PostVrolijkHauserLarameeDoleischFeatureExtractionandVisualisationofFlowFieldsintdata acquuser percdatacompabcvisualizationdata acquuser percdata acquuser percflow intdirect visvis ID: 213432
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EUROGRAPHICS2002STARStateofTheArtReportFeatureExtractionandVisualisationofFlowFieldsFritsH.Post,BenjaminVrolijkandHelwigHauser,RobertS.Laramee,HelmutDoleisch{F.H.Post,B.Vrolijk}@its.tudelft.nl{Hauser,Laramee,Doleisch}@vrvis.atDelftUniversityofTechnology,TheNetherlandsVRVisResearchCenter,Austriahttp://visualisation.tudelft.nl/http://www.vrvis.at/Abstractvisualisationhasalreadybeenaveryattractivepartofvisualisationresearchforalongtime.Usuallyverylargedatasetsneedtobeprocessed,whichoftenconsistofmultivariatedatawithalargenumberofsamplelocations,oftenarrangedinmultipletimesteps.Recently,thesteadilyincreasingperformanceofcomputersagainhasbecomeadrivingfactorforanewboominowvisualisation,especiallyintechniquesbasedonfeatureextraction,vectoreldclustering,andtopologyextraction.Inthisstate-of-the-artreport,anattemptwasmadeto(1)provideausefulcategorisationofFlowVissolutions,(2)giveanoverviewofexistingsolutions,and(3)focusonrecentwork,especiallyintheeldoffeatureextraction.Inseparatesectionswedescribe(a)directvisualisationtechniquessuchashedgehogplots,(b)visualisationusingintegralobjects,suchasstreamlines,(c)texture-basedtechniques,includingspotnoiseandlineintegralconvolution,and(d)techniquesbasedonextractionoffeaturesorowtopology.CategoriesandSubjectDescriptors(accordingtoACMCCS):I.3[ComputerGraphics]:visualisation,owvisuali-sation,computationalowvisualisation1.IntroductionComputershavebecomeincreasinglyimportantinmanyas-pectsofsocietyinscience,businessandeconomics,ed-ucationandpolitics,aswellasinmanyotherelds,com-putersareusedtoacquire,store,process,andcommunicatedata,notintheleasttousers.Visualisation,asaseparateeldofresearchanddevelopmentincomputerscience,addressesexactlythisbridgebetweendataanduser:visualisationso-lutionshelpuserstoexplore,analyse,andpresenttheirdata.Inowvisualisation(FlowVis)oneofthetraditionalsubeldsofvisualisationarichvarietyofapplicationeldsisgiven,formtheautomotiveindustry,aerodynam-ics,turbomachinerydesign,weathersimulationandmete-orology,climatemodelling,groundwaterow,medicalap-plications,etc.,withsignicantlydifferentcharacteristicsre-latingtothedataandusergoals.Consequently,thespectrumofFlowVissolutionsisveryrich,spanningmultipledimen-sionsoftechnicalaspects,e.g.,2Dvs.3Dsolutions,tech-niquesforsteadyandtime-dependentdata,etcetera.1.1.AspectsofFlowVisualisationBringingmanyofthosesolutionsinalinearorder(asneces-saryforatextlikethis),isnotatalleasyorintuitive.Severaloptionsofsubdividingthisbroadeldofliteraturearepos-sible.Hesselinketal.,forexample,addressedthedifcultproblemofhowtocategoriseFlowVistechniquesintheir1994overviewof(atthattime)recentresearchissues33.Inthefollowingsubsectionsseveralofthoseaspectsaredis-cussedonahigherlevel,beforeliteratureisaddresseddi-rectlylater.Directvs.integration-basedvs.feature-basedowvisualisationtothedifferentneedsoftheuserstherearediffer-entapproachestoowvisualisation(cf.Figure1c).Oneistododirectowvisualisationbyusinganasdirectaspossibletranslationoftheowdataintovisualisationcues,suchasbydrawingarrows.FlowVissolutionsofthiskindallowimme-diateinvestigationoftheowdata,withoutalotofmentaltranslationeffort.Forabettercommunicationofthelong-termbehaviourin-ducedbyowdynamics,integration-basedapproachesrstc\rTheEurographicsAssociation2002. Post,Vrolijk,Hauser,Laramee,Doleisch/FeatureExtractionandVisualisationofFlowFieldsint.data acqu.user perc.datacomp.(a)(b)(c)visualizationdata acqu.user perc.data acqu.user perc.flow int.direct vis.vis.Figure1:Directowvisualisation(a)vs.FlowVisbasedonowintegration(b)vs.FlowVisbasedonderiveddatasuchasowfeaturesorowtopology(c).Thisclassicationre-latestotherst-levelstructureofthisreport.integratetheowdataanduseresultingintegralobjectsasbasisforvisualisation,e.g.,usingstreamlinesforvisualisa-tion.Anotherapproachforvisualisingowdataisthefeature-basedapproach,inwhichanabstractionstepisperformedrst.Fromtheoriginaldataset,interestingobjectsareex-tracted,suchasimportantphenomenaortopologicalinfor-mationoftheow.Theseowfeaturesareanabstractionofthedata,andcanbevisualisedefcientlyandwithouttheoriginaldata.Becausetheoriginaldataisnotneededany-more,ahugedatareductionisachieved,ofafactor1000ormore.Thismakesthisapproachverysuitableforlarge(time-dependent)datasets,originatingfromcomputationaluiddynamicssimulations.Thesedatasetsaresimplytoolargetovisualisedirectly,andtherefore,alotoftimeisre-quiredinpreprocessing,forcomputingthefeatures(featureextraction).Butoncethispreprocessinghasbeenperformed,visualisationcanbedoneveryquickly.Inthisoverviewweuseseparatechaptersfortheafore-mentionedclassesofapproaches:directowvisualisationisdiscussedinSection2,integration-basedFlowVisinSec-tions3and4,andfeature-basedFlowVisisdescribedinSec-tions5through8.Figure1illustratesthedifferencebetweentheaforementionedclassesnotetheincreasingamountofcomputationspentwithinthevisualisationstepwhenchang-ingfromdirectFlowVis(a)tofeature-basedFlowVis(c).Spatialdimensionsvs.timeInowvisualisation,availablesolutionssignicantlydifferwithrespecttothegivendimensionalityoftheowdata.Techniqueswhichareusefulfor2Ddata,likecolourcod-ingorarrowplots,sometimeslacksimilaradvantagesin3D.Also,thequestion,whethertheowdataissteadyortime-dependent,usuallymakesabigdifferencewithrespecttotheFlowVissolutionofchoice.Inthisstate-of-the-artreport,we(atleastpartially)sub-structurethesectionsaboutdifferentclassesofFlowVisso-lutionsintosubsectionswithrespecttodifferentspatialdi-mensionsinvolved.Althoughtherearelotsofinterestingworksabout1DFlowVisaswellasnDFlowVis(withn�3),thisreportclearlyfocusesontwoandthreespatialdimen-sions.Below,thetop-levelsectionsstartwithasubsectionon2DFlowVistechniques(Sectionsn.1),i.e.,coveringsolutionswhichfocuson2Dowdata(in2Ddomains).Sincethe2Ddomaininherentlycorrespondstothe2Dscreen,goodoverviewsarepossibleforthesekindsoftechniqueslikewiththeuseof2DLIC(seebelowfordetails).However,thereadershouldbeaware,thatreal-worldows(atleastwhentalkingaboutuidsorgases)arerarelytwo-dimensionaldatasetsthereforeareoftenslicesoutofastackofthose,orstemfromsimplicationsoftheunderlyingmodel.Asecondsubsection(Sectionsn.2)discussesFlowVisso-lutionsforboundaryowsorsectionalsubsetsof3Dows,forexample,owdataonplanarcrosssections.Thissubsec-tionthereforedealswith2Dowdata,atleastwithrespecttothelocaldimensionalityofthedata,butwhichisembeddedwithin3Dspace.Whereasboundaryowsoftenareprimar-ilyinterestingtotheuseranyway(forexampleinaerospacedesign),thevisualisationofsectionalsubsetsof3Dowusu-allyneedsspecialcare(notattheleastbecauseoftheusuallymissingthirdowcomponent).Especiallytheuseofintegralcurvesacrossowcrosssectionsisquestionableasthesug-gestedparticlepaths(ingeneral)donotcorrespondtoactualowtrajectorieswhichnaturallyextendto3Dinthiscase.Finally,athirdsubsection(Sectionsn.3)discussestruly3DFlowVissolutions,i.e.,visualisationtechniques,whichapplytotrue3Dowdata.Withtrue3DFlowVis,renderingbecomesacentralissueinmanycasescompromisesareneeded,tradingvisibilityforcompleteness.Solutionsrangefromclippingandopacitymodulationstofeature-basedse-lections.Inadditiontothespatialdimensionsasaddressedabove,alsodimensionalitywithrespecttotimeisofgreatimpor-tanceinowvisualisation.Firstofall,owdataitselfin-corporatesanotionoftimeowsoftenareinterpretedasdifferentialdatawithrespecttotime,i.e.,whenintegratingthedata,pathsofmovingentitiesareobtained.Additionally,theowitselfcanchangeovertime(likeinturbulentows,forexample),resultingintime-dependentorunsteadydata.Inthiscase,twodimensionsoftimearepresentandthevi-sualisationmustcarefullydistinguishbetweenbothinordertopreventtheuserfrombeingconfused.Thisisespeciallytrue,whenanimationshouldbeusedforowvisualisation.Then,evenathirdtemporaldimensioncanshowupinavi-sualisation,requiringspecialcaretoavoidconfusionalongwithinterpretationoftheanimations.AlthoughthedistinctionbetweensteadyandunsteadyowscouldopenanotherdimensionwhensortingFlowVisliterature,inthisreportsolutionsfortime-dependentdataareputbesiderelatedtechniquesforsteadydata.c\rTheEurographicsAssociation2002. Post,Vrolijk,Hauser,Laramee,Doleisch/FeatureExtractionandVisualisationofFlowFieldsComputationalvs.experimentalandempiricalFlowVisFlowvisualisation,asdiscussedinthisliteratureoverview,isconsideredtobeequivalenttowhatotherscallcompu-tationalowvisualisationjusttodistinguishitfromthelargeandoldeldsofexperimentalandempiricalowvisu-alisation.AlthoughwedonothavespacetoalsofocusonthoseothervariantsofFlowVis,itisinterestingtorecognisethatmanycomputationalFlowVissolutionsmoreorlessmimicthevisualappearanceofwell-acceptedtechniquesinexper-imentalvisualisation(cf.particletraces,dyeinjection,etcetera).fromsimulationvs.measurementsormodelsComputationalFlowVis,ingeneral,dealswithdatathatex-hibittemporaldynamicssuchasresultsfromowsimulation(e.g.,thesimulationofuidowthroughaturbine),owmeasurements(possiblyacquiredthroughlaser-basedtech-nology),oranalyticmodelsofows(e.g.,dynamicalsys-tems,givenassetofdifferentialequations).Inthisreportwemainlyfocusonowvisualisationdeal-ingwithdatafromowsimulation,i.e.,owdatagivenasasetofsamplesonsomekindofgrid,whereassolutionsfordatafromowmeasurementsorowmodellingareonlyad-dressedinlessdetail.Technicalissuesfrequentlyariseduetothecombinationofextremelylargedatasetsanddemandinguserrequirementssuchasinteractivevisualisationoftime-dependentdata.Therefore,solutionsintheeldofparallelcomputing11;60;138;170,out-of-corerendering147,andrender-ingofcompresseddata166areoftendiscussedintheFlowVisliterature.andinteractionManyFlowVissolutionsbuildontheuseofindividualvisu-alisationobjects,forexample,streamlines.Foratleastthreereasons,theplacementofthosevisualisationcuesisanis-suewithinFlowVisliterature:(1)whenusingintegralob-jectssuchasstreamlines,anevendistributionofseedloca-tionsusuallydoesnotresultinanevendistributionofin-tegralobjectsseparatealgorithmsneedtobeemployed;(2)whendealingwith3Dowdata,occlusionand/orvi-sualisationcomplexityraisesspecialchallengesdenseplacementoftenresultsinsevereclutteringwithinrenderedimages;(3)whenusingfeature-basedstrategies,placementneedstobecoupled(andaligned)withthefeatureextractionpartsofthevisualisation.Inadditiontoplacement,userinteractionplaysanim-portantrole,especiallyincaseofowanalysis.Usersre-quiresystemswhichallow(1)navigation,includingzoom-ing,panning,etc.,(2)interactiveplacementofvisualisationcues,forexample,usinganinteractiverakeforstreamlineseeding,aswellasothermeanstoinuencethevisualisa-tion,oreven(3)optionsofinteractingwiththeowdata,forexample,throughsteering.Lastbutnotleasthuman-computerinteractionchallengespresentthemselvesthroughoutowvisualisationresearch,especiallyinthecategoriesofperceptionin3D,andinter-action.Forthereisstrongevidencethatboth3Dvisualisa-tion154andinteraction34areveryimportantcomponentsfortheuserinunderstandingthedata.1.2.FlowVisFundamentalsBeforeoutliningsomeofthemostimportantFlowVistech-niquesinthemainpartofthispaper,ashortoverviewaboutthecommonmathematicalbackgroundaswellassomegen-eralconceptswithregardtothecomputationofFlowVisre-sultsarediscussed.FlowdataAninherentcharacteristicofowdataisthatderivativein-formationisgivenwithrespecttotime,whichislaidoutacrossann-dimensionaldomainWRn,forexample,forrepresenting3Duidow(n=3).Inthecaseofmultidi-mensionalowdata(n�1),temporalderivativesvofnDlo-cationspwithintheowdomainWaren-dimensionalvec-tors:v=dp=dt;p2WRn;v2Rn;t2R(1)Inanalyticmodels(likedynamicalsystems),vectorsvof-tenaredescribedasfunctionsoftherespectivespatialloca-tionsp,saylikev=ApforsteadylinearowdataifAisaconstantnn-matrix.Ageneralformulationof(possiblyunsteady,i.e.,time-dependent)owdatavwouldbev(p;t):WP!Rn(2)wherep2WRnrepresentsthespatialreferenceoftheowandt2PRrepresentsthesystemtime.Iftisconsideredtobeconstant,i.e.,forsteadyowdata,themoresimplecaseofv(p):W!Rnisgiven.IncasesofresultsfromnDowsimulation,likeinauto-motiveapplicationsorairplanedesign,vectordatavusuallyisnotgiveninanalyticform,butneedstobereconstructedfromthe(discrete)simulationoutput.Asusuallynumericalmethodsareusedtoactuallydotheowsimulationsuchasniteelementmethods.Theoutputofowsimulationusu-allyisalarge-sizedgridofmanysamplevectorsvi;t,whichdiscretelyrepresentthesolutionofthesimulationprocess(attimestepst).Forfurtherprocedure,itisassumedthattheowsimulationwasbasedonan(atleastlocally)continuousmodeloftheow,thusallowingforcontinuousreconstruc-tionoftheowdatavduringvisualisation.Oneoptionfordoingsowouldbetoapplyareconstructionlterh:Rn!Rtocomputev(p;t)=åih(p