Fitting Parameterized Three-dimensional Models to Images(3)

Model-based recognition and motion tracking depends upon the ability to solve for projection and model parameters that will best fit a 3-D model to matching 2-D image features. This paper extends current methods of parameter solving to handle objects with

parametersusingNewton’smethodfornonlinearleast-squaresminimization.Sincethattimethemethodhasbeenusedsuccessfullyinanumberofapplications,anditalsoprovidesthestartingpointfortheworkpresentedinthispaper.Theapplicationofthemethodtorobustmodel-basedrecognitionhasbeendescribedbyLowe[20,21,22],McIvor[26],andWorrall,Baker&Sullivan[34].Bray[2]hasappliedthemethodtomodel-basedmotiontrackingofrigidobjects.Ishiietal.[14]describetheapplicationofthisworktotheproblemoftrackingtheorientationandlocationofarobothandfromasingleviewofLEDtargetsmountedonthewrist.Theirpaperprovidesadetailedanalysisthatshowsgoodaccuracyandstability.Goldberg&Lowe[8]describetheapplicationandtestingofanumberofmoreadvancednumericalmethodsforthisproblem.

Inrecentyears,therehasbeenaconsiderableincreaseinthenumberofpublicationsonparametersolvingformodel-basedvision,withmostoftheworkaimedatsolvingforviewpointparametersofrigidobjects.Liuetal.[18]andKumar[15]haveexaminedalternativeiterativeapproachestosolvingfortheviewpointparametersbyseparatingthesolutionforrotationsfromthosefortranslations.However,Kumarshowsthatthisapproachleadstomuchworseparameterestimatesinthepresenceofnoisydata.Therefore,headoptsasimilarsimultaneousminimizationasisusedintheworkabove.AquitedifferentapproachbasedontheuseofeliminationmethodstoprovidetheinitialproblemformulationhasbeenproposedbyPonceandKriegman[29].ThisalsousesNewton’smethodforthe nalparameterdeterminationbasedonleast-squaresminimization.

Haralicketal.[11]haveexperimentedwithrobustmethodssuchasiterativereweightinginordertoallowforoutlierscausedbyincorrectmatches.However,theirresultsshowthatevenoneoutlieramong20correctmatchesleadstoalargeincreaseinexpectederrorfollowingreweighting.Thealternativethatisusedinthispaperistoprovideahigher-levelsearchprocessthatconsidersothersetsofmatcheswhenthe rstsetfailstoresultinanaccurate tofthemodel.

2.1Theproblemofmultiplesolutions

Muchworkhasbeenpublishedoncharacterizingtheminimumamountofdataneededtosolveforthesixviewpointparameters(assumingarigidobject)andonsolvingforeachofthemulti-plesolutionsthatcanoccurwhenonlythisminimumdataisavailable.FischlerandBolles[6]showthatuptofoursolutionswillbepresentfortheproblemofmatching3modelpointsto3imagepoints,andtheygiveaprocedureforidentifyingeachofthesesolutions.Asolutionforthecorresponding4-pointproblem,whichcanalsohavemultiplesolutionsundersomecir-cumstances,isgivenbyHoraudetal.[12].HuttenlocherandUllman[13]showthatthe3-pointproblemhasasimplesolutionfororthographicprojection,whichisasuf cientlycloseapprox-imationtoperspectiveprojectionforsomeapplications.Theyusetheterm“alignment”torefertothesolutionforviewpointparametersduringthemodel ttingprocess.Inthemostvaluabletechniqueformanypracticalapplications,Dhomeetal.[4]giveamethodfordeterminingallsolutionstotheproblemofmatching3modellinesto3imagelines.Theyshowthatthisispar-ticularlyusefulforgeneratingstartingpositionsfortheiterativetechniquesusedinthispaper

3