bmaxbminfmaxfminfmincmaxbmin bmaxbminfmaxfmin1istheinterpolatedclippedboundingboxattiwhereallvectoroperationsareperformedelementwiseThena ID: 446863
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t=0:5t=1Figure3:Ontheleft,twotightboundingboxes(inblueandred)areshownthatresultfromsplittingatrianglebyaplane.Ontheright,thetrianglehasbeenmovedaccordingtoitsmotionvector(ingreen).Sincemotionandclippingarenotcommutative,thetrans-formationofthetightboundingboxesisnotguaranteedtocoverthetransformedtriangle(ingray).3.1.2Axis-AlignedBoundingBoxesClippingmaynotbepracticalforsomeprimitives.Insuchcasesonecantransformtheboundingboxoftheprimitiveintersectedwiththeclippingbox.Theresultingboundingboxesmaynotbeastightcomparedtoclippingtheprimitive.Giventheprimitiveboundingbox[bmin;bmax]att=0:5,itstrans-formedcounterpart[fmin;fmax]atti,andtheboundingboxoftheprimitiveintersectedwiththeclippingbox[cmin;cmax],fmin+cmin�bmin bmax�bmin(fmax�fmin);fmin+cmax�bmin bmax�bmin(fmax�fmin)(1)istheinterpolatedclippedboundingboxatti,whereallvectorop-erationsareperformedelementwise.Thena¨veapproachoftransformingtheboundingboxoftheclippedprimitiveisruledoutduetothefactthattransformationandclippingarenotcommutativeasillustratedinFigure3.3.1.3ComplexShapesResortingtousingonlyboundingboxes(asdescribedinthepre-vioussection)isnotefcientforparametricsurfacesandmulti-resolutionsurfaceslike,forexample,displacementmappedsub-divisionsurfaces.Inthesecases,theelementsresultingfrompara-metricorregulartessellationcanbebounded,asillustratedinFig-ure4similarto[Hanikaetal.2010;Munkbergetal.2010].Theresultingboundingboxesarehandledasintheprevioussec-tion.Finally,theboundingboxofanodeisdeterminedbytheboundsoftheclippedboxesofallelementsinthatnode.Alterna-tively,itispossibletoavoidclippingtheelements'boundingboxes.However,thisgenerallydoesnotresultinanexactspatialsplitany-moreandwouldconsequentlyleadtolargerboundingboxes.Often,theboundingboxesobtainedwithrenementaremuchtighterthanwithoutandenableamorepreciseapproximationoftheSAH.4InstancesInstancesareacommontooltoreducescenecomplexityandhier-archyconstructiontimes.Insteadofatree,adirectedacyclicgraph Figure4:Approximatingacomplicatedpatchbymorethanoneboundingboxallowstoconservativelydeterminepartsofthepatchthatbelongtotheleftandrightchildasaconsequenceofaspatialsplit.isused,wherepartsofthehierarchycanbereferencedmultipletimes.SuchareferencecontainsatransformationmatrixA(t)thatdependsonthetimetandistheframeofthereferencedgeome-try.Whenarayisintersectedwithsuchanode,itistransformedtotheframeoftheinstancebyapplyingtheinversetransformationmatrix.Inthecontextofanimation,theimportantquestionishowtondatightboundingboxofananimatedinstanceduringhierarchycon-struction.Forexample,[PharrandHumphreys2010]sampleA(t)atdiscretetimestitodetermineaboundingbox,butapplythetem-porallycontinuousmappingA�1(t)totransformtheray.Theirsamplingdoesnotguaranteeaconservativeboundingboxandcon-sequentlygeometrycanbemissedduringrayintersection.Wethereforeintroduceanapproachthatdeterminesconservativeboundingboxesforhierarchyconstruction.Inordertoenablein-terpolation,then+1boundingboxesofanMSBVHnodearede-terminedbyboundingtheinstanceusingtheframeAi:=A(ti).Tighterboundingboxesareachievedbytransformingtheinstancegeometry,whiletransformingtheboundingboxoftheinstanceisfaster,butnotastightingeneral.TheresultinginstanceboundingboxesareprocessedasdescribedinSection3.1.2.NotethattheprimitivesofindividualinstancehierarchiesarestillclippedandtransformedasdescribedinSection3.1,Conservativerayintersectionthenrequirestoapproximatethetransformation:Duringhierarchytraversal,theboundingboxesareinterpolatedinexactlythesamewayasdescribedinthealgo-rithmoverview.Whenreachingtheleafthatholdstheinstance,thematricesareinterpolatedanalogouslytotheboundingboxes:A0(t)=Pni=0wi(t)Ai.Therayistransformedusingtheinverseofthisinterpolatedtrans-formationmatrixbeforetraversingtheinstance-levelMSBVH.Notethatwearenotinterpolatingindividualfactoredmatricesre-sultingfromamatrixdecompositionintoscale,rotation,andtrans-lationcomponents[PharrandHumphreys2010].Instead,wein-terpolatetheindividualmatrixelements,resultinginsplinesthatmatchtheinterpolationoftheinstanceboundingboxes.Thisguar-anteesthatnogeometryoftheinstancecanbemissed.Fortheexampleoflinearinterpolationwithequidistanttimesti:=i n,rstthefraction=nt�iinsidethemotionsegmenti=bntcisdetermined.Thentherayistransformedusingtheinverse