Efficient Importance Sampling Techniques for the Photon Map
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Efficient Importance Sampling Techniques for the Photon Map Alexander Keller and Ingo Wald † Abstract In global illumination computations the photon map is a powerful tool for approximating the irradiance, which is stored independent from scene geometry. By presenting a new algorithm, which uses novel importance sampling techniques, we improve the memory footprint of the pho- ton map, simplify the caustic generation, and allow for a much faster sampling of direct illumination in complicated models as they arise in a production environment. 1 Introduction As introduced in [Jen96a, JC95b] the photon map algo- rithm generates an approximation of irradiance by storing information about the collisions of a random walk simula- tion of radiant light transport. Its basic implementation is strikingly simple and storing the power independent from scene geometry is a big advantage as compared to finite element approaches. In the sequel we present three im- portance sampling techniques, which are combined in one very simple algorithm, which improves and is complemen- tary to the current photon map techniques: 1. Importance driven photon deposition: Originally the photon map is generated using the von Neumann- Ulam scheme, i.e. a pure forward simulation of the particle nature of light [PM92, PM95]. In [PP98] an importance driven method for generating photon maps has been introduced, which directed the photon paths into the areas of high visual importance. How- ever the core problem remained: Photons are stored all over the scene, even in regions, in which the im- portance is almost zero. In addition the energy of the photons varies due to the weighting by the emission and scattering probabilities, increasing the variance of the estimate. Instead of controlling emission and scattering as in [PP98], we control the deposition of the photons, re- sulting in a more precise photon map and reduced memory requirements. 2. Automated generation of caustics: Caustics, which are generally speaking the lighting effects that arise from converging light paths on a diffuse surface com- ing from a specular surface, require a photon map with a much higher resolution than the global photon map used for ambient lighting. In [Jen96b, JC95b] objects generating caustics were stored in projection University of Kaiserslautern, [email protected] † University of Saarbr¨ ucken, [email protected] maps in order to shoot an increased number of pho- tons into the direction of the caustics generators. This however only captures caustics caused by direct light- ing, whereas caustics from indirect illumination have to be added by manual user intervention. An example of such a caustic, which also cannot be rendered by a gathering step, is a very bright diffuse patch, which is very close to a light source, that causes a caustic through a singular surface, as found in situations of strong indirect illumination. By a simple extention of our approach of importance driven photon deposition, we enable the automatic generation of all visually important caustic effects. 3. Faster direct illumination computations: Since the direct visualization of the photon map is often too blurry, the calculation of direct illumination is sepa- rated, and the indirect illumination is computed using a gathering step [Jen96a]. This has the advantage that the main contribution to an image is sampled very ac- curately, while only the minor effect of indirect il- lumination is indirectly estimated using the photon map. Only if the ray to be shaded is of very small im- pact on the image, a direct estimate from the global photon map is used. The disadvantage of separating the direct illumination, however, is that the sampling effort depends on the number of light sources, which can be large. We present an importance sampling scheme, which enables a more accurate and faster computation of di- rect illumination in settings with a large number of occluded light sources. 2 Importance driven Photon Map Generation The basic idea of our approach is to control the deposition of photons by visual importance. We store photons only in areas of high visual impact using a representation of the visual importance by an importon map as introduced in [PP98]. It such is possible to obtain the same quality as in previous approaches, but at reduced memory cost and increased query speed. 2.1 Probabilistic Photon Deposition The photon map can be seen as a snapshot of a ran- dom walk simulation of radiance transport from the light sources. A convenient algorithm for the collision esti- mate with Russian Roulette path termination including de-
Transcript of Efficient Importance Sampling Techniques for the Photon Map
Efficient Importance SamplingTechniquesfor the PhotonMap
AlexanderKeller�
andIngoWald†
Abstract
In global illumination computationsthe photonmapis apowerful tool for approximatingthe irradiance,which isstoredindependentfrom scenegeometry. By presentinga new algorithm,which usesnovel importancesamplingtechniques,we improve thememoryfootprint of thepho-ton map,simplify the causticgeneration,andallow for amuchfastersamplingof directilluminationin complicatedmodelsasthey arisein a productionenvironment.
