Motion Estimation Based on Unit Quaternion Decomposition of the Rotation Matrix

Transcription

1 Moto Estmato Based o Ut Qatero Decomposto of the Rotato Matrx Hag Y Ya Baozog (Isttte of Iformato Scece orther Jaotog Uversty Bejg PR Cha Abstract Based o the t qatero decomposto of rotato matrx ths paper pts forward a algorthm to estmate moto parameters from the space posto vectors of D featre pots Rotato matrx's represetato wth the t qatero has o sglar pots so the t qatero-based estmato method s of more practcal mportace ad the algorthm ths paper does ot eed terato comptato compared to those t qatero-based method proposed by Hor (987) ad Fageras etal (987) Solto's qeess aalyss of the algorthm ad smlato expermet reslts are also preseted t ca be see that performace of or method s satsfactory Key words Ut qatero decomposto estmato of moto parameters sglar pots Itrodcto Dscerg the moto of objects from the seqece of mages s mportat for compter vso applcatos If we extract the depth formato drectly e the dstace from the cameras to the objects by the stereo vso system the as opposed to a seqece of -D mages ths addtoal depth formato greatly redces the complexty of the moto estmato task; Otherwse we get the absolte vale of traslato vectors So the stereo vso-based method moto aalyss by ow has bee more emphaszed[] whch the ma problem s how to determe the rotato matrx R where the traslato vector ca be obtaed afterwards Several methods had bee proposed to fer R drectly sch as SVDbased method[] orthogoal decomposto-based method[] ad RS decomposto-based method[4] etc bt we kow that R has oly degrees of freedom (DOF) so more mercal comptato may decrease the estmato's accracy ad the estmated matrx ca ot be garateed to be a rotato matrx hs researchers start to fd those estmato methods wth R expressed by sch fewer parameters as rotato axs ad ts agle skew-symmetrc matrx three Eler agles ad t qatero etc Becase rotato expresso wth the t qatero ot oly does ot cofrot sglar pots bt also descrbes the comptato of several rotato matrxes brefly[5] t wold be of more practcal mportace Hor etal[5] ad Fageras etal[6] had respectvely pt forward smlar moto estmato methods wth t qatero represetato whch s compted by a teratve algorthm of determg the egevectors correspodg to the smallest egevale to solve a mmzato problem of a qadratc form Here based o the heorem of Ut Qatero Decomposto (UQD) a lear least sqares algorthm of moto estmato wthot teratos s proposed the t rs faster Evetally qeess aalyss of ts solto ad reslts of smlato expermets are also gve Rotato Represetato wth Ut Qatero By the kematcs theory the object moto ca be dvded to a rotato compoet ad a traslato compoet Let -D posto vectors of the object's featre pots tme stat t ad t + δ t be p p' respectvely = the we have p' = Rp + + = () where R s the rotato matrx s the traslato vector s the addtve ose DOF of R as a orthogoal rotato matrx s so R ca be expressed terms of sch as three Eler agles (oretato agles abot x y z axes respectvely by the rght-had rle) axs of rotato ad rotato agle skew-symmetrc matrx correspodg to the Carley vector ad t qatero Bt t seems that: whe oe Eler agle s eqal to 90 the other two ca ot be determed qely; whe the rotato agle s eqal to 0 the rotato axs ca take ay drecto; there s o Carley vector correspodg to the rotato matrx whe rotato agle s eqal to 80 At the same tme represetato wth t qatero does ot have sglar pots detaled aalyss sees below:

