matlab相机定标工具箱程序的算法原理上课讲义

1、Good is good, but better carries it.精益求精，善益求善。matlab相机定标工具箱程序的算法原理CameraCalibrationToolboxforMatlabDescriptionofthecalibrationparametersAftercalibration,thelistofparametersmaybestoredinthematabfileCalib_ResultsbyclickingonSave.Thelistofvariablesmaybeseparatedintotwocategories:Intrinsicparametersande

2、xtrinsicparameters.Intrinsicparameters(cameramodel):TheinternalcameramodelisverysimilartothatusedbyHYPERLINKhttp:/www.ee.oulu.fi/jth/Heikkil伯aandHYPERLINKhttp:/www.ee.oulu.fi/olli/SilvenattheHYPERLINKhttp:/www.ee.oulu.fi/UniversityofOuluinFinland.VisittheironlineHYPERLINKhttp:/www.ee.oulu.fi/jth/cal

3、ibr/Calibration.htmlcalibrationpage,andtheirHYPERLINKhttp:/www.ee.oulu.fi/jth/doc/publicationpage.WespecificallyrecommendtheirCVPR97paper:HYPERLINK/bouguetj/calib_doc/papers/heikkila97.pdfAFour-stepCameraCalibrationProcedurewithImplicitImageCorrection.Thelistofinternalparameters:Focallength:Thefocal

4、lengthinpixelsisstoredinthe2x1vectorfc.Principalpoint:Theprincipalpointcoordinatesarestoredinthe2x1vectorcc.Skewcoefficient:Theskewcoefficientdefiningtheanglebetweenthexandypixelaxesisstoredinthescalaralpha_c.Distortions:Theimagedistortioncoefficients(radialandtangentialdistortions)arestoredinthe5x1

5、vectorkc四个内参，包括焦距fc，原点坐标cc，相机坐标系中x和y轴夹角alpha_c（接近90），几何畸变系数kc.Definitionoftheintrinsicparameters:LetPbeapointinspaceofcoordinatevectorXXc=Xc;Yc;Zcinthecamerareferenceframe.Letusprojectnowthatpointontheimageplaneaccordingtotheintrinsicparameters(fc,cc,alpha_c,kc).Letxnbethenormalized(pinhole)imagepro

6、jection:Letr2=x2+y2.Afterincludinglensdistortion,thenewnormalizedpointcoordinatexdisdefinedasfollows:wheredxisthetangentialdistortionvector:Matlab中计算几何畸变还考虑了切向畸变Therefore,the5-vectorkccontainsbothradialandtangentialdistortioncoefficients(observethatthecoefficientof6thorderradialdistortiontermisthefi

7、fthentryofthevectorkc).ItisworthnoticingthatthisdistortionmodelwasfirstintroducedbyBrownin1966andcalledPlumbBobmodel(radialpolynomial+thinprism).Thetangentialdistortionisduetodecentering,orimperfectcenteringofthelenscomponentsandothermanufacturingdefectsinacompoundlens.Formoredetails,refertoBrownsor

8、iginalpublicationslistedintheHYPERLINK/bouguetj/calib_doc/htmls/ref.htmlreferencepage.Oncedistortionisapplied,thefinalpixelcoordinatesx_pixel=xp;ypoftheprojectionofPontheimageplaneis:Therefore,thepixelcoordinatevectorx_pixelandthenormalized(distorted)coordinatevectorxdarerelatedtoeachotherthroughthe

9、linearequation:whereKKisknownasthecameramatrix,anddefinedasfollows:对应于书上的线性映射矩阵AFc(1)和fc(2)分别对应于矩阵A中的f/k和f/l，即转化为按照x和y方向上的像素尺寸为单位Inmatlab,thismatrixisstoredinthevariableKKaftercalibration.Observethatfc(1)andfc(2)arethefocaldistance(auniquevalueinmm)expressedinunitsofhorizontalandverticalpixels.Bothc

10、omponentsofthevectorfcareusuallyverysimilar.Theratiofc(2)/fc(1),oftencalledaspectratio,isdifferentfrom1ifthepixelintheCCDarrayarenotsquare.Therefore,thecameramodelnaturallyhandlesnon-squarepixels.Inaddition,thecoefficientalpha_cencodestheanglebetweenthexandysensoraxes.Consequently,pixelsareevenallow

11、edtobenon-rectangular.Someauthorsrefertothattypeofmodelasaffinedistortionmodel.Inadditiontocomputingestimatesfortheintrinsicparametersfc,cc,kcandalpha_c,thetoolboxalsoreturnsestimatesoftheuncertaintiesonthoseparameters.Thematlabvariablescontainingthoseuncertaintiesarefc_error,cc_error,kc_error,alpha