1 Intr oduction
As introducedin [Jen96a, JC95b] the photonmap algo-rithm generatesanapproximationof irradianceby storinginformationaboutthecollisionsof a randomwalk simula-tion of radiantlight transport.Its basicimplementationisstrikingly simpleandstoringthepower independentfromscenegeometryis a big advantageascomparedto finiteelementapproaches.In the sequelwe presentthreeim-portancesamplingtechniques,which arecombinedin oneverysimplealgorithm,whichimprovesandis complemen-tary to thecurrentphotonmaptechniques:
1. Importance dri ven photon deposition: Originallythephotonmapis generatedusingthevonNeumann-Ulam scheme,i.e. a pureforward simulationof theparticle natureof light [PM92, PM95]. In [PP98]an importancedriven methodfor generatingphotonmapshasbeenintroduced,whichdirectedthephotonpathsinto theareasof high visual importance.How-ever the coreproblemremained:Photonsarestoredall over thescene,even in regions,in which the im-portanceis almostzero.In additiontheenergy of thephotonsvariesdueto the weightingby theemissionand scatteringprobabilities,increasingthe varianceof theestimate.Insteadof controlling emissionand scatteringas in[PP98], we control thedepositionof thephotons,re-sulting in a more precisephotonmap and reducedmemoryrequirements.
2. Automatedgenerationof caustics:Caustics,whicharegenerallyspeakingthe lighting effects that arisefrom converginglight pathsonadiffusesurfacecom-ing from a specularsurface, requirea photonmapwith a muchhigherresolutionthantheglobalphotonmapusedfor ambientlighting. In [Jen96b, JC95b]objectsgeneratingcausticswerestoredin projection�
Universityof Kaiserslautern,[email protected]†Universityof Saarbrucken,[email protected]
mapsin orderto shootan increasednumberof pho-tonsinto thedirectionof thecausticsgenerators.Thishoweveronlycapturescausticscausedby directlight-ing, whereascausticsfrom indirectilluminationhaveto beaddedby manualuserintervention.An exampleof sucha caustic,which alsocannotbe renderedbya gatheringstep,is a verybrightdiffusepatch,whichis very closeto a light source,that causesa causticthrougha singularsurface,as found in situationsofstrongindirectillumination.By a simpleextentionof ourapproachof importancedriven photon deposition,we enablethe automaticgenerationof all visually importantcausticeffects.
3. Fasterdir ect illumination computations: Sincethedirect visualizationof the photonmap is often tooblurry, the calculationof direct illumination is sepa-rated,andtheindirectilluminationis computedusingagatheringstep[Jen96a]. Thishastheadvantagethatthemaincontributionto animageis sampledveryac-curately, while only the minor effect of indirect il-lumination is indirectly estimatedusing the photonmap.Only if theray to beshadedis of verysmallim-pacton the image,a direct estimatefrom the globalphotonmapis used.Thedisadvantageof separatingthedirect illumination,however, is that thesamplingeffort dependson thenumberof light sources,whichcanbelarge.We presentan importancesamplingscheme,whichenablesamoreaccurateandfastercomputationof di-rect illumination in settingswith a large numberofoccludedlight sources.
2 Importance dri ven Photon MapGeneration
Thebasicideaof ourapproachis to controlthedepositionof photonsby visual importance.We storephotonsonlyin areasof highvisualimpactusingarepresentationof thevisual importanceby an importon map as introducedin[PP98]. It suchis possibleto obtainthe samequality asin previous approaches,but at reducedmemorycostandincreasedqueryspeed.
2.1 Probabilistic PhotonDeposition
The photon map can be seenas a snapshotof a ran-dom walk simulationof radiancetransportfrom the lightsources. A convenientalgorithm for the collision esti-matewith RussianRoulettepathterminationincludingde-
Figure 1: Photonmap in a sceneof 10x10 rooms(eachwith onelight source)seenfrom the top. (top) Standardphotonmapwithout importance,(middle) distribution ofvisualimportance,i.e. theimportons,and(bottom)photonmapgeneratedusingvisual importancecontrolleddeposi-tion.
