Adaptive sampling based on the motion

Transcription

1 Adaive samlig based o he moio Soglao, Whoi-Yul Kim School of Elecrical ad Comuer Egieerig Hayag Uiversiy Seoul, Korea sliao@visio.hayag.ac.kr wykim@hayag.ac.kr Absrac Moio based adaive samlig mehod is reseed i his aer. The disribuio of aricles ca be adaively chaged accordig o moio iformaio so ha aricles could rack objec i higher accuracy whe he sae rasiio fucio is ukow. The advaage of roosed mehod is examied by doig sric simulaio ess i D, D ad 3D sace. The resul was comared wih he resul from he sadard aricle filer i each se. The simulaio resuls show ha i would imrove he redicio accuracy usig he roosed mehod. Keywords-aricle filer; adaive; moio; simulaio I. INTRODUCTION aricle filer, oe of dyamic sae esimaio echiques, is commoly used i may egieerig alicaios, esecially i sigal rocessig ad objec rackig. Various forms of he aricle filers ad heir alicaios have bee roosed. Hadel roved ha boosra-ye Moe Carlo aricle filers aroximae he oimal oliear filer i a ime average sese uiformly wih resec o he ime horizo whe he sigal is ergodic ad he aricle sysem saisfies a ighess roery []. Olsso ad Rydé did he research o he asymoic erformace of aroximae maximum likelihood esimaors for sae sace models obaied via sequeial Moe Carlo mehod []. Sakaraarayaa e el. roosed a ew ime-savig mehod for imlemeig aricle filer usig he ideede Meroolis Hasigs samler [3]. Rahi e el. formulaed aricle filerig algorihm i he geomeric acive coour framework ha ca be used for rackig movig ad deformig objecs [4]. Clark ad Bell roosed a aricle HD filer which roagaes he firs-order mome of he muli-arge oserior isead of he oserior disribuio so ha he racker could rack muli-arge i real ime [5]. Feamhead e el. iroduced ovel aricle filers for class of arially-observed coiuous-ime dyamic models where he sigal is give by a mulivariae diffusio rocess [6]. Bolic e el. roosed ovel re-samlig algorihms wih archiecures for efficie disribued imlemeaio of aricle filers [7]. Kha e el. iergraded Markov radom field io aricle filer o deal wih rackig ieracig arges [8]. Särkkä roosed a ew Rao-Blackwellized aricle filerig based algorihm for rackig a ukow umber of arges [9]. Scho e el. imlemeed margialized aricle filer by associaig oe kalma filer o each aricle so ha he roosed mehod could reduce ime comlexiy [0]. The remaider of his aer is orgaized as follows. Secio Ⅱ iroduces coce of aricle filer. Secio Ⅲ discusses he roosed mehod from he asecs of D, D, 3D ad geeral case. Secio Ⅳ shows he simulaio resuls of roosed mehod ad Secio Ⅴ gives he coclusio ad fuure work. II. ARTICLE FILTER aricle filer is a echique used for filerig oliear dyamical sysems drive by o-gaussia oise rocesses. The urose of aricle filer is o esimae he saes { S,, S } recursively usig he samlig echique. aricle filer aroximaes he oserior disribuio ( ( S ( Z: wih a se of samles { S,, S } ad a sequece of oisy observaios { Z,, Z }. The aricle filer cosiss of wo comoes, sae rasiio model ad observaio model. They ca be wrie as: Trasiio Model : S F ( S, N ( Observaio Model : Z H ( S, W The rasiio fucio F aroximaes he dyamics of he objec beig racked usig revious sae S ad he sysem oise N. The measureme H models a relaioshi amog he oisy observaio Z, he hidde sae S ad he W observaio oise. We ca characerize rasiio robabiliy S ( S wih he sae rasiio model ad likelihood Z ( S wih he observaio model. III. ROOSED METHOD I aricle filer framework, each ime sae has is ow disribuio for aricles ad i could be used o esimae he exac sae i coordiaio wih he observed daa. The disribuio is chagig a every ime se. I he revious work, eole assume ha he disribuio of aricles equals o he disribuio of sae rasiio added by he cosa Gaussia oise while doig redicio. I his aricle, a ew framework is roosed, which meas he disribuio of he aricles is chagig accordig o some meaigful feaures, moio vecor i his case.

