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SeminarausarbeitungzumSeminar

InteraktiveVisualisierung

(WS06/07)

Two-LevelVolumeRendering

Westf¨alischeWilhelms-Universit¨atM¨unsterFachbereichMathematikundInformatik

Institutf¨urInformatik

vorgelegtvon

KayHenningBrodersen

(Kay.Brodersen@gmx.de,Matrikelnummer308603)

am12.01.2007

CONTENTSCONTENTS

Contents

1Introduction

1.1TheChallengeof3DVisualization................1.2CommonRequirementsinVisualization.............2352RecentResearch

2.1VisualizationMethods......................2.2VisualizationParameters.....................3VolumeRenderingTechniques3.1DirectVolumeRendering...

..................3.1.1Method..........................3.1.2Discussion.........................3.2Maximum-IntensityProjection

..................3.2.1Method..........................3.2.2Discussion.......

..................

4Two-LevelVolumeRendering4.1Motivation.............................4.2RenderingModel.........................4.3Implementation..........................5Applications5.1MedicalDataVisualization....................5.2DynamicalSystemVisualization.................6Discussion

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1INTRODUCTION

1Introduction

Humanshavealwaysintuitivelymadeuseoftheirgenuinementalcapabil-ities.Devisingaconclusionfromnoisyinformation,recognizingapatternfromloosedata,revealingahiddenstructurewherechaosseemstoreign—inmanyareas,eventhemostrecenttechnologyhasbyfarnotreachedthecomputationalpowerofhumancognition.

Theinformationconcernedwithareal-worldproblemofoneofthekindsoutlinedaboveismostlyvisualinnature[Udu00].Thus,visualizingdatawhereitisnon-visualintheﬁrstplacecouldextendtheuseofhumancog-nitivepotential.

TheGoalofVisualizationThegoalofvisualizationistotranslateab-stractquantitativedataintoavisualsensationsoastoenablethehumanmindtogaininsightintotheunderlyingsystemthedatahasbeentakenfrom.Infact,suitablevisualizationmethodsopenupawholerangeofapplicationsinscienceandengineering.ThisessayreviewsthefundamentalprinciplesofvisualizationandeventuallyfocusesonanextensionrecentlyproposedbyHauseretal.knownastwo-levelvolumerendering[HMBG01].

Visualizationmustbeunderstoodasapartofimagingwhosepurposeitistoyieldqualitativeorquantitativeinformationabouttheobjectsystemunderstudy.Itcanbethoughtofascomprisingfoursub-processes[Udu00]:1.Preprocessingtakesinasetofobjects,anobjectsystem,andextractsvolumesofinterest,appliessegmentation,interpolation,ﬁlteringandmasking.2.Visualizationistheprocessofproducingarenditionfromtheprepro-cesseddatasuchthatthestructureorthedynamicsoftheunderlyingobjectsystemcanbestudied.3.Manipulationallowsforvirtualmodiﬁcationoftheobjectsthathavebeenvisualizedbycuttingaway,separating,movingoranimatingcer-tainpartsofthem.4.Analysis,ﬁnally,isusedtoquantifycertainmorphologicalorfunctionalinformationabouttheobjectsunderconsideration.Thisessayfocusesonthevisualizationaspectofimaging.However,sincethefourprocessesareofteninterleavedandrarelyappliedinstrictsequentialorder,someaspectsofprocessesotherthanvisualizationinitsstrictestsense,suchassegmentationoranalysis,willbetoucheduponaswell.

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1.1TheChallengeof3DVisualization1INTRODUCTION

TheproblemofdimensionalityAnykindofvisualizationrequiresaviewingmedium,andeverysuchmediumischaracterizedbythenumberofdimensionsitfacilitates[Udu00].Acomputerscreenisaﬂexible,low-cost2Dviewingmediumusedubiquitously.Holographypromotesa3Dimpression,butismuchmorecostlyanddoesnotallowforinteraction.Ahead-mounteddisplayisapairoftinyscreens,mountedinfrontofone’seyes,thatfacilitatesstereoscopicviewing,butagainismuchmoreexpensivethanasinglemonitor.Thestandardcomputerscreenremainsthemostpervasiveviewingmediumand,inthesequel,allconsiderationshallberestrictedtoitstwodimensions.Conventionalvisualizationhasbeentheprocessofrenderinga2Ddatasetontoa2Dscreen.Butoftendataistobeconsideredwithmorethantwodimensions.Forinstance,onemightwanttoinquireintoavolumeofdata(3D),orevenstudythechangeofsuchavolumeovertime(4D).

A2Dviewingmediumcannotvisualizeadatasetofsuchhigherdimen-sionality.Infact,evenifoneweretousea3Dviewingmediumsuchasahead-mounteddisplay,onewouldberestrictedtostereoscopicviewingandcouldnotgraspallofagivenvolumeatonce.

Consequently,visualizationtechniques,withtheirtechnicaltaskbeingthegenerationof2Drenditions,needtomakeuseofauxiliarycues,suchastransparency,shadowingormotionparallax,inordertoconveyadatasetofhigherdimensionality.

StructureofthisessayThisessaydiscussesarecentlyproposedapproachto3Dvisualizationcalledtwo-levelvolumerendering.Giventhelimitationsandﬂawsofconventionallycompetingrenderingtechniques,two-levelvolumerenderingfusestworenderingapproachesinordertoproducerenditionsthatcombinetheadvantagesofeachofthem.

Inordertounderstandthereasoningbehindtwo-levelvolumerendering,thefollowingintroductorysubsectionsoutlinethechallengeof3Dvisualiza-tionanditsmainrequirements.Insection2,aselectionofrecentsuggestionsfortacklingtypicalproblemsinvisualizationispresented.Section3looksindetailatthetwovolumerenderingtechniquesthatformthebasisoftwo-levelvolumerendering,anddiscussestheirrespectivestrengthsandweaknesses.Asanovelapproachtocombiningthesetechniques,thecoreoftwo-levelvol-umerenderingitselfisdescribedinsection4.Twoapplicationscenariosaredescribedinsection5.Theresultsareﬁnallydiscussedinsection6.

1.1TheChallengeof3DVisualization

ConveyingmultidimensionalinformationTherearenumerousappli-cationsinwhichdatasetsarenaturallygiveninmorethantwodimensions.

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1.1TheChallengeof3DVisualization1INTRODUCTION

Theﬁeldsthesestemfromareasdiverseasmedicine,biotechnology,geo-science,engineering,meteorology,astrophysics,orﬂuidmechanics.

Traditionalcomputergraphicshasbeenconcernedwithsurfacerendering,applicablefortaskssuchascreatingavirtualmazeconstructedfrompolygonsorsplines.Volumerendering,incontrast,aimsatgeneratingimagesfrom3Dscalardata.Thisdatahasmostlybeenacquiredfromphysicalmeasurements,butmayalsointurnbearesultofacomputersimulation[EHK+06].

A3Ddatasetisreferredtoasavolume.Itisspatiallydiscretizedintocubodialvolumeelements,voxels,eachofwhichcarriesacertaindatavalue,itsdensity.Sometimesavoxelisconsideredtodenotethecentreofthiscubodialelement,thatis,apointin3Dspace.Fortheremainderofthisessay,though,avoxelshalldenotetheactualsmallvolume[EHK+06].Thevoxel’sdensitiesstemfrommeasurementsofobject-diﬀerentiatingpropertiesinthephysicalobject.Animportanttask,eventhoughoftenaccomplishedimplicitly,istoreconstructthein-betweendensities.Involumerendering,thisistypicallydoneinanad-hocfashion,e.g.byinterpolation.

