Monte-Carlo Methods In Global Illumination

123 Pages·1.789 MB·English

by  Szirmay-Kalos L.

MONTE-CARLO METHODS
IN GLOBAL ILLUMINATION
Script writtenby
Szirmay-Kalos La´szlo´
inWSof1999/2000
Institute of Computer Graphics
Vienna University of Technology
i
Contents
1 Introduction 1
1.1 Globalpass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Localpass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Tonemapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Globalilluminationproblem 5
2.1 Therenderingequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Measuringtheradiance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Thepotentialequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Measuringthepotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.5 Therenderingproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.5.1 Geometryofthesurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.5.2 Bi-directionalReflectionDistributionFunctions. . . . . . . . . . . . . . . . . . . . . . . 11
2.5.3 Lightsources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5.4 Measuringdevices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.6 Numericalsolutionoftherenderingequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.6.1 Errormeasuresfornumerictechniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.6.2 Propertiesoftherenderingequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.7 Classificationofthesolutiontechniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Opticalmaterialmodels 18
3.1 Diffusereflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Ideal,mirror-likereflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Idealrefraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 Non-ideal,specularreflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4.1 Phongreflectionmodelanditsmodifications . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4.2 Cook-Torrancemodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4 Solutionstrategiesfortheglobalilluminationproblem 28
4.1 Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2 Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2.1 Expansionoftherenderingequation:gatheringwalks. . . . . . . . . . . . . . . . . . . . 28
4.2.2 Expansionofthepotentialequation:shootingwalks . . . . . . . . . . . . . . . . . . . . 30
4.2.3 Meritsanddisadvantagesofexpansionmethods . . . . . . . . . . . . . . . . . . . . . . . 31
4.3 Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.3.1 Analysisoftheiteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.4 Analyticalsolutionoftherenderingequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.4.1 Sceneswithconstantradiance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.4.2 Sceneswithconstantreflectedradiance . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5 Finite-elementmethodsfortheGlobalIlluminationProblem 35
5.1 Galerkin’smethod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.2 Pointcollocationmethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.3 Finiteelementmethodsforthediffuseglobalilluminationproblem . . . . . . . . . . . . . . . . . 37
5.3.1 Geometricmethodsforformfactorcomputation . . . . . . . . . . . . . . . . . . . . . . 39
ii
CONTENTS iii
6 Numericalquadratureforhighdimensionalintegrals 41
6.1 Monte-Carloquadrature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
6.2 Quasi-MonteCarloquadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
6.2.1 ErrorAnalysisforintegrandsoffinitevariation:Koksma-HlawkaInequality. . . . . . . . 43
6.2.2 Generationofthesamplepoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.2.3 Generationoflow-discrepancysequences . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6.3 Importancesampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.3.1 Generationofarandomvariablewithaprescribedprobabilitydensity . . . . . . . . . . . 49
6.3.2 Importancesamplinginquasi-MonteCarlointegration . . . . . . . . . . . . . . . . . . . 50
6.3.3 Metropolissampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.3.4 ApplicationoftheVEGASalgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
7 Randomwalksolutionoftheglobalilluminationproblem 53
7.1 WhyshouldweuseMonte-Carloexpansionmethods?. . . . . . . . . . . . . . . . . . . . . . . . 53
7.2 Quasi-MonteCarloquadraturefortherenderingequation . . . . . . . . . . . . . . . . . . . . . . 54
7.2.1 Integratingfunctionsofunboundedvariation . . . . . . . . . . . . . . . . . . . . . . . . 54
7.3 Importancesamplingfortherenderingequation . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
7.3.1 BRDFsampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
7.3.2 Lightsourcesampling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
7.3.3 Samplingthelightsourcesingatheringrandomwalks . . . . . . . . . . . . . . . . . . . . 60
7.3.4 Importancesamplingincoloredscenes . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
7.3.5 Multipleimportancesampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
7.4 Handlinginfinite-dimensionalintegrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
7.4.1 Russianroulette. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
7.4.2 Russianrouletteinquasi-MonteCarloquadrature . . . . . . . . . . . . . . . . . . . . . . 63
8 Reviewofrandomwalkalgorithms 66
8.1 Gathering-typerandomwalkalgorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