2 (a.5.8 (b.0.8 (c Required DF Figure. A. Basis coce (d Samlig resul Samlig i D We deoe sae vecor a ime as V { v, v,, v} ad he observaio sae vecor a ime as Z { z, z,, zk}, where is he umber of sae dimesio ad k is he umber of observaio dimesio, usually k is smaller ha. The he derivaive of he each z z z observaio sae ca be rereseed{,,, k }. For a vecor Z { z, z,, zk}, we defie he disace fucio of Z as z z zk DZ ( ( (,, ( ( Eve hough he dimesio of he observaio vecor is k, s ossible o use oly ar of his. Tha is o say, he value of DZ ( could be relaced by he value of DZ (, where Z { z, z,, z q} ad { z, z,, z q } is a subse of { z, z,, z k }, where { z, z,, z q } are he ieresig feaures we wa o use. The relaioshi bewee S could be S ad rereseed usig S f ( S, where is assumed o a cosa Gaussia disribuio i mos of cases. I he case of redicig curre sae S wih he velociy iformaio v from he revious sae, he relaioshi could be wrie as S S v. We chage he disribuio of aricles accordig o he value of v, more secifically, if he absolue value of v becomes larger, ad he he robabiliy of he aricles disribuio would become higher, oo. B. roosed mehod i D I he case of D, here are oly wo oios for he agle. Oe is o raslae alog he osiive direcio; he oher is o raslae alog he egaive direcio. If assume ha he moio iformaio is kow, he osiio of he curre sae is a 0 ad he sae rasiio saisfies S S v, he he robabiliy of S could be rereseed as, ( osiiox moio s ex (3, osiive direcio (4, egaive direcio (5 The sae rasiio disribuio could be see i Fig. (a ad Fig. (b. I is show ha here is a ga a osiio 0, which is he osiio of he curre sae. I order o solve his roblem, we used cubic slie [] o smooh he seleced five ois so ha he ga is elimiaed. The five ois seleced are wo eak ois ad wo half middle ois from lef ad righ Gaussia disribuio ad he oi a osiio 0. The smoohed curve is draw wih gree color as i Fig. (a ad Fig. (b. For N aricles a ime se, we deoe hem N as {,,, }. They would be filered firsly by resamlig se ad he roduce ew locaio of each aricle usig he roosed robabiliy desiy fucio like i Fig. (c. Fig. (d shows he samlig resul. For examle, q, where q is he roosed robabiliy desiy fucio(df a ime se. All he moio iformaio has already bee cosidered iside he DF, ad he DF would chage accordig o he moio iformaio. The coce is very differe from he ormal aricle filer, where v, v is he seed of objec a ime se ad is usually a Gaussia oise. C. roosed mehod i D I he case of D, moio vecor could be rereseed as a D vecor wih legh ad hase. We use ad o deoe hese wo variables. Similar wih he case of D, moio based disribuio would be used o redic he ex sae of each aricle isead of Gaussia oise. The roosed disribuio has a eak a he locaio (,, where deoes he legh of he moio vecor ad deoes he hase of moio vecor. We assume ad are ideede wih each oher ad he kurosis of each disribuio is chagig adaively. Oe of he examles is show i Fig. (a. The shae of he samlig robabiliy desiy fucio is much more like a rile o he waer whe a objec is movig iside he waer.

3 (a roosed DF i D Figure. (b roosed DF i 3D roosed DF i D ad 3D We deoe sae vecor as S { x, y} ad velociy vecor as V { v, v } a he ime se, he we geerae x y he ew sae of each samle usig S S F( V, where F is he roosed samlig fucio based o he moio iformaio a each ime se. I he case of ormal aricle filer, he equaio is like S S v N. So he roosed samlig mehod is differe from origial samlig mehod i he redicio sae. The roosed samlig mehod has a big advaage ha i ca chage DF or iegrae oher facors io he samlig sae. D. roosed mehod i 3D I he case of 3D, he sae vecor is S { x, y, z}. S is he locaio of he objec a he curre ime se. The moio vecor could be rereseed as sherical coordiae sysem. The relaioshi bewee he sherical coordiae ( r,, of a oi ad is Caresia coordiaes ( x, y, z is like as x rsicos y rsisi z rcos If we deoe curre sae as S ad curre velociy V, he he rasiio robabiliy fucio becomes ( S S f ( V, where fv ( is he roosed DF which deeds o he