Frequently,a3Ddatasetiscreatedfromasetof2Dslices,thatis,fromplanesthroughthedatasetwithacertainamountofspacinginbetween.Theymayhavebeenacquiredfromadigitalradiograph,computerizedto-mography(CT),magneticresonanceimaging(MRI),positronemissionto-mography(PET),single-photonemissioncomputedtomography(SPECT),ultrasound(US),orfunctionalMRI(fMRI).Mostexamplescanbefoundinmedicine,inwhichtypicalclinicalapplicationsincluderadiodiagnostics,radiationtherapyplanning,andcomputeraidedsurgery.

Fundamentally,datavaluescarryinformationaboutthestructureoftheobjectunderstudy(e.g.x-rayattenuationinCT,orrelaxationtimesuponmagneticallyexcitingthetissuesinMRI),oraboutitsfunction(e.g.ﬂuidﬂowinPET).Giventherestrictionthatitisimpossibletocompletelyshoweachvolumesamplesimultaneously,thekeychallengeof3Dvisualizationthereforeistoconveythevaluedistributionofagiven3Ddatasetasclearlyandaccuratelyaspossible[ER00].Morespeciﬁcally,Hadwigeretal.de-scribethegoalastobeable“tovisuallyseparateandselectivelyenablespe-ciﬁcobjectsofinterestcontainedinasinglevolumetricdataset.”[HBH03]Generatinga3DsensationAtypicalconventionalapproachtovolumevisualizationistosimplydisplay2Dslicesastheyhavebeenrecordedindi-vidually.Theymightbeshowninthewaytheyhavebeenobtainedfromthephysicaldevice,thatis,alignedtooneoftheplanesofthecoordinateaxes;ortheymightbecomputedasarbitraryslicesthroughthevolume.Atypicalwayofprovidingcontextistovisualizethreeperpendicularslicesthrough

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1.2CommonRequirementsinVisualization1INTRODUCTION

Figure1:Inaconventionalvolumevisualizationapproach,threeperpendic-ularplanesthroughthesamevoxelareshown.Here,theyrepresentsaggital,axialandcoronalslicesthroughanMRIscanofahumanbrain.

thesamesharedvoxelatonce,asfrequentlyusedinmedicalapplications(seeFig.1).

Althoughanimationsthatstepthroughtheslicesmayhelp,evenfortrainedobserversittendstobediﬃculttofullygraspthestructureofavolumebymerelybeingpresented2Dslices.Thechallengeof3Dvisual-ization,fromthisperspective,istogenerateanaccurate,yetintuitive,3Dsensationofthedatasettobeanalyzed.

1.2CommonRequirementsinVisualization

Objective-orientedvisualappearanceTheconceptualideabehind3Dvisualizationistosimulatelightasittravelsthroughamediuminordertoproduceanimageasrecordedbyacamera[EHK+06].Whenmodellingemission,absorption,andscatteringoflight,theopticalpropertiesofthematerialunderstudyneedtobetakenintoaccount.However,realismisoftentradedagainsttheuseofmoretractablemodels.

Imagequalityin3Dvisualizationdependsheavilybothontherenderingmethodandontheparametersusedtotunethemethod.Therefore,animportantaspectin3Dvisualizationistodeterminetheobjectivesofthevisualizationﬁrst.Isthegoaltoseparatediﬀerentkindsoftissueinananatomicalimage?Orisittofocusononestructurewhiledimlyprovidingthecontextofthesurroundingsatthesametime?

Visualizationobjectivesguidetwocanonicalquestions:•Whichrenderingtechniqueistobeused,andwilltherepotentiallybediﬀerenttechniquesfordiﬀerentsegmentsoftheoverallvolume?

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1.2CommonRequirementsinVisualization1INTRODUCTION

•Whataresuitableparametersforthetechniques,andhowcantheparameterspacebeexploredsoastogiveoptimalresults?

Information-per-pixelratioAssoonasa3Ddatasetisrenderedfora2Dviewingmedium,therewillinallprobabilitybepixelsthatcarryin-formationfrommorethanonecorrespondingvoxel.Thenotionofhavingamultitudeofobjectsinanimage,oranumberofvoxelsperpixel,islooselyreferredtoasaninformation-per-pixelratiooftherenditiongreaterthanunity.

Expressiveresultsdependonasuitableinformation-per-pixelratio.For-tunately,humansareabletodistinguishbetweendiﬀerentobjectsina3Dscenetoacertainextent,evenwhenobjectsarelayeredontopofeachother.Forexample,animagemightcontainabonesegmentsurroundedbytissue.Iftheinformation-per-pixelratioistoolow,thefullpotentialofvol-umevisualizationisnotexploited.Ifitistoohigh,theimagebecomesoverloaded.Empiricalresultshaveshownthatthereisalimitofaboutthreesemi-transparentstructuresperimagethatshouldnotbeexceeded[HMBG01].

InteractiveframeratesIthastakenalongtimeuntilvolumevisualiza-tionhasbecomeappliedwidely.Oneofthemainobstaclesusedtobethelackofinteractivity[Pﬁ04].However,withrenderingtechniquesthatareabletocreaterenditionsininteractivetime,thatis,atratesofseveralframespersecond,awholerangeofnewopportunitiesunfolds:

•Visualsensationcanbeimprovedhugelythroughmotionparallax.Forexample,theeyepositionmayrotatearoundtheobject,ideallycon-trolledbydirectuserinteraction.

•Afruitfulwayofﬁndingsuitablerenderingparametersistobeabletosettheminteractively.

•Afourthdimensioncanbeincorporatedintothevisualizationsensationbychangingoneparameterovertimeandimmediatelyupdatingtheimageaftereachstep.Forinstance,thisallowsforvisualizationofdynamicalsystems(seesection5.2).

Interactivityistypicallyachievedbyasmuchpreprocessingaspossible,fastrenderingimplementationsandsophisticatedprogrammablegraphicsprocessingunits(GPUs).

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2RECENTRESEARCH

2RecentResearch

Section1.2posedtwocanonicalquestionsaboutvisualization:Whichtech-niquesshouldbeused?Andwhatparametersaremostsuitabletoyieldthedesiredresults?Inordertounderstandwheretwo-levelvolumerenderingwillsetin,aselectionofrecentimprovementsuponconventionaltechniquesaredescribedhere.

2.1VisualizationMethods

Recentresearchonmethodsdealswithhowtochooseasuitablerenderingtechniqueandhowtochoosefromthehugenumberofvariantsthathavebeenproposed.

VolumeillustrationTheaimofvolumeillustrationistocreaterenditionsthatare,inprinciple,basedonarealisticphysics-basedilluminationmodel,butmayatthesametimehighlightparticularobjectfeatures.Thisemphasisisnotrestrictedtotheusageofaspeciﬁctransferfunction,afunctionthatmapsasamplingpoint’sdensitytocolourandopacityvalues.Instead,non-photorealisticeﬀectsmaybeusedaswell[ER00].