8.1.1 Ray-casting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
8.1.2 Visibilityray-tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
8.1.3 Distributedray-tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
8.1.4 Path-tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
8.2 Shooting-typewalksmethods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
8.2.1 Photontracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
8.2.2 Light-tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
8.2.3 Randomwalksfortheradiositysetting . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
8.3 Bi-directionalrandomwalkalgorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
8.3.1 Bi-directionalpath-tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
8.3.2 Metropolislighttransport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
8.3.3 Photon-map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
8.3.4 Instantradiosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
8.4 Globalmethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
8.4.1 Multi-pathmethodusingglobalrandomlines . . . . . . . . . . . . . . . . . . . . . . . . 82
8.4.2 Globalray-bundletracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
8.4.3 Preprocessingthepointlightsources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
9 Iterationsolutionoftheglobalilluminationproblem 85
9.1 WhyshouldweuseMonte-Carloiterationmethods?. . . . . . . . . . . . . . . . . . . . . . . . . 86
9.2 Formaldefinitionofstochasticiteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
9.2.1 Otheraveragingtechniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
9.2.2 Canweusequasi-MonteCarlotechniquesiniteration? . . . . . . . . . . . . . . . . . . . 88
10 Reviewofstochasticiterationalgorithms 90
10.1 Stochasticiterationforthediffuseradiosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
10.1.1 Stochasticradiosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
10.1.2 Transilluminationradiosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
10.1.3 Stochasticray-radiosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
10.2 Definitionoftherandomtransportoperatorforthenon-diffusefinite-elementcase . . . . . . . . . 92
10.2.1 Singleraybasedtransportoperator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
10.2.2 Stochasticiterationusingray-bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
CONTENTS iv
11 Implementationofthepath-tracingalgorithm 97
11.1 Vectormodule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
11.1.1 Point3Dclass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
11.1.2 Transformationclass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
11.2 Containermodule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
11.3 Colormodule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
11.4 Materialmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
11.4.1 Diffusematerialclass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
11.4.2 Idealmirrorclass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
11.4.3 Idealrefractingmaterialclass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
11.4.4 Specularmaterialclass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
11.4.5 Generalmaterialclass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
11.5 Lightmodule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
11.5.1 Emitterclass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
11.5.2 Positionallightclass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
11.6 Modelmodule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
11.6.1 Primitiveclass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
11.6.2 Objectclass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
11.6.3 Virtualworldclass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
11.7 Cameramodule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
11.8 Scenemodule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
11.8.1 Sceneclass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
11.9 Dynamicmodelofpathtracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
11.9.1 Findingthevisibleprimitive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
11.9.2 Detectingthevisiblelightsources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
11.9.3 Directlightsourcecomputation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
11.9.4 Pathtracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
11.9.5 Renderingcompleteimages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
BIBLIOGRAPHY 111
SUBJECTINDEX 116
Chapter 1
Introduction
Theultimateobjectiveofimagesynthesisorrenderingisto providetheuser withtheillusionofwatchingreal
objects on the computerscreen (figure1.1). Theimage is generatedfrom an internalmodelwhich is called the
virtualworld. Toprovidetheillusionofwatchingtherealworld,thecolorsensationofanobserverlookingatthe
artificialimagegeneratedbythegraphicssystemmustbeapproximatelyequivalenttothecolorperceptionwhich
would be obtained in the real world. The color perceptionof humansdependson the light power reaching the
eyefromagivendirectionandontheoperationoftheeye. Thepower,inturn,isdeterminedfromtheradiance
ofthevisiblepoints. Theradiancedependsontheshapeandopticalpropertiesoftheobjectsandontheintensity
ofthelightsources. Inordertomodelthiscomplexphenomenon,boththephysical-mathematicalstructureofthe
light-objectinteractionandtheoperationoftheeyemustbeunderstood.
