An extended target tracking method with random finite set observations

Transcription

1 4th Intenational Confeence on Infomation Fusion Chicago Illinois USA July An extended taget tacing method with andom finite set obsevations Hongyan Zhu Chongzhao Han Chen Li Dept. of Electonic & Infomation Engineeing Xi an Jiaotong Univesity Xi an Shaanxi P. R. China hyzhu@mail.xtu.edu.cn czhan@mail.xtu.edu.cn lynnlc@6.com Abstact-A taget is denoted extended when the taget extent is lage than the senso esolution. A tacing algoithm should be capable of estimating the taget extent in addition to the state of the centoid. This pape addesses the poblem of tacing an extended taget with unnown paamete about the taget extent. The extended taget is egaded as a spatial distibution model and the taget extent is consideed as a mixtue of multiple pobability distibutions in this pape. The EM algoithm is utilized to estimate the unnown paametes about taget extent and also a paticle implementation of the pesented method is given. Simulation esults validate the effectiveness of the pesented method. Keywods: Extended taget taget extent andom finite sets. Intoduction In geneal tacing applications the taget to be obseved is usually consideed as a point souce obect. That is the taget extent can be neglected compaed with the senso esolution. With the inceasing senso esolution the senso can detect moe than one measuement fom the same taget. What one should concen includes both the state of the centoid and the taget extent. Fo an extended taget tacing poblem thee exist a vaiety of appoaches fo incopoating the taget extent into taget tacing pocess [-3]. A natual way is to egad the extended taget as a set of igid points each of which can be a souce of senso measuements. In this way one needs to constuct the explicit assignments between the measuements and the souces. Howeve if the numbe of measuement souce is lage such a method based on data association is most challenging and also unnecessay. Anothe altenative appoach is to model the extended taget as an intensity distibution athe than a set of points. Each taget-elated measuement is an independent sample fom the spatial distibution. Such an appoach maes it possible to compute the lielihood function without constuct the explicit association hypothesis between the measuements and the souces. The taget extent is closely elated to the taget shape. One typically appoximates the shape by means of a basic geometic shape such as a line a ectangle o an ellipse [ 6 ]. Howeve fo the taget with slightly complex shape such as an aiplane o a ship modeling the taget extent with a basic geometic shape is not enough. Geneally the motion pf a point taget is modeled by its centoid s dynamic model. Fo an extended taget an intuitive idea is to model its centoid as done in point taget case and to teat its extent as andomly distibuted samples in the taget plane which is egaded as a finite mixtue model so as to incopoate the complex shape of the extended taget. And also thee ae some unnown paametes about the finite mixtue model. The andom finite sets (RFSs) theoy [4-5] povides a pomising tool to implement a mathematically consistent genealization of Bayesian ecusion fomula fom singletaget single-measuement case to set-valued case. In this pape what we do fistly is to estimate those paametes about the taget extent based on a sequence of measuements and then update the centoid state unde the RFS theoy fame. This pape is oganized as follows. A poblem fomulation of extended taget tacing using andom finite set theoy is fomulated in section. In Section 3 we intoduced the EM algoithm fo extended taget tacing with unnown paametes. A paticle implementation of the pesented method is given in section 4. Simulation esults ae pesented in section 5. Conclusions and discussions ae given in section 6. Poblem fomulation In extended taget tacing application the senso can detect moe than one measuement oiginating fom the same taget. In this pape we ty to addess the extended taget tacing poblem fom the view of geneal Bayesian ecusion.. Geneal Bayesian ecusion Let x and z denote the taget state and measuement at time. The cumulative measuement set up to time is denoted by z :. The taget state x follows a fist-ode ISIF 73