Inmanyrenderingtechniques,opticalpropertiesoftherenditionaremainlydeterminedbyatransferfunction.Involumeillustration,apartfromitssamplingpoint’slocationandinterpolateddensity,thecolourofapixelisalsodeterminedbyotherproperties.Thesemayincludethelocalgradient,theviewingdirection,andlightingsettings.

Volumeillustrationhasbeensuccessfullyusedforimagesintextbooksandpresentations.

OtherapproachesAmajorpartofcurrentresearchfocusesontheinter-activevisualizationofunstructuredvolumes[Pﬁ04].Otherapproachesareconcernedwiththevisualizationofdynamicalsystems,withvolumecompres-sion,orimage-basedvolumerendering.

2.2VisualizationParameters

Researchonparametersisconcernedwithhowtoconﬁgureagivenrenderingtechniqueappropriately(e.g.bysettingsuitablesegmentationthresholds,orapplyingcertaintransferfunctions)inordertoobtainthedesiredresults.Itisabouteﬃcientlysearchingahugeparameterspace.

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2.2VisualizationParameters2RECENTRESEARCH

InversedesignAnapproachinspiredbyrelationalalgebraisinversede-sign.Ifanobjectivefunctioncanbespeciﬁedthatquantiﬁesthedesiredresultoftherendition,then,byinvertingthisfunction,thecorrectparame-terscanbeobtained.

Inpractice,however,itisusuallyimpossibletospecifyanobjectivefunc-tionandestablishatractableanalyticlinkbetweeninputparametersandoutputquantities.

InteractiveevolutionIfitisillusivetosuﬃcientlyformaliseanobjec-tivefunction,onemightresorttoatrial-and-errorprocedure.Ininteractiveevolution,theevolutionarytechniquesofmutationandrecombinationareusedtoconstructnewvectorsofinputparameters.Thephaseofselectionisperformedbytheuser.Thus,ratherthanhavingtoworryaboutallthedetailsoftheindividualrenderingparameters,theusermerelyjudgesupontheresultandimplicitlyforcestheexplorationprocesstowardsthedesireddirection.

Thisparadigmhasoriginallybeenproposedas‘simulatedevolution’,in-troducedtocreatecomplex3Dstructureswithanelementofrandomness,suchasplantsortrees[Sim91],butmayequallywellbeappliedtorenderingparameters.

Themaindownsideofinteractiveevolutionisthatitcruciallyreliesoninteractiveframerates.Ifparametersaretobefoundforhigh-qualityrendi-tions,thisisnotensuredanylonger.

DesigngalleriesIfrenderingiscomputationallytooexpensivetoallowforinteractiveframerates,designgalleriesproposeawayofuntanglingthetemporalsuccessionofrenderingandperceptualchoiceasitisrequiredininteractiveevolution[MAB+97].

Adesigngalleryinterfaceﬁrstaskstheusertospecifywhichinputparam-etersaretobevaried,andtoroughlyspecifyanoutputvectorthatindicatesthesubjectivelyimportantpropertiesoftheﬁnalimage.Thesystemthentemporallycreatesalargenumberofrenditionsthatarebasedondiﬀerentparametercombinations,andcomputestheoutputvectorforeachofthese.Adistancemetricisusedtoﬁndtheperceptualsimilaritiesbetweentheren-ditions.Thesystemdeterminesasubsetofrenditionsthatcorrespondstoawell-distributedsetofoutputvectors.Finally,theuserispresentedwithatreestructurethatmakesitpossibletobrowsethrough,pick,andpotentiallycombinethedesiredrenditions.

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3VOLUMERENDERINGTECHNIQUES

3VolumeRenderingTechniques

Renderingtechniquesingeneraltransforma3Ddatasetintoa2Dimage.Two-levelvolumerenderinginparticularfusesotherrenderingtechniques.Themosttypicalonesforthis,directvolumerendering(DVR)andmaximum-intensityprojection(MIP),arediscussedinsections3.1and3.2,respectively.Aswithallrenderingtechniques,DVRandMIPcanbeconceptuallysplitupintoapreprocessingandarenderingstage.Theirsharedcharacteristicsaredescribedinthesequel.

PreprocessingRenderingalgorithmsoftentakeinrawdataandperformallnecessaryprocessingimplicitly.Often,however,theyworkwithaprepro-cesseddatasetinstead.

Preprocessingmostlyinvolvessomeformofsegmentation.Itspurposeistoextractobjectinformationfromagivenvolume,andoutputanobjectsystem,i.e.asetofdistinctobjectstheoverallsceneiscomposedof[Udu00].Forinstance,propersegmentationiscrucialforarealisticdisplayoftissue.Perfectsegmentationcansometimesbeachievedwhenavolumehasbeenregisteredfrommorethanonemodality.

Furthermore,preprocessingofteninvolvesaugmentingthegivendatasetbymeansofadditionalcomputations.Forexample,surfacerenderingre-quiresinformationaboutthegradientofeachofthevoxels,andthisgradientmaywellbeprecalculatedandstoredalongwiththevolume.

RenderingBothDVRandMIPcoverthreecloselyrelatedtasks,allofwhichareimportanttoconstructagoodsenseof3D[Udu00]:

1.Projectiondeterminesthemappingbetweensamplingpointcoordinatesandpixelcoordinates.Inparallelprojection,theprojectionlinesfromallsamplingpointsontotheviewingplaneareparalleltoeachother.Thisimpliesthatthesizeofallobjectsisindependentoftheirdistancefromtheviewingplane.Thisbadlyaﬀectsrenditionswheretheeyepo-sitionisinsidethevolume.Aminoreﬀectisthatitbecomesimpossibletotellthedirectionofrotationifnoadditionalcuesareprovided.Inperspectiveprojection,theprojectionlinesconvergeintheeyeposition.Objectsthatarefurtherawayappearsmaller.2.Hiddenpartremovalensuresthatpartsthatareentirelyobscuredbyothersarenotprojectedontotheviewingplane.

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3.1DirectVolumeRendering3VOLUMERENDERINGTECHNIQUES

(a) 2D textureslicing(b) 3D textureslicing(c) Ray casting(d) Splatting(e) Nearest-neighbourinterpolation(f) Optimizedaccumulation(g) Shear-warprendering(h) VoxelprojectionFigure2:Commonprojectionmethodsinvolumerendering(a,b,c,d,andgtakenfrom[Pﬁ04]).

3.Shading,orcompositing,ultimatelyassignsacertaincolourtoeachpixelintheviewingplane.Thecolourdependsonthecolourofthesamplingpoint,itsopacity,gradientorlightingparameters.

3.1DirectVolumeRendering

Directvolumerendering(DVR)isarenderingtechniquethatcreatesaren-ditionfromtheentiresetofvoxelsofa3Ddataset.3.1.1

Method

PreprocessingDuringpreprocessing,segmentationismostlyusedtosep-aratediﬀerentobjectsintherawdataset.Furthermore,wholeportionsofuninterestingregionsmaybesectionedoutofthevolume.

RenderingThemostpervasiverenderingparadigmonmodernGPUsistextureslicing(seeFig.2a/b).Thegivenvolumeisstoredasasetof2Dtextureslicesorasasingle3Dtexture.Polygonsspanningtheslicesofthevolumewiththesetexturesarethenrenderedontotheprojectionplaneusingascanlinealgorithm.

