matlab相机定标工具箱程序的算法原理

1、CameraCalibrationToolboxforMatlabDescriptionofthecalibrationparametersAftercalibration,thelistofparametersmaybestoredinthematabfileCalib_ResultsbyclickingonSave.Thelistofvariablesmaybeseparatedintotwocategories:Intrinsicparametersandextrinsicparameters.Intrinsicparameters(cameramodel):Theinternalcam

2、eramodelisverysimilartothatusedbyHeikkil伯aandSilvenattheUniversityofOuluinFinland.Visittheironlinecalibrationpage,andtheirpublicationpage.WespecificallyrecommendtheirCVPR97paper:AFour-stepCameraCalibrationProcedurewithImplicitImageCorrection.Thelistofinternalparameters:Focallength:Thefocallengthinpi

3、xelsisstoredinthe2x1vectorfc.Principalpoint:Theprincipalpointcoordinatesarestoredinthe2x1vectorcc.Skewcoefficient:Theskewcoefficientdefiningtheanglebetweenthexandypixelaxesisstoredinthescalaralpha_c.Distortions:Theimagedistortioncoefficients(radialandtangentialdistortions)arestoredinthe5x1vectorkc.D

4、efinitionoftheintrinsicparameters:LetPbeapointinspaceofcoordinatevectorXXc=Xc;Yc;Zcinthecamerareferenceframe.Letusprojectnowthatpointontheimageplaneaccordingtotheintrinsicparameters(fc,cc,alpha_c,kc).Letxnbethenormalized(pinhole)imageprojection:Letr2=x2+y2.Afterincludinglensdistortion,thenewnormaliz

5、edpointcoordinatexdisdefinedasfollows:wheredxisthetangentialdistortionvector:dx=2直xy+kcF+2/kc(3)(r2+2y2)+2kc(4)xyTherefore,the5-vectorkccontainsbothradialandtangentialdistortioncoefficients(observethatthecoefficientof6thorderradialdistortiontermisthefifthentryofthevectorkc).Itisworthnoticingthatthis

6、distortionmodelwasfirstintroducedbyBrownin1966andcalledPlumbBobmodel(radialpolynomial+thinprism).Thetangentialdistortionisduetodecentering,orimperfectcenteringofthelenscomponentsandothermanufacturingdefectsinacompoundlens.Formoredetails,refertoBrownsoriginalpublicationslistedinthereferencepage.Onced

7、istortionisapplied,thefinalpixelcoordinatesx_pixel=xp;ypoftheprojectionofPontheimageplaneis:%=氏l：xd(l)+alpha_c*xd(2)i+cc(l)耳=fc(2)心(2)+m(2)Therefore,thepixelcoordinatevectorx_pixelandthenormalized(distorted)coordinatevectorxdarerelatedtoeachotherthroughthelinearequation:whereKKisknownasthecameramatr

8、ix,anddefinedasfollows:Inmatlab,thismatrixisstoredinthevariableKKaftercalibration.Observethatfc(1)andfc(2)arethefocaldistance(auniquevalueinmm)expressedinunitsofhorizontalandverticalpixels.Bothcomponentsofthevectorfcareusuallyverysimilar.Theratiofc(2)/fc(1),oftencalledaspectratio,isdifferentfrom1ift

9、hepixelintheCCDarrayarenotsquare.Therefore,thecameramodelnaturallyhandlesnon-squarepixels.Inaddition,thecoefficientalpha_cencodestheanglebetweenthexandysensoraxes.Consequently,pixelsareevenallowedtobenon-rectangular.Someauthorsrefertothattypeofmodelasaffinedistortionmodel.Inadditiontocomputingestima

10、tesfortheintrinsicparametersfc,cc,kcandalpha_c,thetoolboxalsoreturnsestimatesoftheuncertaintiesonthoseparameters.Thematlabvariablescontainingthoseuncertaintiesarefc_error,cc_error,kc_error,alpha_c_error.Forinformation,thosevectorsareapproximatelythreetimesthestandarddeviationsoftheerrorsofestimation

