计算机视觉教学课件

Course12CalibrationCourse12Calibration1.IntroductionIntheoreticdiscussions,wehaveassumed:-----Cameraislocatedattheoriginofcoordinatesystemofscene.-----Opticaxisofcameraispointinginz-directionofscenecoordinates.-----Imageplaneistoperpendiculartoz-axiswithoriginat(f,0,0)-----XandYaxes

ofimagecoordinatesareparalleltoxandyaxesofscenecoordinates,respectively.-----Noanydistortionsforcamera,i.e.pinholecameramodel.1)Whatweneedtocalibrate:-----Absoluteorientation:betweentwocoordinatesystems,e.g.,robotcoordinatesandmodelcoordinates.-----Relativeorientation:betweentwocamerasystems.-----Exteriororientation:betweencameraandscenesystems-----Interiororientation:withinacamera,suchascameraconstant,principalpoint,lensdistortion,etc.2)Coordinatesystems:-----scenecoordinates(globalcoordinates,worldcoordinates,absolutecoordinates).-----Cameracoordinates.-----Imagecoordinatesystems-----PixelcoordinatesForaimageofsizem×n,imagecenter2.Absoluteorientation:Tofindoutrelationshipbetweentwocoordinatesystemsfrom3ormore3Dpointsthathaveexpressedinthetwocoordinatesystems.Leta3Dpointpisexpressedbyinmodelcoordinatesystem.inabsolutecoordinatesystem.Then:whereRisrotationofmodelcoordinatecorrespondingtoabsolutecoordinates;istheoriginofmodelcoordinatesystemintheabsolutecoordinatesystem.Given:Wewanttofind:Itisthesameexpressionas3Dmotionestimatesfrom3Dpoints.Therefore,allthealgorithmsofmotionestimation(using3Dpoints)canbeusedtosolveforabsoluteorientation.Oneshouldremembertheconstraintofrotationmatrixthatisorthogonalmatrix,i.e.Thisadds6additionalequationsinsolvingforrotation.1)Solverotationwithquaternion:Orthonornalconstraintofrotationmatrixisabsorbedinquaternionexpression.Inabsolutecoordinatesystem,asetofpointsata3Dobjectare:Inmodelcoordinatesystem,thesamesetofpointsarecorrespondinglymeasuredas:theysatisfyInabsolutecoordinatesystem:Centroidofthepointsetis

Andtherayfromsetcentroidtopointis:

Inthesameway,inmodelcoordinatesystem:Sinceandareparallel,Wecansolveforrotationbyleast-squares.Usingquaterniontoexpressrotationandrecallthatwehave:Wherei.e.whereThisequationcanbesolvedbylinearleastsquares,suchasSVD.Afterisfound,camerapositioncanbeeasilycalculatedby2)ScaleProblemIfabsolutecoordinatesystemandmodelcoordinatesystemmayhavedifferentmeasurements,scaleproblemisintroduced.Ifonenoticesthefactthatthedistancesbetweenpointsof3Dscenearenotaffectedbychoiceofcoordinatesystems,wecaneasilysolveforscalefactor:Oncescalefactorisfound,theproblembecomesordinaryabsoluteorientationproblem3.RelativeOrientationTodeterminetherelationshipbetweentwocameracoordinatesystemsfromtheprojectionsofcorrespondingpointsinthetwocamera.(e.g.,instereocases).Thisistosay,givenpairsofimagepointcorrespondences,tofindrotationandtranslationofthecamerafromonepositiontotheotherposition.1)Solverelativeorientationproblembymotionestimation.Inmotionestimation,cameraisstationary.Objectismoving.UsingtofindInstereocase,wecanimaginethatacamerafirsttakeanimageofsceneatposition,andthenmovesto

totakethesecondimageofthesamescene.Thesceneisstationary.Cameraismoving.Usetofind,itisthesameasstationarycameracasewithUse8-pointmethod,onecansolveforrotationandtranslation.2)Iterativemethod:Letandbethedirectionfromcameracenterstoscenepoint,respectively.

Sincebaselineofstereosystemandthenormalofepipolarplaneareperpendicular,wehave:Itcanbeseenthatbaselinecanonlybesolvedwithascalefactor.Weputaconstraint:Byleastsquares:

fiveormorestereoimagepointsareneededtosolvefortherelativeorientationproblem.Inupdating,weusetheincrementforbaselineandforrotation,whichconstrainstherotationmatrixtobeorthonormal;i.e.where:forbaseline,iterativeformulaiswhere subjectto4.ExteriorOrientationUsingimagepoint(X,Y)inimagecoordinatesandthecorresponding3Dpoint(x,y,z)ofsceneinabsolutecoordinatesystemtodeterminetherelationshipofcamerasystemandtheabsolutesystem.Weassumethatimageplaneandcameraarewellcalibrated.Letascenepointisexpressedinabsolutecoordinatesystem:Incameracoordinatesystem:Expressincameracoordinatesystem:i.e.(1)/(3)and(2)/(3),andconsidering

Wehave:OrAsareknown,thereare12unknowns,9forrotationand3fortranslation,atleast6pointareneededtoprovide12equations.However,ifconsidering6constraintsforrotation,theminimumrequiredpointis3tocalibratetheexteriororientationproblem.5.InteriorOrientationTodeterminetheinternalgeometryofthecamera,suchas:-----Cameraconstant:thedistancefromimageplanetoprojectioncenter.-----Principalpoint:theoriginlocationofimageplanecoordinatesystems.-----Lensdistortioncoefficients:opticpropertyofcamera.-----Scalefactors:thedistancebetweenrowsandcolumns.1)Calibrationmodelofcamera:-----Uncorrectedimagecoordinates -----Trueimagecoordinates -----Principalpointofimage,expressedinsystemwherelensdecentering:where2)Calibrationmethod-----Straightlinesinsceneshouldbestraightlinesinimage.Putstraightlinesinscene,suchasapaperprintingwithlines.TakeimagefromthestraightlinesinsceneWithimage,useHoughtransformtogroupedgelinesSubstitutecalibrationmodelintotheequationofstraightlines:Wheredenotesthepointatline.Uselease-squarestofindunknownparameters3)Examples:Affinemethodforcameracalibration-----Combineinteriorandexteriorcalibrations-----Scaleerror,translationerror,rotations,skewerroranddifferentialscaling.-----Cannotcorrectlensdistortion-----2D-3Dpointcorrespondencesofcalibrationpointsaregiven.Correctionmodel(affinetransformation)Sinceisanarbitrarymatrix,thetransformationinvolvesrotation,translation,scaling,skewtransform.Usingprojectionmodel:Affinetransformationbecomes:Fromexteriororientationofcamera:Substituteexteriororientationrelationintoaffinetransformationmodel:wherecoefficientsareabsorbedintoWithknown3D-2Dpointcorrespondencesof

Coefficients

canbecomputedbySVD.After

areobtained,