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3.1DirectVolumeRendering3VOLUMERENDERINGTECHNIQUESThemostpopularalternativeparadigmisraycasting(seeFig.2c).Inaﬁrststep,raysarecastfromtheeyethroughthepixelsintheviewingplaneintotheviewingfrustum.Second,theintersectionoftherayandthevol-umetobevisualizeddeﬁnesaraysegment.Alongthissegment,anumberofequallyspacedsamplingpointsaredetermined.Foreachofthesepoints,atrilinearinterpolationiscomputedfromthesurroundingvoxels’densityandgradientvalues[Pﬁ04].Third,atransferfunctionisappliedtothesam-plingpoint,assigningitacolourandanopacity.Thetransferfunctionmayalsodependonthesegmentationinformationfrompreprocessingandscenelighting.Technically,ityieldsanRGBA4-tuplewhereAspeciﬁesthevoxel’sassignedopacity.Notethatdiﬀerentresultswillbeobtaineddependingonwhetherthetransferfunctionisappliedbeforeorafterinterpolation.Fourth,thebackgroundcolour(e.g.black)andallsamplingpointsalongtherayareaccumulated.Thesinglecolourvalueobtainedistransferredtothecorre-spondingpixelinaframebuﬀer,andtheprocessrepeatsforeachpixelintheviewingplane[Udu00].

Bothtextureslicingandraycastingcanberegardedasimplementationsoftheabsorption-emissionmodeloflighting—aslightlysimpliﬁedlightingmodelwhoseonlyopticalfeaturenotbeingaccountedforisscattering.Atechniquethatdoesincludethiseﬀectisraytracinginwhicheachraymaybesplitupandpursuedinmultipledirections,butitcanvastlyincreasecomputationalcomplexity.

Manyvariantshavebeenproposedtotherenderingalgorithmdescribedabove.Inparticular,thetransferfunctionusedforshadingprovidesmuchroomforvariation.Ittakestheimportantroleofdeterminingwhichphysicaldatavaluesaremappedtowhatkindofvisualproperties.Thus,therender-ingprocessmayeitheraccuratelysimulateaprocesssuchastheilluminationofagaseousvolumebylightrays[ER00];oritmayuseanarbitrarytransferfunctionthatspeciﬁcallyemphasizesdensityvaluescorrespondingtocertainphysicalpropertiesinordertohighlightregionsofinterest.

Withtransferfunctionspotentiallytakingintoaccountgradientsaswell,surfacenessbecomesacontinuousmeasure,ratherthanaBooleanindicator,ofthedegreetowhichthevoxelrepresentsaboundarybetweendisparateregions[ER00].

Aboveandbeyondtheseexamples,asrenderingtechniqueshavematured,transferfunctionshavebecomemoreandmorecompelling.Theyhavetrans-formedfromsimplecolourassignmentstopowerfultoolsforpatternidenti-ﬁcationandsegmentation.

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3.1DirectVolumeRendering3VOLUMERENDERINGTECHNIQUESVariantsTherearenumerousvariantstotextureslicingandraycastingthatarealsorelevanttotwo-levelvolumerendering.

Splatting.Insteadofinterpolatingsamplingpoints,thevoxelsofthedata

volumethemselvesareprojected,or‘splattered’,ontotheviewingplane(seeFig.2d).Becauseofthelackofequidistantsamplingpoints,thedensityofthedatapointsin‘depthdirection’variesamongrays.Sinceitsrenderingloopiteratesontheobjectsratherthanontheviewingplanepixels,splattingisanexampleofanobject-orderapproach,asop-posedtoanimage-orderapproach.Theresultingimagequalityisnotcomparablewithproperraycasting,butinteractivitymaybeaccom-plishedmuchmoreeasilysincecomputationallyexpensiveinterpolationisdropped.Nearest-neighbourinterpolation.Insteadofcomputingacostlytrilin-earinterpolationfromalleightvoxelsthat,asidefrombordersandcorners,surroundeachsamplingpoint,itmaysuﬃcetosimplydeter-minethesinglevoxelwhosecentreisclosesttothesamplingpoint(seeFig.2e).Thisreducesimagequalitybyinducinganti-aliasingartifacts,butgreatlyincreasesrenderingspeed.Optimizedaccumulation.Thepursuitofaraytravellingthroughthevol-umeinfront-to-backordercanbeabortedassoonasacertainopacitythresholdhasbeenreached,e.g.95%(seeFig.2f).Furthervoxelsarenotexpectedtomakeasigniﬁcantcontributiontotheoverallcumu-latedvalueanymore.Conversely,allvoxelswithanopacitybelowacertainthreshold,e.g.10%,maybetaggedduringpreprocessingandignoredduringopacityaccumulation.Furthermore,low-opacityvoxelswhichareembracedbyvoxelswithhighopacitiesmightbeneglectedaswell[Udu00].Shear-warptransformation.Inparallelprojection,hiddenpartremoval

canbespeededupvastlybyaddingashearandawarpsteptotheover-allviewingtransformation(seeFig.2g).First,thesceneisshearedbyshiftingtheslicesofthevolumesuchthatthepointsofintersec-tionofeachraywiththesliceplaneareinrowsparalleltooneofthescenecoordinateaxes.Theshearedsceneisprojectedontoanauxiliaryviewingplane.Second,theauxiliarybuﬀeriswarpedintothecorrectorientationandscale,andﬁnallycopiedintothemainimagebuﬀer.Voxelprojection.Shadingcanbespeededupinconjunctionwithashear-warptransformation.Insteadofapplyingtheconventionalraycasting

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3.1DirectVolumeRendering3VOLUMERENDERINGTECHNIQUES

routine,voxelsaretraversedinback-to-frontordersuchthatthefrontplaneofthevolumefacestheviewingplane(seeFig.2h).Theinputlightoftherearplaneofvoxelsisgivenbythebackgroundlight.Itsoutputlightiscomputedbytakingintoaccounttransmission(depend-ingonitsopacity),emission(dependingonitsdensity),andreﬂection(dependingonitsgradient),andinturnbecomestheinputlightforthesuccessorvoxel.Thelightthatisoutputbythefrontlayerofvoxelsisthenprojectedontotheviewingplane.3.1.2

Discussion

StrengthsInitsstandardcompositingstep,DVRtakesintoaccountallinformationcontainedinthevolume.Accordingly,ithasbeensuccess-fullyappliedtovisualizeclearstructuresthatdonotresultintoohighaninformation-per-pixelratio.Typicalsuchstructuresincludebloodvesselsandbones.ParticularstrengthsofDVRare:

•Completeness.Ifatransferfunctionisusedthatassignsnon-opaquetransparencyvaluestovoxels,everysinglevoxelinthevolumepoten-tiallycontributestotheﬁnalimage.Thus,thestructureofthefullvaluedistributioncanbevisualized,providedthatthishighlevelofsophisticationisrequired.

•Clarity.DVRrenditionstypicallyproviderealisticandfamiliarviewswithacomparativelygood3Dimpression[HMBG01],especiallywhenaperspectiveprojectionisemployed.

•Flexibility.Usingamultidimensionaltransferfunctionsacriﬁcessomeeaseofinterpretation,butopensupawholespectrumofpossibleap-pearancesforthevolume[ER00].