terministiclow discrepancy samplinghasbeenderived in[KMS94]. The simulationyields a cloud of particles
�,
whereeachparticlek � � consistsof its pointof incidencexk, its directionof incidence�ωk andits power Φk. UsingtheDirac-δ-distributiontheparticlek is localizedby
δk�x ���ω � : δ
�x xk ��� δ � �ω �ωk ���
We now modify thediscretedensityapproximationof thepower Φ by introducinganarbitraryacceptanceprobabil-ity pk ��� 0 � 1� :
Φ�x � �ω ��� ∑
k ��� δk�x � �ω � Φk ∑
k ��� δk�x � �ω � Φk
pk
pk ∑k ��� δk
�x ���ω � Φk
pk
� 1
0χ � 0 � pk � � t � dt �
where χ is the characteristicfunction of the interval� 0 � pk � . This transformationverymuchresemblestheRus-sianRouletteabsorptionmechanism[KMS94, AK90], andsimilarly the integral now can be evaluatedby a one-sampleMonteCarlointegration,yielding
pk � 1
0χ � 0 � pk � � t � dt � χ � 0 � pk � � ξ ���� 1 ξ � pk
0 else�
For the actual implementationthis meansto simply in-sert the randomdecision,whetherto storeor discardthephoton. Thus, it is possibleto concentratethe photonsin regionsof high probability pk, sinceonly that fractionof photonsis storedfor which ξ � pk for someuniformrandomvariableξ � � 0 � 1� . This is illustratedin figure1,wherewefirst show thestandardphotonmap,thentheim-portonsandfinally thesameamountof photonsdepositedusingtheimportanceinformation.Figure2 shows theim-provedphotondistributionthatis obtainedby tracingmoretrajectoriesin thepreprocessingstepbut storingthesamenumberof photonsaswithouttheimportancedrivendepo-sition.
2.2 The DepositionProbability
Identical to the approachin [PP98], we shootimportonsfrom a generalcamerasensorinto the scene.The proba-bility
pk �� f � ∑i ��! n " xk # Wi if f � ∑i ��! n " xk # Wi $ 1
1 else
thenis determinedby performinga queryfor the n near-est importons % n
�xk � to the query point xk, whereWi is
thevisualimportancecarriedby theimportoni. Sincethephotonmapperformsbestif thephotonsareof aboutequalpower, thecollectedvisualimportanceis scaledby afactorf andclippedif above1, where f is chosensothat pk 1in theareasof high visual importance,leaving the powerΦk unchanged.Only for the regions where pk $ 1, i.e.for regionsof almostno visual impact, the power growsto Φk
pkincreasingthevarianceof theirradianceapproxima-
tion. Thescalingfactor f canbeestimatedby tracingsome
Figure2: 10x10roomslit by 100light sources.In thetoprow 6 � 105 photonshave beenstored,whereasthebottomrow 6 � 106 photonshavebeendeposited.Bothcolumnsdi-rectlyvisualizethephotonmap,whichhasbeengeneratedwith thestandardmethodon the left andour importance-basedmethodon theright.
test rays from the camerainto the scenesensingfor thevisual importance. Choosingthe reciprocalvalueof theminimumof thesequeriesto be f fulfils therequirements.
Numericalproblemscanarisein regionswhere Φkpk
be-comesvery largedueto a smalldepositionprobability pk.This canbe remediedin two waysby eitherboundingpk
from below by someε & 0, or by discardingall photonswith pk $ ε, which,althoughhardlyperceivable,of courseyieldsabiasedalgorithm.
2.3 Automatic CausticsGeneration
In [JC95b] projectionmapsareusedto shootanincreasednumberof photonsin thedirectionsof thesolidanglecov-eredby objectsthatcouldgeneratecausticssuchasspec-ular surfaces. This approachcovers direct caustics,i.e.L � S �(' D paths.However secondarylight sources(like e.g.bright diffusereflectionsasoftenencounteredin architec-tural visualizations)might causeindirect causticswhicharenotcoveredby thepreviousapproach,but whichareofthesamedetailedvisual impactin thefinal image1. In or-derto automaticallygeneratetheverydetailedcausticpho-ton mapandthe lessdetailedglobal photonmap[JC98],wepartitionthesetof all photons
�into two sets
∑k ��� δk
�x ���ω � Φk ∑
k ��� cau
δk�x �)�ω � Φk * ∑
k ���,+-� cau
δk�x �)�ω � Φk �
where�
cau containsall photonsscatteredby a non-diffusesurface2, i.e. a potentialcausticgenerator. Using the in-tegral transformationfrom the previous sectionwith the
1Imaginethecausticsformedby aglassof waterilluminatedby awalllit by sunlight.
2Causticscouldbecharacterizedmorepreciselyby investigatingfocalpropertiesof wavefrontsandsurfacederivatesasusedin e.g.[Ige99].