curre moio of he objec. The relaioshi bewee roosed DF ad moio could be exressed like V X iˆ Y ˆj Z kˆ X Y Z Z acos(, g( X, Y f ( V (,, (,, I is show ha he roosed DF is he joi robabiliy of (,,. If we assume he robabiliy of,, is ideede wih each oher ad hey comly he Gaussia disribuio, he he joi robabiliy of hese hree variables looks similar o he shae of shockwave. Fig. (a is he roosed DF whe X, Y 3, Z 4, i ca be see ha he shae of he DF is similar o he shockwave. I meas he robabiliy of he osiio which locaes alog he moio direcio has higher value. Of course, he (6 (7 disribuio of, ad is o ecessarily he Gaussia disribuio, hey could be ay disribuio you wa defie. E. roosed mehod i geeral case We have roosed ew samlig mehodology i D, D ad 3D sace. The roosed mehod could be exeded o higher dimesioal sace. We deoe sae vecor a ime as S { s, s,, sm} ad he corresodig observaio sae vecor as Z { z, z,, zk}, where m is he umber of sae dimesio ad k is he umber of observaio dimesio ad k is smaller ha m. The observaio sae vecor could be obaied usig he corresodig sae vecor, we deoe his relaioshi as Z H( S. We eed o firs calculae he gradie of he curre observaio vecor ad he calculae he disace DZ ( usig (. The we calculae gradie of S from he gradie of Z. Z H( S Z H ( S H ( S S ( S Z S (8 gradie( Z, gradie( S H( S gradie( S gradie( Z ( ( S As log as he gradie of sae S is esimaed, he we could calculae he agle bewee gradie vecor of S ad ui base vecor i he sae sace as: gradie( S, S i i acos (9 gradie ( S Si Here i is he i h ui base vecor i he sae sace. I he case of m dimesioal sace, we oly eed m arameers. We roose a joi robabiliy as: ( D( S,,,, i (0 Each radom variable i his joi robabiliy chages adaively accordig o he value of DS (, which meas shae of he samlig robabiliy chages adaively. Fially, he joi robabiliy would be used o redic he sae of objec i he ex ime se. IV. EXERIMENT A. Simulaio i D We will show wo kids of simulaios i D sace, oe is he flucuaio case ad he oher is he o-flucuaio case. I order o ake he es i he flucuaio case, we chose he sae rasiio model ad observaio model as X X 8cos(. (-+N ( Z X W

4 (a DF esimaio (c RMSE for 50 ieraios Figure 3. (b Comared wih aricle filer (d No-flucuaio case Simulaio i D Where N ad W are boh defied as a ormal Gaussia disribuio wih he sadard deviaio.0. I order o chage rasiio desiy robabiliy adaively accordig o he moio iformaio, i (3 ad, i (4 are defied as: 8.8* ex( moio *0.7 (.4 moio * moio *0.64 (3 (4 = 0.5, = 0.5 (5 The above hree equaios defie ad whe moio is osiive, egaive ad zero, resecively. The roosed filer is used o esimae he sae ad he resuls are comared wih ha of a usual aricle filer ad exeded kalma filer (EKF. From he defiiio of i (, he larger he moio is, he smaller is. Fig. 3 shows he simulaio resul. Fig. 3(a shows he robabiliy desiy a he ime se 0; Fig. 3(b shows he comariso wih he usual aricle filer ad Fig. 3(c shows he comariso of he average RMSE amog he filers over 50 ime samles. I is show ha he esimaio resul comared wih ormal aricle filer is very close ad RMSE error is also similar wih he ormal aricle filer bu much smaller ha he EKF. The badwidh of he DF of he roosed filer is arrower ha he oher wo filers. I is also show i Fig. 3(b ha here exiss esimaio errors whe he direcio of he seed chages. This is because we did o cosider he acceleraio i his case. Acually, we oly cosidered he firs derivaive of he sae bu o he secod derivaive of he sae. The usual aricle filer esimaio is a lile more accurae ha he roosed mehod. I is because rasiio model is kow while geeraig ew aricles i he ormal aricle filer mehod, bu we do eve kow he rasiio model, we oly redic he sae usig he velociy (derivaive of he sae iformaio i he roosed mehod. I real cases, he rasiio model is ukow; we ca oly esimae sae usig revious exeriece like he firs ad secod derivaive of he sae ad so o. Besides, he arameer fucios used i his case could be chaged o ay oher forms. Fig. 3(d shows he es resul of he o-flucuaio case. The