ThreestepimagerecticationPascalMONASSE1monasseimagineenpcfrJeanMichelMOREL2morelcmlaenscachanfrZhongweiTANG2tangcmlaenscachanfr1IMAGINELIGMUniversit ID: 151473
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MONASSEetal.:THREE-STEPIMAGERECTIFICATION1 Three-stepimagerecticationPascalMONASSE1monasse@imagine.enpc.frJean-MichelMOREL2morel@cmla.ens-cachan.frZhongweiTANG2tang@cmla.ens-cachan.fr1IMAGINE,LIGM/UniversitéParis-Est/ÉcoledesPontsParisTechMarne-la-Vallée,France2CMLA,ENS-CachanCachan,France AbstractImagestereo-recticationistheprocessbywhichtwoimagesofthesamesolidsceneundergohomographictransforms,sothattheircorrespondingepipolarlinescoincideandbecomeparalleltothex-axisofimage.Apairofstereo-rectiedimagesishelpfulfordensestereomatchingalgorithms.Itrestrictsthesearchdomainforeachmatchtoalineparalleltothex-axis.Duetotheredundantdegreesoffreedom,thesolutiontostereo-recticationisnotuniqueandactuallycanleadtoundesirabledistortionsorbestuckinalocalminimumofthedistortionfunction.Inthispaperarobustgeometricstereo-recticationmethodbyathree-stepcamerarotationisproposedandmathematicallyex-plained.Unlikeothermethodswhichreducethedistortionbyexplicitlyminimizinganempiricalmeasure,theintuitivegeometriccamerarotationangleisminimizedateachstep.Forun-calibratedcameras,thismethodusesanefcientminimizationalgorithmbyoptimizingonlyonenaturalparameter,thefocallength.Thisisincontrastwithallformermethodswhichoptimizebetween3and6parameters.Comparativeexperimentsshowthatthealgorithmhasanaccuracycomparabletothestate-of-art,butndstherightminimumincaseswhereothermethodsfail,namelywhentheepipolarlinesarefarfromhorizontal.1IntroductionThestereorecticationofanimagepairisanimportantcomponentinmanycomputervisionapplications.Theprecise3Dreconstructiontaskrequiresanaccuratedensedisparitymap,whichisobtainedbyimageregistrationalgorithms.Byestimatingtheepipolargeometrybetweentwoimagesandperformingstereo-rectication,thesearchdomainforregistrationalgorithmsisreducedandthecomparisonsimplied,becausehorizontallineswiththesameycomponentinbothimagesareinonetoonecorrespondence.Stereo-recticationmethodssimulaterotationsofthecamerastogeneratetwocoplanarimageplanesthatareinadditionparalleltothebaseline.Fromthealgebraicviewpoint,therecticationisachievedbyapplying2Dprojectivetransformations(orhomographies)onbothimages.Thispairofhomographiesisnotunique,becauseapairofstereo-rectiedimagesremainsstereo-rectiedunderacommonrotationofbothcamerasaroundthebaseline.Thisremainingdegreeoffreedomcanintroducean c 2010.Thecopyrightofthisdocumentresideswithitsauthors.Itmaybedistributedunchangedfreelyinprintorelectronicforms. MONASSEetal.:THREE-STEPIMAGERECTIFICATION3 2.1RecticationgeometryThefundamentalmatrixcorrespondstotwostereo-rectiedimagesifandonlyifithasthespecialform(uptoascalefactor)[i]=2410035=2400000�101035:(1)Havingbothcameraspointingtothesamedirectionwiththeirimageplanesco-planarandparalleltothebaselineisstillnotsufcienttoachieverectication.AssumethecamerastohavetheformP=K[Ij0]andP0=K0[Iji]withthemotionbetweenbothcamerasbeingonlythetranslationalongthex-axis.Thenthefundamentalmatrixisproportionalto[i]ifandonlyifKandK0havethesamesecondrow.Theorientationofthecameracanbeadjustedbyapplyingahomographyontheimage,whichhastheform:H=KRK�1(2)whereRistherelativerotationbeforeandafterrectication.Istherecticationachievedbyndingapairofhomographieswhichsendstheepipolesineachimageto(1;0;0)T(x-direction)?TheanswerisNO.Havingtheepipolesat(1;0;0)Tonlymeanstherelationshipbetweentwocamerasisarotationaroundthebaselineandimagesaregenerallyunrectied.2.2Three-steprecticationAssumecamerasarenotcalibratedbuthavethesamesimplecalibrationmatrixKbeforeandafterrectication:K=24f0w 20fh 200135(3)withw,hthewidthandheightoftheimageandftheunknownfocallength.Thefunda-mentalmatrixFiscomputedfromagroupofnon-degeneratecorrespondencesbetweentwoimages.Theepipolesfortheleftimagee=(ex;ey;1)Tandrightimagee0=(e0x;e0y;1)TcanbecomputedasrightandleftnullvectorsofF:Fe=0ande0TF=0.Theideaistotransformbothimagessothatthefundamentalmatrixgetstheform[i].UnliketheothermethodswhichdirectlyparameterizethehomographiesfromtheconstraintsHe=i,H0e0=iandH0T[i]H=Fandndanoptimalpairbyminimizingameasureofdistortion,weshallcomputethehomographybyexplicitlyrotatingeachcameraarounditsopticalcenter.Thealgorithmisdecomposedintothreesteps(Fig.1):1.ComputehomographiesH1andH01byrotatingbothcamerasrespectivelysothattheleftepipole(ex;ey;1)istransformedto(ex;ey;0)andtherightepipole(e0x;e0y;1)to(e0x;e0y;0).2.Rotatebothcamerassothat(ex;ey;0)istransformedto(1;0;0)and(e0x;e0y;0)to(1;0;0).ThecorrespondinghomographiesaredenotedbyH2andH02.3.Rotateonecameraorbothcamerastogethertocompensatetheresidualrelativerota-tionbetweenbothcamerasaroundthebaseline.ThecorrespondinghomographiesaredenotedbyH3andH03. MONASSEetal.:THREE-STEPIMAGERECTIFICATION7 Figure2:Imagepair“Arch”rectiedbydifferentmethods.Fromtoptobottom:originalimages,proposedmethod,Hartleymethod,Fusielloetal.methodandMallonetal.method.Ahorizontallineisaddedtoimagestochecktherectication.Thethirdcolumnrepresentsanimageaverageofeachpair. 8MONASSEetal.:THREE-STEPIMAGERECTIFICATION Figure3:Imagepair“Building”rectiedbydifferentmethods.Fromtoptobottom:originalimages,proposedmethod,HartleymethodandFusielloetal.method.Ahorizontallineisaddedtoimagestochecktherectication.Thethirdcolumnrepresentsanimageaverageofeachpair.