WeaknessesTheopticallimitationsofDVRbecomeapparentwhenvisu-alizingobjectswithacomplexinteriorstructureorwithoutaclearcorre-spondencebetweendatavaluesandobjecttype[HMBG01].Forexample,inMRIvolumes,airandbonetypicallyhaveverysimilardensities;hence,theyarenotdistinguishableintheﬁnalrendition.ThestrengthsofDVRareopposedbythefollowingweaknesses:

•Computationalcomplexity.Textureslicingcanmeanahugememoryoverhead,andyetdoesnottakeintoaccountthecomplexityoftheoutputimage.Raycasting,ontheotherhand,doesaddressthisis-sueand,moreover,isopticallycorrect.Obviously,itscomputational

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3.2Maximum-IntensityProjection3VOLUMERENDERINGTECHNIQUES

complexityismuchhigherasitneedstocomputethecoordinatesof

samplingpointsalongtheray,interpolatedensityvalues,andstoretheentiresceneinmemory.Butthevariantsdescribedabovecanspeeduptheprocesssigniﬁcantly.Infact,recentimplementationsdoreachinteractiveframeratesonordinaryGPUs.

•Opticallimitations.DVRrenditionsona2Dviewingmediumtendtolookoverloadedandblurry,becausemanyvoxelsareprojectedontoasinglepixelintheviewingplane[HMBG01].Forthesamereason,theyareintrinsicallysubjecttoambiguityastothedepthofvoxels.Thebestwaytoovercomethisisbymotionparallax.

•Setup.Itisnottrivialtoﬁndatransferfunctionthatyieldsthedesiredresults.Usuallysubstantialhand-tuningisindispensable[ER00].

3.2

3.2.1

Maximum-IntensityProjection

Method

Maximum-intensityprojection(MIP)isarenderingtechnique,verymuchakintoDVR,thatcreatesarenditionfromaspeciﬁcselectionofvoxelsofa3Ddataset.

InDVR,therenderingstageinvolvedcompositingallinterpolatedsam-plingpointsalongaraythathadbeencastthroughthevolume.MIPreliesonthesameprinciplesinpreprocessingandrendering,butitscompositingstepisdiﬀerent.

Insteadofusingatransferfunctionthatassigneachdensityvalueacertaincolourandopacity,MIPisusuallybasedonasimpleisotonicrelationbetweendensityandcorrespondingintensity,thatis,brightness.Further,insteadofaccumulatingcolourandopacityvaluesalongtheray,MIPmerelyprojectsthatsamplingpointontotheviewingplanethathasthehighestintensity.Inqualitativeterms,MIPcanbesaidtorenderthe‘mostimportant’voxelsonly.Itdoesnotpileupseverallayersofvoxelsontopofeachother,butmerelychoosesthemostoutstandingvoxelalongeachray.

InordertoillustratehowMIPcontraststoDVR,consideraformaldepic-tionofthecompositionprocess.Ifcarriedoutinback-to-frontorder,DVRshadingcanbedescribedas

Cdst←(1−αsrc)Cdst+Csrc,

whereCsrcdenotesthesourceradiancecomputedforthecurrentvoxel,αsrcitsopacity,andCdsttheaccumulatedradiance.Inotherwords,theupdate

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3.2Maximum-IntensityProjection3VOLUMERENDERINGTECHNIQUESequationtakesintheradiancethathasbeenaccumulatedsofar,Cdst,ﬁltersitaccordingtothetransparencyofthecurrentvoxel,αsrc,andaddstheradianceemittedbythecurrentvoxel,Csrc.InMIP,thisiterativeassignmentreducesto

Cdst←max(Cdst,Csrc),

whichalsoshowsthatthecompositingschemeisnowindependentfromtheorderoftraversalindepthdirection[EHK+06].

MIPisalmostalwaysusedinconjunctionwithparallelprojection.Fur-ther,atransferfunctionissometimesappliedthatmodulatesbothlightnessandcoloursaturationofthepixelsinordertomakeuseofcolour,e.g.inmedicalimages[HMBG01].AllotherstepsaretechnicallyidenticaltoDVR,andmostofitsvariantscanbeappliedlikewise.3.2.2

Discussion

StrengthsMIPdrasticallyreducestheinformation-per-pixelratioto1.Accordingly,itismosteﬀectivewhenthisreductioncanbemadeuseof.Thismaybethecasewhentheobjectsofinterestaresparse,haveasimpleshape,and,intermsoftheirdensityvalues,arebotheasytoseparatefromtheirsurroundingsandfairlyhomogeneouswithinthemselves[Udu00].MIPmayalsoprovepowerfulwhenthevolumecontainsverycomplexstructureswhichwouldoverloadtheﬁnalimageifitwasrenderedwithDVR.TheparticularstrengthsofMIPare:

•Noaccumulation.ThekeyadvantageofMIPisthatnoexpensivecompositingofRGBAvaluesisrequiredduringrendering.

•Nosegmentation.Anotheradvantageisthatitdoesnotnecessarilyrequiresegmentationofanysort.Onecouldarguethataformofsegmentationisimplicitlyincludedinchoosingthemaximumamongthevoxeldensitiesalongaviewingray.Butasubsetofsamplingpointschosenthiswayhasonlyaveryindirectrelationtotheboundariesoftheunderlyingobjectsystem.

MIPhasbeensuccessfullyusedforskin,softtissue,andothercom-plexinnerstructures.Afrequentclinicalapplicationisvirtualangiography[EHK+06].

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4TWO-LEVELVOLUMERENDERING

Weaknesses

•Poor3Dimpression.SinceMIPismostlyusedwithparallelprojection,itsrenditionslackareasonable3Dimpression.Infact,projectingonlythemaximum-intensityvoxelalongparallelraysimpliesthatrenditionsfromoppositeviewpointsaresymmetrical.Interactionisrequiredtoreconciletherenditionwiththeunderlyingobjectsystem[HMBG01].Moreover,sinceonlyasinglevoxelalongeachrayisprojectedontotheviewingplane,itisdiﬃculttocomprehendthegeometricrelationsamongtheshapesofobjects[Udu00].

•Lostcontext.SinceMIP,unlikeDVR,doesnottakeintoaccountsev-erallayersofvoxels,therenditionislimitedtothemostprominentstructuresinthevolume.Thecontextofweakerstructuresislost.•Noocclusion.Sincevoxelsalongaviewingrayareeﬀectivelyshowninanarbitraryz-order,nomutualhidingisaccountedfor[HMBG01].TherearefewsituationsinwhichtheidealconditionsforMIParemet.Frequently,otherbrightobjects,suchasbonesinCTimagesornoisefrommetalobjectsinMRI,havethehighestdensitiesandobscuretheoriginalregionsofinterest.Eveniftheseregionshavethehighestintensity,theymaynotbesparse,leadingtoanimageofplainhighbrightnessinwhichdetailsarelost[Udu00].AtypicalexampleforthisconditionistheMIPrenditionofaskull(seesection5.1).Thesituationcansometimesbeeasedbyapplyingamedianﬁlterthatremoves‘shotnoise’fromtherawdatabeforevisualization.

4Two-LevelVolumeRendering

Insection2,anumberofrecentimprovementsweredescribed,withap-proachesbeingconceptuallydividedintothoseforexploringmethodsandthoseforﬁndingsuitableparameters.

Two-levelvolumerenderingcannowbeunderstoodasanapproachthattacklesbothmethodsandparameterspaceexplorationbyfusingDVRandMIP.Itisconcernedwiththenotionofcombiningdiﬀerenttechniquesandparametervectors,andmaytherebyhelpuntangletheproblemofﬁndingasinglemethodandasetofparametersthatsuitawholevolume.

