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Introduction to Global IlluminationIntroduction to Global IlluminationIntroduction to Global IlluminationIntroduction to Global Illumination

OverviewOverview RadiometryRadiometry The rendering equationThe rendering equation Monte CarloMonte Carlo

OverviewOverview RadiometryRadiometry The rendering equationThe rendering equation Monte CarloMonte Carlo

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Image synthesisImage synthesisImage synthesisImage synthesis

scenedescription

scenedescription

surfaceradiancesurface

radiance

Deterministicand/or

stochasticsimulation.

Deterministicand/or

stochasticsimulation.

3333

Stages of light transportStages of light transportStages of light transportStages of light transport

luminaireluminaire

blockerblocker blockerblocker

Direct illuminationDirect illumination Indirect illuminationIndirect illumination

4444

An example of global illuminationAn example of global illuminationAn example of global illuminationAn example of global illumination

Lischinski, Tampieri, and Greenberg 1993Lischinski, Tampieri, and Greenberg 1993

5555

Photo-realistic renderingPhoto-realistic renderingPhoto-realistic renderingPhoto-realistic rendering

6666

7777

8888

Types of surface scatteringTypes of surface scatteringTypes of surface scatteringTypes of surface scattering

diffusediffuse directional diffusedirectional diffuse

specularspecular

9999

Directional dependenceDirectional dependenceDirectional dependenceDirectional dependence

highly directionalhighly directional

10101010

Defining radianceDefining radianceDefining radianceDefining radiance

dAdAdd

rr

uu

f (r,u) cos dA dpower crossing surface

f (r,u) cos dA dpower crossing surface

Classical DefinitionClassical Definition Measure-TheoreticMeasure-Theoretic

Radiant energy definesa measure on R3 x S2.Radiant energy definesa measure on R3 x S2.

The associated densityfunction is radiance.The associated densityfunction is radiance.

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Definition of radianceDefinition of radianceDefinition of radianceDefinition of radiance

xx

(x,)(x,)ffis a scalar

density functionis a scalar

density function

radianceradiance

12121212

dAdA

Definition of radianceDefinition of radianceDefinition of radianceDefinition of radiance

wattswatts

m2 srm2 sr

dd

wattswatts m2m2 srsr

(x,) dA d(x,) dA dffpower =power =

13131313

Power from radiancePower from radiancePower from radiancePower from radiance

dAdA

Integrate oversolid angle...

Integrate oversolid angle...

and surfaceand surface

14141414

IrradianceIrradianceIrradianceIrradiance

dAdA

power perunit area

power perunit area

15151515

IrradianceIrradianceIrradianceIrradiance

dAdA

weighted integralover solid angle

weighted integralover solid angle

wattswatts

m2m2

16161616

Simulating reflected lightSimulating reflected lightSimulating reflected lightSimulating reflected light

irradianceirradiance

dd

xx

17171717

Simulating reflected lightSimulating reflected lightSimulating reflected lightSimulating reflected light

irradianceirradiance radianceradiance

dfdfdd

xx

18181818

dfdfdd

df ddf d

xx

19191919

dfdfdd

df (’,) ddf (’,) d

xx

’’

20202020

dfdfdd

df (’,) ddf (’,) d

xx

wattwatt

m2m2wattwatt

m2 srm2 sr

21212121

dfdfdd

df (’,) ddf (’,) d

xx

BRDF [sr -1]BRDF [sr -1]

22222222

dfdf

f ’f ’

df = (’,) f’cos’ d’df = (’,) f’cos’ d’

d’d’’’

xx

23232323

ff

f = (’,) f’cos’ d’f = (’,) f’cos’ d’

xx

24242424

ff

xx

25252525

Formulating a balance equationFormulating a balance equationFormulating a balance equationFormulating a balance equation

light leavinga surface

light leavinga surface

reflectedlight

reflectedlight

emittedlight

emittedlight== ++

EasyEasyEasyEasy HardHardHardHard

26262626r’r’

r’’r’’

u’u’uu

Classical formulationClassical formulationClassical formulationClassical formulation