pi)vi;t.Asforpracticalreasonslterhusuallyhasonlylocalextent(aroundtheorigin),efcientproceduresforndingthoseowsam-plesvi;t,whicharenearesttothequerypointp,areneededtodoproperreconstruction.c\rTheEurographicsAssociation2002. Post,Vrolijk,Hauser,Laramee,Doleisch/FeatureExtractionandVisualisationofFlowFieldsGridsowsimulation,thevectorsamplesvi;tusuallyarelaidoutacrosstheowdomainaccordingtoacertaintypeofgrid.GridtypesrangefromsimpleCartesiangridsovercurvilineargridstocomplexunstructuredgrids.Typically,simulationgridsalsoexhibitlargevariationsincellsizes.Thisvarietyofgridsstemsfromthehighinuenceofgriddesignontotheowsimulationprocessandthetherebyde-rivedneedtomodeltheowgridasoptimalaspossiblewithrespecttothesimulationinmind.Althoughtheprincipaltheoryoffunctionreconstructionfromdiscretesamplesdoesnotexhibittoomanydifferenceswithrespecttogridtypesinvolved,thepracticalhandlingdoes.WhileneighboursearchingmightbetrivialinaCarte-siangrid,itusuallyisnotinatetrahedralgrid.Similardiffer-encesaregivenfortheproblemsofpointlocationandvectorreconstruction.InthefollowingweshortlydescribeseveralfundamentaloperationswhichformthebasisforFlowViscomputationsonsimulationgrids.Startingwithpointlocation,i.e.,theproblemofndingthegridcellwhichagivennD-pointliesin,usuallytwocasesaredistinguished.Forthegeneralpointlocationprob-lemspecialdatastructurescanbeusedwhichsubdividethespatialdomaintospeedupthesearch.Thesecondcaseofiterativepointlocation,whichoftenisneededduringinte-gralcurvecomputation,usuallyallowsforquiteefciental-gorithmsduetoexploitationofspatialcoherence.Onekindofalgorithmstartswithaninitialguessforthetargetcell,checksforcontainmentthenandreningaccordinglyafter-wards.Thisprocessisiterateduntilthetargetcellisfound.Moredetailscanbefoundinoldertextsaboutowvisuali-sationfundamentals129;95.Besidepointlocation,owreconstructionwithinacelloftheowdatasetisacrucialissueinowvisualisation.Of-ten,oncethecellwhichcontainsthequerylocationisfound,onlythesamplevectorsatthecell'sverticesareconsideredforowreconstruction.Themostoftenusedapproachisrst-orderreconstructionbyperforminglinearinterpolationswithinthecell.Withinahexahedralcellin3D,forexample,trilinearowreconstructioncanbeused.Usingpointlocationandowreconstruction,owvisu-alisationcanalreadystart:vectorscanberepresented(forexample,byarrows),virtualparticlescanbeinjectedandtracedacrosstheowdomain.Nevertheless,thecomputa-tionofderiveddatamightbenecessarytodomoresophis-ticatedFlowVis.Usually,therststepistorequest(second-order)gradientinformationforarbitrarypointsintheowdomain,i.e.,rvjp,whichgivesinformationaboutlocalpropertiesoftheow(atpointp)suchasowconvergenceanddivergence,owrotationandshear,etcetera.Forfea-tureextraction,alsoowvorticityw=rvcanbeofhighinterest.Furtherdetailsaboutlocalowpropertiescanbefoundinpreviouswork96;77.FlowintegrationRecallingthatowdatainmostcasesisderivativeinforma-tionwithrespecttotimetheideaofintegratingowdataovertimeisnaturaltoprovideanintuitivenotionof(long-term)evolutioninducedbytheowdata.Anexamplewouldbeowvisualisationbytheuseofparticleadvection.Are-spectiveparticlepathp(s)herethroughunsteadyowwouldbedenedbyp(s)=p0+ st=0v(p(t);t+t0)dt(3)wherep0representstheseedlocationoftheparticlepathandt0equalsthetimewhentheparticlewasseeded.Note,thatEquations2and3aremoreorlesscomplimentarytoeachother.Forothertypesofintegralcurvessuchasstreamlines,streaklines,etc.,refertolaterpartsofthistextorpreviousworks129;61.Inadditiontothetheoreticalspecicationofintegralcurves,itisimportanttonote,thatrespectiveintegralequa-tionslikeEquation3usuallycannotberesolvedforthecurvefunctionanalytically,andtherebynumericalintegra-tionmethodsneedtobeemployed.Themostsimpleap-proachistousearst-orderEulermethodtocomputeanapproximationpEoneiterationofthecurveintegrationisspeciedasbypE(t+Dt)=p(t)+Dtv(p(t);t)(4)whereDtusuallyisaverysmallstepintimeandp(t)de-notesthelocationtostartthisEulerstepfrom.Amoreaccuratebutalsomorecostlytechniqueisthesecond-orderRunge-Kuttamethod,whichusestheEulerapproxi-mationpEasalook-aheadtocomputeabetterapproxima-tionpRK2oftheintegralcurve:pRK2(t+Dt)=p(t)+Dt(v(p(t);t)+v(pE(t+Dt);t))=2(5)Higher-ordermethodsliketheoftenusedfourth-orderRunge-Kuttaintegratorutilisemoresuchstepstobetterap-proximatethelocalbehaviouroftheintegralcurve.Also,adaptivestepsizesareusedtomakesmallerstepsinregionswherelotsofchangestakeplaceintheow.Inthefollowing,fourclassesofapproachesintheeldofowvisualisationarediscusseddirectowvisuali-sationisdescribedinSection2,texture-basedFlowVisinSection3,geometricFlowVisisdiscussedinSection4andnally,feature-basedowvisualisationisdescribedinSec-tions5through8.2.DirectowvisualisationDirect,orglobal,owvisualisationtechniquesattempttopresentthecompletedataset,oralargesubsetofit,atalowlevelofabstraction.Themappingofthedatatoavi-sualrepresentationisdirect,withoutcomplexconversionorc\rTheEurographicsAssociation2002. Post,Vrolijk,Hauser,Laramee,Doleisch/FeatureExtractionandVisualisationofFlowFieldsFigure2:Examplesofdirectowvisualisationaninteractiveslicingprobewithcolouredslicesandscalarclipping(left)122;directvolumerenderingbasedonresampling(middle)160;texture-based,colouredspotnoise(right)65.extractionsteps.Thesetechniquesareperhapsthemostin-tuitivevisualisationstrategiesastheypresentthedataasis.Difcultiesarise,whenthelong-termbehaviourinducedbyowdataisinvestigated,ifdirectFlowVisisusedthismayrequirecognitiveintegrationofvisualisationresults.2.1.DirectFlowVisin2DInthissubsectionweshortlyaddresswidelydistributed,standardtechniquesfor2DFlowVis,i.e.,colouringandar-rowplots.Colourcodingin2DAcommondirectowvisualisationtechniqueistomapowattributessuchasvelocity,pressure,ortemperaturetocolour.Sincecolourplotsarewidelydistributed,thisapproachre-sultsinveryintuitivedepictions.Ofcourse,thecolourscalewhichisusedformappingmustbechosencarefullywithrespecttoperceptualdifferentiation.Colourcodingfor2DFlowVisextendstotime-dependentdataverywell,resultinginmovingcolourplotsaccordingtochangesoftheowpropertiesovertime.Arrowplotsin2DAnaturalvectorvisualisationtechniqueistomapaline,arrow,orglyphtoeachsamplepointintheeld,orientedaccordingtotheoweld,asinFigure6(left).Usuallyaregularplacementofarrowsisusedin2D,forexample,onanevenly-spacedCartesiangrid.Twovariantsofarrowplotsareoftenused:(1)normalisedarrowsofunitlengthwhichvisualisethedirectionoftheowonlyand(2)arrowsofvaryinglengththatisproportionaltotheowvelocity.KlassenandHarrington59andSchroederetal.121callthistechniqueahedgehogvisualisation(becauseofthebristlyresult).2Dhedgehogplotscanbeextendedtotime-dependentdata,althoughbiggertimestepsmightresultinjumpingar-rows,diminishingthequalityofsuchavisualisation.HybriddirectFlowVisin2DKirbyetal.proposesimultaneousvisualisationofmultiplevalues(of2Dowdata)byusingalayeringconceptrelatedtothepaintingprocessofartists57.Arrowplotsaremixedwithcolourcodingtoprovidevisualisationresultsrichofinformation.DirectFlowVisonslicesorboundariesWhendealingwith3Dowdata,visualisationnaturallyfacesadditionalchallengessuchas3Drendering.Actingasamiddlegroundbetween2DFlowVisandthevisualisationoftruly3Dowdataistherestrictiontosubdimensionalpartsofthe3Ddomain,e.g.,sectionalslicesorboundarysur-faces.Thereby,techniquesknownfrom2DFlowVisusuallyareapplicablewithoutmajorchanges(atleastfromatechni-calpointofview).Whenworkingwithsectionalslices,thetreatmentofowcomponentsorthogonaltoslicesrequiressomespecialcare.ColourcodingonslicesorboundariesColourcodingisveryeffectiveforvisualisingboundaryowsorsectionalsubsetsof3Dowdata.Agoodexam-pleisNASA'sFieldEncapsulationLibrary85,whichallowstoeasilyusebothtechniquesforvariousowdata.Schulzetal.alsousecolourcodingofscalarson2Dslicesin3Dautomotivesimulationdata122asshowninFigure2(left).Theyintroduceaninteractiveslicingprobewhichmapsthevectorelddatatohue.Theuseofscalarclipping,i.e.,thetransparentrenderingofsliceregionswherethecorrespondingdatavaluedoesnotliewithinaspecicdatarange,allowstousemultiple(coloured)sliceswithreducedproblemsduetoocclusion.2DarrowsonslicesorboundarysurfacesUsing2Darrowsonslicesfrom3Dowdataisalsoanef-fectivevisualisationtechnique19.However,resultsofsuchavisualisationshouldbeinterpretedcarefully,asowcom-ponentswhichareorthogonaltothesliceareusuallynotde-picted.c\rTheEurographicsAssociation2002. Post,Vrolijk,Hauser,Laramee,Doleisch/FeatureExtractionandVisualisationofFlowFieldsAbovementioneddifcultieswith2Darrowsandsec-tionalslicesthrough3Dowarebasicallynegligible,whentalkingaboutboundarysurfaces,sinceinthesecases,rarelycross-boundaryowsaregiven.Thereforetheuseofarrowsspreadoutoverboundarysurfacesusuallyisveryeffective,asusedbyTreinishforweathervisualisation145.2.3.DirectFlowVisin3DAfterdiscussingdirectFlowVisonslicesandboundarysur-faces,directFlowVisofreal3Dowsisdiscussedinthissubsection.Incontrasttopreviouslymentionedtechniques,hererenderingbecomesthemostcriticalissue.Occlusionandcomplexitymakeitdifcult(ifpossibleatall)togetanimmediateoverviewofanentireowdatasetin3D.Volumerenderingfor3DFlowVisThenaturalextensionofcolourcodingin2D(oronslices,etc.)iscolourcodingin3D.This,however,posesspecialrequirementsontorenderingduetoocclusionproblemsandnontrivialcomplexityvolumerenderingisneeded.Vol-umerenderingiswell-knownintheeldofmedical3Dvi-sualisation,i.e.,volumevisualisation.However,thosechal-lenges,whichcloselycorrespondtoowvisualisationarebrieyaddressedhere:(1)owdatasetsareoftensig-nicantlysmootherthanmedicaldataanabsenceofsharpandclearobjectboundaries(likeorganboundaries)makesmappingtoopacitiesmoredifcultandlessintuitive.(2)owdataisoftengivenonnon-Cartesiangrids,e.g.,oncurvilineargridsthecomplexityofvolumerenderinggetssignicantlymoretrickyonthosekindsofgrids,startingwithnontrivialsolutionsrequiredforvisibilitysortingandblending.