4.1Motivation

Despitetheirdiﬀerences,DVRandMIPshareacommonweakness:Ifthevolumecontainsseveraldistinctobjectsallofwhichhavearathercomplex

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4.2RenderingModel4TWO-LEVELVOLUMERENDERING

interiorstructure,thentheresultingrenditionislikelytobeoverloadedwithdetails.Whatisoftenrequiredisaclearrenditionofsomeinterestingstruc-turewhilethesurroundingcontextisonlyfaintlylit.Thisnotionoffocusandcontexthasbeensubjecttorecentresearch,andgreatlyexempliﬁesthecentralmotivationfortwo-levelvolumerendering.

AshasbecomeapparentinthediscussionofDVRandMIP,itheavilydependsonthenatureoftheunderlyingdatawhichrenderingtechniqueismostsuitabletoyieldthedesiredresults.Ifsegmentationinformationwasgiventhatseparatesdistinctobjectsinthevolume,thenonemightnotonlyemployindividually-tunedtransferfunctionsbutevenusedistinctrenderingtechniquesfordiﬀerentkindsofobjects.Therenderingprocesswouldturnintoanestedprocessoftwolevels:alocallevelforindividualobjectsandagloballevelfortheoverallrendition.Thisistheideaoftwo-levelvolumerendering[HMBG01].

4.2RenderingModel

Therenderingmodeloftwo-levelvolumerenderingisbasedonraycasting.However,forperformancereasons,theactualrealizationemploystheshear-warptransformationdescribedinsection3.1.1.Thismeansthateveryvoxelismappedontoexactlyonepixelintheviewingplane,anditimpliesthatvoxelscanbeprocessedinback-to-frontorder.Conceptually,raysareshotandpursuedthroughthevolume;practically,voxelsaretraversedfrombacktofrontinparallel.

Shear-warprenderingrequiresthattheboundingplaneofthevolumewhichisclosesttotheviewingplaneisknown.Thisinformationcanbecapturedbygivingthecoordinateaxisthatisclosesttobeingparallelwiththeviewingaxis.Callthistheprincipalviewingdirection,orpvd.

Theonlypreliminaryrequirementfortwo-levelvolumerenderingisa3Dsegmentationmaskwhichassignsauniqueobjectidentiﬁertoeveryvoxel.Asaresult,themaskpotentiallypartseveryrayshotintothevolumeintoseveralsegments.

Asthetwo-levelvolumerenderingraycastingroutinetraversesalongaray,ittakesaccountoftheobjectIDofthecurrentsamplingpoint.AslongastheobjectIDdoesnotchange,thesamplingpointsareaccumulatedintoalocalbuﬀer,dependingonthatobject’slocalrenderingtechnique,DVRorMIP.Assoonasanewobjectisentered,however,thislocalbuﬀerisaccu-mulatedintoaglobalbuﬀer,dependingontheglobalrenderingtechnique,typicallyDVR.Astherayexitsthevolume,thelastlocalbuﬀerisaddedtotheglobalbuﬀer,whosecolourvalueinturnisassignedtothecorrespondingpixelintheviewingplane.

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4.3Implementation4TWO-LEVELVOLUMERENDERINGvoxelArray[z]x,y,lastVisibleR,G,B,Alast...firstz,objId,rendMode,opacityrenderList[z]...Figure3:Datastructurescreatedduringpreprocessingfortwo-levelvolumerendering.Eachboxrepresentsasinglearrayelement,withitsﬁnalﬁeldsindicated.Thevoxelarrayandtherenderlistaredepictedforthezaxis;theyarecreatedfortheothertwoaxesaswell.

4.3Implementation

Theimplementationoftwo-levelvolumerendering,asproposedbyHauseretal.[HMBG01],realizestherenderingmodeldescribedinsection4.2.PreprocessingThepreprocessingstageinvolvestheconstructionoftwosetsofdatastructuresthatwillbeusedduringrendering(seeFig.3).•Voxelarray.Threearraysarecreated,calledvoxelarrays,oneforeachprincipalviewingdirection(pvd).

Consider,withoutlossofgenerality,thezvoxelarray.Thearrayisinitializedwiththevoxelsofthegivendataset.Initially,eacharrayelementisan(x,y,z,density)-tuple.AnysegmentationinformationavailableisthenusedtoassignauniqueobjectidentiﬁerasanobjIdﬁeldtoeachvoxel.Similarly,thegradientofeachvoxelisprecomputedandaddedasagradientﬁeld.Sinceparallelprojectionwithshear-warprenderingwillbeused,nocomputationofsamplingpointswillberequired.Thus,thetransferfunctioncanbeapplieddirectlytothevoxelsbeforerendering.Thisway,thedensityandgradientﬁelds

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4.3Implementation4TWO-LEVELVOLUMERENDERING

inthearrayarereplacedby(R,G,B,A)ﬁeldsspecifyingcolourandopacityofthevoxel.

Atthisstage,thearraycontainstheﬁeldsx,y,z,objId,R,G,B,andA.Itissortedaccordingtoitszcoordinate.

•Renderlist.Threefurtherarraysareconstructed,calledrenderlists,oneforeachpvd.

Eachrenderlistisinitializedwithentriesforallobjectsinthescene.Atthisstage,anentrycontainstheobjIdofitsrespectiveobjectonly.Consideragainthezrenderlist.If,say,segmentationhasseparatedbone,vesselsandtissue,thelistcontainsthreeentries.ThevoxelsassignedtoeachobjIdarenowtraversedinthevoxelarray.Foreachobject,thesinglerenderlistentryisturnedintoanumberofentrieseachofwhichcontainsadistinctzcoordinatethatoccurssomewhereinthatobject’svoxelarray,suchthatallappearingzcoordinatesarecoveredbyoneandonlyonerenderlistentry.

TheobjIdandzﬁeldsinthezvoxelarrayentriesarenowredundant.Forthatreason,insteadofstoringtheminthevoxelarray,theyareremovedandstoredonlyonceintheircorrespondingrenderlisten-try.Thisentryholdstwopointers,firstandlast,whichindextheﬁrstandthelastvoxelbelongingtoit,respectively.Athirdpointer,lastVisibleindicatesthelastvoxeltobeconsideredduringrender-ing.Thisway,clippingcanbeeﬃcientlyintegratedintopreprocessingbysimplymovingallvoxelstobeclippedawaytothebottomofeachzsegmentinthevoxelarray,andhavingthelastVisibleﬁeldpointtothelastvisiblevoxel.

Finally,eachrenderlistentryalsocontainsﬁeldsindicatingitsobject’slocalrenderingmode(e.g.DVRorMIP)andtheobject’sgeneralbase-lineopacitysettingwhichallowstheusertogenerallyincreaseorre-duceanobject’stransparency.Therenderlistnowcontainstheﬁeldsz,objId,rendMode,opacity,first,lastVisible,andlast.Thepreprocessingphaseverballydescribedabovecanbeformalisedasfollows(adaptedfrom[HMBG01]):

PreProcessRenderLists{

FOREACHpvdIN{x,y,z}{FOREACHobjId{

FOREACHpvdValue(depthvaluesconcerningpvd){first=indexoffirstvoxelINvoxelArray[pvd]

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4.3Implementation4TWO-LEVELVOLUMERENDERING

Figure4:Conceptually,arayisshotfromtheeyepositionthroughtheviewingplaneintothevolume.Thesegmentationboundarybetweenthetwosegments,skullandvessels,maybecrossedmanytimes.

