Lie Groups for 2D and 3D Transformations

1 LieGroupsfor2 Dand3 DTransformationsEthanEadeUp datedMay20,2017*1 Intro ductionThisdo ologicalgroupthatisalsoasmo othmanifold,withsomeotherniceprop ciatedwitheveryLiegroupisaLiealgebra,whi chisavectorspacediscussedb ortantly,aLiegroupanditsLiealgebraareint imatelyrelated,allowingcalculationsinone tob emapp cumentdo esnotgivearigorousintro ductiontoLiegroups,nordo esattempttoprovideenoughinformationthatt heLiegroupsrepresentingspatialtransforma tionscanb eemployedusefullyinrob (3)3 DRotations33 DrotationmatrixSE(3)3 DRigidtransformations6 Lineartransformationonhomogeneous4-vecto rsSO(2)2 DRotations12 DrotationmatrixSE(2)2 DRigidtransformations3 Lineartransformationonhomogeneous3-vecto rsSim(3)3 DSimilaritytransformations(rigidmotion+s cale)7 Lineartransformationonhomogeneous4-vecto rsForeachofthesegroups,therepresentation isdescrib ed,andtheexp oticsorcomputervision?

2 Manyproblemsinrob ,*Added erentiationof/bygroupelementp ecomp osed,inverted,di erentiatedandinterp ciatedmachineryaddressalloftheseop erations,anddosoinaprincipledway,sothato nceintuitionisdevelop ed,itcanb efollowedwithcon ertiesEveryLiegrouphasanasso ciatedLiealgebra, ,theLiealgebraisavectorspacegeneratedbyd i erentiatingthegrouptrans-formationsalong chosendirectionsinthespace, ,thoughtangentvectorsundergoaco (andthusofthetangentspace)arecalledgener atorsinthisdo ortantly,thetangentspaceasso ciatedwithaLiegroupprovidesan optimal spaceinwhichtorepresentdi ,velo cities,Jacobians.

3 Optimal spaceinwhichtorepresentdi erentialquantitiesb ecause Thetangentspaceisavectorspacewiththesame dimensionasthenumb erofdegreesoffreedomofthegrouptransforma tions Theexp onentialmapconvertsanyelementofthetangen tspaceexactlyintoatransformationinthegro up Theadjointlinearlyandexactlytransformsta ngentvectorsfromonetangentspacetoanother Theadjointprop ertyiswhatensuresthatthetangentspacehast hesamestructureatallp ointsonthemanifold,b ecauseatangentvectorcanalwaysb edb ,3 Drigidtransfor-mationshavetheactionofrot atingandtranslatingp (3) ,SO(3), ositionandinversioninthegroupcorresp ,inversionisequivalenttotransp SO(3)(1)R 1=RT(2)2 TheLiealgebra,so(3),isthesetof3 (3)corresp ondtothederivativesofrotationaroundtheea chofthestandardaxes,evaluatedattheidenti ty:G1= 0 0 00 0 10 1 0 , G2= 0 0 10 0 0 1 0 0 , G3= 0 1 01 0 00 0 0 (3)Anelementofso(3)isthenrepresentedasal inearcombinationofthegenerators.

4 R3(4) 1G1+ 2G2+ 3G3 so(3)(5)Wewillsimplywrite so(3)asa3-vectoroftheco e cients,anduse torepresentthecorre-sp onentialMapTheexp onentialmapthattakesskewsymmetricmatrice storotationmatricesissimplythematrixexp onentialoveralinearcombinationofthegener ators:exp ( ) exp 0 3 2 30 1 2 10 (6)=I+ +12! 2 +13! 3 + (7)Writingthetermsinpairs,wehave:exp ( ) =I+ i=0[ 2i+1 (2i+ 1)!+ 2i+2 (2i+ 2)!](8)Nowwecantakeadvantageofaprop ertyofskew-symmetricmatrices: 3 = ( T ) (9)Firstextendthisidentitytothegeneralca se: 2 T (10) 2i+1 = ( 1)i 2i (11) 2i+2 = ( 1)i 2i 2 (12)Nowwecanfactortheexp onentialmapseriesandrecognizetheTaylorex pansionsintheco e cients:3exp ( ) =I+( i=0( 1)i 2i(2i+ 1)!)

5 +( i=0( 1)i 2i(2i+ 2)!) 2 (13)=I+(1 23!+ 45!+ ) +(12! 24!+ 46!+ ) 2 (14)=I+(sin ) +(1 cos 2) 2 (15)Equation15isthefamiliarRo onentialmapyieldsarotationby radiansaroundtheaxisgivenby .PracticalimplementationoftheRo driguesformulashouldusetheTaylorexpansio nsoftheco e cientsofthesecondandthirdtermswhen onentialmapcanb einvertedtogivethelogarithm,goingfromSO( 3)toso(3):R SO(3)(16) = arccos(tr(R) 12)(17)ln (R) = 2 sin (R RT)(18)Thevector isthentakenfromtheo -diagonalelementsofln (R).Again,theTaylorexpansionoftheco e cient 2 sin shouldb eusedwhen , ,theadjointiswrittenAdjX: so(3),R SO(3)(19)R exp ( ) = exp (AdjR ) R(20)Theadjointcanb , :exp (AdjR ) =R exp ( ) R 1(21)Then,withoutlossofgenerality,let =t v,fort R,anddi erentiatebytatt= 0:4ddt t=0exp (AdjR t v) =ddt t=0[R exp (t v) R 1](22)ddt t=0[I+ (AdjR t v) +O(t2)]=R ddt t=0[I+ (t v) +O(t2)] R 1(23)(AdjR v) =R v R 1= (Rv) (24)= AdjR=R(25)InthecaseofSO(3), moves erentiatingtheactionofSO(3)onR3 ConsiderR SO(3)andx.