Figure3: Causticsandclose-upat (top) ratio q 1, (mid-dle) q 1
5, and (bottom) q 120. Note that the amount
of photonsin theglobalphotonmapremainsroughlyun-changed,i.e. thesmallerq is, themorephotonsarestoredin thecausticphotonmap.
constantprobabilityq, weapproximate
Φi�x ���ω ��� ∑
k ��� cau
δk�x �)�ω � Φk
pk
pk* ∑k �.�,+-� cau
δk�x �)�ω � Φk
q � pk
q � pk� ∑k ��� cau
δk�x � �ω � Φk
pk� χ � 0 � pk � � ξk �
* ∑k �.�,+-� cau
δk�x � �ω � Φk
q � pk� χ � 0 � q / pk � � ξk ���
wherethe ξk are independent,uniform randomvariableson � 0 � 1� . The improvedquality of theautomaticallygen-eratedcausticphotonmapfor differentchoicesof theprob-ability q canbeseenin figure3.
2.4 The Algorithm
Theimplementationof theideasof theprevioussectionsisobtainedby replacingthestandardStore-call for photon
a) b)
c) d)
Figure4: Theveryproblematicsettingof a roombeinglitby a doorslit seenfrom atop.Theroomon theleft hasnolight sourcesbut thecamerawithin; theroomon therightcontainsonelight source.a) Theimportancedistribution,b) the standardphotonmap, c) the unbiasedimportancedriven photonmap,andthe biasedversionusinga mini-mumcutoff probabilityε.
k by thefollowing codefragment:
if(k � � cau)0if(ξk $ pk)
CausticMap->Store(xk, �ωk,Φkpk);1
else if(ξk $ q � pk)
GlobalMap->Store(xk, �ωk,Φk
q / pk);
If the photonunderconsiderationcould generatea caus-tic, it is storedin the detailedcausticphotonmap if theprobability inducedby visual importanceis high enough.Otherwiseonly the fraction q of the photonsis storedinthe global photonmap,which in consequenceis lessde-tailed.Thisof courseintroducestheoverheadof tracing 1
qtimesmoretrajectoriesthanrequiredto fill theglobalpho-ton map. Sincethe computationof pk is very expensive,it is moreefficient to usetwo randomvariablesξk � 1 � ξk � 2,first testingwhetherξk � 1 $ q andthenif necessarywhetherξk � 2 $ pk. This indicatesa third alternative: Only thefrac-tion of q trajectoriesis usedfor the global photonmap.Usinga total of N trajectories,we usethefirst 2 q � N 3 tra-jectoriesto fill thecausticandglobalphotonmapby onlytestingfor ξk $ pk andthenusetheremainingtrajectories2 q � N 3 * 1 �4�-� N to completethecausticphotonmap,yield-ing themostefficient implementation.
Discardingphotonsrequiresto tracemoretrajectories,i.e. having a moreexpensive preprocessingstep,in orderto obtainthe samenumberof photonsasin the purefor-wardsimulation,but is paidoff by themuchbetterphotondistribution in theregionsof highvisualimportanceat thesamememorycost, which is illustratedin figure 2. Onthe other handstoring photonsonly in areasof high vi-sualimpactcandramaticallyreducethememoryfootprintof the photonmapsby preservingtheir quality. Assum-ing thescalef of visualimportanceto bedeterminedasinsection2.2,ournew importancedrivenphotonmapgener-ationalgorithmis controlledby only two parameters,i.e.the numberN of photonsto trace,andthe ratio q of theglobal andcausticphotons. The tupel
�N � q � is intuitive
andincreasesusabilitysincenomanualinterventionis re-quiredfor completecausticsgeneration.
2.5 RemainingProblems
Our approachis complementaryto [PP98], where theemissionratesandshootingdirectionsaresampledfromthevisualimportance.In settingsof multiple light sourcesthealgorithmof [PP98] causesproblems:Theimportanceof a light sourceis determinedby a small randomwalk,andthefractionof trajectoriesstartedfromonelight sourceis givenby the ratio of its importanceto the total sumofimportances.Thusthephotonshave very differentpowervalues. Now if a photonof an unimportantlight sourcescattersintoaregionof highvisualimportance,it increasesthe varianceof the photonmapestimates.This effect isfurther amplifiedby the importancesamplingof scatter-ing directions.Theproblemis not encounteredby ourap-proach,sincethephotonsin the importantregionsremainof aboutequalpowerasexplainedin section2.2.
Solely selectingthe scatteringdirectionsby visual im-portanceasin [PP98] canbeproblematic,too. This is il-lustratedin figure4, whereseveralphotondistributionsofa room lit by a doorslit arecompared.The standardap-proachwould depositthe majority of the photonsin thevisually unimportantpart of the scene,since the smalldoorslit is hardto hit, even whenusingimportancesam-pling. Theprobabilitycontrolleddepostionrequiresmuchmoretrajectoriesto betraced,but storesthephotonsonlywherethey are important,yielding a betterbasisfor thesuccessive renderingstep.