Figure5:EvolutionoftheSAHcostfunctionappliedtotheMSBVHforthehairballscenefromFigure1:asexpected,thecostdifferencebetweenrettingandrebuildingthehierarchyeachframegraduallyincreasesoverthecourseoftheanimation.ofA0():=(1�)Ai+Ai+1.Forarbitrarytiabinarysearchcouldbeusedtolocatethecorrespondingmotionsegment.Mail-boxing[AmanatidesandWoo1987]canbeappliedinordertoavoidintersectingaraymorethanoncewiththesameinstance.Alternatively,itispossibletoqueueaxednumberofinstancein-tersectionsalongtheray[Hanikaetal.2010].Thisqueuethenissortedtoremoveduplicatesbeforeperformingfurtherintersections.Inthiscontextitisrecommendedtoaccountforthemail-boxingwhenevaluatingtheSAH,similarto[Hunt2008].Whileinstancingcansubstantiallyreducememoryrequirementsandhierarchyconstructiontime,itcannotreducetheoverlapamonginstances.Thus,therearecaseswherereplacingthedirectedacyclicgraphbyitscorrespondingtreeresultsinsuperiorperformance.5RettingBoundingvolumehierarchyconstructioncostcanbeamortizedacrossframesbykeepingthehierarchytopology,updatingleafboundingboxes,andpropagatingtheresultsupthehierarchy[Waldetal.2007].UsingtheSAHcostfunctionasameasure,rettingcanresultinadegradationofthetreequality,asillustratedinFigure5.Therefore,rettingusuallyiscomplementedbyaheuristic[Lauter-bachetal.2006]whichtriggersrebuildsfordegeneratedpartsofthehierarchy.InordertoretaleafboundingboxfortheMSBVH,wetakeintoaccounttheclippedprimitives.Forasubsequentframe,were-clipeachprimitiveagainstthecorrespondingleafboundingboxofthehierarchyatthepreviousframe'sshutterclosingtime.Thisway,noadditionalgeometryorhierarchydataneedstobestored.Duetothere-clippingofprimitives,theMSBVHrettingpassisslightlymoreexpensivecomparedtoclassicBVHretting.6ImplementationDetailsFollowingtheSBVHimplementation[Stichetal.2009],animple-mentationoftheMSBVHasdescribedintheprevioussectionsisstraightforward.Inthefollowing,complementaryaspectsarede-tailed.6.1UpdatingtheReferenceSortingOrderOnecommonwaytospeedupaccelerationhierarchyconstructionistopre-sorttheprimitiveindexarray,forexampleaccordingtoprimitivecentroids[WaldandHavran2006].However,whenprim-itivesaresplitspatially,thecentroidsoftheresultingreferencesneedtoberecomputed.Thus,caremustbetakentonotinvalidatethesortingorderofthereferenceindexarrayduetospatialsplits.Consequently,foreachspatialsplit,wekeepalistofthoseprimitivesinthenodethatareduplicatedduetoaspatialsplit.Werecomputecentroidsforthoseprimitives,sortthisarray,andthenuseamerge-sortsteptocombinethisarraywiththearrayofunsplitprimitives,similarto[WaldandHavran2006].6.2HierarchyTraversalOrderRelyingononespatialsplitatasingletimeinstantforthewholeshutterintervalrequirescoherentmotion.Evenwiththisassump-tionfullled,primitivesinonenodecanmoveintooppositedi-rectionswithrespecttoaspatialsplitplaneandchangebound-ingboxes.Thus,insteadofusingthesplitplanefordeterminingthehierarchytraversalorder,itcanbemoreefcienttodeterminethetraversalorderaccordingtotheparametricdistanceoftheray-boundingboxintersection.6.3MultipleMotionSegmentsUsingasinglemotionsegmentisnotasufcientapproximationfornon-linearmotionlike,forexample,arotatingpropeller.Insuchcasesmultiplemotionsegmentsneedtobeusedforasingleframe.Restrictingthenumberofmotionsegmentsto2mwithequidistanttimesti=i 2mallowsfortheefcientdeterminationofbound-ingboxesincasechildreninthehierarchyusedifferentnumbersoftimesamples.Thepower-of-tworestrictionguaranteesthattheexistingmotionsegmentendpointsforthechildrencoincide,asweonlyneedtotakethemaximumnumberofsegmentsofthechil-drenandresamplethechildrenwithasmallernumberofmotionsegments.Thisavoidsincreasingthenumberofmotionsegmentsarbitrarilywhenpropagatingboundingboxesfromchildrentoparents.Forexample,ifarbitrarynumbersofmotionsegmentswereallowed,theparentnodeoffourchildrenwith2,3,5,and7motionseg-mentswouldneed2357=210motionsegmentstoaccuratelyrepresentthechildren'stransformationinterpolationswithoutap-proximations.Ingeneral,thisnumberisequaltotheleastcom-monmultiple,whichmotivatesrestrictingthenumberofmotionsegmentstopowersoftwo.7ResultsandDiscussionWeimplementedtheMSBVHintheNVIDIAOptiXraytracingframework[Parkeretal.2010]andcomparedtheefciencyoftheresultingtreestoregularBVHswithnodeinterpolation.Ta-ble1showsvariousperformancestatisticsobtainedbyrenderinganumberofdifferenttestsceneswithambientocclusionshading.Throughoutalltestcases,areductionofoverallintersectionand/ortraversalstepscanbeobserved.Asin[Stichetal.2009],theSAHcostsofthehierarchiesinalltestcasesisreducedbyintroducingspatialsplits.Ourimplementationfocusesexclusivelyontreequalityratherthanoptimizingbuildperformance.Thus,constructiontimesoftheMSBVHexceedthoseofaregularBVHbyafairamount.Anop-timizedimplementationoftheMSBVHwouldlikelynarrowthegapsignicantly,butduetotheinherentadditionalcomputation,itsconstructionspeedcannotreachthatofaBVH.Nevertheless,assoonasenoughraysaretraversedthroughthehier-archy,theadditionaloverheadtobuildahierarchyofhigherquality