2 Maov tansition pocess accoding to a tansition density f ( x x ). The measuement lielihood is given by g( z x ). Accoding to the classical Bayesian ecusion the posteio density P ( x z: ) about taget state is popagated as follows. P ( x z ) f ( x x) P ( x z ) dx () : : P ( x z ) : g ( z x ) P ( x z ) : () g ( z x ) P ( x z ) dx :. RFS measuement model fo multiple measuements case In many cases such as extended taget tacing and multi-path eflections one taget can poduce multiple measuements. The geneal Bayesian ecusion is extended to accommodate multiple measuements cases in [4]. The ey step is to deive a consistent lielihood function based on andom finite set theoy. The elated esults ae given as follows. In a multiple measuement case the measuement is a set-valued vaiable. Let Z denote the measuement space m be the numbe of measuements at time. Then all the measuements at time ae zi Z ( i... m ).The measuement set at time can be descibed as the andom finite set vaiables: Z { z z... z m } Z ( ) F Z. whee F ( Z ) epesent the espective collections of all finite subsets geneated byz. Accoding to the measuement oigin the total measuement RFS at time can be classified into the following thee pats: Z Θ( x) E( x) W Θ ( x) : the RFS of pimay taget-geneated measuement which coesponds with the one fom the taget centoid. Φ with pobability Pd Θ ( x) { z * } ( * with pobabilitydensity Pdg z x) whee g (..) and Pd ae the lielihood and the pobability of detection fo the pimay measuement espectively. W : the clutte RFS which is modelled as a Poisson RFS with the intensity vw ( z ). E( x ) : the RFS of extaneous taget-geneated measuement coesponding with the measuements fom the taget extent which is also modelled as a Poisson RFS with the intensity v ( z x ). E And also we assume that these thee RFS ae independent. By integating two RFS W and E( x ) togethe we have K( x) E( x) W (3) The intensity of RFS K( x ) : vk ( z ) ( ) ( ) x ve z x + vw z (4) The cadinality distibution ρ K ( n x ) of K( x ) is Poisson with mean v ( z x ) dz K. Fo multiple measuement cases the ey step is to deive the pobability density η( Z x) that the state x poduce the measuement set Z which coesponds with the single measuement lielihood g( z x ) in fomula (). The pobability density η( Z x) is given in [4] by η( Z x) P ρ ( Z x ) Z! c ( z x ) (5) [ ] d K z Z ( ) + Pρ ( Z x ) Z! g ( z x ) c ( z x ) * d K * * z Z z Z whee c means the pobability density of measuement z in RFS K( x) given the system state x vk ( z ) x c( z x) (6) v ( z x ) dz 3 Extended taget tacing with andom finite set obsevations 3. The RFS Bayesian ecusion fo single extended taget. As descibed above if all the paametes in the intensity vk ( z ) x ae nown then the lielihood function η ( Z x) can be obtained diectly by fomula (5). Unfotunately the measuements fom the taget extent ae usually poduced by an unnown distibution. In this case we need to appoximate the unnown distibution fistly. In this pape we assume that the extaneous tagetgeneated measuements oiginate fom the mixtue of multiple pobability density function and each component Eq ( x)( q.. M) is a Poisson RFS with intensity v ( z x ) with the mean fo the Eq K λe q cadinality distibution and M is the numbe of mixtue components. The whole RFS fo all extaneous taget-geneated measuements ae modelled as the union of multiple Poisson RFS. To unify the notations we define 74