In figures4cand4dweseethephotondistributionof theimportancedrivenphotonmapgeneration.Dueto energybleeding[KW00] theimportanceshinesthroughthewallsandphotonsaredepositedin areasof actuallyno impor-tance,i.e. behindthewall. Thiscouldbeavoidedby usingthe Metropolislight transportalgorithm[VG97, PKK00]to depositphotonsby the techniquesof theprevioussec-tions. This however indicatesthat usingMetropolislighttransportalgorithmaloneis moresophisticated.
3 Efficient Dir ect Illumination Com-putation
The efficient computationof direct illumination still isa core problem of photorealisticimage synthesis. Ex-act shadow computationsrequirea lot of shadow raystobe shot, especiallyin environmentswith a large numberof light sources. The performanceof shadow cachesisnot worth the effort for currentscenes,sincethe coher-enceis destroyedby thefine tesselationor the incoherentsecondaryraysof globalilluminationalgorithms[Smi98].Approachesasin [HDG99] indicatethatneverthelessco-herencecanbeexploited,but fail for secondaryrays,gen-eralscenegeometryanddueto asubstantialmemoryfoot-print. A very powerful approachis dueto Ward [War91],wherethe most importantlight sourcesaresampledfirstandshadow raysaresavedby estimatingthecontributionof thelessimportantlight sources.Theapproachhasbeenconsideredfurther by Shirley in [SWZ96], motivatedbyanarticleof Kirk on unbiasedsampling[KA91]. Similarto theobservationsin [HDG99], in [PMS' 99] it hasbeenshown thatreasonableapproximationscanbeobtainedbyshootingonly oneshadow ray in theregion of penumbra.In [JC95a]Jensenintroducedtheconceptof shadow pho-tons,wherehoweverthequestionwhereto depositshadowphotonsremainsanopenproblemor is impracticalduetoexcessivememoryconsumption.Evenassumingthisprob-lemto besolvedsatisfactory, theshadow estimatesaretooimpreciseandfail for finely structuredshadow details.
We now combinethe ideaof Ward of partitioningthesetof lightsources5 into two sets,definingprobabilitiesfor the lightsourcesin bothsetslike Shirley andsamplingfrom thesesetsin an unbiasedway similar to Arvo to aunified conceptbasedon the ideasalreadypresentedin[Kel98]: We extendthe photondatastructureby storingthe light sourceidentificationwith eachprimary photon,i.e. eachphotonthatdirectlycamefrom alight source.Foreachpoint to be illuminateddirectly, we performa pho-ton mapquery. Thecollectedsetof primaryphotonsnowvery roughly indicateswhich light sourcemay be visiblefrom thequerypoint. Thelight sources,whoseidentifica-tion is notamongthecollectedphotonsaremostprobablyoccluded.Sowehavethetwo sets5 con of potentiallycon-tributing light sourcesand 5 6�5 con of probablyoccludedlight sources.
We needthestochasticevaluationof sums,which is de-rivedby theintegral transformation
M 7 1
∑j 8 0
s j � M
0 9 M 7 1
∑j 8 0
siχ � j � j ' 1# � x �;: dx
� MS
S 7 1
∑i 8 0 9 M 7 1
∑j 8 0
siχ � j � j ' 1# � Mxi �;:<� (1)
whereS randomsamplesx0 �4�-�-�=� xS 7 1 � � 0 � 1� areusedtoevaluatethe sum of M summandssi. We now fix theamountof Scon andSnon 7 con shadow rays,which is inde-pendentof theactualnumberof light sources,in orderto
evaluatethe lighting contributionsof 5 con and 5>6?5 con,respectively.
If Scon $A@ 5 con @ , weuse(1) to randomlyselectScon loca-tionson the @ 5 con @ light sourcesfor thedirectilluminationcomputations.Otherwisethe variancecanbe further re-ducedby stratifying the total of Scon directly to the lightsourcesin 5 con by their emissionarea3. Concerningthelight sourceswhich probablydo not contribute,themini-malchoiceSnon 7 con 1 guaranteesanunbiasedalgorithm,i.e. weshootoneshadow ray to a randomlyselectedloca-tion on a randomlyselectedlight sourcefrom 5>6?5 con.From(1) weinfer thatthissampleis scaledby B C + C con B1 andin consequenceany unoccludedlight sourcebelongingto5<6�5 con cancausea very high variance. Thus it is veryimportantto select5 con securely, i.e. to usea largeradiusin thephotonmapquery, in orderto guaranteeanalmostzeroprobabilityof missingany contributing light source.In that case,however, omitting that singleshadow ray isbiased,but hardlynoticeableasshown in figure5c.