Balance equation in terms of Balance equation in terms of radianceradiance [Polyak, 1960][Polyak, 1960]Balance equation in terms of Balance equation in terms of radianceradiance [Polyak, 1960][Polyak, 1960]

solid anglesolid anglesolid anglesolid anglesource termsource termsource termsource term

ff ((rr'' ,,uu)) ff00 ((rr'' ,,uu)) kk((rr'' ;;uu'' uu)) ff ((rr'' '' ,,uu'' )) coscos dd ((uu'' ))

measure on spheremeasure on spheremeasure on spheremeasure on sphere

27272727

Classical formulationClassical formulationClassical formulationClassical formulation

Important features of the classical formulation:Important features of the classical formulation:Important features of the classical formulation:Important features of the classical formulation:

ff ((rr'' ,,uu)) ff00 ((rr'' ,,uu)) kk((rr'' ;;uu'' uu)) ff ((rr'' '' ,,uu'' )) coscos dd ((uu'' ))

new measurenew measurenew measurenew measure

implicit functionimplicit functionimplicit functionimplicit function

r’r’

r’’r’’

u’u’uuThe point The point rr’’ ’’ depends on thedepends on the

point point rr’’ and the direction and the direction uu’’..The point The point rr’’ ’’ depends on thedepends on thepoint point rr’’ and the direction and the direction uu’’..

only part of the domainonly part of the domainonly part of the domainonly part of the domain

28282828

Two linear operatorsTwo linear operatorsTwo linear operatorsTwo linear operators

f (r,u’)f (r,u’) dd ((uu’)’)k(r;u’ u)k(r;u’ u)( K f ) (r,u) ( K f ) (r,u)

( G f ) (r,u) f (r’,u) ( G f ) (r,u) f (r’,u)

““cosine weighted”cosine weighted”measuremeasure

““cosine weighted”cosine weighted”measuremeasure

implicit functionimplicit functionimplicit functionimplicit function

r’r’

r’’r’’

u’u’uu

29292929

Linear operatorsLinear operatorsfor global illuminationfor global illumination

Linear operatorsLinear operatorsfor global illuminationfor global illumination

Field RadianceField RadianceOperatorOperator

Field RadianceField RadianceOperatorOperator

GGGG KKKKLocal ReflectionLocal Reflection

OperatorOperatorLocal ReflectionLocal Reflection

OperatorOperator

surface radiancesurface radiancesurface radiancesurface radiance field radiancefield radiancefield radiancefield radiance surface radiancesurface radiancesurface radiancesurface radiance

30303030

Another way to writeAnother way to writethe rendering equationthe rendering equationAnother way to writeAnother way to write

the rendering equationthe rendering equation

ff = = ss + + KG KG f f ff = = ss + + KG KG f f

Local ReflectionLocal ReflectionOperatorOperator

Local ReflectionLocal ReflectionOperatorOperator

SourceSourceSourceSourceRadianceRadianceRadianceRadiance

““Global”Global”OperatorOperator““Global”Global”OperatorOperator

31313131

Operator normsOperator normsOperator normsOperator norms

|| K ||p < 1|| K ||p < 1

|| G ||p = 1|| G ||p = 1

1) First law of thermodynamics1) First law of thermodynamics1) First law of thermodynamics1) First law of thermodynamics

2) Second law of thermodynamics2) Second law of thermodynamics2) Second law of thermodynamics2) Second law of thermodynamics

3) Constancy of radiance along rays3) Constancy of radiance along rays3) Constancy of radiance along rays3) Constancy of radiance along rays

32323232

IrradianceIrradianceIrradianceIrradiance

dAdA

weighted integralover solid angle

weighted integralover solid angle

wattswatts

m2m2

33333333

A vector form of irradianceA vector form of irradianceA vector form of irradianceA vector form of irradiance

Integrate vectorsover solid angle

Integrate vectorsover solid angle

vector irradianceor light vector

vector irradianceor light vector

rr

34343434

Lambert’s formula for irradianceLambert’s formula for irradianceLambert’s formula for irradianceLambert’s formula for irradiance

ii

ii

Vector IrradianceVector Irradiance

rr MM

22 ii ii

rr

polygonalpolygonalLambertianLambertianluminaireluminaire

polygonalpolygonalLambertianLambertianluminaireluminaire

PP

35353535

Ideal diffuse reflectionIdeal diffuse reflectionIdeal diffuse reflectionIdeal diffuse reflection