WHERE((voxel.objId==objId)AND(voxel.[pvd]==pvdValue));

lastVisible=indexoflastvoxelinvoxelArray[pvd]WHICHisnottobeclippedawayWHERE((voxel.objId==objId)AND(voxel.[pvd]==pvdValue));

last=indexoflastoverallvoxelinvoxelArray[pvd]WHERE((voxel.objId==objId)AND(voxel.[pvd]==pvdValue));

renderList[pvd].addEntry(objId,pvdValue,first,lastVisible,last);}}}}

RenderingItistheinterleavedworkingofthelocalandtheglobalbuﬀerthatconstitutesthecoreideaoftwo-levelvolumerendering(seeFig.4).

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4.3Implementation4TWO-LEVELVOLUMERENDERING

•Localbuﬀer.Thelocalbuﬀer,orobjectbuﬀer,isusedforwithin-objectrendering.Foreachpixelintheviewingplane,itholdsacolour,anopacity,andanobjIdvalue.Itisinitializedwithallzeros.

•Globalbuﬀer.Inter-objectrenderingisaccomplishedwiththeglobalbuﬀer.Liketypicalbuﬀersinconventionalrenderingtechniques,itholdsasimplecolourvalueforeachpixelintheviewingplane.Itisinitializedwiththebackgroundcolourofthescene.

Dependingontheviewingdirection(pvd),therenderingroutineworkswithtwoofthesixpreprocesseddatastructures,voxelArray[pvd]andrenderList[pvd].Foreachitemintherenderlist,itaccumulatesthepro-jectionsofthecorrespondingvoxelsinthelocalbuﬀer.IfthecurrentobjIdofthecorrespondingpixelinthelocalbuﬀerisdiﬀerenttotheobjIdofthecurrentrenderlistentry,thenthatpixelisremovedandmergedwiththerespectivepixelintheglobalbuﬀer.Thisway,theimplementationwalksthroughallviewingraysinparallel,workingitswaythroughthevolumeinback-to-frontorder.Afterthelastentryhasbeenprocessed,thelocalbuﬀerhastobemergedwiththeglobalbuﬀeronelasttime.Theglobalbuﬀeristhenwarpedtoyieldtheﬁnalimage.

Theprojectionstepassignseverypixelthatjoinsthelocalbuﬀeracolour.Typically,distinctcoloursareusedfordiﬀerentobjectsinascene,withthetransferfunctionmerelyalteringsaturationasdeterminedbythecurrentdensityandgradient.Otherimplementationsareequallypossible,though.Formally,therenderingprocedurecanbesummarisedasfollows(adaptedfrom[HMBG01]):

RenderVolume{

FOREACHentryINrenderList[pvd]{FOREACHvoxelINvoxelArray[pvd]{

IF(entry.first<=voxel.id<=entry.lastVisible){pixel=project(voxel,pvd);

IF(localBuffer[pixel].objId==entry.objId){

localBuffer[pixel].include(pixel,entry.rendMode);}ELSE{

globalBuffer[pixel].include(localBuffer[pixel]);localBuffer[pixel].clear();

localBuffer[pixel].include(pixel,entry.rendMode);}}}

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5APPLICATIONS

}

FOREACHpixelinlocalBuffer{

globalBuffer[pixel].include(localBuffer[pixel]);}

image=globalBuffer.warp();}

PerformanceTwo-levelvolumerenderingnotonlyallowsforusingindi-vidualrenderingtechniques,transferfunctions,andcompositingmodesforsegmentedobjects,butalsostrivesforinteractivity.Severalpartsoftheim-plementationenhancetheperformanceoftherenderingprocesssigniﬁcantly:•Clipping.Asindicatedabove,thestructureofthevoxelarraysallowsforstraight-forwardclipping:VoxelstobeclippedawaycanbemovedbeyondthelastVisibleboundarywithineachrenderlistsegment.•Emptyspaceskipping.Objectsmaynotspanthewholeofthevolume.Theresultingemptyvoxelscanbeskippedbyexcludingthemfromthepreprocessedvoxelarrays.

•Projection.Insteadofusingexpensiveraycasting,theshear-warpap-proximationisused.Voxelsarenotinterpolated,butprojectedontoexactlyonepixel.

•Shading.Duringpreprocessing,thegradientofeachvoxelinpolarcoordinatesiscomputedandquantisedto12bit.Alookuptableisthenassembledthatmapsagivengradienttoitslightingintensity.ThisintensitymaybetheresultofPhongshadingwithaﬁxedroughnesscoeﬃcientandambientlighting.

5Applications

Hauseretal.[HMBG01]considertwoexamplesofvolumevisualizationinwhichtwo-levelvolumerenderingresultsinimprovedimagequalitywhilemaintainingareasonablelevelofinteractivity.Theirﬁndingsarebrieﬂydescribedinthefollowingtwosubsections.

5.1MedicalDataVisualization

Experimentsonmedicalvolumedatahaveshownthatdiﬀerentkindsoftissuerequirediﬀerentrenderingmethods.Two-levelvolumerenderingmaybeof

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5.2DynamicalSystemVisualization5APPLICATIONS

Figure5:Left:ConventionalDVRrendition.Right:Two-levelvolumeren-dering,withDVRbeingusedfortheskullandMIPforthevessels.Onlythelatterclearlyresolvesbothvessels(focus)andskull(context).

particularhelpwhenrenderingcomplexvolumescontainingseveraldistinctobjectsofdiﬀerentphysicaltype.

DVRhasbeensuccessfullyappliedtotissuetransitions.Itemphasizesthe3Dstructureandshapeofanobject,aswellasprovidingsuﬃcientdepthinformation.

MIP,incontrast,hasprovenespeciallyusefulwhenrenderingmorecom-plexsystemssuchaswholeorgans.Inthesecases,DVRtendstooverloadimageswhereasMIPfocusesontheimportantpartsofthedata.

Two-levelvolumerenderingnowallowstheusertoselectindividualun-derlyingrenderingtechniquesforthediﬀerenttypesofobjects.TheexampleinFig.5showshowtherenditionofaskullwasimprovedbytwo-levelvol-umerendering[HMBG01].Bothimageswererenderedinlessthan200msonordinaryconsumerhardwarein2001.

5.2DynamicalSystemVisualization

Inspiredbyeconomics,Hauseretal.haveinvestigatedadiscretedynamicalsystem.Itisdescribedbytheﬁnitediﬀerenceequationxt+1=fp(xt),wherext∈X⊆Rn,t∈N0,isastatevectorwhoseevolutionisspeciﬁedbyatransitionfunctionfp:X→Xbasedonaparametervectorp∈Π⊆Rm.Analysisisconcernedwiththesequenceofstates(xt)t∈N0.

•Forgivenx0andp,theattractingsetisthestate,orsetofstates,thesystemarrivesatinthelongrun.

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6DISCUSSION

•Foragivenattractingsetandaparametervectorp,thebasinofat-tractionisthesetofcorrespondinginitialconditionsx0.Abasinisabasinofattractionwhoseattractingsetliesinsideitself.