(3)owdataisalsotime-dependentinmanycases,imposingadditionalloadsontherenderingprocess.Intheearlynineties,Crawsetal.15,aswellasEbertetal.18appliedvolumerenderingtechniquestovectorelds.Littlelater,Frühaufappliedraycastingtovectorelds22.Recently,Westermann,presentedarelativelyfast3DvolumerenderingmethodusingaresamplingtechniqueforvectorelddatafromunstructuredtoCartesiangrids160.AresultfromthistechniqueisillustratedinFigure2(middle).Recently,ClyneandDennis14aswellasGlau24presentedvolumerenderingfortime-varyingvectoreldsusingalgo-rithmswhichmakespecialuseofgraphicshardware.Onoetal.usedirectvolumerenderingtovisualisethermalowsinthepassengercompartmentofanautomobile90.Theirgoalistoattaintheabilitytopredictthethermalcharacteristicsoftheautomotivecabinthroughsimulation.Swanetal.ap-plydirectvolumerenderingtechniquesinowvisualisationinasystemthatsupportscomputationalsteering137.TheirvisualisationresultsareextendedtotheCAVEenvironment.Recently,EbertandRheingansdemonstratedtheuseofnonphotorealisticvolumerenderingtechniquesfor3Dowdata17.Theyapply,forexample,silhouetteenhancementortoneshadingtoimproverenderingsof3Dows.Arrowplotsin3DTheuseofarrowsfordirect3DFlowVisposesatleasttwoproblems:(1)thepositionandorientationofavectorisof-tendifculttounderstandbecauseofitsprojectionontoa2Dscreenusing3Drepresentationsofarrows(likeacylin-derplusacone)decreasestheseproblemswithperceptionand(2)glyphsoccludingoneanotherbecomeaproblemcarefulseedingisrequired(incontrasttothedefaultofdensedistributions).Inactualapplications,arrowplotsareusuallybasedonselectiveseeding,forexample,allarrowsstartingfromoneoutofafewsectionalslicesthroughthe3Dow.BoringandPangaddresstheproblemofclutterin3Ddi-rectFlowVisbyhighlightingthosepartsofa3Darrowplot,whichpointinasimilardirectioncomparedtoauser-deneddirection8.Theirmethodologyreducestheamountofdatabeingdisplayedthusresultsinlessclutter.Theirmethodscanbecombinedwithothertechniquesthatuseglyphrep-resentationsandowgeometriessuchasstreamlinesforFlowVis.Theyapplythemethodstobothanalyticandsim-ulationdatasetstohighlightowreversals.3.Texture-basedVisualisationWemakeadistinctionbetweengeometricowvisualisation(seeSection4)anddense,texture-basedowvisualisation,however,thesetwotopicsarecloselycoupled:conceptually,thepathfromusinggeometricobjectstotexture-basedvi-sualisationisobtainedviaadenseseedingstrategy.Thatis,denselyseededgeometricobjectsresultinanimagesimilartothatobtainedbydense,texture-basedtechniques.Like-wise,thepathfromdense,texture-basedvisualisationtovi-sualisationusinggeometricobjectsisobtainedusingsome-thingsuchasasparsetexturefortextureadvection.Texture-basedtechniquesinowvisualisationcanpro-videdensespatialresolutionimages.Texture-basedalgo-rithmsareeffective,versatile,andapplicabletoawidespec-trumofapplications.Sannaetal.presentasummaryofthisresearchinasurveypaper117.3.1.Texture-basedFlowVisin2DInthissubsection,wedescribetexture-basedFlowVissolu-tionsfor2Dowdata,i.e.,spotnoise,lineintegralconvolu-tion(LIC),andrelatedapproaches.Spotnoisein2DSpotnoise,introducedbyVanWijk162,wasamongstthersttexture-basedtechniquesforvectoreldvisualisation.Spotnoisegeneratesatexturebydistributingasetofintensityfunctions,orspots,overthedomain.Eachspotrepresentsaparticlemovingoverasmallstepintimeandresultsinastreakinthedirectionofthelocalowfromwheretheparticleisseeded.c\rTheEurographicsAssociation2002. Post,Vrolijk,Hauser,Laramee,Doleisch/FeatureExtractionandVisualisationofFlowFieldsOnelimitationoftheoriginalspotnoisealgorithmwasthelackofvelocitymagnitudeinformationintheresultingtexture.Enhancedspotnoise70,byDeLeeuwandVanWijkwasintroducedtoaddressthisproblem.Spotnoisehasalsobeenappliedtothevisualisationofturbulentow67byDeLeeuwetal.Aspotnoisealgorithmforinteractivevisuali-sationisproposedbyDeLeeuw65,also.DeLeeuwandVanLierealsocomparespotnoisetoLIC68.Spotnoisein2DcombinedwithcolourcodingisshowninFigure2(right).Lineintegralconvolutionin2DLineintegralconvolution(LIC),rstintroducedbyCabralandLeedom12isaverypopulartechniqueforthedensecov-erageofvectoreldswithowvisualisationcues.Theorig-inalmethodologybehindLICtakesasinputavectoreldonaCartesiangridandawhitenoisetextureofthesamesize.Thenoisetextureislocallyltered(smoothed)alongthepathofstreamlinestoacquireadensevisualisationoftheoweld.SeeFigure6(middle)foranexample.TheresearchinowvisualisationbasedonLICdescribedhereextendsLICinseveralways:(1)addingdirectionalcues,(2)showingvelocitymagnitudes,(3)addedsupportfornon-Cartesiangrids,(4)allowingreal-timeandinteractiveexploration,(5)extendingLICto3D,and(6)extendingLICtounsteadyvectoreldvisualisationwithtimecoherency.Shenetal.addresstheproblemofdirectionalcuesinLICbycombininganimationandintroducingdyeadvectionintothecomputation126.KiuandBanksproposedtouseamul-tifrequencynoiseforLIC58.Thespatialfrequencyofthenoiseisafunctionofthemagnitudeofthelocalvelocityintheeld.Khouasetal.synthesiseLIC-likeimagesin2Dwithfur-liketextures56.Theirtechniqueisabletolocallycontrolattributesoftheoutputtexturesuchasorientation,length,density,andcolour.MuchresearchhasbeendedicatedtobringingLICcom-putationtointeractiverates.StallingandHegepresentsig-nicantimprovementsinLICperformancebyexploitingco-herencealongstreamlines135;29.ParallelimplementationsofLICarepresentedbyCabralandLeedom11,andZöckleretal.170.OLICfor2DFlowVisWegenkittletal.alsoaddresstheproblemoforientationofowwiththeirOLIC(OrientedLineIntegralConvolution)approach157.Conceptually,theOLICalgorithmmakesuseofasparsetextureconsistingofmanyseparatedspotswhicharemoreorlesssmearedinthedirectionofthelocalvec-toreldthroughintegration.AfastversionofOLIC(calledFROLIC)ispresentedbyWegenkittlandGröller156viaatrade-offofaccuracyfortime.BergerandGröllerpresentanalgorithmforanimating2DFROLICimagesovertheworldwideweb7.Löffelmannetal.usevirtualinkdroplets,likestreamlets,tovisualise2Ddynamicalsystems74.Similartoorientedlineintegralconvolution(OLIC),thevirtualinkdropletmethodiscapableofvisualisingnotonlydirectionandve-locityofow,butalsotheorientationofvectors.SeeFig-ure6foracomparisonbetweenstreamlets(right)andLIC(middle).TextureAdvectionJobardandLeferuseamotionmapdatastructureforanimat-ing2D,steady-stateowelds47.Themotionmapcontainsbothadenserepresentationoftheowandtheinformationrequiredtoanimatetheow.Itofferstheadvantageofsav-ingmemoryandcomputationtimesinceonlyoneimageoftheowhastobecomputedandstoredinthemotionmapdatastructure.Jobardetal.proposeatechniquetovisualisedenserep-resentationsofunsteadyvectoreldsbasedonwhattheycallaLagrangian-EulerianAdvectionscheme45.Thealgo-rithmcombinesadense,time-dependent,integration-basedrepresentationofthevectoreldwithinteractiveframerates.SomeresultsfromthetechniqueareshowninFigure3.Unsteadyowvisualisationtechniquesmayaddresstheproblemofinteractiveperformancetimethroughtheuseoftexturemappingsupportedbythegraphicshardware.BeckerandRumpfillustratehardware-supportedtexturetransportfor2D,unsteadyowdata6.Jobardetal.43;44presentadditional2D,unsteadyowvi-sualisationtechniques.Theyachievehighperformanceviatheuseofgraphicshardware.Theyalsodetailspatialandtemporalcoherencetechniques,dyeadvectiontechniques,andfeatureextraction.3.2.Texture-basedFlowVisonsurfacesorboundariesTexture-basedtechniquesare,ingeneral,bettermethodsforconveyingowinformationonsectionalslicesthantech-niquesusing(long)geometricobjects.Thisisbecausetheconnectionalongthepathofwhatwouldbeastreamlineislostwithdense,texture-basedtechniques.Thusthedepictionoftheowisnotmisleadingintermsofapotentialsugges-tionsofparticlepaths.Letusrecallthatthevectorcompo-nentorthogonaltothesliceisremovedwhenusingtexture-basedandgeometricmethodsforvisualisationresults.SpotnoiseonboundariesorslicesDeLeeuwetal.extendthespotnoisealgorithmtosurfacesinastudythatcomparesexperimentalsurfaceowvisuali-sation(withoil)tothatofspotnoiseonsurfaces66.Acombinationofbothtexture-basedFlowVis(onslices)and3Darrowsfor3DFlowVisisemployedbyTeleaandVanWijk144wherearrowsdenotethemaincharacteristicsofthe3Dow(afterclustering)anda2Dslicewithspotnoiseorc\rTheEurographicsAssociation2002. Post,Vrolijk,Hauser,Laramee,Doleisch/FeatureExtractionandVisualisationofFlowFieldsFigure3:ThreeimagestakenfromananimationofanunsteadyvectoreldcreatedwiththeLagrangian-Eulerianadvectionalgorithm45.LICisusedtovisualisetherestofthevectoreld(onasliceonly).forboundaryowsAlargebodyofresearchliteratureisdedicatedtotheexten-sionofLIContoboundarysurfaces,surveyed,forexample,byStalling134.TheextensionofLICtonon-CartesiangridsandsurfacesispresentedbyresearcherssuchasForssell20.ForssellandCohen21extendLICtocurvilinearsurfaceswithanimationtechniques,addmagnitudeanddirectioninformation,andshowhowtouseLICtodepicttime-dependentows.Theiralgorithmalsoutilisestexturemappinghardwaretoimproveperformancetimetowardsinteractiverates.Teitzeletal.142presentanapproachthatworksonboth2Dunstructuredgridsanddirectlyontriangulatedsurfacesinthree-dimensionalspace.Maoetal.82presentanalgo-rithmforconvolvingsolidwhitenoiseontrianglemeshesin3Dspace,andextendLICforvisualisingavectoreldonarbitrary3Dsurfaces.Battkeetal.4describeanextensionofLICforarbitrarysurfacesin3D.Someapproachesarelimitedtocurvilinearsurfaces,i.e.,surfaceswhichcanbeparameterisedbyusing2D-coordinates.Theirmethodalsohandlesthecaseofgen-eral,multiplyconnectedsurfaces.Scheuermannetal.presentamethodforvisualising3Dvectoreldsthataredenedona3Dmanifold118.Theirworkaddressesthenormalvectorcomponenttothesurfacethatothermethodsdonot.AproblemwithmanycurvilineargridLICalgorithmsisthattheresultingLICtexturesmaybedistortedafterbeingmappedontothegeometricsurfaces,sinceacurvilineargridusuallyconsistsofcellsofdifferentsizes.Maoetal.proposeasolutiontotheproblembyusingmultigranularitynoiseastheinputimageforLIC81.UFLIC,PLIC,etceteraShenandKaopresentUFLIC(UnsteadyFlowLIC),whichincorporatestimeintotheconvolution127;125.SeeFigure4(left).Theiralgorithmaddressesproblemswithtemporalcoherencybysuccessivelyupdatingtheconvolutionresultsovertime.TheyalsoproposeaparallelUFLICalgorithm.Vermaetal.presentamethodforcomparativeanalysisofstreamlinesandLICcalledPLIC149.Avisualcompari-sonbetweenELIC(enhancedLIC)89,PLIC,andUFLICisshowninFigure4.3.3.Texture-basedFlowVisin3DHighcomputationalcosts,demandingmemoryrequire-ments,occlusion,andvisualcomplexitycanallbeinhibi-tantsfortexture-basedowvisualisationin3D.Figure5:3DLIC38.LICin3DOcclusionandinteractiveperformancearenontrivialchal-lengeswhenimplementingLICin3D(showninFigure5).Rezk-Salamaetal.tackletheproblemofinteractiveperfor-manceusinga3D-texturemappingapproachcombinedwithc\rTheEurographicsAssociation2002. Post,Vrolijk,Hauser,Laramee,Doleisch/FeatureExtractionandVisualisationofFlowFieldsFigure4:Acomparisonof3LICtechniques:(left)UFLIC,(middle)ELIC,and(right)PLIC149.Figure6:ExampleofcomparingFlowVistechniquesfromSections2,3,and472.FlowVisbytheuseofarrows(left)iscomparedtotexture-basedFlowVisbytheuseofLIC(middle)andFlowVisbasedongeometricobjects(right).aninteractiveclippingplanetoaddresstheproblemsofoc-clusionandinteraction103.AcombinedapproachofdirectvolumerenderingandLICistakenbyInterrante40forextendingLICto3D.InterranteandGroschaddresssomeperceptualdifcultiesencounteredwithdense,3Dvisualisations38;39;40.Techniquesforselec-tivelyemphasisingimportantregionsofinterestintheow,enhancingdepthperception,andimprovingorientationper-ceptionofoverlappingstreamlinesarediscussed.Textureadvectionin3DKaoetal.discusstheuseof3Dand4Dtextureadvectionforthevisualisationof3Duidows51.Formidablechal-lengesareintroducedbythememoryrequirementsinvolvedinusing3Dand4Dtextures.Theyalsoapplyasteady-stateanimationtothese3Dand4Dtextures.4.GeometricFlowVisualisationGeometricFlowVisentailsextractinggeometricobjectsforwhichtheirshapeisdirectlyrelatedtotheunderlyingdata.Inwhatfollows,wediscussgeometricowvisualisationtech-niquessuchascontouringinboth2Dand3Daswellasgeo-metricFlowVisusingintegralobjects(suchasstreamlines).Contouringin2DContouringisanaturalextensiontocolourcodingin2D.Acontourisaboundarybetweentwodistinctregions.Often,theuserishighlyinterestedintransitionareasinthevectoreld.Inacolourplot,transitionsareshownbyachangeofcolour.Withcontouring,anexplicitlineorcurveisdrawn.Isosurfacesfor3DFlowVisExtendingcontouringfrom2Dto3D,resultsintheuseofisosurfacesfor3Dowvisualisation.Specialcareneedstobetakenwithisovalueselection,mostlybecauseoftheusu-allysmoothnatureofowdataincasesofnosharptran-sitionswithinthedata,anyisovaluelacks(atleastpartially)c\rTheEurographicsAssociation2002. Post,Vrolijk,Hauser,Laramee,Doleisch/FeatureExtractionandVisualisationofFlowFieldsintuitiveinterpretation.Neverthelessthereareusefulappli-cationsofisosurfacestoowdata,e.g.,inthevisualisationofshockwaves139orburningfrontsinsimulatedcombus-tiondata.Furthermore,whenscalarclippingisusedtogetherwithcolourcodingofslices,thisnaturallycombineswithisosurfacesaslongasisovalueandclippingvaluecoincide.Röttgeretal.presentahardwareacceleratedvolumeren-deringtechniquewhichallowstousemultiple(semitrans-parent)isosurfacesforvisualisation109.Treinishappliesiso-surfacingtovisualise(unsteady)weatherdata145.Weberetal.155presentcrack-freeisosurfaceextractionforadaptive(multiresolution)grids.LarameeandBergeronprovideiso-surfacesforsuperadaptiveresolutiongrids63.4.1.Geometric2DFlowVisusingintegralobjectsInthissubsectionweshortlydiscussgeometricFlowVistechniquesin2Dbasedonintegralobjectssuchasstream-lets,streamlines,andtheirrelativeswithinunsteadyows.Also,theseedingproblemisaddressed,whichrequiresaso-lutioninordertorealisebetterdistributionsofintegralob-jects.eamletsin2DIfowvectorsareintegratedforaveryshorttime,streamletsaregenerated.Eventhoughshort,streamletsalreadycom-municatetemporalevolutionalongtheow.Figure6showsanexample,whereseveralstreamletsareusedtovisualisea2Doweld.Streamlinesin2DIflongerintegrationisperformed(ascomparedtostream-lets),streamlinesaregained.Theyareanaturalextensionofglyph-basedtechniquesandofferintuitivesemantics:userseasilyunderstandthatowsevolvealongintegralobjects.Streaklines,timelines,andpathlinesWhenunsteadyowdataareinvestigated,severaldistinctintegralobjectsareusedforowvisualisation.Apathlineorparticletraceisthetrajectorythataparticlefollowsinauidow121.Atimelinejoinsthepositionsofparticlesreleasedatthesameinstantintimefromdifferentinsertionpoints,i.e.,joinspointsataconstanttimet88.Astreaklineistracedbyasetofparticlesthathavepreviouslypassedthroughauniquepointinthedomain121.Streaklinesrelatetocontinuousin-jectionofforeignmaterialintorealow.Sannaetal.presentanadaptivevisualisationmethodusingstreaklineswheretheseedingofstreaklinesisafunctionoflocalvorticity116.Streamlineseedingin2DOneimportantaspectofstreamlines,orintegralcurves,whenusedforvisualisingcontinuousvectoreldsisthebestchoiceofinitialconditions.Since,ingeneral,evenlydis-tributedseedpointsdonotresultinevenlyspacedstream-lines,specialalgorithmsneedtobeemployed.TurkandBanks146aswellasJobardandLefer46developedtech-niquesforautomaticallyplacingseedpointstoachieveauni-formdistributionofstreamlinesona2Dvectoreld.Streamlineseedingstrategiesin2Dmayalsobetopology-based.Vermaetal.150presentaseedplacementstrategyforstreamlinesbasedonowfeaturesinthedataset.Theirgoalistocaptureowpatternsnearcriticalpointsintheoweld.Buildingontheirpreviouswork,JobardandLeferpre-sentedamultiresolution(MR)methodforvisualisinglarge,2D,steady-statevectorelds49.TheMRhierarchysupportsenrichmentandzooming.Theuserisabletointeractivelysetthedensityofstreamlineswhilezoominginandoutofthevectoreld(Figure7).Thedensityofstreamlinescanbecomputedautomaticallyasafunctionofvelocityorvorticity.Seedingofintegralobjectsbecomesaspecialchallengewhendealingwithtime-dependentdata.JobardandLeferpresentedanunsteadyFlowVisalgorithmbycorrelatingin-stantaneousvisualisationsofthevectoreldatthestream-linelevel48.Foreachframe,afeedforwardalgorithmcom-putesasetofevenly-spacedstreamlinesasafunctionofthestreamlinesgeneratedforthepreviousframe.Theirmethodalsoprovidesfullcontroloftheimagedensitysothatsmoothanimationsofarbitrarydensitycanbeproduced.4.2.FlowVisusinggeometricobjectsonslicesorboundariesAfterdiscussing2DFlowVisbasedongeometricobjects,thissubsectionshortlyaddressessimilarapproachesonsub-setsof3Dowssuchasboundaryows.Interpretationofin-tegralcurvesonsectionalslicesrequiresspecialcare,again.IntegratedtuftsWegenkittletal.useintegratedtufts(similartostreamlets),seededonspecicequilibriumsurfaces,forthevisualisationofacomplexdynamicalsystem158,alsoovervariationsofthatsysteminafourthdimension.GeometricobjectsonslicesorboundariesSimilarto2DFlowVis,geometricobjectssuchasstream-linesarealsousedforvisualisingboundaryowsorsec-tionalslicesthrough3Dow19.However,itisimportanttonotethattheuseoftheseobjectsonslicesmaybemislead-ing,evenwithinsteadyowdatasets.Astreamlineonaslicemaydepictaclosedloop,eventhoughnoparticlewouldevertraversetheloop.Thereasonagainliesinthefact,thatowcomponentswhichareorthogonaltothesliceareomittedduringowintegration.c\rTheEurographicsAssociation2002. Post,Vrolijk,Hauser,Laramee,Doleisch/FeatureExtractionandVisualisationofFlowFieldsFigure7:ThreeimagesfromaninteractiveexplorationofavectoreldusingtheMRviewer49.Asuitablelevelofresolutioncanbechosenwhilemaintainingaroughlyconstantstreamlinedensity.StreamlineseedingonboundarysurfacesMaoetal.80extendthestreamlineseedingofTurkandBanks146inordertogenerateevenlydistributedstreamlinesonboundarysurfaceswithincurvilineargrids.4.3.3DFlowVisusinggeometricobjectsWhendealingwith3Dow,arichvarietyofgeometricobjectsisavailableforowvisualisation.Thissubsectionaddressesaseriesofobjects,fromstreamletstoowvol-umes,primarilysortedaccordingtotheirdimensionality,andwithinequaldimensionalityroughlywithrespecttowhichtechniqueextendstoanother.Streamletsin3DStreamletseasilyextendto3D,althoughperceptualprob-lemsmayariseduetodistortionsresultingfromtheren-deringprojection.Also,seedingbecomesmoreimportantin3D,again.LöffelmannandGrölleruseathreadofstreamletsalongcharacteristicstructuresof3Dowtogainselective,butimportance-basedseedingaswellasanenhancementofabstractowtopologythroughdirectvisualisationcues73.Streamlinesin3DAtNASAtheFlowAnalysisSoftwareToolkit(FAST)1isusedtovisualiseCFDdatabasedonstreamlinesin3D.Care-fulseedingisnecessarytoobtainusefulresults,sincevisualcluttercaneasilybecomeaproblem.IlluminatedstreamlinesZöckleretal.presentilluminatedstreamlinestoimproveperceptionofstreamlinesin3Dbytakingadvantageofthetexturemappingcapabilitiessupportedbygraphicshard-ware169.Theirshadingtechniqueincreasesdepthinforma-tion.Bymakingthestreamlinespartiallytransparent,theyalsoaddresstheproblemofocclusion,asshowninFigure8(left).Forseeding,theauthorsproposeaninteractiveseed-ingprobewhichcanbemovedaroundtostartstreamlinesatspecicplacesofinterest.Also,seedingnearpotentialob-jectsofinterestsisdemonstrated.Particletracingin3DKenwrightandLanepresentanefcient,3Dparticletracingalgorithmthatisalsoaccurateforinteractiveinvestigationoflarge,unsteady,aeronauticalsimulations55.Aperformancegainisobtainedbyapplyingtetrahedraldecompositiontospeeduppointlocationandvelocityinterpolationincurvi-lineargrids.Teitzeletal.analysedifferentintegrationmethodsinordertoevaluatethetrade-offbetweentimeandaccuracy141;143.Theypresenta3Dparticletracingalgorithmtargetedatsparsegridsthatisveryefcientwithrespecttostoragespaceandcomputingtime.Theauthorsrecommendusingsparsegridsasadatacompressionmethodinordertovisu-alisehugedatasets.Nielsonpresentsefcientandaccuratemethodsforcom-putingtangentcurvesfor3Dows87.Themethodsworkdirectlywithphysicalcoordinates,eliminatingtheneedtoswitchbackandforthtocomputationalcoordinates.EfcientparticletracingmethodologiesarealsoaddressedbySadar-joenetal.111.Sincestreamlinesareusuallyeasilycomputedinrealtime,theyoffer(togetherwiththeirintuitivesemantics)anoftenchosentoolforinteractiveowanalysis.BrysonandLevit10demonstrateseedingofintegralobjectsinavirtual3Denvironmentbyuseofaso-calledrake.StreamribbonsandstreamtubesArstextensionofstreamlinesin3Darestreamribbonsandstreamtubes.Astreamribbonisbasicallyastreamlinec\rTheEurographicsAssociation2002. Post,Vrolijk,Hauser,Laramee,Doleisch/FeatureExtractionandVisualisationofFlowFieldswithawinglikestripadded,toalsovisualiserotationalbe-haviourofthe3Dow(whichisnotpossiblewithstream-linesalone)148.Astreamtubeisathickstreamlinethatcanbeextendedtoshowtheexpansionoftheow148.Streamribbonsandstreamtubesofferadvantagesoverstreamlinesinthatway,thattheycanencodemoreproperties,suchasdivergenceandconvergenceofthevectoreld,inthegeo-metricpropertiesoftherespectiveintegralobjects.Uengetal.presenttechniquesforefcientstreamline,streamribbon,andstreamtubeconstructionsonunstructuredgrids148.AspecialisedRunge-Kuttamethodisemployedtospeedupstreamlinecomputation.Explicitsolutionsarecal-culatedfortheangularrotationratesofstreamribbonsandtheradiiofstreamtubes.Theresultingspeedupinoverallperformanceaidsintheexplorationoflargeowelds.FuhrmannandGröller23useso-calleddashtubes,i.e.,animated,opacity-mappedstreamtubes,asavisualisationicon.Analgorithmisdescribedwhichplacesthedashtubesevenlyin3Dspace.Theyalsoapplyamagiclensandmagicboxasinteractiontechniquesforinvestigatingdenselylledareaswithoutllingtheimagewithvisualdetailandcom-plexity.Larameeintroducesthestreamrunnerasanextensionofstreamtubesaninteractivelycontrolled3Dowvi-sualisationtechniquethatattemptstominimiseocclusion,minimisevisualcomplexity,maximisedirectionalcues,andmaximisedepthcuesbylettingtheusercontrolthelengthofthestreamtubes62.StreampolygonsAnotherextensionofstreamlinesarestreampolygonsusedbySchroederetal.120.Streampolygonsaretoolstovisu-alisevectorsandtensorsusingtubeswithapolygonalcrosssection.Thepropertiesofthepolygonssuchastheradius,thenumberofsides,theshapeandtherotationreectprop-ertiesofthevectoreldincludingstrain,displacement,androtation.eamballsandstreakballsStreamballsareausefulowvisualisationtechniqueusedbyBrilletal.9,whichvisualisesdivergenceandaccelerationinuidow.Streamballssplitormergedependingoncon-vergence/divergenceandacceleration/deceleration,respec-tively.TeitzelandErtlintroducestreakballswhentheypresentandcomparetwodifferentapproachestoaccelerateparticletracingonsparsegridsandcurvilinearsparsegridsforun-steadyowdata140.StreamsurfacesYetanotherextensiontostreamlinesarestreamsurfaces,whicharesurfacesthatareeverywheretangenttoavectoreld.Astreamsurfacecanbeapproximatedbyconnectingasetofstreamlinesalongtimelines(andvaryingthenumberofstreamlinesusedaccordingtoconvergenceordivergenceoftheow)36.Streamsurfacesareverygoodfortexture-basedvisualisationtechniquessuchasSpotNoiseandLIC,becausethereisnocross-owcomponentnormaltothesur-faces,i.e.thevectoreldisnotprojectedlikeitisfor2Dslicesthrougha3Ddomain.Streamsurfacespresentchal-lengesrelatedtoocclusion,visualcomplexity,andinterpre-tation.Hultquistpresentsaninteractiveowvisualisationtech-niqueusingstreamsurfaces35.VanWijkpresentstwofollow-uptechniquesforgeneratingimplicitstreamsur-faces164.CaiandHeng13addresstheissuesassociatedwiththeplacementandorientationofstreamsurfacesin3D.Löffelmannetal.presentstreamarrows(seeFigure8,middle)asanenhancementofstreamsurfacesbyseparatingarrow-shapedportionsfromastreamsurface76;75.Streamarrowsaddresstheproblemofocclusionassociatedwith3Dowvisualisation,butespeciallywithstreamsurfaces.Streamarrowsalsoprovideadditionalinformationabouttheow,usuallynotseenwithstreamsurfaces,suchasowdi-rection,convergence/divergence,etcetera.VanWijksimulatesstreamsurfacesbyalargesetofso-calledsurfaceparticles163.Surfaceparticlesexhibitlessoc-clusionwhencomparedtostreamsurfaces.Interestingly,VanWijk'sapproachinawayanticipatedrecentadvancesinpixel-basedrenderingtechniques.Timesurfacesin3DAnaturalextensionoftimelines(in2Dor3D)aretimesurfaces,whenconstant-timeinstantsofmovingparticlesareassumed,whichpreviouslyhavebeenreleasedfromatwo-dimensionalpatch.Anexampleofanapplicationofthisprinciple,arelevel-setsurfacesusedbyWestermannetal.161.FlowvolumesThelast(direct)extensionofastreamlineinto3Ddescribedhereareowvolumes(seeFigure8,right).Aowvolumeisaspecicsubsetofa3Dowdomain,whichistracedoutbyaparticularinitial2DpatchovertimeasdescribedbyMaxetal.83.Theresultingvolumeisdividedupintoasetofsemitransparenttetrahedra,whicharevolumerenderedinhardwareinawayderivedfromthemethodofShirleyandTuchmann128.Beckeretal.extendowvolumestounsteadyow5.Theresultingunsteadyowvolumesarethe3Danalogueofstreaklines.Considerationsaremadewhenextendingthevisualisationtechniquetounsteadyowssinceparticlepathsmaybecomeconvolutedintime.Theauthorspresentsomesolutionstotheproblemswhichoccurinsubdivision,ren-dering,andsystemdesign.Theresultingalgorithmsareap-pliedtoavarietyofowtypesincludingcurvilineargrids.c\rTheEurographicsAssociation2002. Post,Vrolijk,Hauser,Laramee,Doleisch/FeatureExtractionandVisualisationofFlowFieldsFigure8:Examplesofowvisualisationusinggeometricobjectsilluminatedstreamlines(left)169,streamarrows(middle)76,andowvolumes(right)83.5.FeatureExtractionFeature-basedowvisualisationisanapproachforvisual-isingtheowdataatahighlevelofabstraction.Theowdataisdescribedbyfeatures,whichrepresenttheinterest-ingobjectsorstructuresinthedata.Theoriginaldatasetisthennolongerneeded.Becauseoften,onlyasmallper-centageofthedataisofinterest,andthefeaturescanbede-scribedverycompactly,anenormousdatareductioncanbeachieved.Thismakesitpossibletovisualiseevenverylargedatasetsinteractively.Therststepinfeature-basedvisualisationisfeatureex-traction.Thegoaloffeatureextractionisdetermining,quan-tifyinganddescribingthefeaturesinadataset.Afeaturecanbelooselydenedasanyobject,structureorregionthatisofrelevancetoaparticularresearchproblem.Ineachapplication,ineachdatasetandforeachresearcher,adifferentfeaturedenitioncouldbeused.Commonexam-plesinuiddynamicsarevortices,shockwaves,separationandattachmentlines,recirculationzonesandboundarylay-ers.Inthenextsectionanumberoffeature-specicdetectiontechniqueswillbediscussed.Althoughmostfeaturedetec-tiontechniquesarespecicforaparticulartypeoffeature,ingeneralthetechniquescanbedividedintothreeapproaches:basedonimageprocessing,ontopologicalanalysis,andonphysicalcharacteristics.5.1.ImageProcessingImageprocessingtechniqueswereoriginallydevelopedforanalysisof2Dand3Dimagedata,usuallyrepresentedasscalar(greyscale)valuesonaregularrectangulargrid.Theproblemofanalysinganumericaldataset,representedonagrid,issimilartoanalysinganimagedataset.Therefore,ba-sicimageprocessingtechniquescanbeusedforfeatureex-tractionfromscienticdata.Afeaturemaybedistinguishedbyatypicalrangeofdatavalues,justasdifferenttissuetypesaresegmentedfrommedicalimages.Edgesorbound-ariesofobjectsarefoundbydetectingsuddenchangesinthedatavalues,markedbyhighgradientmagnitudes.Thus,basicimagesegmentationtechniques,suchasthresholding,regiongrowing,andedgedetectioncanbeusedforfeaturedetection.Also,objectsmaybequantitativelydescribedus-ingtechniquessuchasskeletonisationorprincipalcompo-nentanalysis.However,aproblemis,thatincomputationaluiddynamicssimulations,oftengridtypesareusedsuchasstructuredcurvilineargrids,orunstructuredtetrahedralgrids.Manytechniquesfromimageprocessingcannotbeeasilyadaptedforusewithsuchgrids.5.2.VectorFieldTopologyAsecondapproachtofeatureextractionisthetopologicalanalysisof2Dlinearvectorelds,asintroducedbyHel-manandHesselink30;32,whichisbasedondetectionandclassicationofcriticalpoints.Thesearethepointswherethevectormagnitudeiszero.Bycomputingtheeigenvaluesandeigenvectorsofthevelocitygradienttensor,thecriticalpointscanbeclassiedandtangentcurvescanbecomputed.