Compute using Lambert’s formulaCompute using Lambert’s formulaCompute using Lambert’s formulaCompute using Lambert’s formula

36363636

Ideal diffuse reflectionIdeal diffuse reflectionIdeal diffuse reflectionIdeal diffuse reflection

Boundaryintegral

Boundaryintegral

EyeEye

37373737

Ideal specular reflectionIdeal specular reflectionIdeal specular reflectionIdeal specular reflection

Compute using ray tracingCompute using ray tracingCompute using ray tracingCompute using ray tracing

38383838

Ideal specular reflectionIdeal specular reflectionIdeal specular reflectionIdeal specular reflection

EyeEye

39393939

Glossy reflectionGlossy reflectionGlossy reflectionGlossy reflection

Use extended Lambert’s formula Use extended Lambert’s formula Use extended Lambert’s formula Use extended Lambert’s formula

40404040

Glossy reflectionGlossy reflectionGlossy reflectionGlossy reflection

NumericalquadratureNumericalquadratureEyeEye

41414141

Glossy reflectionGlossy reflectionGlossy reflectionGlossy reflection

Monte CarloMonte Carlo

EyeEye

42424242

Boundary integral for glossy reflectionBoundary integral for glossy reflectionBoundary integral for glossy reflectionBoundary integral for glossy reflection

boundaryintegral

boundaryintegral

EyeEye

43434343

Applications of directional scatteringApplications of directional scatteringApplications of directional scatteringApplications of directional scattering

glossyglossytransmissiontransmission

glossyglossytransmissiontransmission

glossyglossyreflectionreflectionglossyglossy

reflectionreflection

directionaldirectionalemissionemission

directionaldirectionalemissionemission

luminaireluminaireluminaireluminaire

44444444

A range of glossy reflectionsA range of glossy reflectionsA range of glossy reflectionsA range of glossy reflections

10th-ordermoment

10th-ordermoment

45th-ordermoment

45th-ordermoment

400th-ordermoment

400th-ordermoment

45454545

Comparison with Monte CarloComparison with Monte CarloComparison with Monte CarloComparison with Monte Carlo

Region used forcomparison

Region used forcomparison

order 65order 65 order 300order 300 order 1000order 1000

46464646

Comparison with Monte CarloComparison with Monte CarloComparison with Monte CarloComparison with Monte Carlo

order 65order 65 order 300order 300 order 1000order 1000

47474747

Monte Carlo integrationMonte Carlo integrationMonte Carlo integrationMonte Carlo integration

estimateestimateirradianceirradianceestimateestimate

irradianceirradiance

luminaireluminaireluminaireluminaire

blockerblockerblockerblocker

48484848

Advantages of Monte CarloAdvantages of Monte CarloAdvantages of Monte CarloAdvantages of Monte Carlo

Arbitrarily complex environmentsArbitrarily complex environmentsArbitrary reflectance functionsArbitrary reflectance functionsSmall memory requirementsSmall memory requirementsEasily to distributeEasily to distributeRelatively easy to implementRelatively easy to implement

Arbitrarily complex environmentsArbitrarily complex environmentsArbitrary reflectance functionsArbitrary reflectance functionsSmall memory requirementsSmall memory requirementsEasily to distributeEasily to distributeRelatively easy to implementRelatively easy to implement

49494949

Monte Carlo sampling methods Monte Carlo sampling methods Monte Carlo sampling methods Monte Carlo sampling methods

HemisphereHemisphereHemisphereHemisphere PolygonPolygonPolygonPolygon PhongPhongdistributiondistribution

PhongPhongdistributiondistribution

50505050

Light-ray tracingLight-ray tracingLight-ray tracingLight-ray tracing

luminaireluminaire

Rays represent photons that deposit energy on surfaces.Rays represent photons that deposit energy on surfaces.No inverse-square law here!No inverse-square law here!Rays represent photons that deposit energy on surfaces.Rays represent photons that deposit energy on surfaces.No inverse-square law here!No inverse-square law here!

