White-light-heterodyne-principle-for-3D-measuremen

Whitelightheterodyneprinciplefor3D-measurementCarstenReich,ReinholdRitter,JanThesingInstituteofMetrologyandExperimentalMechanicsABSTRACT3D-measurementsystemsbasedonactivetriangulationusingprojectionofstructuredlightandrecordingbyCCD-camerasaregainingmoreandmoreimportanceinindustrialapplication.Theiradvantagescomparedtoconventionalcoordinatemeasurementmachinesarethenon-contactmeasurementandthefastacquisitionofdensepointclouds.Thispaperpresentsanalternativemethodtothewell-knowncombinationofGraycodeandphase-shiftprojection.Itwillbeshownhowtousetheheterodyneprincipleforovercomingtheunwrappingproblemofphasefunctionsandforestimatingtheparametersforareliableunwrappingprocedure.Theredundantphaseinformationisusedtoincreasetheaccuracyofthephotogrammetrical3D-coordinatedetermination.Finally,theheterodyneprincipleisusedtorealizeon-linemeasurementbasedoncolor-codedfringeprojection.Keywords:3D-measurement,color-coded,fringeprojection,heterodyneprinciple,on-line,photogrammetry,structuredlight,topometricsensor1.INTRODUCTIONSeveraltypesofstructuredlightarecurrentlyusedandthemostrelevantcharacteristicsarethemeasuringtimeandtheaccuracy.Thephase-shiftprincipleisbasedonsinusoidalintensitydistribution,whichisprojectedontheconsideredobjectsurface.ForeachpixeloftheCCD-chiptheactualphasecanbecomputed.Problemsariseduetothefactthatthephasevaluesareperiodicperpendiculartothefringedirection.ThemostcommonmethodtosolvethisambiguityproblemistheusageofGraycodepatterns.7Theyprovideareliabledeterminationofthephaseorders,butmoretime-consumingprojectionsareneeded,whichdonotraisetheaccuracy.Amethodusingonlyphase-shiftingtechniqueswasproposedbyZumbrunn.9Here,thefirstmeasurementismadebyafringespacingcoveringthefieldofview,sothatonlyonefringeorderappears.Thephasevaluescanbeusedtoobtainthephaseordersofasecondfringepatternwithasmallerfringespacing.Malz3showedhowthebeatoftwoperiodicfunctionswithhighfrequenciescanbeusedtocalculatetheorders.2.FUNDAMENTALPRINCIPLESThewhitelightheterodyneprincipleconsistsofseveralphase-shiftsandisbasedonthebeatoftwoperiodicfunctionswithdifferentfrequencies.Theresultisanewperiodicfunction.Thefrequenciesoftheinitialfunctionshavetobeselectedinawaythatthebeatfrequencyislowenoughtobeunambiguousoverthefieldofviewofthecameras.Then,thephasevaluesofthebeatfrequencycanbeusedtounwraptheperiodicphasevaluesoftheinitialfunctions.236SPIEVol.3100.0277-786X/97/$10.00DownloadedFrom::(1)hastobesolved:1(x)=Io+A(cos(cb(x)+Lçb))(1)1(x)Intensity1IntensityofbackgroundAAmplitudeofsinusdistributionq5(x)PhaseLçbconst.phaseshiftThiseqn.containsfourunknownparameters(Io,A,q(x),Zc).Awellknownmethodfordeterminingtheseparametersisthesocalled4-step-phase-shift.'Bythis,foursinusoidalfringepatternswiththesameamplitudeAandbackgroundintensity10,butwithaconstantangularphasedifferenceof90degrees,areneededtoobtainthefourintensitiesIi(x),12(x),13(x),14(x),eqns.(2)-(5),fig.1(a):11(x)=Io+A(cos(q(x)))12(x)=Io+A(cos(çb(x)+))13(x)=Io+A(cos(q5(x)+ir))14(X)=Io+A(cos((x)+))Combiningthesefourequationsleadstothephasefunctionq5(x),eqn.(6),fig.1(b):14(x)—12(X)q(x)=arctan(6)I,(x)—13(x)(b)x:z0xFigure1.4-step-phase-shiftFigure1demonstratesthatthereisnoone-to-onecorrespondencebetweenthephaseqandthelocalcoordinatex.237(2)(3)(4)(5)I(a)1234DownloadedFrom::(x)andc2(x),whicharesuperposed(fig.2).Thesuperpositionmeansthephase-correctsubstractionofthesetwofunctionsgettingthedesiredbeatfunction(x)withthefrequencyA,,,eqn.(7):A1A2A,,—(7)Ai—A2Figure2.Beatfunction2.3.PhaseunwrappingbyheterodyneprincipleTheheterodyneprinciplecanbeusedtosolvetheproblemofphaseunwrapping.Thefrequenciesofthephasefunctionscl5i(x),q(x)havetobechoseninawaythattheresultingbeatfunction(x)isunambiguousoverthefieldofview,fig.3.Figure3.PhaseunwrappingphaseLJW2phasexxphasecphasex238DownloadedFrom::(8),(9):tanai=fi(x)(8)tanabfb(x)(9)Duetotheconstructionofthebeatfunction,theratioofthetwoanglesa,bisalwaysconstant,eqn.(1O):tan1==R1=const.Vx(10)tanabfb(x)Ifthebeatfunction1ismultipliedbytheconstantratioR1(fig.4),eqn.(11),4M(X)=(x)Ri(11)Figure4.MultipliedfunctionMandphasefunctionq(x)andthephasefunctionçbi(x)issubstractedfrom?M(x),thenormalizedresultisthefunctionoftheorders0i(x)forthephasefunctionci(x)(fig.5),eqn.(12):Oi(x)='IM(X)—q5i(x)=(x)Ri—q5i(x)(12)2ir2irorder020xFigure5.Functionoftheordersforthephasefunctionçb1(x)239phasexDownloadedFrom::(x)iscalculatedbyaddingthephasefunctionq5i(x)a