


• This is still unbiasedThis is still unbiased

M

kii

N

iiN

FNN

xfN

F

1

1

1

)(1

M

kii

N

iiN

FNN

xfN

F

1

1

1

)(1

xx11 xxNN

EEkk(f(x))(f(x))


• Less overall variance if less variance Less overall variance if less variance in subdomainsin subdomains

M

kiiN FVarN

NFVar

12

1

M

kiiN FVarN

NFVar

12

1

xx11 xxNN

EEkk(f(x))(f(x))

Importance SamplingImportance Sampling

• Put more samples where f(x) is biggerPut more samples where f(x) is bigger

)(

)(

1)(

1

i

ii

N

ii

xp

xfY

YN

dxxf

)(

)(

1)(

1

i

ii

N

ii

xp

xfY

YN

dxxf

xx11 xxNN

E(f(x))E(f(x))


• This is still unbiasedThis is still unbiased

xx11 xxNN

E(f(x))E(f(x))

dxxf

dxxpxp

xf

dxxpxYYE i

)(

)()(

)(

)()(

dxxf

dxxpxp

xf

dxxpxYYE i

)(

)()(

)(

)()(

for all Nfor all N


• Zero variance if p(x) ~ f(x)Zero variance if p(x) ~ f(x)

x1 xN

E(f(x))

Less variance with betterLess variance with betterimportance samplingimportance sampling

0)(

1

)(

)(

)()(

YVar

cxp

xfY

xcfxp

i

ii

0)(

1

)(

)(

)()(

YVar

cxp

xfY

xcfxp

i

ii

Generating Random PointsGenerating Random Points

• Uniform distribution:Uniform distribution:– Use pseudorandom number generatorUse pseudorandom number generator

Pro

bab

ility

Pro

bab

ility

00

11

Pseudorandom NumbersPseudorandom Numbers

• Deterministic, but have statistical properties Deterministic, but have statistical properties resembling true random numbersresembling true random numbers

• Common approach: each successive Common approach: each successive pseudorandom number is function of pseudorandom number is function of previousprevious

• Linear congruential method:Linear congruential method:– Choose constants carefully, e.g.Choose constants carefully, e.g.

a = 1664525a = 1664525b = 1013904223b = 1013904223c = 2c = 232 32 – 1– 1

cbaxx nn mod1 cbaxx nn mod1

Pseudorandom NumbersPseudorandom Numbers

• To get floating-point numbers in [0..1),To get floating-point numbers in [0..1),divide integer numbers by divide integer numbers by cc + 1 + 1

• To get integers in range [u..v], divide byTo get integers in range [u..v], divide by(c+1)/(v–u+1), truncate, and add u(c+1)/(v–u+1), truncate, and add u– Better statistics than using modulo (v–u+1)Better statistics than using modulo (v–u+1)

– Only works if u and v small compared to cOnly works if u and v small compared to c


• Uniform distribution:Uniform distribution:– Use pseudorandom number generatorUse pseudorandom number generator

Pro

bab

ility

Pro

bab

ility

00

11


• Specific probability distribution:Specific probability distribution:– Function inversionFunction inversion

– RejectionRejection

)(xf )(xf


• ““Inversion method”Inversion method”– Integrate Integrate ff((xx): Cumulative Distribution ): Cumulative Distribution

FunctionFunction

dxxf )( dxxf )(1111

)(xf )(xf


• ““Inversion method”Inversion method”– Integrate Integrate ff((xx): Cumulative Distribution ): Cumulative Distribution

FunctionFunction

– Invert CDF, apply to uniform random Invert CDF, apply to uniform random variablevariable dxxf )( dxxf )(

1111

)(xf )(xf


• Specific probability distribution:Specific probability distribution:– Function inversionFunction inversion

– RejectionRejection

)(xf )(xf


• ““Rejection method”Rejection method”– Generate random (x,y) pairs,Generate random (x,y) pairs,

y between 0 and max(fy between 0 and max(f((xx))) )


• ““Rejection method”Rejection method”– Generate random (x,y) pairs,Generate random (x,y) pairs,

y between 0 and max(fy between 0 and max(f((xx))))

– Keep only samples where y < f(x)Keep only samples where y < f(x)

Monte Carlo in Computer Monte Carlo in Computer GraphicsGraphics

or, Solving Integral Equationsor, Solving Integral Equationsfor Fun and Profitfor Fun and Profit

or, Ugly Equations, Pretty or, Ugly Equations, Pretty PicturesPictures

AnimationAnimationAnimationAnimation

Computer Graphics PipelineComputer Graphics Pipeline

RenderingRenderingRenderingRendering

LightingLightingandand

ReflectanceReflectance

LightingLightingandand

ReflectanceReflectance

ModelingModelingModelingModeling

xx

''

