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

thatthederivativescanbecomputedinanextremelyef cientform.Forexample,thedirec-tionalderivativeofapointwithrespecttoanincrementalrotationaboutthe-axisisthevector

,whereandrefertothecoordinatesofthevectorfromtheoriginofrotationtothepoint.

Oncethedirectionalderivativesofeachmodelpointhavebeencomputed,itissimplya

.Perspectiveprojectionofamodelpointmatterofprojectingtheseintoimagecoordinates

incamera-centeredcoordinatestoproduceanimagepointisgivenasfollows:

and

whereisaconstantproportionaltothefocallengthofthecameralens.Weincludeanotherconstant,,specifyingthewidth-to-heightaspectratioofeachpixelintheoriginalimage,asmostcurrentvideostandardshavenon-squareaspectratios.Takingthepartialderivativeofeachoftheabovefunctionswithrespecttoaparameter,weget

and

Herethepartialderivativesof,andwithrespectto

directionalderivativescalculatedearlier.aresimplythecomponentsofthe

4.3Measuringperpendicularerrorsforcurves

Themethodsabovewouldbesuf cientifwehadmatchesbetweenpointsonthemodelandpointsintheimage.However,inmostcasesthematcheswillactuallybebetweenprojectedcontoursofthemodelandpartialedgesintheimage.Sincetheprecisepositionoftheend-pointsofimageedgesareunknown(andmaybedisplacedduetoocclusion),itisnecessarytominimizeonlytheperpendiculardistancefrompointsonanimageedgetotheprojectedmodelcurve.

Itmightbethoughtthatself-occludingedgesofcurvedsurfaceswouldrequirespecialtreat-ment,astheactualmodeledgethatformssuchanoccludingcontourwillshiftwithchangesinviewpoint.However,thesurfacenormalatsuchanoccludingpointisexactlyperpendiculartotheviewingdirection,andthereforetheinstantaneousmotionofthecontourprojectedintotheimageiszeroasnearbypointsonthesurfacereplaceit.Forlargerrotations,theerrorintro-ducedbynon-linearityiseasilyhandledthroughthesameiterationsthatcompensateforothernon-linearities.

Inordertomeasuretheperpendiculardistancefromanimagepointtoaprojected2-Dmodelline,itisusefultoexpresstheprojectedmodellineinthefollowingform:

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