12、_c_error.Forinformation,thosevectorsareapproximatelythreetimesthestandarddeviationsoftheerrorsofestimationFc_error,cc_error,kc_error,alpha_c_error这四个变量记录四个内参的不确定度，大小设定为三倍标准差.Hereisanexampleofoutputofthetoolboxafteroptimization:Inthiscasefc=657.30254;657.74391andfc_error=0.28487;0.28937,cc=302.71656;

13、242.33386,cc_error=0.59115;0.55710,.ImportantConvention:Pixelcoordinatesaredefinedsuchthat0;0isthecenteroftheupperleftpixel注意：图像左上角默认为原点0;0左下角为0;ny-1右上角为nx-1;0右下角为nx-1;ny-1oftheimage.Asaresult,nx-1;0iscenteroftheupperrightcornerpixel,0;ny-1isthecenterofthelowerleftcornerpixelandnx-1;ny-1isthecentero

14、fthelowerrightcornerpixelwherenxandnyarethewidthandheightoftheimage(fortheimagesofthefirstexample,nx=640andny=480).Onematlabfunctionprovidedinthetoolboxcomputesthatdirectpixelprojectionmap.Thisfunctionisproject_points2.m.Thisfunctiontakesinthe3Dcoordinatesofasetofpointsinspace(inworldreferenceframeo

15、rcamerareferenceframe)andtheintrinsiccameraparameters(fc,cc,kc,alpha_c),andreturnsthepixelprojectionsofthepointsontheimageplane.Seetheinformationgiveninthefunction函数project_points2.m将相机坐标系下的三维点阵按照内参转化成图像平面上的二位点阵.Theinversemapping:Theinverseproblemofcomputingthenormalizedimageprojectionvectorxnfromth

16、epixelcoordinatex_pixelisveryusefulinmostmachinevisionapplications.However,becauseofthehighdegreedistortionmodel,thereexistsnogeneralalgebraicexpressionforthisinversemap(alsocallednormalization).Inthetoolboxhowever,anumericalimplementationofinversemappingisprovidedintheformofafunction:normalize.m.He

17、reisthewaythefunctionshouldbecalled:xn=normalize(x_pixel,fc,cc,kc,alpha_c).Inthatsyntax,x_pixelandxnmayconsistofmorethanonepointcoordinates.Foranexampleofcall,seethematlabfunctioncompute_extrinsic_init.m.Reducedcameramodels:Currentlymanufacturedcamerasdonotalwaysjustifythisverygeneralopticalmodel.Fo

18、rexample,itnowcustomarytoassumerectangularpixels,andthusassumezeroskew(alpha_c=0).It一般情况下，相机坐标系中的坐标轴正交，alpha_c就可以默认为0；在非广角镜头中，一般径向畸变模型到4阶就足够了（好的镜头2阶就够了），而且切向畸变可以忽略isinfactadefaultsettingofthetoolbox(theskewcoefficientnotbeingestimated).Furthermore,theverygeneric(6thorderradial+tangential)distortionm

19、odelisoftennotconsideredcompletely.Forstandardfieldofviews(nonwide-anglecameras),itisoftennotnecessary(andnotrecommended)topushtheradialcomponentofdistortionmodelbeyondthe4thorder(i.e.keepingkc(5)=0).Thisisalsoadefaultsettingofthetoolbox.Inaddition,thetangentialcomponentofdistortioncanoftenbediscard

20、ed(justifiedbythefactthatmostlensescurrentlymanufactureddonothaveimperfectionincentering).The4thordersymmetricradialdistortionwithnotangentialcomponent(thelastthreecomponentofkcaresettozero)isactuallythedistortionmodelusedbyHYPERLINK/bouguetj/calib_doc/htmls/ref.htmlZhang.Anotherverycommondistortion

21、modelforgoodopticalsystemsornarrowfieldofviewlensesisthesecondordersymmetricradialdistortionmodel.Inthatmodel,onlythefirstcomponentofthevectorkcisestimated,whiletheotherfouraresettozero.Thismodelisalsocommonlyusedwhenafewimagesareusedforcalibration(toolittledatatoestimateamorecomplexmodel).Asidefrom

22、distortionsandskew,othermodelreductionsarepossible.Forexample,whenonlyafewimagesareusedforcalibration(e.g.one,twoorthreeimages)theprincipalpointccisoftenverydifficulttoestimatereliably.Itisknowntobeoneofthemostdifficultpartofthenativeperspectiveprojectionmodeltoestimate(ignoringlensdistortions).Ifth

23、isisthecase,itissometimesbetter(andrecommended)tosettheprincipalpointatthecenteroftheimage(cc=(nx-1)/2;(ny-1)/2)andnotestimateitfurther.Finally,infewrareinstances,itmaybenecessarytorejecttheaspectratiofc(2)/fc(1)fromtheestimation.Althoughthisfinalmodelreductionstepispossiblewiththetoolbox,itisgenera

