Fitting Parameterized Three-dimensional Models to Images(13)

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

wealsotakeaccountofpotentialambiguitiesinthematchtoincreasethestandarddeviation(i.e.,reducetheweighting)formatchesthatexhibitmorethanonenearbyalternative,sothatuncertaintiesinthecorrectmatchfornearbyalternativestranslateintotheappropriateuncer-taintiesinposition.

Themoreimportantnormalizationistoweightthepriormodelaccordingtothestandardde-viationsinthepriorestimatesforeachparameter.Thisisrelativelystraightforwardinthecaseofmotiontracking,wherelimitsontheaccelerationofeachparameterfromframetoframecanbeexpressedasastandarddeviation.However,inthecaseofmodel-basedrecognitionfromanyviewpoint,itmayseemthattherangeofexpectedvaluesisin nite.Infact,eachparameterislimitedduringconvergencebecauseweareassumedtobestartingfromsomeinitialapprox-imationtotheviewpoint.Therefore,therotationparameterswillhaveastandarddeviationof

,andthetranslationswillbelimitedtomaintainingthepositionoftheobjectwithinatmost

theimageframe.Internalmodelparameterswillhavestandarddeviationscorrespondingtoalargefractionoftheirvalidrangeofmovement.Thesedeviationsmaybelargeincomparisontothosearisingfromtheimagemeasurements,buttheystillplayasubstantialroleinstabilizingthesolutionforill-conditionedproblems.Infactthestandarddeviationscanbemadeseveraltimessmallerwithoutanadverseeffectonthedegreetowhichthe nalsolution tsthedatameasurements,becausethenon-lineariterativesolutioncanresetthestartingpointofthepriormodeltotheresultsofeachpreviousiteration.

5.2Ef cientcomputationofstabilization

Thepriorestimatesoftheparametervalueswillbeweightedbyadiagonalmatrix

eachweightisinverselyproportionaltothestandarddeviation,,forparameter:inwhich

Thismatrixisusedtoscaleeachrowofthepriormodelinthelowerpartofequation(2).Weassumethattheconstraintsbasedonimagemeasurementsintheupperpartoftheequationarealreadyscaledtohaveunitstandarddeviation.

Wewillminimizethissystembysolvingthecorrespondingnormalequations:

Whichmultipliesoutto

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