rendering
observer of the
computer screen
monitor virtual world
measuring window
R device
G
Tone
B
mapping
power power
l l radiance
l
color perception
in the nerve cells
power
radiance
l
observer of the l
real world real world
Figure1.1:Tasksofrendering
Theimagesynthesisusesaninternalmodelconsistingofthegeometryofthevirtualworld,opticalmaterial
propertiesandthedescriptionofthelightinginthescene(figure1.2). Fromthese,applyingthelawsofphysics
(e.g. Maxwellequations)the realworld opticalphenomenacan besimulatedto find the lightdistributionin the
scene.Thisstepiscalledtheview-independentsteportheglobalpassofrendering.Thenameasurementdevice,
calledtheeyeorcamera,isplacedintothesceneandthelightdistributionismeasuredfromagivenlocationand
orientation.Thisiscalledtheview-dependentsteporthelocalpass. Notethatnotallrenderingalgorithmsmakea
cleardistinctionbetweenthedeterminationoftheview-independentlightdistributionandthemeasurementofthis
1
1.1. GLOBALPASS 2
distributionbythecamera,butsimultaneouslycomputethelightdistributionanditseffectonthecamera.
Renderingresultsinarepresentationoftheperceivedimage,whichisusuallythecollectionofpixelcolorsor
somediscretesamplingoftheradiancefunction. Theexactsimulationofthelightperceivedbytheeyeisimpos-
sible,sinceitwouldrequireendlesscomputationalprocess.Ontheotherhand,itisnotevenworthdoingsincethe
possibledistributionswhichcanbeproducedbycomputerscreensarelimitedincontrasttotheinfinitevarietyof
realworldlightdistributions. Consequently,colorperceptionisapproximatedinsteadofhavingacompletelyac-
curatesimulation.Theaccuracyofthisapproximationisdeterminedbytheabilityoftheeyetomakeadistinction
betweentwolightdistributions.
Computerscreenscanproducecontrollableelectromagneticwaves,orcoloredlight,mixedfromthreeseparate
wavelengthsfortheirobservers.Thusinthefinalstepofimagesynthesistonemappingisneededwhichconverts
thecomputedcolororradianceinformationtothe , , intensitiesthatcanbeproducedbythecolormonitor.
R G B
geometry of
the virtual world
global image
material rendering radiance of calculation radiance of tone R,G,B of pixels
properties (global pass) surface points (local pass) pixels mapping
lighting
camera
Figure1.2:Dataflowofrendering
1.1 Global pass
Theglobalpassdeterminesthelightreflectedoffthesurfacepointsatdifferentdirections.Sincelightisanelectro-
magneticwave,lightdistributioninapointandatagivendirectioncanberepresentedbyawavelength-dependent
function [A´br97, Ko´n85]. Rendering algorithms usually evaluate this functions at a few representative wave-
lengths.Onagivenwavelengththeintensityofthelightisdescribedbytheradiance.Inscenesnotincorporating
participatingmediaitisenoughtocalculatetheradianceatsurfacepoints. Theradiancereflectedoffasurface
pointisaffectedbytheemissionofthispoint(lighting),theilluminationprovidedbyothersurfacepointsandthe
opticalpropertiesofthematerialatthispoint(materialproperties).Formallythisdependenceischaracterizedby
aFredholmtypeintegralequationofthesecondkind,whichiscalledtherenderingequation.Frommathematical
pointofview,globalpassisthesolutionofthisintegralequationfortherepresentativewavelengths.
1.2 Local pass
Thelocalpassmeansthemeasurementoftheglobalradiancefunctionbyacamera. Acameraisacollectionof
lightmeasuringdeviceswhichusuallycorrespondtopixelsintheimage. Acertainmeasuringdeviceischaracter-
izedbyasensitivityfunctionthatdescribeswhichpointsanddirectionsmayaffectthedevice.
1.3 Tone mapping
Lightisanelectromagneticwave,anditscolorisdeterminedbytheeye’sperceptionofitsspectralenergydistri-
bution. Due to its internalstructure, theeyeis averypoorspectrometersinceit actuallysamplesand integrates
theenergyinthreeoverlappingfrequencyrangesbythreetypesofphotopigmentsaccordingtoawidelyaccepted
(butalsoargued)model.Asaconsequenceofthis,anycolorperceptioncanberepresentedbythreescalars(called
tristimulusvalues)insteadofcompletefunctions[A´br97,Ko´n85,Nem90].
