基于平面二次曲线的摄像机标定

23520005CHINESEJ1COMPUTERSVol.23No.5May20001)1),2)1)1)(100080)2)(100041):1999207214;:1999212201.(69775006)(863251229820203).,,1974,,.,,1956,,.,,1961,,,,..2233,.,.,,.,,.,,,:TP391PlanarConicBasedCameraCalibrationYANGChang2Jiang1)SUNFeng2Mei1),2)HUZhan2Yi1)1)(NationalLaboratoryofPatternRecognition,InstituteofAutomation,ChineseAcademyofSciences,Beijing100080)2)(CollegeofFundamentalResearch,NorthChinaUniversityofTechnology,Beijing100041)AbstractInspiredbythetechniqueproposedbyZhang,thispaperproposesaneweasycameracalibrationtechnique.Thetechniqueonlyrequiresthecameratoobserveamodelplaneshownatafew(atleasttwo)differentorientations.Themodelplanecontainsafew(atleastthree)concentricconics.Eitherthecameraorthemodelplanecanbefreelymovedandthemotionsneednotbeknown.Allcomputationsinvolvedarematrixoperationsinthelinearalgebra,andaclosed2formsolutioncanbeobtained.Sincetheconicsareusedasthecalibrationprimitives,whichareglobalfeatures,theproposedtechniqueismorerobustandstable.Bothcomputersimulationandrealdataareusedtotesttheproposedtechnique.Comparedwiththeclassicaltechniqueswhereanexpensivecalibrationpatternisalwaysused,thistechniqueiseasytouseandmoreflexible.Keywordscameracalibration,modelplane,conic,homography1.[1,2],,[35].,,,.,.,Zhang[6].Zhang,,,(),,.,,,,.:Triggs[7]5,SturmMaybank[8],.,Zhang,,.,.,[911],.,,,.2.:m=[u,v]T,M=[X,Y,Z]T,m=[u,v,1]TM=[X,Y,Z,1]T,,Mmsm=K[Rt]M(1)s,(Rt),K,K=ACu00Bv0001,(u0v0),Au,Bv,C.,X2Y,Z=0.Riri,(1),suv1=K[r1r2r3t]XY01=K[r1r2t]XY1,M,M=[X,Y]T,M=[X,Y,1]T,,MmH:sm=HM(2),H=KK[r1r2t]33,K.H=[h1h2h3],[h1h2h3]=KK[r1r2t],t,r1,r2,tr1,r2,r1r2,det([r1r1t])0.det[K]0,det(H)0.3C,,Q,MTCM=0(3)mTQm=0(4)(2)(4),MTHTQHM=0(5)(3)(5),sC=HTQH(6)det(C)=det(Q),det(H)=1,(6)C=HTQH(7),,.,().,,C=1a20001b2000-1,(7),Q-1=HC-1HT(8)2452000H=[h1h2h3],Q-1=a2Hx1+b2Hx2-Hx3(9)Hx1=h1hT1,Hx2=h2hT2,Hx3=h3hT3.Hx1,Hx2,Hx3,,,3.Hx1,Hx2,Hx3,3h1,h2,h3,Hx1,Hx2,Hx33.Hx1,Hx2,Hx3,.[12](matrixfactorization)h1,h2,h3.h1.Hx1=h1hT11,:Hx1=U£UT,£=R100000000,U.,Rdi(i1),Rd1,.Hx1=h1hT1URd100000000UT=URd100000000Rd100000000TUT,G=URd100000000=g1100g2100g3100,,h1=[g11,g21,g31]T.h2h3.H,r1r2,,hT1K-TK-1h2=0(10)hT1K-TK-1h1=hT2K-TK-1h2(11).8,6(3,3),2.K-TK-1(absoluteconic)[13].4B=K-TK-1=B11B12B13B12B22B33B13B23B33(12)B,b=[B11,B12,B22,B13,B23,B33]T(13)Hihi=[hi1,hi2,hi3]T,,hTiBhj=vTijb(14)vij=[hi1hj1,hi1hj2+hi2hj1,hi2hj2,hi3hj1+hi1hj3,hi3hj2+hi2hj3,hi3hj3]T.,(10)(11)b:vT12(v11-v22)Tb=0(15)n,n,Vb=0(16),V2n6.nE3,,b.n=2,C=0,[0,1,0,0,0,0]b=0,(16).(16)VTV(,V)[12].b,Cholesky[12]K-1,K.K,,(2),r1=LK-1h1,r2=LK-1h2,r3=r1r2,t=LK-1h3,L=1öK-1h1=1öK-1h2.,,:(1)(3),.