Fitting Parameterized Three-dimensional Models to Images(11)

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

whereistheorientationofthelinewithrespecttothe-axisandisthesignedperpendicular

intotheleftsideofdistanceofthelinefromtheorigin.Ifwesubstituteanimagepoint

thisequationandcalculateanew,thenthesignedperpendiculardistanceofthispointfrom

.Thepartialderivativeofthisperpendicularerrormeasureisjustalinearthelineis

combinationofthepartialderivativesofand:

Inpractice,wecalculateandfrom2points,

bethelengthofthelinebetweenthesepoints:and,ontheline.Let

then

and

Theperpendicularerrorismeasuredbetweenselectedpointsontheimagecurveandtheperpendicularprojectionofthispointontotheclosestsegmentoftheprojectedmodelcurve.Thisdeterminationoftheclosestmatchingpointisupdatedoneachiterationofconvergence.

4.4Determiningastartingpositionforconvergence

Worrall,Baker&Sullivan[34]havestudiedtherangeofconvergencefortheauthor’searlierversionofthisalgorithmusingMonteCarlotechniques.Theyfoundthatthealgorithmwouldconvergetothecorrectsolutioninvirtuallyeverycaseforrotationerrorsoflessthan90degrees(translationerrorshavealmostnoeffect).Thenumberofiterationsriseswithincreasingerrorsuptoanaverageofabout6iterationsat90degrees.Withthestabilizationmethodsdescribedinthenextsection,convergenceissigni cantlyimprovedovereventheselevels.

Therefore,theaccuracyrequirementsfordeterminingtheinitialstartingpositionarequiteminimal.Forthemotiontrackingproblemwhichservesasourinitialfocus,wesimplyusetheparameterestimatesfromthepreviousframeaddedtoavelocityestimateforeachparameterobtainedfromtheprevious2frames.Forageneralrecognitionproblem,propertiesoftheimagematchesthatarebeing ttedcanbeusedtodetermineinitialparameterestimates.Forrotationindepth,eachmatchcanvoteforameandirectionfromwhichitisvisible(veryfewmodelfeaturesarevisiblefromallviewpoints)andthesedirectionvectorscanbeaveraged.Forrotationintheimageplane,wecanprojectthemodelfromtheestimatedrotationindepthandtaketheaverageimagerotationbetweenprojectedmodeledgesandthematchingimageedges.Estimatesfortranslationcanbemadebymatchingthecentersofgravityandstandarddeviationsfromthecentersofgravityfortheprojectedmodelfeaturesandimagefeatures.See

[21]foranexampleofcalculatinginitialestimatesforarecognitionproblem.

Ifthereareonlyaboutasmanymatchesasareneededtosolveforthedegreesoffreedom,thenitispossiblethatthereismorethanonelocalminimum.Thisproblemcanbeovercome

11