51515151

Path tracingPath tracingPath tracingPath tracing

At each scattering event, estimate indirect irradianceAt each scattering event, estimate indirect irradiancewith a single ray; continue recursively.with a single ray; continue recursively.At each scattering event, estimate indirect irradianceAt each scattering event, estimate indirect irradiancewith a single ray; continue recursively.with a single ray; continue recursively.

luminaireluminaireeyeeye

52525252

Bidirectional path tracingBidirectional path tracingBidirectional path tracingBidirectional path tracing

Simultaneously follow paths from the light and theSimultaneously follow paths from the light and theeye, looking for points that can “see” each other.eye, looking for points that can “see” each other.Simultaneously follow paths from the light and theSimultaneously follow paths from the light and theeye, looking for points that can “see” each other.eye, looking for points that can “see” each other.

luminaireluminaireeyeeye

53535353

Metropolis path tracingMetropolis path tracingMetropolis path tracingMetropolis path tracing

Start with a path from eye to luminaire, then integrateStart with a path from eye to luminaire, then integrateover others by perturbing to nearby paths.over others by perturbing to nearby paths.Start with a path from eye to luminaire, then integrateStart with a path from eye to luminaire, then integrateover others by perturbing to nearby paths.over others by perturbing to nearby paths.

luminaireluminaireeyeeye

54545454

A Taxonomy of ErrorsA Taxonomy of ErrorsA Taxonomy of ErrorsA Taxonomy of Errors

Discrete EquationDiscrete EquationDiscrete EquationDiscrete Equation

ApproximationApproximationApproximationApproximation

Exact EquationExact EquationExact EquationExact Equation

Perturbed EquationPerturbed EquationPerturbed EquationPerturbed Equation PerturbationsPerturbationsPerturbationsPerturbations

DiscretizationDiscretizationDiscretizationDiscretization

ComputationComputationComputationComputation

Radiance Function SpaceRadiance Function SpaceRadiance Function SpaceRadiance Function Space

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Features of surface illuminationFeatures of surface illuminationFeatures of surface illuminationFeatures of surface illumination

luminaireluminaireluminaireluminaire

blockerblockerblockerblocker

gradientsgradientsgradientsgradients

isolux contoursisolux contoursisolux contoursisolux contours

extremaextremaextremaextrema

56565656

An example of meshingAn example of meshingAn example of meshingAn example of meshing

A simpleenvironment

A simpleenvironment

The underlyingmesh

The underlyingmesh

57575757

Classical balance equationClassical balance equationClassical balance equationClassical balance equation

(x,’) f(x’,’) cos d’(x,’) f(x’,’) cos d’∫∫ f(x,) = s(x,) + f(x,) = s(x,) +

A point on aA point on adistant visible surfacedistant visible surface

A point on aA point on adistant visible surfacedistant visible surface

radianceradianceradianceradiance

58585858

The change is a “pullback”The change is a “pullback”The change is a “pullback”The change is a “pullback”

The 2-form on thesphere is pulled

back to the surface

The 2-form on thesphere is pulled

back to the surface

xx

59595959

Change of variablesChange of variablesChange of variablesChange of variables

d = dAd = dAcos’cos’

r 2r 2

differentialdifferentialsolid anglesolid angledifferentialdifferentialsolid anglesolid angle

differentialdifferentialareaarea

differentialdifferentialareaarea

60606060

Kajiya’s rendering equationKajiya’s rendering equationKajiya’s rendering equationKajiya’s rendering equation

e(x,x’) + (x,x’,x’’) I(x’,x’’) dx’’e(x,x’) + (x,x’,x’’) I(x’,x’’) dx’’∫∫I(x,x’) = g(x,x’)I(x,x’) = g(x,x’)

SS

x, x’, x’’x, x’, x’’ are points on surfacesare points on surfacesare points on surfacesare points on surfaces

I =I = unknown intensity functionunknown intensity functionunknown intensity functionunknown intensity function

