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

includetheprojectedlocations,tangents,curvatures,anddiscontinuitiesofedges.Ifshadingorsurfacepropertieswerebeingmatched,thensurfacecurvaturesmustalsobeapproximated.Wehavedevelopedamodelingsystemthatallowsthesequantitiestobemodeledasafunctionofviewpointandinternalmodelparameterstoanydesireddegreeofaccuracyandusedforef cientparametersolving.

Althoughmodel-basedvisioncanlearnmuchfromcomputergraphics,themodelingre-quirementsalsohaveimportantdifferences.Inmodel-basedmatchingto2-Dimages,themod-elsarematchedtoderivedimagefeaturesratherthanbeingusedtogeneratedensesurfacedescriptions.Forexample,itisimportanttobeabletodirectlycalculatethepositionsofoc-cludingcontours,whichisnotpossibleinmanymodelingsystemsdevelopedforcomputergraphics.Sincethemodelsareprojectedandmanipulatedintheinner-loopofthematchingprocess,itisimportantthatallpossiblesourcesofef ciencyparticulartothevisiondomainbeexploited.Inaddition,certainquantitiesthatdonotoccuringraphicsapplications,suchasderivativeswithrespecttomodelparameters,mustbeef cientlyrepresentedandcomputed.Forallthesereasons,itisnecessarytodevelopamodelingsystemaimedatvisionratherthanadoptingexistingsystemsdevelopedforgraphics.

Asimplemodelinglanguagehasbeendevelopedthatcanbeusedtodescribearbitrarymod-elsandtheirinternalparametersforuseinmodel-basedvision.Thelanguageisusedtode neandname3-Dpoints,edgesandsurfacepatches,aswellasmodelparametersde ningrotationsandtranslations.Eachmodelpointisaleafinatreeofcoordinatetransformationsthatrep-resentanycombinationofpreviousrotationsandtranslationsspeci edbydifferentparameters(seeFigure1).Thesamemechanismisusedtohandlebothviewpointparametersandinternalmodelparameters,sotherootofthistreerepresentsthecameracoordinateframe.

Whenaninternalmodelisbuiltfromadescriptioninthislanguage,adensepointernetworkisconstructedthatlinkseachedgeelementtoitsadjoiningsurfacepatchesandendpoints.Acachingmechanismisusedsothatthevisibilityofeachsurfacepolygonandtheprojectionofeachpointiscalculatedonlyonce,unlikeinmostgraphicsmodelingsystems.Becauseeachpointonasurfaceorlinemaymoveindependentlybybeingattachedtodifferentframes,itispossibletospecifyarbitrary exiblemotionsofmodels.Forexample,thelineconnectingthepointsFandGinFigure1canstretchunderthein uenceofparameterandrotateinconjunctionwithanothermodelsubpartunderthein uenceofparameter(eachtranslationframespeci esatranslationdirectionandeachrotationspeci esarotationaxis).

Edgesarelabeledaccordingtowhethertheylieonasmoothsurfaceorformadiscontinuity.Bycachingarecordofthesurfacenormalforthepatchoneachsideofeachedge,thevisibil-ityandlocationsoftheoccludingboundariesandsurfacediscontinuitiescanbegeneratedveryef ter,wewilldescribehowthemodelrepre-sentationenablestheef cientcomputationofpartialderivativesofimagefeatureswithrespecttoeachparameter.

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