2 Assme a t qatero Qater=( q) wth as a vector ad q as a scalar wth the costrat that + q = he we have the followg Lemma: Lemma A 4 4 sqare matrx Q costrcted wth a t qatero Qater=( q) wold be a orthogoal matrx e 0 qi + S Q = q S = 0 0 () where I s the t matrx S s a skew-symmetrc matrx correspodg to the vector = ( ) Proof[7] QQ q I S + = 0 0 Sce q I S + = q I + I = I + q = so QQ = I 4 heorem For a rotato matrx R there are oly two t qateros wth the cotrary sgs e ( q) ( q) satsfyg the followg decomposto: R 0 Q 0 = () Proof[7] Mltply the rght-had sde of the eqato () wth ( q) we have R 0 QQ R I q q q 0 0 q (4) here mst be o zero solto of ( q) for the above eqato so we ca get at least oe t qatero to satsfy the eqato () ow we assme that two t qateros ( q) ( q) satsfy the eqato () ad each of them costrcts respectvely the correspodg sqare matrx Q Q () Let = 0 the q = ± ow R=I Q = ± I 4Q Q = I 4 so Q Q = we ca fd that = 0 q = ± ()Let 0 the Q Q q q [ ] = [ ] ad S = 0 so =k (k 0) sbsttto to the eqato () ca make sre that k=± So q = ± q = ± heorem gves the t qatero decomposto (UQD) of the rotato matrx ow we ca fer the relatoshp betwee the t qatero ad R as below: Lemma A rotato matrx R s expressed wth t qatero Qater = ( q) = ( q) as below q + q + q ( ) ( ) R = ( + q ) q + ( q) ( q + q q + ) ( ) (5) Proof[7] From the eqato () we ca fer that R=( qi + S ) + = ( q I) I + + qs Represetato wth t qatero ca sbsttte for sch represetato as axs of rotato ad rotato agle ( r θ ) the Carley vector ( a b c) as below: [ a b c] θ = = r s θ + a + b + c q = cos q = + a + b + c (6) From the eqato (6) sglar pots ca be fod: whe q=0 there s oe of ( ) a b c correspodg to the rotato matrx; whe =0 the rotato axs r ca take ay vale + q

3 Estmato of Rotato Matrx wth the Ut Qatero Sce the t qatero ca represet rotato wthot ay sglar pots so research o algorthms of moto estmato based o ths represetato s more emphaszed Hor[5] ad Fageras[6] respectvely pt forward smlar moto estmato methods wth t qatero represetato whch s compted by a teratve algorthm of determg the egevectors (e t qatero) correspodg to the smallest egevale to solve a mmzato problem of the qadratc form Here we oly smply descrbe the method the referece [6]: Frst the moto estmato problem s to fer R mmzg the sm of sqared resdes as follows m J = p' Rp = (7) Let p = p / p' = p' / q = p p q' = p' p' the the eqato (7) s restated as m J = q' Rq = (8) Whle R s obtaed wold be calclated by: = p' Rp (9) ow we defe as below [8] : P = [ q q q ] P' = [ q' q' q' ] M = PP' S = M + M β = P + P' γ = tr( M) ω = ( m m m m m m ) (0) where m j s all the elemets of matrx M ad the sg tr refers to trace of matrx he we gve a symmetrc matrx as below β γ ω H = ω ( β + γ ) I S so the problem (8) ca be chaged to[6] m J = Q HQ () ow solto to the problem s the 4-D t egevector Q correspodg to the smallest egevale of H to mmze the qadratc form Q HQ Comptato of the egevector s a teratve process here we chage the problem to be a lear LS mmzato based o UQD the fer a o-teratve lear algorthm below s detals: Sbsttto of eqato () to eqato (8) geerates m J Q 4 = ( q' 0) Q( q 0) = () t ca be restated as a smplfed form as m J5 = ( qv S + v ) = () wth v = q' q S as a skew-symmetrc matrx correspodg to = q' + q ow we dscss how to solve the problem (): () Whe q 0 the problem () chage to the followg form m J5 = ( v S / q + v / q ) = (4) he problem (4) s eqvalet to solto of the followg reglar eqato:

4 A / q = b (5) wth A = ( S S + v v ) b = S v Oce /q s obtaed we ca determe two soltos of q ( + q = ) wth cotrary sgs From heorem each of them ca correspod to the same rotato matrx () Whe q=0 the problem () chage to the followg form m J5 = ( S + v ) = (6) wth = Smlarly the problem (6) s eqvalet to solto of a eqato as below: A = 0 = (7) Here the (parallel) stereo vso system lets z axs' drecto of the coordate system be the optcal axs' drecto ad posto of the object s lmted always the frot of cameras; sce the rotato ceter eqato () s chose as org of the coordate system ths we ca assme the z compoet of rotato axs = ( ) 0 e the axs of rotato wold tersect wth x-y plae ow let A = [ A A A ] the eqato (7) ca be restated as: / [ A A ] A / = (8) Oce / / are determed oly two soltos of wth the cotrary sgs ( = ) ca be obtaed Sce the axs of rotato does ot cosder the sg so we choose ay oe of them ether postve or egatve Uqeess of solto abot eqato (5) ad eqato (8) s aalyzed as below: Frst let V = [ v v v ] = P' P U = [ ] = P' + P Us = [ S S S ] X = [ Us V] α = tr( UU ) (9) It yelds A = XX = U su s + VV = α I UU + VV Besdes f the estmated t qatero ( q) correspods to the rotato matrx R the from Lemma we have q 0; q ± ; rak( R + I) = rak(r - I) = q = 0; 0q = ± ; ad U = ( R + I) P V = ( R I) P () Cosder the eqato (5) whe q 0: Frst we have Lemma rak ( U) rak(u s ) = Proof UU s sem-postve-defte where λ s ts ege vale λ 0 wth λ = α So rak ( U) rak( UU ) λ < α rak (U su s ) = rak( U s ) = heorem If rak ( P) the a qe rotato matrx ca be obtaed from the eqato (5) Proof If q 0 rak(u)=rak(p) the rak(a)=rak[u s V] rak( U s ) so rak ( P) ad rak(a)= rak(u s )= by Lemma ths the qe solto of /q ca be garateed Wth the costrat = we ca obta two t qateros wth the cotrary sgs e the qe rotato matrx () Cosder the eqato (8) whe q=0 we have: heorem If qe rotato matrx ca be obtaed from the eqato (8) Proof rak(a)=rak[u s V] rak( U s ) ad two axes of rotato wth the cotrary sgs ca be obtaed from the eqato (7) whe rak(a)= so ether of them s eogh; Uqe LS solto of / / ca also be determed from eqato (8) whe rak(a)= smlarly we ca fer a qe rotato matrx he fal solto of R s chose from the two soltos of the eqato (5) ad the eqato (8) to mmze the eqato (8) 4 Reslts of smlato expermets 4

5 Smlato data s geerated from the followg scearo: A set of -D featre pots {p } = 0 s obtaed at radom from (--) to (4) the axs of rotato s (00) ad the rotato agle θ s geerated from [ 0 80 ] three compoets of the traslato vector s chose radomly as a real mber betwee ad ; Wth these moto parameters above t yelds { p' } =0 by the eqato () ow the stereo vso system s descrbed as below: two les ceters o the x axs wth ther mddle pot at the org of the camera cetered coordate system the two les axs parallel cocdet wth the z axs both of the two mage plaes parallel to the x-y plae wth the focal legth f = ad the basele legth B =5 Secodly throgh projectve magg model we ca obta left ad rght mage pot l r pars {P P } = 0 the add the Gassa ose to the -D mage pot pars ; Fally throgh the stereo traglato we recostrct cotamated -D featre pots Sce the algorthm of moto estmato based o the Carley heorem proposed the referece [8] has some smlarty wth that ths paper here we compare estmato reslt of the latter wth that of the former SR s vared from 0dB to 40dB estmato error of both methods see Fg ad Fg : Fg llstrates the estmato error of the traslato vector wth d= 0 ; Fg for the rotato agle ( ) wth dθ = θ θ 0 From the estmato reslts we ca fd that accracy of the former method s close very mch to that of the latter oe sometmes estmato of or method s more accrate a lttle (Sce ormalzato of the compoets the t qatero s coveet for mercal comptato especally for the rotato agle more larger or smaller tha 90 ) What s more the method the Referece [8] may cofrot the problem of sglarty represetato of rotato matrx (see Chapter detal) 5 Coclso I ths paper a o-teratve lear algorthm of moto estmato based o the heorem of Ut Qatero Decomposto (UQD) s proposed whch s smpler ad faster tha that referece [5 6]; Otherwse represetato method wth t qatero does ot have sglar pots so t s more mportat tha the other represetato methods (sch as skew-symmetrc matrx Eler agles axs of rotato ad rotato agle etc) Fally solto's qeess ad reslts of smlato expermets are gve as well Referece [] Sabata B Aggarwal J K CVGIP: Image Uderstadg 99 54(): 09-4 [] Ar K etal IEEE ras o PAMI 987 PAMI-9(5): [] Hor B etal 9885(7): 7-5 [4] S R J etal Proc of the It Symp MS-89 etherlads: [5] Hor B J Opt Soc Ame (A) 987 4(4): [6] Fageras O D etal Proc of IEEE It Cof o Compter Vso Eglad: [7] Fa Hog Ph D Dssertato orther Jaotog U Bejg: 999 [8] X Wel L Wehag Acta Atomatca Sca 99 8(4): ( Chese) Carley UQD Carley UQD SR(dB) Fg raslato vector estmato error SR(dB) Fg Rotato agle estmato error 5