11 introduction to global illumination overview overview radiometry radiometry the rendering...
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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.
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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
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Photo-realistic renderingPhoto-realistic renderingPhoto-realistic renderingPhoto-realistic rendering
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Types of surface scatteringTypes of surface scatteringTypes of surface scatteringTypes of surface scattering
diffusediffuse directional diffusedirectional diffuse
specularspecular
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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
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
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
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IrradianceIrradianceIrradianceIrradiance
dAdA
weighted integralover solid angle
weighted integralover solid angle
wattswatts
m2m2
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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
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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
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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
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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
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Comparison with Monte CarloComparison with Monte CarloComparison with Monte CarloComparison with Monte Carlo
order 65order 65 order 300order 300 order 1000order 1000
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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