Fitting Parameterized Three-dimensional Models to Images(12)

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

bystartingwithsomeextramatches(thesolutionadoptedintheauthor’sapplications),byattemptingtoconvergefromseveralstartingpositions,orbyusingananalyticmethodappliedtosubsetsofthematches(asinDhomeetal.[4])tocomputeacandidatesetofstartingpositions.Yetanotherapproachistoconstructaninverselookuptablethatmapsfeaturemeasurementsintoapproximateviewpointparameterestimates.SuchanapproachhasbeenusedbyThompsonandMundy[32]forverticesandbyGoad[7]forawiderangeofarbitrarymodelfeatures.5Stabilizingthesolution

Aslongastherearesigni cantlymoreconstraintsonthesolutionthanunknowns,Newton’smethodasdescribedabovewillusuallyconvergeinastablemannerfromawiderangeofstart-ingpositions.However,inbothrecognitionandmotiontrackingproblems,itisoftendesirabletobeginwithonlyafewofthemostreliablematchesavailableandtousethesetonarrowtherangeofviewpointsforlatermatches.Evenwhentherearemorematchesthanfreeparameters,itisoftenthecasethatsomeofthematchesareparallelorhaveotherrelationshipswhichleadtoanill-conditionedsolution.Theseproblemsarefurtherexacerbatedbyhavingmodelswithmanyinternalparameters.

5.1Specifyingapriormodel

Alloftheseproblemscanbesolvedbyintroducingpriorconstraintsonthedesiredsolutionthatspecifythedefaulttobeusedintheabsenceoffurtherdata.Inmanysituations,thedefaultso-lutionwillsimplybetosolveforzerocorrectionstothecurrentparameterestimates.However,forcertainmotiontrackingproblems,itispossibletopredictspeci c nalparameterestimatesbyextrapolatingfromvelocityandaccelerationmeasurements,whichinturnimplynon-zeropreferencesforparametervaluesinlateriterationsofnon-linearconvergence.

Anyofthesepriorconstraintsonthesolutioncanbeincorporatedbysimplyaddingrowstothelinearsystemstatingthevaluethatwewishtoassigneachparameter:

Theidentitymatrixaddsonerowforspecifyingthevalueofeachparameter,andspeci esthedesireddefaultvalueforparameter.

Theobviousproblemhereisthatthereisnospeci cationofthetrade-offsbetweenmeetingtheconstraintsfromthedataversusthoseofthepriormodel.Theappropriatesolutionistoweighteachrowofthematrixequationsothateachelementoftheright-handsidehasthesamestandarddeviation.Therefore,asweminimizetheerrorvector,eachconstraintwillcontributeinproportiontothenumberofstandarddeviationsfromitsexpectedvalue.

Wewillnormalizeeachrowofthesystemtounitstandarddeviation.Iftheimagemea-surementsareinpixels,thenleavingthesewithastandarddeviationof1isalreadyagood rstestimatefortheerrorinmeasuringthepositionofimagefeatures.Inourmatchingalgorithm,

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