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

whentherearemultiplesolutions.

Thisworkondeterminingallpossibleexactsolutionswillnodoubtbeimportantforsomespeci cvisionapplications,butitisprobablynotthebestapproachforpracticalparameterdeterminationingeneralmodel-basedvision.Oneproblemwiththesemethodsisthattheydonotaddresstheissueofill-conditioning.Evenifaproblemhasonlyoneanalyticsolution,itwilloftenbesuf cientlyill-conditionedinpracticetohaveasubstantialnumberandrangeofsolutions.Secondly,allthesemethodsdealwithspeci cpropertiesofthesixviewpointparameters,andthereislittlelikelihoodthattheycanbeextendedtodealwithanarbitrarynumberofinternalmodelparameters.Finally,thesemethodsfailtoaddresstheproblemofwhattodowhenthesolutionisunderconstrained.Thestabilizationmethodsdescribedinthispaperallowanapproximatesolutiontobeobtainedevenwhenaproblemisunderconstrained,aswilloftenbethecasewhenmodelscontainmanyparameters.

Possiblythemostconvincingreasonforbelievingthatitisnotnecessarytodetermineallpossiblesolutionsisthefactthathumanvisionapparentlyalsofailstodoso.Thewell-knownNeckercubeillusionillustratesthathumanvisioneasilyfallsintoalocalminimuminthedeter-minationofviewpointparameters,andseemsunabletoconsidermultiplesolutionsatonetime.Rock[31],pp.22ffsummarizesthewayinwhichhumanperceptionseemstoalwaysadoptoneparticularperceptionatanytimeeveninthefaceofcompletelyindeterminatecontinuousvariables.Theperceptioncansuddenlychangetoanewstablepositioninthefaceofnewin-formation,whichmaycomeinternallyfromothercomponentsofthevisualsystem(attention)aswellasfromtheexternalstimulus.Thisbehaviorisconsistentwithastabilizedminimiza-tionapproachfordeterminingtheparametervalues,inwhichtheprocesscanbeinitiatedfromnewstartingpointsasnewinformationbecomesavailable.Theextremelygoodperformanceofhumanvisioninmostrecognitionproblems,inspiteofitspotentialforgettingstuckinfalselocalminima,indicatesthatlocalminimamaynotbeamajorproblemwhendeterminingmodelparameters.

Itisworthnotingthattheparametersolvingproblemissimpli edwhenaccurate3-Dim-agedataisavailable(asfromascanninglaserrange nder),sincethisavoidssomeofthenon-linearitiesresultingfromprojection.ExamplesofsolutionstothisproblemaregivenbyFaugeras&Hebert[5]andGrimson&Lozano-P´erez[10].However,inthispaperwerestrictourattentionto tting3-Dmodelsto2-Dimagefeatures.

3Objectandscenemodeling

Mostresearchinmodel-basedvisionhasbeenbasedonmodelsofsimplepolyhedral3-Dob-jects.Whiletheyaresimpletoworkwith,theyareclearlyinadequateforrepresentingmanyreal-worldobjects.Someresearchhasbeenbasedonmodelsbuiltfromcertainclassesofvol-umetricprimitives,mostnotablygeneralizedcylinders[1,3]andsuperquadrics[27].Whiletheseareattractivebecauseoftheirabilitytocapturecommonsymmetriesandrepresentcertainshapeswithfewparameters,theyareill-suitedformodelingmanynaturalobjectsthatdonotexhibitthesetofregularitiesincorporatedintotheprimitives.

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