Rendering EquationRendering Equation

SurfaceSurfaceLightLight

dnxfxLxLxL rieo )()',,(),()',()',(


dd

[Kajiya 1986][Kajiya 1986]

ViewerViewer

nn


• This is an This is an integral equationintegral equation

• Hard to solve!Hard to solve!– Can’t solve thisCan’t solve this

in closed formin closed form

– Simulate complexSimulate complexphenomenaphenomena

HeinrichHeinrich




• This is an This is an integral equationintegral equation

• Hard to solve!Hard to solve!– Can’t solve thisCan’t solve this

in closed formin closed form

– Simulate complexSimulate complexphenomenaphenomena

JensenJensen



Monte Carlo IntegrationMonte Carlo Integration

f(x)f(x)

N

iixfN

dxxf1

1

0

)(1

)(

N

iixfN

dxxf1

1

0

)(1

)(

ShirleyShirley

Monte Carlo Path TracingMonte Carlo Path Tracing

Estimate integral Estimate integral for each pixel for each pixel

by random samplingby random sampling

Estimate integral Estimate integral for each pixel for each pixel

by random samplingby random sampling

Monte Carlo Global IlluminationMonte Carlo Global Illumination

• Rendering = integrationRendering = integration– AntialiasingAntialiasing

– Soft shadowsSoft shadows

– Indirect illuminationIndirect illumination

– CausticsCaustics






SurfaceSurface

EyeEye

PixelPixel

xx

dAe)L(xLS

P dAe)L(xLS

P






SurfaceSurface

EyeEye

LightLight

xx

x’x’

dAxxGxxx)VxL(exxxxfe)(x,xL)wL(x,S

re ),(),(),,(


re ),(),(),,(






HerfHerf


re ),(),(),,(


re ),(),(),,(






wdnw)w(x,Lwwxf)w(x,L)w(x,L ireo

)(),,(


)(),,(

SurfaceSurface

EyeEye

LightLight

xx

SurfaceSurface

’’






DebevecDebevec


)(),,(


)(),,(






Diffuse SurfaceDiffuse Surface

EyeEye

LightLight

xx

SpecularSpecularSurfaceSurface

’’


)(),,(


)(),,(






JensenJensen


)(),,(


)(),,(

ChallengeChallenge

• Rendering integrals are difficult to Rendering integrals are difficult to evaluateevaluate– Multiple dimensionsMultiple dimensions

– DiscontinuitiesDiscontinuities• Partial occludersPartial occluders

• HighlightsHighlights

• CausticsCaustics

DrettakisDrettakis


re ),(),(),,(


re ),(),(),,(

ChallengeChallenge

• Rendering integrals are difficult to Rendering integrals are difficult to evaluateevaluate– Multiple dimensionsMultiple dimensions

– DiscontinuitiesDiscontinuities• Partial occludersPartial occluders

• HighlightsHighlights

• CausticsCaustics

JensenJensen


re ),(),(),,(


re ),(),(),,(


Big diffuse light source, 20 minutesBig diffuse light source, 20 minutes

JensenJensen


1000 paths/pixel1000 paths/pixel

JensenJensen


• Drawback: can beDrawback: can benoisy unless noisy unless lotslots of ofpaths simulatedpaths simulated

• 40 paths per pixel:40 paths per pixel:

LawrenceLawrence


• Drawback: can beDrawback: can benoisy unless noisy unless lotslots of ofpaths simulatedpaths simulated

• 1200 paths per pixel:1200 paths per pixel:

LawrenceLawrence

Reducing VarianceReducing Variance

• Observation: some paths more importantObservation: some paths more important(carry more energy) than others(carry more energy) than others– For example, shiny surfaces reflect more lightFor example, shiny surfaces reflect more light

in the ideal “mirror” directionin the ideal “mirror” direction

• Idea: put more samples where f(x) is Idea: put more samples where f(x) is biggerbigger


• Idea: put more samples where f(x) is Idea: put more samples where f(x) is biggerbigger

)(

)(

1)(

1

1

0

i

ii

N

ii

xp

xfY

YN

dxxf

)(

)(

1)(

1

1

0

i

ii

N

ii

xp

xfY

YN

dxxf

Effect of Importance SamplingEffect of Importance Sampling

• Less noise at a given number of samplesLess noise at a given number of samples

• Equivalently, need to simulate fewer Equivalently, need to simulate fewer paths for some desired limit of noisepaths for some desired limit of noise

Uniform random samplingUniform random sampling Importance samplingImportance sampling

Monte Carlo Integration COS 323 Acknowledgment: Tom Funkhouser.

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Transcript of Monte Carlo Integration COS 323 Acknowledgment: Tom Funkhouser.