11、.Hereisanexampleofoutputofthetoolboxafteroptimization:FocalLength:Fc=657.30254657.74391士0.284870.28937Principalpoint:cc=302.71656242.33386土0.591150.55710Skeu:alphac=0.000420.00019=angleofpixelaxes=89.975950.Distortion:kc=-0.253490.11868-0.000280.000050.00000|0.00231Pixelerror:err=0.117438.11585Calib

12、rationresultsafteroptimization(withuncertainties):Note:Thenumericalerrorsareapproximatelythreetimesthestandarddeviations(ForreferenceInthiscasefc=657.30254;657.74391andfc_error=0.28487;0.28937,cc=302.71656;242.33386,cc_error=0.59115;0.55710,.ImportantConvention:Pixelcoordinatesaredefinedsuchthat0;0i

13、sthecenteroftheupperleftpixeloftheimage.Asaresult,nx-1;0iscenteroftheupperrightcornerpixel,0;ny-1isthecenterofthelowerleftcornerpixelandnx-1;ny-1isthecenterofthelowerrightcornerpixelwherenxandnyarethewidthandheightoftheimage(fortheimagesofthefirstexample,nx=640andny=480).Onematlabfunctionprovidedint

14、hetoolboxcomputesthatdirectpixelprojectionmap.Thisfunctionisproject_points2.m.Thisfunctiontakesinthe3Dcoordinatesofasetofpointsinspace(inworldreferenceframeorcamerareferenceframe)andtheintrinsiccameraparameters(fc,cc,kc,alpha_c),andreturnsthepixelprojectionsofthepointsontheimageplane.Seetheinformati

15、ongiveninthefunction.Theinversemapping:Theinverseproblemofcomputingthenormalizedimageprojectionvectorxnfromthepixelcoordinatex_pixelisveryusefulinmostmachinevisionapplications.However,becauseofthehighdegreedistortionmodel,thereexistsnogeneralalgebraicexpressionforthisinversemap(alsocallednormalizati

16、on).Inthetoolboxhowever,anumericalimplementationofinversemappingisprovidedintheformofafunction:normalize.m.Hereisthewaythefunctionshouldbecalled:xn=normalize(x_pixel,fc,cc,kc,alpha_c).Inthatsyntax,x_pixelandxnmayconsistofmorethanonepointcoordinates.Foranexampleofcall,seethematlabfunctioncompute_extr

17、insic_init.m.Reducedcameramodels:Currentlymanufacturedcamerasdonotalwaysjustifythisverygeneralopticalmodel.Forexample,itnowcustomarytoassumerectangularpixels,andthusassumezeroskew(alpha_c=0).Itisinfactadefaultsettingofthetoolbox(theskewcoefficientnotbeingestimated).Furthermore,theverygeneric(6thorde

18、rradial+tangential)distortionmodelisoftennotconsideredcompletely.Forstandardfieldofviews(nonwide-anglecameras),itisoftennotnecessary(andnotrecommended)topushtheradialcomponentofdistortionmodelbeyondthe4thorder(i.e.keepingkc(5)=0).Thisisalsoadefaultsettingofthetoolbox.Inaddition,thetangentialcomponen

19、tofdistortioncanoftenbediscarded(justifiedbythefactthatmostlensescurrentlymanufactureddonothaveimperfectionincentering).The4thordersymmetricradialdistortionwithnotangentialcomponent(thelastthreecomponentofkcaresettozero)isactuallythedistortionmodelusedbyZhang.Anotherverycommondistortionmodelforgoodo

20、pticalsystemsornarrowfieldofviewlensesisthesecondordersymmetricradialdistortionmodel.Inthatmodel,onlythefirstcomponentofthevectorkcisestimated,whiletheotherfouraresettozero.Thismodelisalsocommonlyusedwhenafewimagesareusedforcalibration(toolittledatatoestimateamorecomplexmodel).Asidefromdistortionsan

21、dskew,othermodelreductionsarepossible.Forexample,whenonlyafewimagesareusedforcalibration(e.g.one,twoorthreeimages)theprincipalpointccisoftenverydifficulttoestimatereliably.Itisknowntobeoneofthemostdifficultpartofthenativeperspectiveprojectionmodeltoestimate(ignoringlensdistortions).Ifthisisthecase,i