61616161

Kajiya’s rendering equationKajiya’s rendering equationKajiya’s rendering equationKajiya’s rendering equation

e(x,x’) + (x,x’,x’’) I(x’,x’’) dx’’e(x,x’) + (x,x’,x’’) I(x’,x’’) dx’’∫∫I(x,x’) = g(x,x’)I(x,x’) = g(x,x’)

transport intensitytransport intensitytransport intensitytransport intensity

geometry termgeometry termgeometry termgeometry term

transporttransportemittanceemittancetransporttransportemittanceemittance

scattering functionscattering functionscattering functionscattering function

62626262

Power from transport intensityPower from transport intensityPower from transport intensityPower from transport intensity

dxdx

Integrate overtwo surfaces

Integrate overtwo surfaces

dx’dx’

sourcesource

receiverreceiver

63636363

Radiance & transport intensityRadiance & transport intensityRadiance & transport intensityRadiance & transport intensity

radianceradiance transport intensitytransport intensity

wattswatts

m2 srm2 sr

wattswatts

m4m4

64646464

Radiance & transport intensityRadiance & transport intensityRadiance & transport intensityRadiance & transport intensity

radianceradiance transport intensitytransport intensity

invariantalong lines

in free space

invariantalong lines

in free space

obeys inversesquare law

obeys inversesquare law

definedeverywhere

definedeverywhere

defined onlyat surfaces

defined onlyat surfaces

65656565

Another way to writeAnother way to writethe rendering equationthe rendering equationAnother way to writeAnother way to write

the rendering equationthe rendering equation

RadianceRadianceRadianceRadiance

SourceSourceSourceSource

TransportTransportOperatorOperator

TransportTransportOperatorOperator

ff = = ss + + M M f f ff = = ss + + M M f f

66666666

The formal “solution”The formal “solution”to the rendering equationto the rendering equation

The formal “solution”The formal “solution”to the rendering equationto the rendering equation

ff = ( I - = ( I - MM ) ) s s ff = ( I - = ( I - MM ) ) s s -1-1

IdentityIdentityoperatoroperatorIdentityIdentityoperatoroperator

67676767

The Neumann seriesThe Neumann seriesThe Neumann seriesThe Neumann series

ff = = ss + + MMs s + + MM22s s ++ ff = = ss + + MMs s + + MM22s s ++ ......

68686868

LLpp-norms for radiance functions-norms for radiance functionsLLpp-norms for radiance functions-norms for radiance functions

ddmm((rr ))| f (r,u) |

p| f (r,u) |

p

s2s2MM

dd ((uu))|| f ||p =|| f ||p = [[ ]]““cosine weighted”cosine weighted”

measuremeasure““cosine weighted”cosine weighted”

measuremeasure

11pp

The collection of all functions withThe collection of all functions withfinite finite LLpp-norm is a Banach space-norm is a Banach space

The collection of all functions withThe collection of all functions withfinite finite LLpp-norm is a Banach space-norm is a Banach space

LLpp mm

69696969

Significance of the Significance of the LLpp-norms-normsSignificance of the Significance of the LLpp-norms-norms

total powertotal power wattswatts

symbolsymbol meaningmeaning unitsunits

11ff

rms radiancerms radiancewattswatts

m srm sr22ff

maximum radiancemaximum radiancewattswatts

m2 srm2 srff

70707070

The The LL11-norm of -norm of KKThe The LL11-norm of -norm of KK

KK11 maxmax

rrmaxmax

uu''kk((rr;;uu'' uu)) dd((uu))

maximal directional-hemisphericalreflectance over all r and u'maximal directional-hemisphericalreflectance over all r and u'

dd

u'u'

rr

71717171

The The LL -norm of -norm of KKThe The LL -norm of -norm of KK

maxmaxrr

maxmaxuu''

kk((rr;;uu '' uu )) dd((uu))

maximal hemispherical-directionalreflectance over all r and u'maximal hemispherical-directionalreflectance over all r and u'

dd

KK

rr

u'u'

72727272

The The LLpp-norms of -norms of KKThe The LLpp-norms of -norms of KK

energy conservationenergy conservationKK11

KK pp

KK11

KK reciprocityreciprocity

KKpp

KK KK11

,,max {max { }} convexityconvexity

73737373

The G operatorThe G operatorThe G operatorThe G operator

Surface radiancefunctionSurface radiancefunction

An enclosure.An enclosure.

Equivalent flowthrough fictitiousboundary

Equivalent flowthrough fictitiousboundary

74747474

Hilbert adjoint operatorsHilbert adjoint operatorsHilbert adjoint operatorsHilbert adjoint operators

K*K* = = KK G*G* = = GG

M M = I = I -- KG KG M* = I M* = I -- GK GK

SinceSinceSinceSince andandandand

it follows thatit follows thatit follows thatit follows that