6 Y=f(R,x) =R x(26)Thendi erentiationbythevectorisstraightforward, asfislinearinx: y x=R(27)Di erentiationbytherotationparametersisp erformedbyimplicitlyleftmultiplyingthero tationbytheexp onentialofatangentvectoranddi erentiatingtheresultingexpressionaroundt hezerop ductbythegenerators. y R= | =0(exp ( ) R) x(28)= | =0exp ( ) (R x)(29)= | =0exp ( ) y(30)=(G1yG2yG3y)(31)= y (32) erentiatingagroup-valuedfunctionbyanargu mentinthegroupConsideraLiegroupGandafunc tionf:G ,butbyintro ducingtangentspacep erturbationsontheargumentandresult,wecan usethedi erentationnotationasashorthandforthemapp ingfrominputtooutputp erturbations:exp ( ) f(g) =f(exp ( ) g)(33) f g | =0(34) anddi erentiatingyieldsanexplicitformulaforthe di erentialoftheoutputp erturbation bytheinputp erturbation.

7 = log(f(exp ( ) g) f(g) 1)(35) f g log(f(exp ( ) g) f(g) 1) | =0(36) ducesalinearmappingfromleft-tangent-spac ep erturbationsoftheargumenttoleft-tangent- spacep ected,applyingthisdi erentiationshorthandtotheidentityfunctio nf(g) = cedure,considerapro ductofelementsinG= SO(3)bythesecondfactorR0:R2=f(R0) R1 R0(37)First,theinputandoutputp erturbationsinthetangentspaceso(3) ( ) R2=R1 exp ( ) R0(38)Di erentationof bytheinputp erturbation isp erformedaround = R2 R0 log((R1 exp ( ) R0) (R1 R0) 1) | =0(39)= | =0[log((exp(AdjR1 ) R1 R0) (R1 R0) 1)](40)= | =0[log(exp(AdjR1 ))](41)= | =0[AdjR1 ](42)= AdjR1(43)=R1(44) (3) deGaussiandistributionsover3 Drotationsbyrepresentingthemeanwithanele mentofSO(3)andthecovarianceasaquadraticf ormovertangentvectorsinso(3).

8 Moreprecisely,consideraGaussiandistribut iongivenbymeanR SO(3)andcovariance R3 : N(0, )(45)S= exp ( ) R(46) ositionofuncertainrotationsGiventwoGauss iandistributionsonrotation,wecancomp e(R0, 0)andtheotherb e(R1, 1).Thenthedistributionofrotationsby rsttransformingbyR0andthenbyR1isgivenby: (R1, 1) (R0, 0) =(R1 R0, 1+R1 0 RT1)(47) ecombinedinaBayesianmannertoyield(Rc, c)by rst ndingthedeviationb etweenthetwomeansinthetangentspace, (inversecovariance)adds,asusual: c=( 10+ 11) 1(48)= 0 0( 0+ 1) 1 0(49)v R1 R0(50)= ln(R1 R 10)(51)Rc= exp( c 11 v) R0(52) (3)Equation47couldb eusedasthedynamicsup dateinanextendedKalman lter(EKF),where(R0, 0)isthepriorstateand(R1, 1)isthedynamicmo dateforthecovarianceandEquation52istheme asurementup dateforthemean,assumingatrivialmeasureme ntJacobian(identitymatrix).

9 ,theKalmangainKisde nedK 0( 0+ 1) 1(53)sothattheKalmanup datecanb ewritteninitsstandardform:Rc=R0 (K v)(54)= exp (K v) R0(55) c= (I K) 0(56)Lab ellingtheab oveinthestandardEKFframework,thestatecov arianceisgivenby 0andthemeasurementnoiseisgivenby , di (3) ,SE(3),iswellrepresentedbylineartransfor mationsonhomogeneousfour-vectors:R SO(3),t R3(57)C=(Rt01) SE(3)(58)Notethat,inanimplementation,onl yRandtneedtob eimplicitlyimp ,asinSO(3),meansthattransformationcomp ositionandinversionarecoincidentwithmatr ixmultiplicationandinversion:C1,C2 SE(3)(59)C1 C2=(R1t101) (R2t201)(60)=(R1R2R1t2+t101)(61)C 11=(RT1 RT1t01)(62)8 Thematrixrepresentationalsomakesthegroup actionon3Dp ointsandvectorsclear.

10 X=(x y z w)T RP3( x'x R)C x=(Rt01) x(63)=(R(x y z)T+wtw)(64)Typically,w= 1,sothatxisaCartesianp ondsto ,enco dedwithw= 0, (3)isthesetof4 4matricescorresp ondingtodi erentialtranslationsandrotations(asinso( 3)).Therearethussixgeneratorsofthealgebr a:G1= 0 0 0 10 0 0 00 0 0 00 0 0 0 , G2= 0 0 0 00 0 0 10 0 0 00 0 0 0 , G3= 0 0 0 00 0 0 00 0 0 10 0 0 0 ,G4= 0 0 0 00 0 1 00 1 0 00 0 0 0 , G5= 0 0 1 00 0 0 0 1 0 0 00 0 0 0 , G6= 0 1 0 01 0 0 00 0 0 00 0 0 0 (65)Anelementofse(3)isthenrepresentedbym ultiplesofthegenerators:(u )T R6(66)