Atypicalrenderingtaskis,giventhatseveralattractingsetscoexist,tovisualizetheboundarybetweentheirsurroundingbasins.Hauseretal.havefoundthatparticularlyexpressiveimagescanbegeneratedwhenusingMIPfortheattractingsetandDVRforthebasins’boundaries.

6Discussion

Two-levelvolumerenderinghasthepotentialofimprovingimagequalityinadistinctway:Itgeneratesimagesinwhichdiﬀerentobjectsaremoreeasilydistinguishablethanwithasinglerenderingmethodonitsown.Inaddition,theimplementationpresentedbyHauseretal.showsreasonableinteractivityduetoanumberofperformanceoptimizations.

Morefundamentally,two-levelvolumerenderingoﬀersaprincipledwayofseparatelyprocessingobjectsofwhichonlyrawsegmentationinformationisknown.Thisway,itcombinesthepartialstrengthsofthetworenderingmethodsdescribedinsection3.ComparedtoMIP,theproblemofdisplayingseveralstructuressimultaneouslyisovercome;multiplestructuresaretrulymerged.ComparedtoDVR,asolutiontotheproblemofoverloadedpicturesisproposed;innerstructuresarefusedwithanouterhullwhichhasafairlyuniformtransparency,makingtheinnerobjectstandoutclearly.

PracticalapplicationshaveshownthatparticularlyconvincingresultscanbeobtainedwithasettinginwhichDVRisusedforinnerstructuresandMIPfortheoutercontext.

Thedisadvantagesoftwo-levelvolumerenderingareduetoitsincreasedcomplexity.Trackingtwobuﬀersinparallelimpliesbothadditionalcompu-tationalcostsandfurthermemoryusage.ItalsomovesthealgorithmfurtherawayfromwhatisnativelysupportedbytypicalGPUs.

Toallowforboth,theideaofalternatingbetweenafullyfeaturedtwo-levelvolumerenderingandtheoptiontostepbacktotheusageofasin-glebuﬀerinwhichvoxelprojectionsareaccumulatedhasbeenproposedbyHadwigeretal.[HBH03].

Two-levelvolumerenderinghasbeendevelopedfurtherinanumberofsubsequent,relatedpublications.Hadwigeretal.takethetechniquetohardwarelevel[HBH03].Mlejnekincorporatesitinaframeworkforin-teractivevolumerenderingofﬂowsimulationdata[Mle03].Doleischet

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REFERENCESREFERENCES

al.considertheinteractivespeciﬁcationofhigh-dimensionalfeaturesinsim-ulationdata[DGH03].Bruckneretal.usetwo-levelvolumerenderinginconjunctionwithvolumeillustration[BGKG05].

Otherresearchers,suchasZhouetal.arguethateventwo-levelvolumerenderinglackstheabilitytountanglestructuressuﬃcientlyforillustrationpurposes[ZHT02].Theyfocusonnon-photorealisticeﬀectsinstead.Nomattertowhatextentrenderingtechniquesareadvanced,however,thequestionwhichmethodstousecanhardlybedecidedsolelyonthegroundsoftheirrespectiveadvantagesanddisadvantages.Instead,italsostronglydependsonwhattypeofdataisgiven(e.g.MRI),whatstructuretheobjectsrepresent(e.g.complextissuevs.simplebones),andwhattherenderingobjectivesare(e.g.overview,speciﬁcdetails,orfocusandcontext).

Thisindicatesthespectrumofgeneraldiﬃcultiesin3Dimaging[Udu00],aboveandbeyondthequestionsofvisualizationmethods(section2.1)andparameters(section2.2).Ontheonehand,diﬃcultiesarerelatedtoobjectdeﬁnition.Thisinvolvesdeterminingandminimizingthevolumeofinterest,ﬁlteringthedata,interpolatingdensitiestoacertainlevelofdiscretization,combiningimagesofthesameobjectsystemfromdiﬀerentmodalities,andsegmentingphysicallydisparatestructures.Ontheotherhand,generaldiﬃ-cultiesarerelatedtoissuesofvalidation.Assessingarenditionqualitativelymeanstorateitsvisualmanifestationofphysicalpropertiesintheunderly-ingvolume.Validatingquantitativelyreferstothemeasurementofphysicalreliability,visualaccuracy,andoveralleﬃciency.

Inthiscontext,two-levelvolumerenderingcanbeunderstoodasatech-niquetoimprovethequalitativevalidationofpreprocessedsegmentation.Two-levelvolumerenderinghasbeenshowntobeasimpleyetpowerfulwayofcombiningtheadvantagesofexistingrenderingtechniqueswhicharenotlimitedatalltoDVRandMIP.Itswiderangeofexistingandpotentialapplicationsmakeitanexcellentcandidateforbothimprovedsupportingraphicsboardsandincreasedusageinpracticalrenderingapplications.

References

[BGKG05]S.Bruckner,S.Grimm,A.Kanitsar,andM.E.Gr¨oller.Illustra-tivecontext-preservingvolumerendering.Proc.ofEuroVis’05,

pages69–76,2005.

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REFERENCES[DGH03]

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HelmutDoleisch,MartinGasser,andHelwigHauser.Interactivefeaturespeciﬁcationforfocus+contextvisualizationofcomplexsimulationdata.InVISSYM’03:Proceedingsofthesympo-siumonDatavisualisation2003,pages239–248,Aire-la-Ville,Switzerland,Switzerland,2003.EurographicsAssociation.

[EHK+06]KlausEngel,MarkusHadwiger,JoeM.Kniss,ChristofRezk-Salama,andDanielWeiskopf.Real-timevolumegraphics.AKPeters,Ltd.,2006.[ER00]

D.EbertandP.Rheingans.Volumeillustration:Non-photorealisticrenderingofvolumemodels.ProceedingsofIEEEVisualization,pages195–202,2000.

M.Hadwiger,C.Berger,andH.Hauser.High-qualitytwo-levelvolumerenderingofsegmenteddatasetsonconsumergraphicshardware.ProceedingsofIEEEVisualization,14:301–308,2003.

[HBH03]

[HMBG01]H.Hauser,L.Mroz,G.-I.Bischi,andE.Gr¨oller.Two-levelvol-umerendering.IEEETransactionsonVisualizationandCom-puterGraphics,7:242–252,2001.[MAB+97]J.Marks,B.Andalman,P.A.Beardsley,W.Freeman,S.Gibson,

J.Hodgins,T.Kang,B.Mirtich,H.Pﬁster,W.Ruml,K.Ryall,J.Seims,andS.Shieber.Designgalleries:Ageneralapproachtosettingparametersforcomputergraphicsandanimation.Pro-ceedingsofACMSTGGRAPH,pages389–400,1997.[Mle03][Pﬁ04][Sim91][Udu00][ZHT02]

MatejMlejnek.Feature-basedvolumerenderingofsimulationdata.Technicalreport,VRVisResearchCenter,Vienna,2003.H.Pﬁster.ModerneVolumenvisualisierung.InformationTech-nology,46:117–122,2004.

KarlSims.Artiﬁcialevolutionforcomputergraphics.ComputerGraphics,25:319–328,1991.

J.K.Udupa.3Dimaging:Principlesandapproaches,chapter1,pages1–74.CRCPress,2edition,2000.

JianlongZhou,ManfredHinz,andKlausD.Tonnies.Hybridfocalregion-basedvolumeredneringofmedicaldata.InBildver-arbeitungf¨urdieMedizin,pages113–116,2002.

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