Dynamic Foveal 3D Sensing Using Affine Models

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DynamicFoveal3DSensingUsingAffineModelsDianeLingrandandThierryVie´villeINRIA,Sophia,BP93,06902Valbonne,FranceE-mail:dlingran@sophia.inria.fr,vthierry@sophia.inria.fr

AbstractThisstudyisaimedatdevelopingamethodforanalysing3Dstructureofasceneconsideringamonocularimagese-quence,withanuncalibratedcamera-asforanactivevisualsystem-andusingacontinuousmodelofmotion,consider-ingthatthesceneisstatic.Thisworkisacontinuationof[9].

1.Problempresentation.1.1.Simplemodel.Inordertoovercomethealgebraiccomplexity[10]oftheequationssolvingourproblem,wehaveattemptedtode-velopasimplifiedparameterizationoftheprobleminthecaseoftwoormoreviews,consideringascenewithasetofstationaryobjects,approximatingthedisplacementbe-tweentwoframestothefirstderivativeorder(fromwhichwededucethedisplacementofthepoint:)andapplyinganorthographicmodeloftheprojectionrelatingthepointofthe3D-spacetothepointoftheimage(see[6]fordetails):

(1)Following[11]theretinalmotionfieldofaplanarpatchdefinedby

(2)

,foratranslationandarotation,isnow:

(3)

with:(4)Asetof3pointsgiveusasystemof6linearequationswith6unknowns,andauniquesolutionifthepointsarenon-collinear.Thus,wemakeuseoftheDelaunaytriangulationasimplementedby[3]onthefirstframeandtriangulatethesecondframebypoints-corresponding.Furthermoreweas-sumethatareconstantineachtriangle.Fromtheseunknowns,boundtothemotionparameters,tothenormalstothefacetsandtotheintrinsicsparameters,weobtainquadraticequationswhichallowustodetermine,uptoaconstant,the2firstcomponentsofthenormaland,byasimilarway,therotationvector:

(5)(6)

Knowingthat,inourcase,therotationvectorisconstant,wefirstcalculatethisvectorbyminimizationforallfacets.Then,wededucethelastcomponentoftherotationvec-torandthe2firstcomponentsofthenormaltoeachfacetwiththefollowingequations,optimalinthesenseofleast-squares:1.2.Self-calibration.Onotherhand,weobtainapseudo-linearequation(linearifweconsider5unknowns,non-linearifweconsider3un-knowns),fromwhichwerecovertheopticalcenterandthefocusofexpansionrepresentedby

(8)Byminimizationofthepreviouslinearequationforallfacets,weobtainaninitializationforthenon-linearequa-tion.Theresidualerrorsobtainedwiththisminimization,allowustoquantifythevalidityofourmodel.Anexperimentalresultisshownin(1)anddetailscanbefoundin[7].

motionresults

002532560.8700.4%0.5%9%00.042722370.8701.6%3.2%3%0.04254264-0.025%5%178%

Table1.Computationof,andwith:(15)whileandarehugeexpressionsofthemotionandstructureparametersbutcanbesimplifiedwithafewalge-bra:(16)Inthecaseof,byeliminationofinthepreviousequations(ifandtheparametersfromwhich,byminimization,wecancomputethelocationoftheopticalcenterand,uptoascalarfactor,thetranslation,asdetailledinthesequel.Asexpected,weobtain-foraplanarpatchunderaper-spectiveprojection-aquadraticmodeloftheretinalmo-tionfield,whichparametersdonotcorrespondtotheaffinemodelobtainedinequation(3),andmustbeestimatedusingatleastfourpoints.However,thesetwomodelscanberelatedintwositua-tions:Ifweareundergoingatranslationparalleltotheretinalplane,andiftherotationaxisisparalleltotheopticalaxis,i.e.ifwehave(17)thetwosetsofequationsareincorrespondencethroughthetransformation:(18)Itmightbenotedthatdependsonthepointsdepthbutthisscalefactordisappearsinallequationssuchasin(6)sothatitwillnotperturbatethesolution.Thereisageometricinterpretationofthissituation.Wearesimplyinacasewherethe3D-translationoftheperspectivemodel“isseenasarotation”bytheortho-graphicmodel.Indeed,arotationaroundtheaxis,a“pan”,inducesahorizontalretinaldisplacementasanhorizontaltranslationwoulddo,whilearotationaroundtheaxis,a“tilt”,inducesaverticalretinaldis-placementasaverticaltranslationuptoasign.

Iftheplaneisafronto-parallelplaneandiftherotationaxisisparalleltotheopticalaxis,i.e.ifwehaveandthemotionfieldisstillanaffinemotionfield.However,itscorrespondenceswithequation(3)doesnotexistunless.

Thissituationwillnotbeconsideredhere,i.e.wewillalwaysimpose.

So,wehavedemonstratedthattheprojectiveandaffinemodelsareequivalentincaseofdisplacementsforwhichtheretinalplaneisconstant(translationparalleltothereti-nalplaneandrotationorthogonaltotheretinalplane).Inthecaseofdisplacementforwhichtheretinalplaneisapproxi-matelyinvariant,theapproximationisbetterifweareclosetothefovea(figure(eflowrt))as,naturally,inmostvisionproblems[8,5,2,4,1].

Figure1.Simulationonsyntheticaldata(sceneofthegrid).Errorsonaffineopticalflow(inpixels)foratranslationof1.5pixelsin,1.3pixelsinandarotationof0.04rad.aroundthe-axisand(a)arotationof0.01rad.aroundtheand-axis,(b)translationof0.5pixelson,relatedtothedistancefromtheopticalcenter(inpixels).