(2),.(3).(4)343455:.5,.5.1.:A=1250,B=1250,C=1.0908,u0=v0=500,10001000.()r,,(:).t(:cm).3:r1=[20,0,0]T,20,t1=[-9,-12.5,100]T;r2=[0,20,0]T,20,t2=[-9,-12.5,110]T;r3=[-30,-30,15]T,20.125,t3=[-10.5,-12.5,120]T.3(),,,30cm,30cm,20cm,90.1000.0R()..1A,Bu0,v0.R0.1pixel3.0pixel,R40,.1,,,.,..3.4,,30.220.,0,0.5pixel,40,.2.,,23(,C=0).44520005.2.CCD,1280960.32,39.5cm,5cm,3cm,12.5cm,7.5cm9cm,6cm,90..5,3.Hough,.4.2,3,4,5,1.3.54,2,4,5,,.355455:1ABCu0v023451410.231379.751383.161385.421415.061390.931393.561394.2801.401.211.29532.60540.33538.42538.68434.08430.78432.24432.93254ABCu0v0(1234)1383.161393.561.21538.42432.24(1235)1379.991390.761.32540.45429.37(1245)1381.011391.931.33541.16433.40(1345)1380.321393.121.29539.49431.91(2345)1385.661396.491.31540.72430.13ABCu0v01382.031393.171.29540.05431.412.382.150.051.101.64,.(5).18,9(),[14],,6..,91.1,.6Zhang,.2233,..,.,,.,,.,,,..64520001BrownDC.Close2rangecameracalibration.PhotogrammetricEngineering,1971,37(8):855-8662FaigW.Calibrationofclose2rangephotogrammetrysystem:Mathematicalformulation.PhotogrammetricEngineeringandRemoteSensing,1975,41(12):1479-14863FaugerasOD,LuongQT,MaybankSJ.Cameraself2calibration:Theoryandexperiments.In:ProceedingsoftheEuropeanConferenceonComputerVision,Santa2Margerita,Italy,1992.321-3344MaybankSJ,FaugerasOD.Atheoryofself2calibrationofamovingcamera.InternationalJournalofComputerVision,1992,8(2):123-1515TsaiRY.Aversatilecameracalibrationtechniqueforhigh2accuracy3Dmachinevisionmetrologyusingoff2the2shelfTVcamerasandlenses.IEEEJournalofRoboticsandAutomation,1987,3(4):323-3446ZhangZ.Aflexiblenewtechniqueforcameracalibration.MicrosoftCorporation:TechnicalReportMSR2TR298271,19987TriggsB.Autocalibrationfromplanarscenes.In:ProceedingsoftheEuropeanConferenceonComputerVision,Freiburg,Germany,1998.89-1058SturmP,MaybankS.Onplane2basedcameracalibration:Ageneralalgorithm,singularities,applications.In:ProceedingsoftheIEEEConferenceonComputerVisionandPatternRecognition,FortCollins,Colorado,USA,1999.432-4379MaSD.Conics2basedstereo,motionestimation,andposedetermination.InternationalJournalofComputerVision,1993,10(1):7-2510QuanL.Conicreconstructionandcorrespondencefromtwoviews.IEEETransactionsonPatternAnalysisandMachineIntelligence,1996,18(2):151-16011MundyJ,ZissermanAeds.GeometricInvarianceinComputerVision.Cambridge,MA:MITPress,199212GolubG,VanLoanC.MatrixComputations.3rdedition.TheJohnHopkinsUniversityPress,199613TriggsB.Autocalibrationandtheabsolutequadric.In:ProceedingsoftheIEEEConf