seed is always osiive i his case, so he acceleraio iformaio is o eeded. I is show ha he esimaio resul is almos he same as he rue sae. B. Simulaio i D I order o make a es i D sace, we have maually defied a secific rajecory which could be exressed as vx 5cos(, vy 5si( 4 4 = 0:: 300 (6.5 X vx 00.4 Y vy 0.4 The he seed ad he hase a ime se ca be exressed as: V vx 0.05 x V vy 0.56 y V V, g( V, V x y x y Ad he disribuio of ad are defied as:, 8.8 ex( 0.3 (7 ( ex( 0.8 Fig. 4 shows he simulaio resul. Fig. 4(a is origial rajecory, Fig. 4(b is he rackig resul usig aricle filer ad Fig. 4(c is he rackig resul usig roosed mehod. Fig. 4(d shows he RMS error comariso bewee roosed mehod ad aricle filer. The resul shows ha he RMS error of roosed mehod is smaller ha he geeral aricle filer hrough he whole ieraio ime. (a Origial rajecory (b aricle filer resul

5 (c roosed samlig resul Figure 4. (d RMS error comariso Simulaio i D V. CONCLUSION Moio based adaive aricle filer is roosed i his aer. This ew mehod uses a ovel samlig mehod which chages aricles robabiliy desiy fucio adaively accordig o he moio iformaio. We show he accuracy of he ew mehod by doig simulaios i D, D ad 3D saces, resecively. The resul has show ha he ew mehod has a good erformace as we execed. This mehod is very useful whe he sae rasiio fucio is ukow. The fuure work icludes exadig his framework o he higher dimesioal sace. ACKNOWLEDGMENT This work was suored by he Brai Korea rojec i 00. (a roosed mehod resul (c True sae Figure 5. (b aricle filer resul (d RMS error comariso Simulaio i 3D C. Simulaio i 3D I order o make a es i 3D sace, we have maually defied a secific rajecory which could be exressed as i 0 : :0 i 00 x 30si( (9 y 0cos( z si(0.6 Ad he disribuios of, ad are defied as:,, 8.8 ex( ex( 0.5 (0 4.8 ex( 0.5 Fig. 5 shows he simulaio resul i 3D, where Fig. 5(a is he resul from roosed mehod; Fig. 5(b is he resul from ormal aricle filer mehod; Fig. 5(c is he rue sae. I is difficul o see he rajecory clearly, so we calculaed roo mea square (RMS error of roosed mehod ad aricle filer, resecively. Fig. 5(d shows he RMS error comariso bewee roosed mehod ad aricle filer. The resul shows ha he RMS error of aricle filer is.9766 ad he RMS error of he roosed mehod is.904, which is smaller ha he geeral aricle filer whe ieraio was fiished. REFERENCES [] R. va Hadel. Uiform ime average cosisecy of moe carlo aricle filers, Sochasic rocesses ad heir Alicaios, 9(: , 009. [] J. Olsso ad T. Rydé. Asymoic roeries of aricle filerbased maximum likelihood esimaors for sae sace models, Sochasic rocesses ad heir Alicaios, 8(4: , 008. [3] A.C. Sakaraarayaa, A. Srivasava, ad R. Chellaa. Algorihmic ad archiecural oimizaios for comuaioally efficie aricle filerig, IEEE Trasacios o Image rocessig, 7(5: , may 008. [4] Y. Rahi, N. Vaswai, A. Taebaum, ad A. Yezzi. Trackig deformig objecs usig aricle filerig for geomeric acive coours, IEEE Trasacios o aer Aalysis ad Machie Ielligece, 9: , 007. [5] D. E. Clark ad J. Bell. Covergece resuls for he aricle hd filer, IEEE Trasacio o Sigal rocessig, 54(7:65 66, 006. [6]. Fearhead, O. aasiliooulos, ad Gareh O. Robers. aricle filers for arially observed diffusios, Techical reor, 006. [7] M. Bolic,. M. Djuric, ad Sagji Hog. Resamlig algorihms ad archiecures for disribued aricle filers, IEEE Trasacios o Sigal rocessig, 53(7:44 450, 005. [8] Z. Kha, T. Balch, ad F. Dellaer. Mcmc-based aricle filerig for rackig a variable umber of ieracig arges, IEEE Trasacios o aer Aalysis ad Machie Ielligece, 7(:805 89, 005. [9] S. Särkkä, A. Vehari, ad J. Lamie. Rao-blackwellized aricle filer for mulile arge rackig, Secial Issue o he Seveh Ieraioal Coferece o Iformaio Fusio. Iformaio Fusio, 8(: 5, 007. [0] T. Scho, F. Gusafsso, ad.-j. Nordlud. Margialized aricle filers for mixed liear/oliear sae-sace models, IEEE Trasacios o Sigal rocessig, 53(7:79 89, 005. [] Seve C. Chara. Alied Numerical Mehods wih MATLAB for Egieerig ad Scieiss, MC Graw Hill, secod ediio.