22、tissometimesbetter(andrecommended)tosettheprincipalpointatthecenteroftheimage(cc=(nx-1)/2;(ny-1)/2)andnotestimateitfurther.Finally,infewrareinstances,itmaybenecessarytorejecttheaspectratiofc(2)/fc(1)fromtheestimation.Althoughthisfinalmodelreductionstepispossiblewiththetoolbox,itisgenerallynotrecomme

23、ndedastheaspectratioisofteneasytoestimateveryreliably.Formoreinformationonhowtoperformmodelselectionwiththetoolbox,visitthepagedescribingthefirstcalibrationexample.CorrespondencewithHeikki雌notation:IntheoriginalHeikkil懺paper,theinternalparametersappearwithslightlydifferentnames.Thefollowingtablegive

24、sthecorrespondencebetweenthetwonotationschemes:OurnotationHeikkil唯notationfcf.Du.sufc(2)f.DvCCu0cc(2)v0alphac0kcf3.k1kc(2)f5.k2kc(3)f2.plkc(4)f2.p2kc(5)0AfewcommentsonHeikkil临model:Skewisnotestimated(alpha_c=0).Itmaynotbeaproblemasmostcamerascurrentlymanufactureddonothavecenteringimperfections.Thera

25、dialcomponentofthedistortionmodelisonlyuptothe4thorder.Thisissufficientformostcases.Thefourvariables(f,Du,Dv,su)replacingthe2x1focalvectorfcareingeneralimpossibletoestimateseparately.Itisonlypossibleiftwoofthosevariablesareknown(forexamplethemetricfocalvaluefandthescalefactorsu).SeeHeikki雌paperformo

26、reinformation.CorrespondencewithRegWillsonsnotation:InhisoriginalimplementationoftheTsaicameracalibrationalgorithm,RegWillsonusesadifferentnotationforthecameraparameters.Thefollowingtablegivesthecorrespondencebetweenthetwonotationschemes:OurnotationWillsonsnotationfc(1)f.sx/dpxfc(2)f/dpyCCCxcc(2)Cya

27、lphac0Willsonusesafirstorderradialdistortionmodel(withanadditionalconstantkappa1)thatdoesnothaveaneasyclosed-formcorespondencewithourdistortionmodel(encodedwiththecoefficientskc(1),.,kc(5).However,weincludedinthetoolboxafunctioncalledwillson_convertthatconvertstheentiresetofWillsonsparametersintoour

28、parameters(includingdistortion).Thisfunctioniscalledinanotherfunctionwillson_readthatdirectlyloadsinacalibrationresultfilegeneratedbyWillsonscodeandcomputesthesetparameters(intrinsicandextrinsic)followingournotation(tousethatfunction,firstsetthematlabvariablecalib_filetothenameoftheoriginalwillsonca

29、librationfile).AfewextracommentsonWillsonsmodel:SimilarlytoHeikkil临model,theskewisnotincludedinthemodel(alpha_c=0).SimilarlytoHeikkil临model,thefourvariables(f,sx,dpx,dpy)replacingthe2x1focalvectorfcareingeneralimpossibletoestimateseparately.Itisonlypossibleiftwoofthosevariablesareknown(forexamplethe

30、metricfocalvaluefandthescalefactorsx).Extrinsicparameters:Rotations:Asetofn_ima3x3rotationmatricesRc_1,Rc_2,.,Rc_20(assumingn_ima=20).Translations:Asetofn_ima3x1vectorsTc_1,Tc_2,.,Tc_20(assumingn_ima=20).Definitionoftheextrinsicparameters:Considerthecalibrationgrid#i(attachedtotheithcalibrationimage

31、),andconcentrateonthecamerareferenceframeattahedtothatgrid.Withoutlossofgenerality,takei=1.Thefollowingfigureshowsthereferenceframe(O,X,Y,Z)attachedtothatcalibrationgid.BacktomaincalibrationpageLetPbeapointspaceofcoordinatevectorXX=X;Y;Zinthegridreferenceframe(referenceframeshownonthepreviousfigure).LetXXc=Xc;Yc;ZcbethecoordinatevectorofPinthecamerareferenceframe.ThenXXand