This basictechniqueis illustratedin figure5, wherewecomparesamplingthe light sourceswith andwithout im-portance. Image5b clearly shows much lessnoisethanthedirect computationwithout importancefunction in 5aat thesamecomputationalcost. We intentionallychosealittle too small radiusfor the importonmapqueriesin or-derto demonstratethelocal overmodulations,which havebeenexplainedin theprevioussection.
This importancesamplingschemeis easily addedtoan existing photonmapimplementation.It is muchsim-pler thanthespatialpartioningschemefrom [SWZ96] andmoreexactthantheglobalvisibility estimatesin [War94].However it canbecompletedby usingWard’s sortingal-gorithm [War94] for the caseScon $D@ 5 con @ , where thenthe Scon samplescan be distributed to the brighter lightsources.In theothercasethesamplesaredistributedby theimportanceof the light source,which is computedidenti-cal to [PP98]. The useronly needsto specifythe overallshadow ray numberanddoesnot needto take careof thesampleratefor thesinglelight sources.
The techniquealsoeasily transfersto the bidirectionalpathtracingalgorithm[LW93, VG94, VG95], whereit isusedto save shadow or connectionrays. Thenthephotonmapwouldserve asa pointapproximationof theradiancelike in [Kel97]. This wouldalsoimprovethebidirectionalmutationsof the Metropolislight transport[VG97] algo-rithm.
4 Conclusion
We presentednew importancesampling techniquesforthe photonmap,which result in a reducedmemoryfoot-print andincreasedrenderingefficiency. Our importancedrivengenerationof the photonmapcontrolsthe deposi-tion, which is moreefficient thancontrolling the scatter-ing directions.It alsoallows to automaticallygeneratethe
3Notethatthecompositionmethodcanraisevariance.Thusit is moreefficient to usebiasedpoststratificationestimators[KW86].
completecausticphotonmapat the requiredhigh resolu-tion. Finally we presentedan efficient samplingschemefor environmentswith a largenumberof light sourcesofwhich a fraction is permanentlyoccluded. Theseimpor-tancesamplingtechniquesreducethe requireduser in-teractionfor e.g. fine tuning of the emissionand sam-pling ratesfor the light sources.This increasesusabilityof the photonmapalgorithmsby replacingthe setof pa-rametersby an intuitive onerequiringlessuserinterven-tion in theactualcomputations.Although the techniquespresentedherearemajorimprovementsovercurrentalgo-rithms,openproblemsremain(seealso[KW00]):
1. Thesamplingof thedirectilluminationefficiently de-tectsocclusions. However it doesnot save shadowrays,if a largenumberof the light sourcesis visible.Spatialcoherencesuchneedsto beexploitedsimilarto [War91].
2. Photon map implementationstypically samplethelight sourcesdirectly. The current trend in designhowever is to usemoreandmore indirect illumina-tion. Due to the lack of direct illumination the pho-ton mapis visualizedusingthegatheringstep,whichexposesblurry artifactsin settingsof strongindirectlight sources.
Ward partially dealt with theseissuesin his Radiancerenderingsystem[War94, LS98] by reclassifyingindirectlightsaslight sourcesandperformingirradianceinterpola-tion [LS98]. Thesetechniqueshowever arecomputation-ally expensive. Simplerand more robust techniquesfortheseproblemsaresubjectof our futureresearchandwillbe basedon the photonmapmechanismfor representingdiscretedensities.
Acknowledgement
The authorslike to thank GeraldMaitschke from Com-paqComputersfor supportingthis work by the donationof an DEC Alpha workstation,andPhilipp Slusallekforhishelpfulcomments.
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Figure5: Direct lighting calculations(top) without, (mit-ddle)unbiasedwith importance,and(bottom)biasedwithimportanceinformationgeneratedfrom the photonmap.Thefirst two imagestook thesametime to compute.Thenoise in the rear of the middle imagedemonstratestheovermodulationeffect dueto a too small importonqueryradius,which is eliminatedby omitting to samplefrom5E6F5 con in thebottomimage.
