Nonlinear Least Squares Optimisation of Unit Quaternion Func(3)

Pose estimation from an arbitrary number of 2-D to 3-D feature correspondences is often done by minimising a nonlinear criterion function using one of the minimal representations for the orientation. However, there are many advantages in using unit quatern

Figure1.Stereoimagepairshowingthelineseg-mentsextractedfromoneimageandtheedgesofthelocalisedobjectsprojectedontotheotherimageOurgoalistodevelopamethodfortheminimisationof(5)over(t,q)∈R3×S3.Toachievethiswewrite(5)as

13 MN2hk(ti+d,exp(ω) qi)2=1

gi(d,ω)Tgi(d,ω).k=1

2(15)

Usingthesameapproachasinthecaseofpureunitquater-nionfunctions,theGauss-NewtoniterationonR3×S3canbeformulatedasfollows

ti+1

=ti+di,

qi+1=exp(ωi) qi,

(16)

dTi

ωi

T

T=

(JTiJi)

1JTigi

(0,0),whereJiistheJacobianofgiat(0,0).Thisiterationgen-eratesasequencewhichisguaranteedtolieinthesearch

spaceR3×S3.Sincetherearenoconversionsoforien-tationinsomeforeignform,suchasEuler’sangles,toaquaternionform,thenon-uniquenessofthequaternionrep-resentationdoesnotcauseanyproblems.SincethemetricstructureofSO(3)isthesameastheoneofS3,theaboveiterationmaybeviewedasaniterationinR3×SO(3).

4.Experimentalresultsandconclusions

Tovalidatetheproposedmethodexperimentally,weuseditforthecalculationofobjects’posesfromlineseg-mentcorrespondencesinasystemforobjectrecognitionandlocalisation(seeFig.1).TheconvergenceofthemethodisshowninTab.1.Evenwhenthestartingpointwasveryinaccurate,theGauss-Newtonmethodconvergedtothetrueobjectposeprovidedthefeaturecorrespondenceswerecorrect.HencethereisnoneedtousemorerobusttechniquesliketheLevenberg-Marquardtmethodforposeveri cationincalibratedstereoimages.Assumingthatfea-turecorrespondencesarecorrectandthemeasurementnoiseismoderate,thecriterionfunction(15)tendstozerointheneighbourhoodofthesolutionandtheconvergenceofthemethodisnearlyasgoodastheconvergenceoftheNewton-Raphsoniteration.

Table1.Convergenceofthemethod.TheEuclideanandtheangularmetricwereusedtomeasurethechangeinthepositionandorientation,respectively.

Step(trans.)Step(orien.)4.962116e+012.827351e-011.095884e+01

7.604011e-022.851983e-012.920441e-037.462760e-041.835331e-052.017427e-06

1.241607e-07

Furtherexperimentsshouldbecarriedouttotestthemethodforthecasewhenonlyonecameraisavailable.Thepresentedmethodisgeneralandworkswithanykindoffea-tures.Theusageoflineorpointcorrespondencesrequiresonlyarede nitionoffunctionf,whichisde nedforthecaseoflinesegmentsinEq.(3).

ComparingourapproachwiththeoneofPhongetal.[4],whoalsousedquaternionstorepresenttheorientation,ouriterationhastheadvantagethatitsearchesfortheoptimalorientationdirectlyinS3andnotinR4asthemethodofPhongetal.Phongetal.hadtointroduceapenaltytermtoforcetheminimumoftheircriteriontotendtowardsS3.However,thispenaltytermrequiresthesettingofauser-de nedparameterwhichisatbestarbitraryandcancauseproblemswiththeconvergenceoftheiteration.Ourmethoddoesnotsufferfromthisproblem.Moreover,itispossibletodeterminethesearchdirectionusingthetrustregionap-proachofPhongetal.inouriterationandthusmakeitlesssensitivetothequalityofastartingpointandwrongcorre-spondences.Thiswasnotnecessaryforourapplication.Acknowledgment:ThemeasurementsweretakenwhiletheauthorwaswiththeInstituteforReal-TimeComputerSystemsandRobotics,UniversityofKarlsruhe,Germany.

References

[1]R.L.CarceroniandC.M.Brown.Numericalmethodsfor

model-basedposerecovery.TechnicalReport659,ComputerScienceDepartment,TheUniversityofRochester,Rochester,NewYork,August1997.

[2]B.K.P.Horn.Closed-formsolutionofabsoluteorientation

usingunitquaternions.J.Opt.Soc.Am.A,4(4):629–642,1987.

[3]M.-H.Kyung,M.-S.Kim,andS.-J.Hong.Anewapproach

tothrough-the-lenscameracontrol.GraphicalModelsandImageProcessing,58(3):262–285,May1996.

[4]T.Q.Phong,R.Horaud,A.Yassine,andD.T.Pham.Object

put.Vis.,15:225–243,1995.