24、llynotrecommendedastheaspectratioisofteneasytoestimateveryreliably.Formoreinformationonhowtoperformmodelselectionwiththetoolbox,visittheHYPERLINK/bouguetj/calib_doc/htmls/example.htmlpagedescribingthefirstcalibrationexample.CorrespondencewithHeikkilnotation:IntheoriginalHYPERLINK/bouguetj/calib_doc/

25、papers/heikkila97.pdfHeikkilpaper,theinternalparametersappearwithslightlydifferentnames.Thefollowingtablegivesthecorrespondencebetweenthetwonotationschemes:OurnotationHeikkilnotationfc(1)f.Du.sufc(2)f.Dvcc(1)u0cc(2)v0alpha_c0kc(1)f3.k1kc(2)f5.k2kc(3)f2.p1kc(4)f2.p2kc(5)0AfewcommentsonHeikkilmodel:Sk

26、ewisnotestimated(alpha_c=0).Itmaynotbeaproblemasmostcamerascurrentlymanufactureddonothavecenteringimperfections.Theradialcomponentofthedistortionmodelisonlyuptothe4thorder.Thisissufficientformostcases.Thefourvariables(f,Du,Dv,su)replacingthe2x1focalvectorfcareingeneralimpossibletoestimateseparately.

27、Itisonlypossibleiftwoofthosevariablesareknown(forexamplethemetricfocalvaluefandthescalefactorsu).SeeHYPERLINK/bouguetj/calib_doc/papers/heikkila97.pdfHeikkilpaperformoreinformation.CorrespondencewithRegWillsonsnotation:InhisoriginalHYPERLINK/afs/user/rgw/www/TsaiCode.htmlimplementationoftheTsaicamer

28、acalibrationalgorithm,HYPERLINK/rgw/RegWillsonusesadifferentnotationforthecameraparameters.Thefollowingtablegivesthecorrespondencebetweenthetwonotationschemes:OurnotationWillsonsnotationfc(1)f.sx/dpxfc(2)f/dpycc(1)Cxcc(2)Cyalpha_c0Willsonusesafirstorderradialdistortionmodel(withanadditionalconstantk

29、appa1)thatdoesnothaveaneasyclosed-formcorespondencewithourdistortionmodel(encodedwiththecoefficientskc(1),.,kc(5).However,weincludedinthetoolboxafunctioncalledwillson_convertthatconvertstheentiresetofWillsonsparametersintoourparameters(includingdistortion).Thisfunctioniscalledinanotherfunctionwillso

30、n_readthatdirectlyloadsinacalibrationresultfilegeneratedbyWillsonscodeandcomputesthesetparameters(intrinsicandextrinsic)followingournotation(tousethatfunction,firstsetthematlabvariablecalib_filetothenameoftheoriginalwillsoncalibrationfile).AfewextracommentsonWillsonsmodel:SimilarlytoHeikkilmodel,the

31、skewisnotincludedinthemodel(alpha_c=0).SimilarlytoHeikkilmodel,thefourvariables(f,sx,dpx,dpy)replacingthe2x1focalvectorfcareingeneralimpossibletoestimateseparately.Itisonlypossibleiftwoofthosevariablesareknown(forexamplethemetricfocalvaluefandthescalefactorsx).Extrinsicparameters:Rotations:Asetofn_i

32、ma3x3rotationmatricesRc_1,Rc_2,.,Rc_20(assumingn_ima=20).Translations:A外部参数Rc和Tc同时注意n_ima好像是图片的数量（对一个镜头进行参数估计的图片数量）setofn_ima3x1vectorsTc_1,Tc_2,.,Tc_20(assumingn_ima=20).Definitionoftheextrinsicparameters:Considerthecalibrationgrid#i(attachedtotheithcalibrationimage),andconcentrateonthecamerarefere

33、nceframeattahedtothatgrid.Withoutlossofgenerality,takei=1.Thefollowingfigureshowsthereferenceframe(O,X,Y,Z)attachedtothatcalibrationgid.LetPbeapointspaceofcoordinatevectorXX=X;Y;Zinthegridreferenceframe(referenceframeshownonthepreviousfigure).LetXXc=Xc;Yc;ZcbethecoordinatevectorofPinthecamerareferenceframe.ThenXXandXXcarerelatedtoeachotherthroughthefollowingrigidmotionequation:XXc=Rc_1*XX+Tc_1XXc=Rc_1*XX+Tc_1世界坐标系到相机坐标系的投影，用两个外部参数矩阵Inparticular,thetransla