2000 Multiple view geometry 讲稿PPT很好需要好好看

MultipleViewGeometryRichardHartleyandAndrewZissermanCVPRJune1999PartI:SingleandTwoViewGeometryThemainpointscoveredinthispartare:•Aperspective(central)projectioncameraisrepresentedbya3×4matrix.•Themostgeneralperspectivetransformationtransformationbetweentwoplanes(aworldplaneandtheimageplane,ortwoimageplanesinducedbyaworldplane)isaplaneprojectivetransformation.Thiscanbecomputedfromthecor-respondenceoffour(ormore)points.•Theepipolargeometrybetweentwoviewsisrepresentedbythefundamentalmatrix.Thiscanbecomputedfromthecorrespondenceofseven(ormore)points.ImagingGeometryPerspectiveprojectionλxyf=XYZwhereλ=Z/f.XyYZOxXxpimageplaneThiscanbewrittenasalinearmappingbetweenhomogeneouscoordinates(theequationisonlyuptoascalefactor):xyf=100001000010XYZ1wherea3×4projectionmatrixrepresentsamapfrom3Dto2D.ImageCoordinateSystemInternalcameraparameterskxxcam=x−x0kyycam=y−y0wheretheunitsofkx,kyare[pixels/length].y00pcamxcamyyxxx=xy1=1fαxx0αyy01xcamycamf=Kxcamycamfwhereαx=fkx,αy=fky.CameraCalibrationMatrixKisa3×3uppertriangularmatrix,calledthecameracalibrationmatrix:K=αxx0αyy01•Therearefourparameters:(i)Thescalingintheimagexandydirections,αxandαy.(ii)Theprincipalpoint(x0,y0),whichisthepointwheretheopticaxisintersectstheimageplane.•Theaspectratioisαy/αx.WorldCoordinateSystemExternalcameraparametersXcamYcamZcam1=Rt01XYZ1XZYR,tYXcamcamOOcamcamZEuclideantransformationbetweenworldandcameracoordinates•Risa3×3rotationmatrix•tisa3×1translationvectorConcatenatingthethreematrices,x=xy1=K100001000010Rt01XYZ1=K[R|t]Xwhichdefinesthe3×4projectionmatrixfromEuclidean3-spacetoanimageasx=PXP=K[R|t]=KR[I|Rt]Note,thecameracentreisat(X,Y,Z)=−Rt.Inthefollowingitisoftenonlythe3×4formofPthatisimportant,ratherthanitsdecomposition.AProjectiveCameraThecameramodelforperspectiveprojectionisalinearmapbetweenhomogeneouspointcoordinatesxy1P(3×4)XYZ1ImagePointScenePointx=PX•Thecameracentreisthenull-vectorofPe.g.ifP=[I|0]thenthecentreisX=(0,0,0,1).•Phas11degreesoffreedom(essentialparameters).•Phasrank3.Whatdoescalibrationgive?•KprovidesthetransformationbetweenanimagepointandarayinEuclidean3-space.•OnceKisknownthecameraistermedcalibrated.•Acalibratedcameraisadirectionsensor,abletomeasurethedirectionofrays—likea2Dprotractor.x=xy1=αxx0αyy01XcamYcamZcam=KdAnglebetweenrayscosθ=d1.d2(d1.d1)12(d2.d2)12θCdd12x1x2cosθ=d1d2(d1d1)1/2(d2d2)1/2=x1(K−K−1)x2(x1(K−K−1)x1)1/2(x2(K−K−1)x2)1/2=x1ωx2(x1ωx1)1/2(x2ωx2)1/2whereω=(KK)−1.CameraCalibration(Resectioning)ProblemStatement:Givenncorrespondencesxi↔Xi,whereXiisascenepointandxiitsimage:ComputeP=K[R|t]suchthatxi=PXi.Thealgorithmforcameracalibrationhastwoparts:(i)ComputethematrixPfromasetofpointcorrespondences.(ii)DecomposePintoK,RandtviatheQRdecomposition.Algorithmstep1:ComputethematrixPxi=PXi.Eachcorrespondencegeneratestwoequationsxi=p11Xi+p12Yi+p13Zi+p14p31Xi+p32Yi+p33Zi+p34yi=p21Xi+p22Yi+p23Zi+p24p31Xi+p32Yi+p33Zi+p34MultiplyingoutgivesequationslinearinthematrixelementsofPxi(p31Xi+p32Yi+p33Zi+p34)=p11Xi+p12Yi+p13Zi+p14yi(p31Xi+p32Yi+p33Zi+p34)=p21Xi+p22Yi+p23Zi+p24TheseequationscanberearrangedasXYZ10000−xX−xY−xZ−x0000XYZ1−yX−yY−yZ−yp=0withp=(p11,p12,p13,p14,p21,p22,p23,p24,p31,p32,p33,p34)a12-vector.Algorithmstep1continuedSolvingforP(i)Concatenatetheequationsfrom(n≥6)correspondencestogenerate2nsimultaneousequations,whichcanbewritten:Ap=0,whereAisa2n×12matrix.(ii)Ingeneralthiswillnothaveanexactsolution,buta(linear)solutionwhichminimises|Ap|,subjectto|p|=1isobtainedfromtheeigenvectorwithleasteigenvalueofAA.OrequivalentlyfromthevectorcorrespondingtothesmallestsingularvalueoftheSVDofA.(iii)Thislinearsolutionisthenusedasthestartingpointforanon-linearminimisationofthedifferencebetweenthemeasuredandprojectedpoint:minPi((xi,yi)−P(Xi,Yi,Zi,1))2Example-CalibrationObjectDetermineaccuratecornerpositionsby(i)ExtractandlinkedgesusingCannyedgeoperator.(ii)Fitlinestoedgesusingorthogonalregression.(iii)Intersectlinestoobtaincornerstosub-pixelaccuracy.Thefinalerrorbetweenmeasuredandprojectedpointsistypicallylessthan0.02pixels.Algorithmstep2:DecomposePintoK,RandtThefirst3×3submatrix,M,ofPistheproduct(M=KR)ofanuppertriangularandrotationmatrix.(i)FactorMintoKRusingtheQRmatrixdecomposition.Thisde-terminesKandR.(ii)Thent=K−1(p14,p24,p34)Note,thisproducesamatrixwithanextraskewparametersK=αxsx0αyy01withs=tanθ,andθtheanglebetweentheimageaxes.WeakPerspectiveTrackback,whilstzoomingtokeepimagesizefixedperspectiveweakperspectiveTheimagingraysbecomeparallel,andtheresultis:P=Kr11r12r13∗r21r22r23∗000∗AgeneralizationistheaffinecameraTheAffineCameraP=m11m12m13t1m21m22m23t20001ThematrixM2×3hasranktwo.Projectionunderanaffinecameraisalinearmappingonnon-homogeneouscoordinatescomposedwithatranslation:xy=m11m12m13m21m22m23XYZ+t1t2•Thepoint(t1,t2)istheimageoftheworldorigin.•Thecentreoftheaffinecameraisatinfinity.•Anaffinecamerahas8degreesoffreedom.•Itmodelsweak-perspectiveandpara-perspective.P