Aconvenientwaytodefinetheaxesofacoordinatesysteminthethree-dimensionalspaceofcolorsensations
istoselectthreewavelengthswhereonetypeofphotopigmentsissignificantlymoresensitivethantheothertwo
[SK99c]. This method has been devised by Grassmann, who also specified a criterion for separating the three
representativewavelengths. Grassmann laws state that the representativewavelengths should be selected such
thatnooneofthemcanbe matchedbythemixtureof theothertwo intermsofcolorsensation(thiscriterionis
1.3. TONEMAPPING 3
similartotheconceptoflinearindependence.)Anappropriatecollectionoftherepresentativewavelengthsis:
nm nm nm (1.1)
(cid:21)red =645 ; (cid:21)green =526 ; (cid:21)blue =444 :
Nowletussupposethatmonochromaticlightofwavelength isperceivedbytheeye.Theequivalentportions
of red, greenandbluelight, or ( , , ) tristimulusvalues, can(cid:21)begeneratedby threecolormatchingfunctions
( , and )whicharebrasgedbonphysiologicalmeasurements.Notethenegativesectionof (andtoa
lers(s(cid:21)e)xgte(n(cid:21)t)in b((cid:21)))infigure1.3.Itmeansthatnotallcolorscanberepresentedbypositive( , , )vra(l(cid:21)u)es.
g((cid:21)) r g b
R=645nm, G=526nm, B=444nm matching functions
3.5
r(lambda)
g(lambda)
b(lambda)
3
2.5
2
b
g, 1.5
r,
1
0.5
0
-0.5
400 450 500 550 600 650 700
lambda[nm]
Figure1.3:Mean10-degcolormatchingfunctionsofStilesandBurch: , , .
r((cid:21)) g((cid:21)) b((cid:21))
Iftheperceivedcolorisnotmonochromatic,butisdescribedbyan distribution,thetristimuluscoordinates
are computed using the assumption that the sensation is produced bLy(a(cid:21)n)additive mixture of the perceptions of
elementalmonochromaticcomponents:
(1.2)
r = L((cid:21)) r((cid:21))d(cid:21); g= L((cid:21)) g((cid:21))d(cid:21); b= L((cid:21)) b((cid:21))d(cid:21):
(cid:1) (cid:1) (cid:1)
Z(cid:21) Z(cid:21) Z(cid:21)
For computer generated images, the color sensation of an observer watching a virtual world on the screen
must be approximately equivalent to the color sensation obtained in the real world. Since color sensations are
representedby ( , , ), it means that the tristimulus values should be identical. If two energydistributionsare
associatedwithtrhegsambetristimuluscoordinates,theyproducethesamecolorsensation,andarecalledmetamers.
Incomputermonitorsandintelevisionscreensthreephosphorlayerscanbestimulatedtoproducered,green
and blue light. The objective, then, is to find the necessary stimulus to produce a metamer of the real energy
distribution of the light [Sch96, BS95]. This stimulus can be controlled by the ( , , ) values of the actual
pixel.The( , , )matchingfunctionsoffigure1.3dependonthewavelengthoftheRseGlectBedprimaries,whichare
notnecessarrilygidbenticaltothewavelengthsonwhichourmonitorcanemitlight. Thisrequirestheconversionof
tristimulusvaluesbyalineartransformation.
Thecalculationofpixel valuesthusconsistsofthefollowingsteps. Firstthespectrumassociatedwith
thepixeliscomputed. ThenRth;eGs;pBectrumismatchedbythreestandardcolormatchingfunctionsdefinedbythree
primaries. Finally, the standard color coordinates are transformed to the monitor color coordinates taking into
accountthe monitor properties. In practice, the standard color system is usually the CIE XYZ system [WS82]
whichusesthreehypotheticalprimariestoallowthedefinitionofanycolorbypositiveweights.
