Nonlinear Least Squares Optimisation of Unit Quaternion Func

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

InProc.14thInt.Conf.PatternRecognition,Brisbane,Australia,pp.425-427,August1998.

NonlinearLeastSquaresOptimisationofUnitQuaternionFunctionsforPose

EstimationfromCorrespondingFeatures

AleˇsUde

JoˇzefStefanInstitute,DepartmentofAutomatics,BiocyberneticsandRobotics

Jamova39,1000Ljubljana,Slovenia,E-mail:ales.ude@ijs.si

Abstract

Poseestimationfromanarbitrarynumberof2-Dto3-Dfeaturecorrespondencesisoftendonebyminimisinganonlinearcriterionfunctionusingoneoftheminimalrep-resentationsfortheorientation.However,therearemanyadvantagesinusingunitquaternionstorepresenttheori-entation.Unfortunately,astraightforwardformulationoftheposeestimationproblembasedonquaternionsresultsinaconstrainedoptimisationproblem.Inthispaperweproposeanewmethodforsolvinggeneralnonlinearleastsquaresoptimisationproblemsinvolvingunitquaternionfunctionsbasedonunconstrainedoptimisationtechniques.Wedemonstratetheeffectivenessofourapproachforposeestimationfrom2-Dto3-Dlinesegmentcorrespondences.

1.Introduction

Theobjectposeisde nedasthedisplacementoftheco-ordinateframerigidlyattachedtotheobjectfromitsini-tialposition,whereitisalignedwiththeworldcoordinateframe,toitscurrentposition.Thereexistanalyticalandlin-earsolutionstotheproblemofposeestimationfrom2-Dto3-Dfeaturecorrespondences[1,2],buttheyaresensitivetonoise.Inthepresenceofnoise,whichisunavoidableinreal-worldapplications,algorithmsbasedonnonlinearop-timisationmethodsgivemoreaccurateresults.

Nonlinearoptimisationtechniqueshavebeenusedforposeestimationbymanyresearchersinthepast.Agoodoverviewisgivenin[1].Inmostoftheseapproaches,Eu-ler’sangleswereusedtoparameterisethegroupofrotationsSO(3)oftheEuclideanspace.However,itiswellknownthatSO(3),whichisathreedimensionalmanifold,cannotbegloballyembeddedinthethreedimensionalEuclideanspace.Itfollowsthatiftherotationgroupisrepresentedbythreerealparameters,theEuclideanmetrictopologyin

Currently,

theauthoriswiththeKawatoDynamicBrainProject,ER-ATO,JapanScienceandTechnologyCorporation,2-2HikaridaiSeika-cho,Soraku-gun,Kyoto619-0288,Japan,e-mail:ude@erato.atr.co.jp.

R3doesnotinduceaglobaltopologyandmetricstructureinSO(3).Thissuggeststhatcommonsolutionsusingminimalrepresentationsoftherotationgrouparenotideal.

Therepresentationoftherotationgroupbyunitquater-nions,whichformasphereS3inR4,hasmanyadvantagesoverminimalrepresentations.Methodsforposeestimationbasedonthequaternionrepresentationoftheorientationhavebeenproposedintheliteraturebefore[1],buttheposeestimationproblemhasbeenformulatedasanoptimisationprobleminR4ratherthanonS3intheseapproaches.

2.Preliminaries

3

Inthefollowingweshallneedtheexponentialmapexp:R→S3,which isgivenexp(r)=

by

cos( r ),sin( r )

r

,r=0.(1)(1,0,0,0),r=0

r

Theexponentialmaptransformsatangentvectorr∈R3T∈S3≡1(S3)intoq,whereqisapointatdistance r from1alongageodesiccurvestartingfrom1inthedirectionofr[3].Geodesicsarede nedasshortestpathsconnectinganytwopointsonamanifold3(sphereS3).Itturnsoutthatforanyotherpointq∈Sandforanyr∈T3S)3≡R3andtheexponentialq∈Tmapatq(S3),rq,expq qT(S3)1(S→,havingtheabovepropertiesisgivenby

q:qexpq(rq)=exp(rq q) q,

(2)

where denotesthequaternionmultiplication.

Letsconsidertheproblemofposeestimationfrom2-Dto3-Dlinesegmentcorrespondences.Letm(x1k,x2

k=

k),k=1,...,N,betheend-pointsofthek-th3-Dlinesegmentbelongingtotheobject’smodelandletAj,j=1,...,M,betheprojectivemappingontothej-thimageplane.Thesemappingsshouldbemadeavailablebyacameracalibrationprocedure.Letfdenotethemappingwhichtransformstheend-pointrepresentationofa2-Dlinesegmentintoitsmid-point f(v,vvTT

representation T12)=1+v22,arctan

y2 y1

x2 x1

,(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

wherevl=[xl,yl]T.Thelengthofalinesegmentisex-cludedfromthismappingbecauseitismorenoisythanothermid-pointparameters.3Weencodetheposepbya3-Dvectort∈Randbyaquaternionq∈S3.Letgjk:R3×S3→R3bethefollowingmappingsgjk(p)=f(Aj(q

x1k

q+t),Aj(q

x2k

q+t)).(4)

Eachgjkmapsthek-th3-Dmodelsegmentontothe2-D

imagesegmentobtainedbymovingthemodelsegmentbyp=(q,t)andbyprojectingtheresultingsegmentontothej-thimageplane.

Weassumenowthatthecorrespondencesbetweenmeasuredimagesegmentsyjk=[uTT

the

andthemjk, jk]

modelsegmentsobjectposekaregiven.Theoptimalestimateforthecurrentcanbecalculatedbyminimisingthefollowingnonlinearcriterionfunction

1 M N

2y jk gjk(t,q) Σ jk

1

yjk g jk(t,q)j=1

k=1=13

MN

2

hk(t,q)2,(5)k=1

whereΣjkarethecovariancesofthemeasuredsegments.Theminimisationof(5)overtandqwouldbeaclassicnonlinearleastsquaresoptimisationproblemifwecouldtreatqasanelementofR4andnotofS3.Sincethisisnotthecase,theclassicapproachwouldbetoaddthecon-straint|q|=1totheabovecriterion.Inthenextsectionweproposeabettersolution.

3.Leastsquaresoptimisationonunitsphere

Letsconsidertheminimisationofsumofsquaresofgen-eralunitquaternion functions

1

n qminF(q)=fk(q)2=1f(q)Tf(q).(6)∈S3

2k=1

2Denotingthen×4Jacobianmatrixoff(q)asJ(q),the

gradient F(q)andtheHessian 2F(q)aregivenby F(q)=J(q)Tf(q), 2

F(q)=

J(q)T

J(q)+

n(7)

fk(q) 2

fk(q).

(8)

k=1

Letqibethecurrentapproximationfortheminimumof(6).

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