(SeeFigure9.)Usingthisinformation,aschematicvisuali-sationofthevectoreldcanbegenerated.(SeeFigure15.)HelmanandHesselinkhavealsoextendedtheiralgorithmto2Dtime-dependentandto3Dows.Scheuermannetal.presentedanalgorithmforvisualis-ingnonlinearvectoreldtopology119,becauseotherknownalgorithmsareallbasedon(piecewiseorbi-)linearinterpo-lation,whichdestroysthetopologyincaseofnonlinearbe-haviour.TheiralgorithmmakesuseofCliffordalgebraforcomputingpolynomialapproximationsinareaswithnonlin-earlocalbehaviour,especiallyhigher-ordersingularities.DeLeeuwandVanLierepresentedatechniqueforvisu-alisingowstructuresusingmultilevelowtopology69.Inhigh-resolutiondatasetsofturbulentows,thehugenum-berofcriticalpointscaneasilyclutteraowtopologyim-age.Thealgorithmpresentedattemptstosolvethisproblembyremovingsmall-scalestructuresfromthetopology.Thisisachievedbyapplyingapairdistancelterwhichremovesc\rTheEurographicsAssociation2002. Post,Vrolijk,Hauser,Laramee,Doleisch/FeatureExtractionandVisualisationofFlowFieldsRepelling Focus\rR1, R2 � 0\rI1, I2 � 0Attracting Focus\rR1, R2 0\rI1, I2 � 0Center\rR1, R2 = 0\rI1, I2 � 0Attracting Node\rR1, R2 0\rI1, I2 = 0Repelling Node\rR1, R2 � 0\rI1, I2 = 0Saddle Point\rR1 * R2 0\rI1, I2 = 0Figure9:Vectoreldtopology:criticalpointsclassiedbytheeigenvaluesoftheJacobian30.pairsofcriticalpoints,thatareneareachother.Thisremovessmalltopologicalstructuressuchasvortices,butdoesnotaf-fecttheglobaltopologicalstructure.Thethresholddistance,whichdetermineswhichcriticalpointsareremoved,canbeadapted,makingitpossibletovisualisethestructureatdif-ferentlevelsofdetailatdifferentzoomlevels.5.3.SelectiveVisualisationAgenericapproachtofeatureextractionisSelectiveVisual-isation,whichisdescribedbyVanWalsum152.Thefeatureextractionprocessisdividedintofoursteps(seeFigure10).Therststepisthesegmentationstep.Inprinciple,anyseg-DataGenerationSelectionClusteringAttributeCalculationIconicMappingDisplayRaw DataSelectedNodesRegions of InterestAttributeSetsIconsSelectionExpressionConnectivityCriteriaCalculationMethodMappingFunctionScientist's knowledge andconceptual modelFigure10:Thefeatureextractionpipeline102.mentationtechniquecanbeused,thatresultsinabinaryseg-mentationoftheoriginaldataset.Averysimplesegmenta-tionisobtainedbythresholdingoftheoriginalorderiveddatavalues;also,multiplethresholdscanbecombined.Thedatasetresultingfromthesegmentationstepisabinarydatasetwiththesamedimensionsastheoriginaldataset.Thebinaryvaluesinthisdatasetdenotewhetherornotthecor-respondingpointsintheoriginaldatasetareselected.Thenextstepinthefeatureextractionprocessistheclusteringstep,inwhichallpointsthathavebeenselectedareclusteredintocoherentregions.Inthenextstep,theattributecalcu-lationstep,theseregionsarequantied.Attributes,suchasposition,volumeandorientation,oftheregionsarecalcu-lated.Wenowspeakofobjects,orfeatures,withanumberofattributes,insteadofclustersofpoints.Oncewehavede-terminedthesequantiedobjects,wedon'tneedtheorigi-naldataanymore.Withthis,wemayaccomplishadatare-ductionfactorof1000ormore.Inthefourthandnalstep,iconicmapping,thecalculatedattributesaremappedontotheparametersofcertainparametricicons,whichareeasytovisualise,suchasellipsoids.6.Feature-basedowvisualisationInthissection,anumberoffeatureextractiontechniqueswillbediscussedthathavebeenspecicallydesignedforcertaintypesoffeatures.Thesetechniquesareoftenbasedonphysi-calormathematical(topological)propertiesoftheow.Fea-turesthatoftenoccurinowsarevortices,shockwavesandseparationandattachmentlines.6.1.VortexextractionFeaturesofgreatimportanceinowdatasets,bothinthe-oreticalandinpracticalresearch,arevortices.(SeeFig-ure11.)Insomecases,vortices(turbulence)havetobeim-pelled,forexampletostimulatemixingofuids,ortore-ducedrag.Inothercases,vorticeshavetobeprevented,forexamplearoundaircraft,wheretheycanreducelift.ThereFigure11:Reconstructionofahairpinvortextube,withgroovesindicatingvelocity2.aremanydifferentdenitionsofvorticesandlikewisemanydifferentvortexdetectionalgorithms.Adistinctioncanbemadeinalgorithmsforndingvortexregionsandalgorithmsthatonlyndthevortexcores.OtheroverviewsofalgorithmsaregivenbyRothandPeikert107andbyBanksandSinger2.Thereareanumberofalgorithmsforndingregionswithvortices:Oneideaistondregionswithahighvorticitymagni-tude.Vorticityisthecurlofthevelocity,thatis,rv,andrepresentsthelocalowrotation,bothinspeedanddirection.However,althoughavortexmayhaveahighc\rTheEurographicsAssociation2002. Post,Vrolijk,Hauser,Laramee,Doleisch/FeatureExtractionandVisualisationofFlowFieldsvorticitymagnitude,thereverseisnotalwaystrue168.Vil-lasenorandVincentpresentanalgorithmforconstructingvortextubesusingthisidea151.Theycomputetheaveragelengthofallvorticityvectorscontainedinsmall-radiuscylinders,andusethecylinderwiththemaximumaver-ageforconstructingthevortextubes.Anotherideaistomakeuseofhelicityinsteadofvortic-ity71;167.Thehelicityofaowistheprojectionofthevorticityontothevelocity,thatis(rv)v.Thisway,thecomponentofthevorticityperpendiculartotheveloc-ityiseliminated.Anothersimpleideaistosearchforregionsoflowpres-sure104.JeongandHussaindeneavortexasaregionwheretwoeigenvaluesofthesymmetricmatrixS2+W2arenegative,whereSandWarethesymmetricandantisymmetricpartsoftheJacobianofthevectoreld,respectively41:S=12(V+VT),andW=12(V VT).Thismethodisknownasthel2method.Theabovemethodsmayallworkincertainsimpleowdatasets,buttheydonothold,forexample,inturbomachineryows107,suchasshowninFigure12.Figure12:Visualisationofaturbomachineryow107.Therearealsosomealgorithmsspecicallyforndingvortexcorelines:GlobusandLevitpresentedamethodforndingcorelinesbyintegratingstreamlinesfromthecriticalpointsinthevelocityeld25.Itisalsopossibletousestreamlinesofthevorticityeld84,butsuchanalgorithmisverysensitivetothestart-inglocation.BanksandSingeralsousestreamlinesofthevorticityeld,withacorrectiontothepressureminimumintheplaneperpendiculartothevortexcore133;3.Combiningtheaboveideas,RothandPeikertsuggestthatavortexcorelinecanbefoundwherevorticityisparalleltovelocity107.Thissometimesresultsincoherentstruc-tures,butinmostdatasetsitdoesnotgivetheexpectedfeatures.Inthesamearticle,RothandPeikertsuggestthat,inlinearelds,thevortexcorelineislocatedwheretheJacobianhasonereal-valuedeigenvector,andthiseigenvectorisparalleltotheow107.However,intheirownapplicationofturbomachineryows,theassumptionofalinearowistoosimple.ThesamealgorithmispresentedbySujudiandHaimes136.Recently,Jiangetal.presentedanewalgorithmforvortexcoreregiondetection42,whichisbasedonideasderivedfromcombinatorialtopology.Thealgorithmdeterminesforeachcellifitbelongstothevortexcore,byexaminingitsneighbouringvectors.Afewofthesealgorithmswillbereviewedindetail,next.BanksandSingerdevelopedapredictor-correctoralgo-rithmforndingvortexcores3.Afterinitialisation,vortexcoresaretrackedbypredictinginthedirectionofthevor-ticityvectorandcorrectingtothepressureminimumintheplaneperpendiculartothatvorticityvector.Next,theycreatevortextubes,bycomputingcrosssectionsofthevortices,inaplaneperpendiculartothevortexcore.Theyuseathresh-oldofthepressureasaselectioncriterion,incombinationwiththerestrictionthattheanglebetweenthevorticityvec-toratanypointonthecrosssectionandthevorticityvectoratthevortexcoreisnomorethanninetydegrees.SujudiandHaimesdevelopedanalgorithmforndingthecentreofswirlingowin3Dvectoreldsandimple-mentedthisalgorithminpV3136.AlthoughpV3canusemanytypesofgrids,thealgorithmhasbeenimplementedfortetrahedralcells.Whenusingdatasetswithothertypesofcells,thesersthavetobedecomposedintotetrahedralcells.Thisisdoneforefciency,becauselinearinterpolationforthevelocitycanbeusedinthecaseoftetrahedralcells.Thealgorithmisbasedoncritical-pointtheoryandusestheeigenvaluesandeigenvectorsofthevelocitygradienttensororrate-of-deformationtensor.Thealgorithmworksoneachpointinthedatasetseparately,makingitverysuitableforparallelprocessing.Thealgorithmsearchesforpointswherethevelocitygradienttensorhasonerealandtwocomplex-conjugateeigenvaluesandthevelocityisinthedirectionoftheeigenvector,correspondingtotherealeigenvalue.Thealgorithmresultsinlargecoherentstructureswhenastrongswirlingowispresent,andthegridcellsarenottoolarge.Thealgorithmissensitivetothestrengthoftheswirlingow,resultinginincoherentstructuresorevennostructuresatallinweakswirlingows.Also,ifthegridcellsarelarge,orirregularlysized,thealgorithmhasdifcultiesndingco-herentstructuresoranystructuresatall.c\rTheEurographicsAssociation2002. Post,Vrolijk,Hauser,Laramee,Doleisch/FeatureExtractionandVisualisationofFlowFieldsKenwrightandHaimesalsostudiedtheeigenvectormethodandconcludedthatithasproventobeeffectiveinmanyapplications53.Thedrawbacksofthealgorithmarethatitdoesnotproducecontiguouslines.Linesegmentsaredrawnforeachtetrahedralelement,buttheyarenotneces-sarilycontinuousacrosselementboundaries.Furthermore,whentheelementsarenottetrahedra,theyhavetobede-composedintotetrahedrarst,introducingapiecewiselin-earapproximationforanonlinearfunction.Anotherprob-lemisthatowfeaturesarefoundthatarenotvortices.In-stead,swirlingowisdetected,ofwhichvorticesareanex-ample.However,swirlingowalsooccursintheformationofboundarylayers.Finally,theeigenvectormethodissen-sitivetoothernonlocalvectorfeatures.Forexample,iftwoaxesofswirlexist,thealgorithmwillindicatearotationthatisacombinationofthetwoswirldirections.Theeigenvectormethodhassuccessfullybeenintegratedintoaniteelementsolverforguidingmeshrenementaroundthevortexcore16.RothandPeikerthavedevelopedamethodforndingcorelinesusinghigher-orderderivatives,makingitpossibletondstronglycurvedorbentvortices108.Theyobservethattheeigenvectormethodisequivalenttondingpointswheretheaccelerationaisparalleltothevelocityv,orequivalently,tondingpointsofzerocurvature.Theaccelerationaisde-nedas:a=DvDt;(6)wherethenotationDfDtisusedforthederivativefollowingaparticle,whichisdened,inasteadyow,asrfv.There-fore:a=DvDt=rvv=Jv;(7)withJtheJacobianofv,thatisthematrixofitsrstderiva-tives.RothandPeikertimprovethealgorithmbydeningvortexcoresaspointswhereb=DaDt=D2vDt2(8)isparalleltov,thatis,pointsofzerotorsion.Themethodinvolvescomputingahigher-orderderivative,introducingproblemswithaccuracy,butitperformsverywell.Incom-parisonwiththeeigenvectormethod,thisalgorithmndsstronglycurvedvorticesmuchmoreaccurately.RothandPeikertalsointroducetwoattributesforthecorelines:thestrengthofrotationandthequalityofthesolution.Thismakesitpossiblefortheusertoimposeathresholdonthevortices,toeliminateweakorshortvortices.PeikertandRothhavealsointroducedanewoperator,theparallelvec-torsoperator93,withwhichtheyareabletomathematicallydescribeanumberofpreviouslydevelopedmethodsunderonecommondenominator.Usingthisoperatortheycande-scribemethodsbasedonzerocurvature,ridgeandvalleylines,extremumlinesandmore.Jiangetal.recentlypresentedanewapproachfordetect-ingvortexcoreregions42.ThealgorithmisbasedonanideawhichhasbeenderivedfromSperner'slemmaincombina-torialtopology,whichstatesthatitispossibletodeducethepropertiesofatriangulation,basedontheinformationgivenattheboundaryvertices.Thealgorithmusesthisfacttoclas-sifypointsasbelongingtoavortexcore,basedonthevectororientationattheneighbouringpoints.In2D,thealgorithmisverysimpleandstraightforward,andhasonlylinearcom-plexity.In3D,thealgorithmissomewhatmoredifcult,be-causeitrstinvolvescomputingthevortexcoredirection,andnext,the2Dalgorithmisappliedtothevelocityvectorsprojectedontotheplaneperpendiculartothevortexcoredi-rection.Still,alsothe3Dalgorithmhasonlylinearcomplex-ity.Theabovedescribedmethodsallusealocalcriterionfordeterminingonapoint-to-pointbasiswherethevorticesarelocated.Thenextalgorithmsuseglobal,geometriccriteriafordeterminingthelocationofthevortices.Thisisaconse-quenceofusinganothervortexdenition.SadarjoenandPostpresenttwogeometricmethodsforextractingvorticesin2Delds112.Therstisthecurva-turecentremethod.Foreachsamplepoint,thealgorithmcomputesthecurvaturecentre.Inthecaseofvortices,thiswouldresultinahighdensityofcentrepointsnearthecentreofthevortex.Themethodworksbuthasthesamelimitationsastraditionalpoint-basedmethods,withsomefalseandsomemissingcentres.Thesecondmethodisthewinding-anglemethod,whichhasbeeninspiredbytheworkofPortela94.Themethoddetectsvorticesbyselectingandclusteringloopingstreamlines.Thewindingangleawofastreamlineisdenedasthesumoftheanglesbetweenthedifferentstreamlinesegments.Streamlinesareselectedthathavemadeatleastonecompleterotation,thatis,aw2p.Asecondcriterionchecksthatthedistancebetweenthestartingandendingpointsisrelativelysmall.Theselectedstream-linesareusedforvortexattributecalculation.Thegeometricmeaniscomputedofallpointsofallstreamlinesbelongingtothesamevortex.Anellipsettingiscomputedforeachvortex,resultinginanapproximatesizeandorientationforeachvortex.Furthermore,theangularvelocityandrotationaldirectioncanbecomputed.Alltheseattributescanbeusedforvisualisingthevortices.(SeeFigure13.)6.2.ShockwaveextractionShockwavesarealsoimportantfeaturesinowdatasets,andcanoccur,forexample,inowsaroundaircraft.(SeeFigure14.)Shockwavescouldincreasedragandcausestructuralfailure,andtherefore,areimportantphenomenatostudy.Shockwavesarecharacterisedbydiscontinuitiesinphysicalowquantitiessuchaspressure,densityandve-locity.Therefore,shockdetectioniscomparabletoedgede-tection,andsimilarprinciplescouldbeusedasinimageprocessing.However,innumericalsimulations,thediscon-c\rTheEurographicsAssociation2002. Post,Vrolijk,Hauser,Laramee,Doleisch/FeatureExtractionandVisualisationofFlowFieldsFigure13:FlowintheAtlanticOcean,withstreamlinesandellipsesindicatingvortices.Blueandredellipsesindi-catevorticesrotatingclockwiseandcounterclockwise,re-spectively113.Figure14:ShockwavesaroundamodelofaX-15inawindtunnelwithanairowatMach3.5.ImagefromtheNASAwebsite.tinuitiesareoftensmearedoverseveralgridpoints,duetothelimitedresolutionofthegrid.Maetal.haveinvesti-gatedanumberoftechniquesfordetectingandforvisual-isingshockwaves79.Detectingshocksintwodimensionshasbeenextensivelyinvestigated64;86;105.However,thesetechniquesareingeneralnotapplicabletoshocksinthreedimensions.Theyalsodescribeanumberofapproachesforvisualisingshockwaves.TheapproachofHaimesandDar-mofal28istocreateisosurfacesoftheMachnumbernormaltotheshock,usingacombineddensitygradient/Machnum-bercomputation.VanRosendalepresentsatwo-dimensionalshock-ttingalgorithmforunstructuredgrids105.Theideareliesonthecomparisonofdensitygradientsbetweengridnodes.Maetal.compareanumberofalgorithmsforshockex-tractionandalsopresenttheirowntechnique79:TherstideaistocreateanisosurfaceofthepointswheretheMachnumberisone.However,thisresultsinthesonicsurface,which,ingeneral,doesnotrepresentashock.Theoretically,abetterideaistocreateanisosurfaceofthepointswherethenormalMachnumberisequaltoone.However,ifthesurfaceisunknown,itisimpossibletocomputetheMachnumber,normaltothesurface.Thisproblemcanberesolved,byapproximatingtheshocknormalwiththedensitygradient,sinceashockisalsoassociatedwithalargegradientofthedensity.There-fore,rris(roughly)normaltotheshocksurface.Thus,thealgorithmcomputestheMachnumberinthedirec-tionof,orprojectedonto,thedensitygradient.TheshocksurfaceisconstructedfromthepointswherethisMachnumberequalsone.ThisalgorithmisalsousedbyLovelyandHaimes78,buttheydenetheshockregionasthere-gionwithintheisosurfaceofMachnumberone,anduselteringtechniquestoreconstructasharpsurface.Pagendarmpresentedanalgorithmthatsearchesformax-imainthedensitygradient91.Therstandsecondderiva-tivesofthedensityinthedirectionofthevelocityarecomputed.Next,zero-levelisosurfacesareconstructedofthesecondderivative,tondtheextremainthedensitygradient.Finally,therstderivativeisusedtoselectonlythemaxima,whichcorrespondtoshockwaves,anddis-cardtheminima,whichrepresentexpansionwaves.Thiscanbedonebyselectingonlypositivevaluesoftherstderivative.However,thesecondderivativecanalsobezeroinsmoothregionswithfewdisturbances.Inthesere-gionstherstderivativewillbesmall,therefore,thesere-gionscanbeexcludedbydiscardingallpointswheretherstderivativeisbelowacertainthresholde.Ofcourse,thisposestheproblemofndingthecorrecte.Whenthevalueistoosmall,erroneousshockswillbefound,butifthevalueistoolarge,partsoftheshockscoulddisappear.Thisalgorithmcanalsobeusedforndingdiscontinu-itiesinothertypesofscalarelds,andthusforndingothertypesoffeatures.Maetal.presentanadaptedversionofthisalgorithm,whichusesthenormalMachnumbertodotheselectioninthethirdstep79.Again,intherstandsecondstep,thezero-levelisosurfacesoftheseconddirectionalderiva-tiveofthedensityareconstructed.Butfordiscriminat-ingshockwavesfromexpansionwavesandsmoothre-gions,thenormalMachnumberisused.Moreprecisely,thosepointsareselectedwherethenormalMachnumberisclosetoone.Herealso,asuitableneighbourhoodofonehastobechosen.6.3.SeparationandattachmentlineextractionOtherfeaturesinowdatasetsareseparationandattach-mentlinesontheboundariesofbodiesintheow.Thesearec\rTheEurographicsAssociation2002. Post,Vrolijk,Hauser,Laramee,Doleisch/FeatureExtractionandVisualisationofFlowFieldsthelineswheretheowabruptlymovesawayfromorreturnstothesurfaceofthebody.Theseareimportantfeaturesinaerodynamicdesignbecausetheycancauseincreaseddragandreducedlift106,andtherefore,theiroccurrenceshouldbepreventedoratleastminimised.HelmanandHesselinkuseFigure15:Vectoreldtopology:atopologicalskeletonofaowaroundacylinder32.vectoreldtopologytovisualiseowelds31;32.Inadditiontothecriticalpoints,theattachmentanddetachmentnodesonthesurfaceofabodydeterminethetopologyoftheow.(SeeFigure15.)Theattachmentanddetachmentnodesarenotcharacterisedbyazerovelocity,becausetheyonlyoccurinowswithano-slipcondition,thatis,allpointsontheboundariesofobjectsareconstrainedtohavezerovelocity.Instead,theyarecharacterisedbyazerotangentialvelocity.Therefore,streamlinesimpingingonthesurfaceterminateattheattachmentordetachmentnode,insteadofbeingde-ectedalongthesurface.Globusetal.designedandimplementedasystemforanalysingandvisualisingthetopologyofaoweldwithiconsforthecriticalpointsandintegralcurvesstartingclosetothecriticalpoints25.Thesystemisalsoabletovisualiseattachmentanddetachmentsurfacesandvortexcores.PagendarmandWalter92andDeLeeuwetal.66usedskin-frictionlinesforvisualisingattachmentanddetachmentlinesinthebluntndataset.Forvisualisingtheselines,thewallshearvectortwiscomputed,whichisthegradientofthevelocitymagnitudejvj,projectedalongthenormalontothewall:tw=rjvj (rjvjn)n;(9)wherenistheunitvectornormaltothewall.Next,astandardstreamlinealgorithmisusedtointegratetheskin-frictionlinesfromtheshearvectoreld.Theseskin-frictionlinesshowthelocationofseparationandattachmentoftheowatthewall.(SeeFigure16.)Kenwrightgivesanoverviewofexistingtechniquesforvisualisingseparationandattachmentlinesandpresentsanewautomaticfeaturedetectiontechniqueforlocatingtheselines,basedonconceptsfrom2Dphaseplaneanalysis52.Somecommonapproachesare:Particleseedingandcomputationofintegralcurves,suchasstreamlinesandstreaklines,whichareconstrainedtoFigure16:Skin-frictiononabluntnfromaowsimulationatMach5,visualisedwithspotnoise66.thesurfaceofthebody.Thesecurvesmergealongsepara-tionlines.Skin-frictionlinescanbeused,analogoustosurfaceoilowtechniquesfromwindtunnelexperiments92.(Seeabove.)Texturesynthesistechniquescanbeusedtocreatecontin-uousowpatternsratherthandiscretelines66.HelmanandHesselinkcanautomaticallygeneratesepa-rationandattachmentlinesfromtheirvectoreldtopol-ogy31.Theselinesaregeneratedbyintegratingcurvesfromthesaddleandnodetypecriticalpointsinthedirec-tionoftherealeigenvector.However,onlyclosedsepara-tionsarefound,thatis,thecurvesstartandendatcriticalpoints.Openseparationdoesnotrequireseparationlinestostartorendatcriticalpoints,andisthereforenotdetectedbyowtopology.Openseparationhasbeenobservedinexper-iments,buthadnotpreviouslybeenstudiedinowsimula-tions.However,thealgorithmpresentedbyKenwrightdoesdetectbothclosedandopenseparationlines.Thetheoryforthisalgorithmisbasedonconceptsfromlinearphaseplaneanalysis.Itisassumedthatthecomputationaldomainonthesurfacecanbesubdividedintotrianglesandthevectorcom-ponentsaregivenatthevertices.Thealgorithmisexecutedforeachtriangle,makingitsuitableforparallelisation.Foreachtriangle,alinearvectoreldisconstructedsatisfyingthevectorsatthevertices.IfthedeterminantoftheJaco-bianmatrixisnonzero,thealgorithmcontinuesbycalculat-ingtheeigenvaluesandeigenvectorsoftheJacobian.Everytrianglehasacriticalpointsomewhereinitsvectoreld.Thelinearvectoreldistranslatedtothiscriticalpointandthecoordinatesystemischangedsothattheeigenvectorsareorthogonal.This(x;y)planeisalsoreferredtoasthePoincaréphaseplane.Bycomputingtangentcurvesinthephaseplane,weobtainthephaseportraitofthesystem.Forc\rTheEurographicsAssociation2002. Post,Vrolijk,Hauser,Laramee,Doleisch/FeatureExtractionandVisualisationofFlowFieldsasaddle,thetangentcurvesorstreamlinesconvergealongthexandyaxes.Forarepellingnode,theyconvergealongthexaxisandforanattractingnode,theyconvergealongtheyaxis.Ifthephaseportraitisasaddleorarepellingnode,theintersectionofthexaxiswiththetriangleiscomputed.Ifitintersects,thelinesegmentwillformpartofanattachmentline.Ifthephaseportraitisasaddleoranattractingnode,theintersectionoftheyaxiswiththetriangleiscomputed,andifitdoesintersect,thelinesegmentwillformpartofaseparationline.Aproblemwiththisalgorithmisthatdisjointedlineseg-mentsarecomputedinsteadofcontinuousattachmentandseparationlines.Otherproblemsoccurwhentheowsepa-rationorattachmentisrelativelyweak,orwhentheassump-tionoflocallylinearowisnotcorrect.Kenwrightetal.presenttwoalgorithmsfordetectingsep-arationandattachmentlines54.Therstisthealgorithmdiscussedabove,thesecondistheparallelvectoralgorithm.Bothalgorithmsuseeigenvectoranalysisofthevelocitygra-dienttensor.However,therstiselement-basedandresultsindisjointedlinesegments,whilethesecondispoint-basedandwillresultincontinuouslines.Intheparallelvectoralgorithm,pointsarelocatedwhereoneoftheeigenvectorseiofthegradientrvisparalleltothevectoreldv,thatis,pointswherethestreamlinecurvatureiszero,orinformula:eiv=0:(10)Thevelocityvectorsandtheeigenvectorscanbedeterminedattheverticesofthegridandinterpolatedwithintheele-ments.Atthevertices,eiviscalculatedforbotheigen-vectors,butonlyifbotheigenvectorsarereal,thatis,theclassicationofrvatthevertexiseitherasaddleoranode.Ifthecrossproducteivchangessignacrossanedge,thatmeansanattachmentorseparationlineintersectstheedge.Theintersectionpointcanthenbefoundbyinterpolationalongtheedge.Theattachmentandseparationlinescanbeconstructedbyconnectingtheintersectionpointsineachel-ement.Thedistinctionbetweenattachmentandseparationcanbemadeeasily,becauseattachmentwilloccurwherevisparalleltothesmallesteiandseparationwherevisparal-leltothelargestei.Anothersetoflinesisdetectedwiththisalgorithm,theinectionlines.Thesecaneasilybelteredoutbycheckingif:r(eiv)v=0:(11)Thiswillnotbetrueforinectionlines.BothalgorithmsdiscussedbyKenwrightetal.correctlyidentifymanyseparationandattachmentlines,butmayfailinidentifyingcurvedseparationlines54.Theparallelvectoralgorithmwillresultincontinuouslines,whereasthephaseplanealgorithmresultsindiscontinuouslinesegments.Bothalgorithmsdodetectopenseparationlines,whichdonotstartorendatcriticalpoints.6.4.OthertypesoffeaturesThereareothertypesoffeatures,suchasrecirculationzonesandboundarylayers.Workhasbeendoneinextractingthesefeatures,forexamplebyHaimes27andbySadarjoenandPost110.Huntetal.givequantitativecriteriafordividingaowintothreeareas,withspeciccharacteristics:eddies,streams,andconvergencezones37.7.FeaturetrackingandeventdetectionIntime-dependentdatasets,featuresareobjectsthatevolveintime.Determiningthecorrespondencebetweenfeaturesinsuccessivetimesteps,thatactuallyrepresentthesameob-jectatdifferenttimes,iscalledthecorrespondenceprob-lem.Featuretrackingisinvolvedwithsolvingthiscorre-spondenceproblem.Thegoaloffeaturetrackingistobeabletodescribetheevolutionoffeaturesthroughtime.Duringtheevolution,certaineventscanoccur,suchastheinterac-tionoftwoormorefeatures,orsignicantshapechangesoffeatures.Eventdetectionistheprocessofdetectingsuchevents,inordertodescribetheevolutionofthefeaturesevenmoreaccurately.Therearetwobasicapproachestosolvingthecorrespon-denceproblem.Therstisbasedonregioncorrespondence,thesecondonattributecorrespondence.7.1.RegioncorrespondenceRegioncorrespondenceinvolvescomparingtheregionsofinterestobtainedbyfeatureextraction.Basically,thebinaryimagesfromsuccessivetimesteps,containingthefeaturesfoundinthesetimesteps,arecomparedonacell-to-cellbasis.Correspondencecanbefoundusingaminimumdis-tanceoramaximumcross-correlationcriterion26orbymin-imisinganafnetransformationmatrix50.Itisalsopossi-bletoextractisosurfacesfromthefour-dimensionaltime-dependentdataset159,wheretimeisthefourthdimension.Thecorrespondenceisthenimplicitlydeterminedbyspatialoverlapbetweensuccessivetimesteps.Thiscriterionissim-ple,butnotalwayscorrect,asobjectscanoverlapbutnotcorrespond,orcorrespondbutnotoverlap.SilverandWangexplicitlyusethecriterionofspatialoverlapinsteadofcre-atingisosurfacesinfourdimensions130;131;132.Theypreventcorrespondencebyaccidentaloverlap,bycheckingthevol-umeofthecorrespondingfeaturesandtakingthebestmatch.Thisisalsotheideaofattributecorrespondence,whichisdiscussednext.Byusingspatialoverlap,certaineventsareimplicitlydetected,suchasabifurcationwhenafeatureinonetimestepoverlapswithtwofeaturesinthenexttimestep.Eventdetectionisalsodiscussedmoreelaboratelylater,inSection7.3.7.2.AttributecorrespondenceWithattributecorrespondence,thecomparisonoffeaturesfromsuccessiveframesisperformedonthebasisoftheat-c\rTheEurographicsAssociation2002. Post,Vrolijk,Hauser,Laramee,Doleisch/FeatureExtractionandVisualisationofFlowFieldstributesofthefeatures,suchastheposition,size,volume,andorientation.Theseattributescanbecomputedinthefea-tureextractionphase,(seeSection5.3,)andcanbeusedfordescriptionandforvisualisationofthefeatures,andalsoforfeaturetracking,asdescribedhere.Theoriginalgriddataisnotneededanymore.Samtaneyetal.usetheattributevaluestogetherwithuser-providedtolerancestocreatecorrespon-dencecriteria115.Forexample,forpositionthefollowingcouldbeused:dist(pos(Oi+1);pos(Oi))Tdist;(12)wherepos(Oi)andpos(Oi+1)arethepositionsoftheob-jectsintimestepsiandi+1,respectively,andTdististheuser-providedtolerance.Forscalarattributes,thedifferenceortherelativedifferencecouldbeused.Forexample,totesttherelativedifferenceofthevolume,thefollowingformulacanbeused:vol(Oi+1) vol(Oi)max(vol(Oi+1);vol(Oi))Tvol;(13)wherevol(Oi)andvol(Oi+1)arethevolumesofthefeaturesinthetwotimesteps,andTvolisthetolerancegivenbytheuser.Eventssuchasabifurcationcanalsobetested.Ifafeatureintimestepisplitsintotwofeaturesintimestepi+1,thetotalvolumeaftertheeventhastobeapproximatelythesameasbeforetheevent.Thesameformulacanbeusedasforthenormalvolumetest,exceptthatvol(Oi+1)inthiscaseequalsthesumofthevolumesoftheseparatefeatures.Thepositioncriterionincaseofabifurcationeventcouldinvolvetheweightedaverageoftheindividualpositionsaftertheevent,wherethepositionsareweightedwiththevolume:dist(pos(Oi);å(vol(Oi+1)pos(Oi+1))å(vol(Oi+1)))Tdist;(14)whereOi+1nowrepresentsallobjectsintimestepi+1thatareinvolvedintheevent.Sethietal.presentamethodforimage-basedmotionanal-ysis,withtheuseofmarkersortokens123.Thebasiccon-ceptissmoothnessofmotionoffeaturepointtrajectoriesinpropertyspace.Propertiesorattributesoffeaturesarerepresentedbypointsinpropertyspace.Thesepointsmovethroughthepropertyspaceovertime,andthealgorithmtriestondthesmoothestpathsortrajectoriesinthispropertyspace.Thenotionofpropertycoherenceisused,thatis,thepropertiesaresupposedtochangegradually.Twoalgorithmsaredescribedtondthesmoothesttrajectories:ModiedGreedyExchangeandSimulatedAnnealing.Wewillde-scribetheformerhere.Thebasicideaofbothalgorithmsistocreateinitialtrajectoriesbyconnectingtheclosestpointsinpropertyspace,andthentoiterativelyrenethetrajecto-riestomaximisethetotalsmoothnessofalltrajectories.IntheGreedyExchangealgorithm,thisoptimisationisdonebyexchangingtokensbetweentrajectoriesandcomputingthegaininsmoothness.Theexchangewiththemaximumgainischosen.Theprocessisrepeated,bothforwardandbackwardintime,untilnomoreexchangesaremade.Thepropertyco-herenceisameasureforthreeconsecutivepointsinpropertyspace.Whenwecallthesepoints,fromthreeconsecutivetimesteps,a,bandc,thepropertycoherenceisdenedas:F(a;b;c)=w11 ~ab~bck~abkk~bck+w2 1 2qk~abkk~bckk~abk+k~bck!(15)Thersttermontherighthandsidegivesameasureforthechangeindirectionbetweenthevectors~aband~bc,andthesecondtermgivesameasureforthechangeinlengthofthesevectors.Boththesemeasuresarecombinedwithuser-providedweightsw1andw2.Becausedifferentpropertiescanhavedifferentcharacteristics,theaxesinthepropertyspacecanbescaledusingsuitablescalingfactors.Thenor-malformulasforcomputingtheinnerproductoftwovectorsandthelengthofavectorarethereforeadaptedasfollows:~ab~bc=kåi=1si(bi ai)(ci bi)(16)k~abk=vkåi=1si(bi ai)2;wheresiisthescalingfactorfortheithaxis,withåsi=1,andaiistheithcomponentofthek-dimensionalpropertyvectora.Reindersetal.describeanalgorithmforfeaturetracking,thatisbasedonpredictionandverication99;100.Thisalgo-rithmisbasedontheassumptionthatfeaturesevolvepre-dictably.Thatmeans,ifapartoftheevolutionofafeature(path)hasbeenfound,apredictioncanbemadeintothenexttimestep(frame).Then,inthatnexttimestep,afea-tureissought,thatcorrespondstotheprediction.Ifafea-tureisfoundthatmatchesthepredictionwithincertainuser-providedtolerances,thefeatureisaddedtotheevolutionandthesearchiscontinuedtothenexttimestep.Whennomorefeaturescanbeaddedtothepath,anewpathisstarted.Inthismanner,allframesaresearchedforstartingpoints,bothinforwardandbackwardtimedirection,untilnomorepathscanbecreated.Apathisstartedbytryingallpossiblecombinationsoffeaturesfromtwoconsecutiveframesandcomputingthepredictiontothenextframe.Then,thepredictioniscom-paredtothecandidatefeaturesinthatframe.Ifthereisamatchbetweenthepredictionandthecandidate,apathisstarted.Toavoidanyerroneousorcoincidentalpaths,thereisaparameterfortheminimalpathlength,whichisusuallysetto4or5frames.Acandidatefeaturecanbedenedintwoways.Allfeaturesintheframecanbeusedascandidates,oronlyunmatchedfeaturescanbeused,thatis,thosefeaturesthathavenotyetbeenassignedtoanypath.Therstdeni-tionensuresthatallpossiblecombinationsaretestedandthatc\rTheEurographicsAssociation2002. Post,Vrolijk,Hauser,Laramee,Doleisch/FeatureExtractionandVisualisationofFlowFieldsthebestcorrespondenceischosen.However,itcouldalsoresultinfeaturesbeingaddedtomorethanonepath.Thishastoberesolvedafterwards.Usingtheseconddenitionismuchmoreefcient,becausethemorepathsarefound,thelessunmatchedfeatureshavetobetested.However,inthiscase,theresultsdependontheorderinwhichthefeaturesaretested.Thisproblemcanbesolvedbystartingthetrack-ingprocesswithstricttolerancesandrelaxingthetolerancesinsubsequentpasses.Thepredictionofafeatureisconstructedbylinearex-trapolationoftheattributesofthefeaturesfromthelasttwoframes.Otherpredictionschemescouldalsobeused,forex-ample,ifa-prioriknowledgeoftheowisavailable.Thepredictionismatchedagainstrealfeaturesusingcor-respondencecriteria,similartotheonesusedbySamtaneyetal.asdiscussedabove115.Foreachattributeofthefeatures,acorrespondencefunctioncanbecreated,whichreturnsapos-itivevalueforacorrespondencewithinthegiventolerance,withavalueof1foranexactmatch,andanegativevaluefornocorrespondence.Eachcorrespondencefunctionisas-signedaweight,besidesthetolerance.Usingthisweight,aweightedaverageiscalculatedofallcorrespondencefunc-tions,resultinginthecorrespondencefactorbetweenthetwofeatures.Forthiscorrespondencefactor,thesameappliesasfortheseparatecorrespondencefunctions,thatis,apositivevalueindicatesacorrespondence,with1indicatingaperfectmatch.Anegativecorrespondencefactormeansnomatch.7.3.EventdetectionAfterfeaturetrackinghasbeenperformed,eventdetectionisthenextstep.Eventsarethetemporalcounterpartsofspatialfeaturesintheevolutionoffeatures.Forexample,ifthepathorevolutionofafeatureends,itcanbeinterestingtodeter-minewhythathappens.Itcouldbethatthefeatureshrinksandvanishes,orthatthefeaturemovestotheboundaryofthedatasetanddisappears,orthatthefeaturemergeswithanotherfeatureandthetwocontinueasone.Samtaneyetal.introducedthefollowingevents:continuation,creation,dissipation,bifurcation,amalgamation115.(SeeFigure17.)Reindersetal.developedafeaturetrackingsystemthatisabletodetecttheseandotherevents100.Theterminologytheyuseisbirthanddeathinsteadofcreationanddissipa-tion,andsplitandmergeforbifurcationandamalgamation.Furthermore,theycandetectentryandexitevents,whereafeaturemovesbeyondtheboundaryofthedataset.Finally,foraspecic,graph-typefeature,thesystemisabletode-tectchangesintopology.Itdiscriminatesloopandjunctionevents.(SeeFigure18.)Manyothertypesofeventscanbeenvisioned,butforeachtypespecicdetectioncriteriahavetobeprovided.Foreventdetection,justasforfeaturetracking,onlythefeatureattributesareused.Analogoustothecorrespondencefunctions,foreventdetection,eventfunctionsarecomputed.1234123amalgamationbifurcationcontinuation21534dissipationcreationabc4Figure17:ThedifferenttypesofeventsasintroducedbySamtaneyetal.115.Forexample,todetectadeathevent,twoconditionsmusthold.First,thevolumeofthefeaturemustdecrease.Andsecond,thevolumeofthepredictionmustbeverysmallornegative.Theeventfunctionforthiseventreturnsaposi-tivevalueifthevolumeofthepredictioniswithintheuser-providedtolerance,and1ifthevolumeofthepredictionisnegative.Ifthevolumeisnotwithinthetolerance,there-turnedvaluewillbenegative.Theeventfunctionsfortheseparateattributesarecombinedintoasinglefactor,whichdeterminesiftheeventisadeathevent.Abirtheventcanbedetectedbydoingthesametestsinthebackwardtimedirection.Similarly,thetestsforsplitandmergeevents,andforen-tryandexiteventsareeachother'sreverseintime.8.VisualisationoffeaturesandeventsThenalstepinthefeatureextractionprocessis,ofcourse,thevisualisationofthefeatures.Anumberoftechniqueswillbecoveredinthissection.Themoststraightforwardvisual-isationistoshowthenodesinthedataset,thathavebeenc\rTheEurographicsAssociation2002. Post,Vrolijk,Hauser,Laramee,Doleisch/FeatureExtractionandVisualisationofFlowFieldsFigure18:Aloopeventhasoccurred.Inthetopgure,thefeaturecontainsaloop,inthebottomgure,thenextframe,theloophasdisappeared97.selectedintherststepofthefeatureextractionpipeline,theselection.Thisstepresultsinabinarydataset,witheachvalueindicatingwhetherthecorrespondingnodehasbeenselectedornot.Thisbinarydatasetcanbevisualised,forexample,withcrossesattheselectednodes.InFigure19,Figure19:Visualisationoftheselectedpointsinthebackward-facing-stepdataset114.suchavisualisationisshown.Thevisualisationisofasim-ulationoftheowbehindabackward-facingstep.Thefea-turethatisvisualisedhereisarecirculationzone,behindthestep.Thepointswereselectedwiththecriterion:normalisedhelicityH�0:6.Anothersimplevisualisationtechniqueistouseisosur-faces.Thiscanbedoneonthebinarydataset,resultingfromtheselectionstep,or,iftheselectionexpressionisasimplethreshold,directlyontheoriginaldataset.Thisresultsinisosurfacesenclosingtheselectedregions.Also,otherstandardvisualisationtechniquescanbeusedincombinationwiththebooleandatasetresultingfromtheselectionstep.Forexample,ina3Dowdataset,usingthestandardmethodsforseedingstreamlinesorstreamtubes,willnotprovidemuchinformationaboutthefeaturesandwillpossiblyresultinvisualclutter.However,iftheselectedpointsareusedtoseedstreamlines,bothbackwardandfor-wardintime,thiscanprovideusefulinformationaboutthefeaturesandtheirorigination.SeeFigure20,foranexample,Figure20:Visualisationwithstreamtubesoftherecircula-tioninthebackward-facing-stepdataset153.wheretwostreamtubesareshowninthebackward-facing-stepdataset.Theradiusofthetubesisinverselypropor-tionaltothesquarerootofthelocalvelocitymagnitude,andthecolourofthetubescorrespondstothepressure.If,insteadoftheseparateselectedpoints,theattributesareused,thathavebeencomputedinthefeatureextractionprocess,thenparametriciconscanbeusedforvisualisingthefeatures.Ifanellipsoidttingoftheselectedclustershasbeencomputed,therearethreeattributevectors:thecentrepo-sition,theaxislengths,andtheaxisorientations,whichcanbemappedontotheparametersofanellipsoidicon.Thisisasimpleicon,butveryefcientandaccurate.Itcanberep-resentedwith9oating-pointvalues,andisthereforespace-efcient.Furthermore,itcanbeveryquicklyvisualised,andalthoughitissimple,itgivesanaccurateindicationofthepositionandvolumeofafeature.InFigure21,anellipsoidttingiscomputedfromtheselectedpointsinFigure19.InFigure22,vorticesareshownfromaCFDsimulationwithturbulentvortexstructures.Thefeatureshavebeenselectedbyathresholdonvorticitymagnitude.Theyarebeingvi-sualisedwithisosurfacesandellipsoids.Itisclearlyvisiblethat,inthisapplication,withthestronglycurvedfeatures,theellipsoidsdonotgiveagoodindicationoftheshapeofthefeatures.But,asmentionedabove,thepositionandvol-umeattributesoftheellipsoidswillbeaccurate,andcanbec\rTheEurographicsAssociation2002. Post,Vrolijk,Hauser,Laramee,Doleisch/FeatureExtractionandVisualisationofFlowFieldsFigure21:Anellipsoidttingcomputedfromtheselectedpointsinthebackward-facing-stepdataset114.Figure22:Vorticesinadatasetwithturbulentvortexstruc-tures,visualisedusingisosurfacesandellipsoids97.usedforfeaturetracking.InFigure23,theowpastata-peredcylinderisshown.Streamlinesindicatetheowdi-rection,androtatingstreamlinesindicatevortices.Thevor-ticesareselectedbylocatingtheserotatingstreamlines,us-ingthewinding-anglemethod112.Ellipsoidsareusedtovi-sualisethevortices,withthecolourindicatingtherotationaldirection.Greenmeansclockwiserotation,redmeanscoun-terclockwiserotation.Thesliceiscolouredwithl2,whichisthesecond-largesteigenvalueofthetensorS2+W2.(SeeSection6.1.)Thetaperedcylinderdatasetconsistsofanumberofhor-izontalslices,suchastheoneinFigure23,butthevor-ticesarenaturallythree-dimensionalstructures.ReindersFigure23:Vorticesbehindataperedcylinder.Thecolouroftheellipsoidsrepresentstherotationaldirection112.etal.createdthesevortexstructuresbyperformingfea-turetrackinginaspatialdimensioninsteadoftrackingintime101.First,featureextractionwasperformedinthetwo-dimensionalslices.Thisresultedintwo-dimensionalvor-tices,whichwererepresentedbyaspecialtypeofellipseicon.Next,these2Dfeaturesweretrackedinthez-direction,forming3Dvortexstructures.Figure24showsanimageoftheresultingfeatures.The2Diconsareellipseswithanum-berofcurvedspokes.Thecurvatureofthespokesindicatestherotationaldirectionofthevortices,andthenumberofspokesrepresentstherotationalspeed.The3Diconsarecon-structedbyconnectingthecentrepointsofthe2Dellipses.Forthe3DvorticesinFigure22,anothertypeoficonhastobeused,ifwewanttovisualisethestronglycurvedshapeofthefeatures.Reindersetal.presenttheuseofskeletongraphdescriptionsforfeatures,withwhichtheycancreateiconsthataccuratelydescribethetopologyofthefeatures,andapproximatelydescribetheshapeofthefeatures98.ComparetheuseofellipsoidiconswiththeuseofskeletoniconsinFigure25.Forvisualisingtheresultsoffeaturetracking,itisofcourseessentialtovisualisethetimedimension.Themostobviouswayistoanimatethefeatures,andtogivetheusertheopportunitytobrowsethroughthetimesteps,bothback-wardandforwardintime.Figure26showstheplayerfromthefeaturetrackingprogram,developedbyReinders100.Ontheleftofthegure,thegraphviewerisshown,whichgivesanabstractoverviewoftheentiredataset,withthetimestepsonthehorizontalaxis,andthefeaturesrepresentedbynodes,ontheverticalaxis.Thecorrespondencesbetweenfeaturesfromconsecutiveframesarerepresentedbyedgesinthegraph,andtherefore,theevolutionofafeatureintime,isrepresentedbyapathinthegraph.Ontherightoftheg-ure,thefeatureviewerisshown,inwhichthefeatureiconsfromthecurrentframearedisplayed.Also,acontrolpanelc\rTheEurographicsAssociation2002. Post,Vrolijk,Hauser,Laramee,Doleisch/FeatureExtractionandVisualisationofFlowFieldsFigure25:Turbulentvortexstructuresrepresentedbyellipsoidicons(left)andskeletonicons(right)97.isvisible,withwhichtheanimationcanbestarted,paused,andplayedforwardandbackward.Thegraphviewercanalsobeusedforvisualisingevents97.Foreachevent,aspeciciconhasbeencreated,whichismappedontothenodesofthegraph,sothattheusercanquicklyseewhicheventsoccurwhere,andhowof-tentheyoccur.InFigure27,thegraphviewerisshown,withapartofthegraph,containinganumberofevents.Eacheventisclearlyrecognisablebyitsicon.InFigure28,twoframesareshown,betweenwhichaspliteventhasoccurred.Inbothframes,thefeaturesareshownwithbothellipsoidandskeletonicons.Theadvantageoftheuseofskeletoniconsinthisapplicationisobvious.Becausetheshapeofthefeaturesismuchmoreaccuratelyrepresentedbytheskeletonicons,changesinshapeandeventssuchasthesearemuchmoreeasilydetected.9.ConclusionsandfutureprospectsAstate-of-the-artreportmustendwithanassessment:whathasbeenachievedinowvisualisationduringthelast15years?Havetheproblemsbeensolved?Aretheresultsap-pliedinpractice?Whataretheremainingchallenges?Alargenumberoftechniqueshasbeendevelopedandrened.Ingeneral,whichtechniquesarethebest,dependsstronglyonthepurposeofthevisualisation:theresearchproblemsaddressed,themethodsandapproachesused,andthepersonalinterestoftheresearcherorengineer.Usersmayalsohavedifferentpurposes,suchasexploration,detailedanalysis,orpresentation.Therefore,webelievethatalargevarietyoftechniquesmustbeavailabletoallowresearcherstochoosethemostsuitabletechniquefortheirpurpose.Inthissense,goodprogresshasbeenmade.Averysuccessfulgroupisthetexture-basedtechniques(seeSection3),mainlyusedfor2Dowsandsurfaceowelds.Theyareverysuitableforanimation,bothofstation-aryandtime-dependentows.Performancelimitationsseemtobeovercome165,andinteractiveusewithunsteadyowsisnowfeasible.However,generalisationto3Doweldsisstillproblematic.Techniquesbasedonintegrationforgener-atinggeometriesandparticleanimation(seeSection4)arealsoverysuccessful,andgeneralisebetterto3Delds.Oneoftheoriginalkeyproblemsinowvisualisationwasthedirectvisualisationofdirectionalstructuresina3Deld,possiblyvaryingintime.Despitesomeheroicattempts,thisproblemhasnotbeensolved,asperceivingthreespatialandthreedatadimensionsdirectlyseemsaverytoughjobforthehumaneyeandbrain.Atthesametime,thescaleofnumer-icalowsimulations,andthusthesizeoftheresultingdatasets,continuestogrowrapidly.Forthesereasons,simplica-tionstrategieshavetobeconceived,suchasspatialselection(slicing,regionsofinterest),datadimensionreduction,ge-ometrysimplication,andfeatureextraction.Slicingina3Deldreducestheproblemto2D,allowinguseofgood2Dtechniques,butcaremustbetakenwithin-terpretation,asthelossofthethirddimensionmayleadtophysicallyirrelevantresultsandwronginterpretation.Tak-ingasingle3Dtimeslicefroma3Dtime-dependentdatasethassimilardangers.Otherspatialselectionssuchas3Dregion-of-interestselectionarelessrisky,butmayleadtolossofcontext.Reductionofdatadimension,suchasre-ducingvectorquantitiestoscalarswillgivemorefreedomofchoiceinvisualisationtechniques(suchasusingvolumerendering),butwillnotleadtomuchdatareduction.Geom-etrysimplicationtechniquessuchaspolygonmeshdeci-mation,levels-of-detail,ormultiresolutiontechniqueswillbeeffectiveinmanagingverylargedatasetsandinteractiveexploration,enablinguserstotradeaccuracywithresponsetime.Featureextraction(Section5)isselectionandsimplica-tionbasedoncontent:extractingimportanthigh-levelinfor-mationfromadataset,visualisingthedatafromaproblem-orientedpointofview.Thisleadstoalargereductionofthedatasize,andtofullyorsemiautomaticgenerationofsim-pleandclearimages.Thetechniquesaregenerallyveryspe-cicforacertaintypeofproblem(suchasvortexdetection),c\rTheEurographicsAssociation2002. Post,Vrolijk,Hauser,Laramee,Doleisch/FeatureExtractionandVisualisationofFlowFieldsFigure28:Asplitevent,before(left)andafter(right).Thefeaturesarevisualisedwithbothanellipsoidandaskeletonicon97.andtherelationwiththeoriginalrawdataisindirect,andthereductionisachievedatthecostoflossofotherinfor-mation,whichisconsideredirrelevantforthepurpose.Butthetechniquesgeneralisewelltoanalysisoftime-dependentdatasets,leadingtocondensedepisodicvisualsummaries.Agoodpossibilityiscombiningfeatureextractiontech-niqueswithdirectorgeometrictechniques.Forexample,se-lectivevisualisationhasbeenusedeffectivelywithstream-linegeneration,toplaceseedpointsinselectedareas,andshowimportantstructureswithonlyasmallnumberofstreamlines.Combiningsimpleadvection-basedtechniqueswithiconicfeaturevisualisationcanalsoclarifytherelationbetweentherawdataandthederivedinformationusedinfeaturedetection.Theworkofvisualisationandsimulationexpertswillinthefuturebecomeinseparable:thedistinc-tionbetweensimulationandvisualisationwillbeincreas-inglyblurred.Agoodexampleisthetrackingofphasefronts(separationbetweentwodifferentuidsinmultiuidows)usinglevelsetmethods124,wherethefeatureextractionisapartofbothsimulationandvisualisation.Howaboutpracticalapplication?Manytechniqueshavebeenincorporatedincommercialvisualisationsystems,in-cludingfeature-basedtechniquesy.Thepracticaluseofowvisualisationismosteffectivewhenvisualisationex-pertscloselycooperatewithuiddynamicsexperts.Thisisespeciallytrueinfeature-basedvisualisation,wheredevel-opingdetectioncriteriaiscloselyconnectedtothephysi-calphenomenastudied.Butalsootherdisciplinescancon-tributetothiscomputationalscienceeffort:mathematicians,artistsanddesigners,experimentalscientists,imagepro-cessingspecialists,andalsoperceptualandcognitivescien-tists96.Someareasthatneedadditionalworkare:yhttp://www.ensight.com/products/flow-feature.htmlcomparativevisualisationandmultisourcecomparativedataanalysisvisualisationofmultivariateoweldswithscalar,vector,andtensordatahandlingandexploringhugetime-dependentowdatasetsdetectionandtrackingofnewtypesoffeatures,suchassurfacefeatures(shockwaves,phasefronts)intime-dependentdatasetstheuseofvirtualenvironmentsforvisualdataexplorationandcomputationalsteering:problemsofperformanceand3Dinteractionuserstudiesforevaluation,validation,andeldtestingofowvisualisationtechniquesvisualisationofinaccuracyanduncertainty.Overlookingthewholelandscapeofowvisualisationtechniques,wecansaythatvisualisationof2Dowshasreachedahighlevelofperfection,andfor3Darichsetoftechniquesisavailable.Inthefuture,wewillconcentrateontechniquesthatscalewellwitheverincreasingdatasetsizes,andthereforeselectionandsimplicationtechniqueswillgetmoreattention.10.AcknowledgementsTheauthorsthankallthosewhohavecontributedtothisresearchincludingAVL(www.avl.com),theAustriangovernmentalresearchprogramcalledKplus(www.kplus.at),andtheVRVisResearchCenter(www.VRVis.at).Furthermore,theauthorsthankallthepermissionholdersofimagesshowninthisreport.ThisprojectwaspartlysupportedbytheNetherlandsOr-ganizationforScienticResearch(NWO)ontheNWO-EWComputationalScienceProjectDirectNumericalSimula-tionofOil/WaterMixturesUsingFrontCapturingTech-niques.c\rTheEurographicsAssociation2002. 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