The linear transformation that converts from the XYZ system to the monitor RGB system can be obtained
fromthe coordinatesoftheemissionsofthethreephosphorsandofthewhitepointofthemonitor. Fora
monitorwXit;hYs;tZandardNTSCphosphorsandwhitepoint,thefollowingtransformationcanbeused[Gla95]:
R 1:967 0:548 0:297 X (1.3)
(cid:0) (cid:0)
G = 0:955 1:938 0:027 Y :
2 3 2 (cid:0) (cid:0) 3(cid:1)2 3
B 0:064 0:130 0:982 Z
(cid:0)
4 5 4 5 4 5
1.3. TONEMAPPING 4
X,Y,Z matching functions
3.5
X(lambda)
Y(lambda)
Z(lambda)
3
2.5
2
Z
Y, 1.5
X,
1
0.5
0
-0.5
400 450 500 550 600 650 700
lambda[nm]
Figure1.4:Mean10-degcolor matchingfunctionsofStilesandBurch: , ,
XYZ X((cid:21)) Y((cid:21)) Z((cid:21))
Thewholecomputationofthe( , , )valuesinorderforthemonitorcolortobeametamerofthecalculated
spectrum is called tone mapping.RThGe (B, , ) values are positive numbersusually in the rangeof [0...255]
if8bitsareavailabletorepresentthem. URnfGortuBnately,notallcolorscanbereproducedonthecomputerscreen,
becauseofthenegativesectionsofthe colormatchingfunctionsanddueto thefactthatthenumberofavailable
intensitylevelsisusuallymuchlessthancanbeperceivedintherealworld.Thustonemappingisalsoresponsible
foroptimallyselectingfromtheavailableintensitylevelsforcolorreproduction.Themappingfromthecomputed
levels to the available ones can be either linear or logarithmic. The latter takes advantage of the logarithmic
characteristicsofthehumanperceptionsystem[PP98].
Chapter 2
Global illumination problem
Inthischapterthemathematicalmodelofthelight-surfaceinteractionispresented.Thismathematicalmodelisan
integralequation,whichhastobesolvedtoobtainphysicallyaccurateimages.
2.1 The rendering equation
Hereinafter, monochromatic light of a representative wavelength will be assumed, since the complete color
calculation can be broken down to these representative wavelengt(cid:21)hs. The parameters of the equations usually
dependonthewavelength,butfornotationalsimplicity,wedonotalwaysincludethe variableinthem.
Inthissection,webrieflyreviewthemeasuresofthelighttransportandthemathem(cid:21)aticalformulationthatcan
computethem.
z w z w dq
q
sin q df
dq
y y
f
df
x x
Figure2.1:Definitionofdirectionsinasphericalcoordinatesystem(left)andcalculationofdifferentialsolidangles(right)
Thedirectionalpropertyofthelightenergyemissionisdescribedinaso-calledilluminationsphere orin
illuminationhemisphere whichcontainthosesolidanglestowherethesurfacepointcanemitenerg(cid:10)y. The
surfaceoftransparentmate(cid:10)riHalscanemitinanydirectionsofasphere,whilethesurfaceofopaquematerialscan
transferenergyonlytothehemispherethatis“above”thesurface.
Settingupasphericalcoordinatesystem(figure2.1),adirection canbedefinedbytwoangles ,where
is theanglebetweenthegivendirectionandthe -axis, and is the!anglebetweentheprojectiono(cid:18)f;t(cid:30)hegiven
(cid:18)directionontothe planeandthe -axis. z (cid:30)
Setsofdirectioxn;syaredefinedbysxolidangles. Bydefinition,asolidangleisaconeorapyramid,withitssize
determinedbyitssubtendedareaofaunitspherecenteredaroundtheapex(figure2.2).Adifferential(infinitesimal)
solidanglecanalsobegivenbyavector ,wherethevectorequalstoadirectionofthedifferentialset.
Adifferentialsolidanglecanalsobed~!expressedbythe angles. Supposethat ismodifiedby and is
by . Duringthisthedirectionalvectorscansadifferential(cid:18)r;e(cid:30)ctanglehaving verti(cid:18)caland dh(cid:18)orizo(cid:30)ntal
sizeds(cid:30)(rightoffigure2.1),thusthesizeofthesolidangleis d(cid:18) sin(cid:18) d(cid:30)
(cid:1)
(2.1)
d! =sin(cid:18) d(cid:30)d(cid:18):
(cid:1)
The solid angle, in which a differential surface can be seen from point , is the projected (visible) area
perthesquareofthe distanceofthesurfaced(Afigure2.2). If theanglebetweenthp~e surfacenormalof andthe
dA
5