3 Θ ( x) i Ω i ( x) W i Ei ( x) < i M + By using of standad measuement theoetic pobability we can deive the pobability density that the state x poduces the measuement set Z as done in [4]. Fo any Boel subset S F ( Z ) decomposing S into S S S is the subset of S with the length. 0 P ( S x ) P ( Z S x ) P ( Z S x ) P ( S x ) 0 0 N Define the events set{ } ξ ξ N ξ is the numbe of all i feasible events. { l... l } ξ (7) l is the component index which indicates that measuement z comes fom component l in eventξ. fom the centoid l fom clutte s( s > ) fom taget extent Accoding to the total pobability fomula we have ξ ξ i P( ξ x) P( Z S ξ x) i i P ( S x ) P( x ) P ( S x ) We intoduce n () s ( s M + ) as the measuement count fom component s in event ξ (8). By combining the detection pobability P d the cadinality distibution fo each Poisson RFS and possible pemutations of measuement set the pobability P( ξ x ) is given by M + δ δ Pd Pd ρs ns x s 3 x n n n3 nm C C nc... n n C n... nm P( ξ ) ( ) ( ) whee δ is a binay vaiable indicating whethe the centoid of extended taget is detected o not. If thee exists a measuement coming fom the centoid in event n thenδ is set to be ; othewise it is set to be zeo. We denotes ρ s be the cadinality distibution of the RFS Ω s ( x) ( s M + ). (9) The density PZ ( S ξ x) can be obtained by appopiately integating ove S the densityη( Z x ξ ) whee η ( Z x ξ ) f ( z x ) (0) ( i ) l {... } P( S ξ x) η( Z x ξ ) λ ( dz... dz ) whee Z (! ( S ) Z χ () λ is the th poduct Lebesque measue on Z is the th Catesian poduct of Z ) χ is a T χ ([ z... z ] ) { z... z }. mapping such that f ( z x ) l g( z x) l () uz ( ) l vr ( z () ) ( ) i x vr z x dz othewise l l whee uz ( i ) is the pobability density fo clutte measuement z i. We have PS ( x) (3) 0! χ ( S ) i Z η ( Z x ξ ) P( ξ x ) λ ( dz... dz ) By applying the Radon-Niodym deivative the pobability density η ( Z x) that the state x poduces the measuement set Z is: Z x Z x P x i η ( ) η ( ξ ) ( ξ ) (4) 3.EM algoithm fo extended taget tacing with unnown paametes When thee ae some unnown paametes ϑ { ϑ } M + i i about the intensity function vω ( z ) i x the lielihood η ( Z x) will be eplaced by its maximum lielihood function ˆ η ( Z x). ˆ η ( Z x ) max η ( Z x ϑ) (5) { } ϑ In this section EM algoithm is used to poduce the estimates fo these unnown paametes. The EM 3 75

4 algoithm is an iteative algoithm fo paamete estimation when the data set is incomplete. It can be applied fo not only ML estimation but also MAP estimation. It caies out two steps altenately (E step and M step) until the algoithm conveges. E step: Find the expected value of the complete-data loglielihood with espect to the miss data { } m l l i i given the obseved data and the cuent paamete estimates. That is ˆ g ( ) { log ( ) ˆg Q ϑ ϑ El G Z l x Z ϑ } (6) M step: Maximum the expectation value ( ˆ g Q ϑ ϑ ) obtained in E step to acquie the updated paamete ˆ ϑ g+. The lielihood fo complete data is given as ( ˆg Qϑϑ) log ( ) ˆg E G Z l x ϑ x Z ϑ + l { } l log ( ) ( ˆg G Z l x ϑ pl x Z ϑ ) l... log( ( ) ( )) ( ˆg G Z x l ϑ pl x ϑ pl x Z ϑ ) l l m m... lm ( log( f ( z x ϑ )) + (7) m l l m l l m l i l log( p( ))) ( ˆg l x ϑ pl x Z ϑ )... log( ( )) ( ˆg f z x ϑ pl x Z ϑ ) l i l... log( ( )) ( ˆg pl x ϑ pl x Z ϑ ) The fist item in equation (7) can be computed as m... log( ( )) ( ˆg f z x ϑ p l x Z ϑ ) l l m m M l l m s m l l... log( ( )) ( ˆg δ f z x ϑ p l x Z ϑ ) sl s s log( f ( z x ϑ))... s s sl s l l m m s whee δ s s ( ˆg pl x Z ϑ ) log( ( )) ( ˆg f z x ϑ p s x z ϑ ) (8) Pl ( x ϑ) P P λω M s ns e ( λ ) δ δ Ω s D ( PD) s ns! n n n3 nm Cm C... m nc m n n C m n... nm δ δ M D ( PD) λωs e λ m! Ωs s ( ) ns (9) whee λ Ω is the mean of the cadinality distibution fo s Poisson RFS Ω s ( s M + ) The second item in equation (7) is... log( ( )) ( ˆg pl x pl x Z ϑ ) l l m δ δ P ( P) D D λes n s log e ( λω ) s... m! s l l m ( ˆg pl x Z ϑ )... ( ) ( ˆg C λω pl ) s x Z ϑ l l m s + + m M n s... ˆg δsl log ( ) ( ) λω pl s x Z ϑ l l m s s... ( ) ( ˆg C+ λ pl x Z ϑ ) + m s whee l l m s Ωs Ωs log ( ˆg λ ps x z ϑ ) m li i li i (0) G ( Z x l ) f ( z x ϑ ) () π f ( z x ˆ ϑ ) g g ˆ s s s z ϑg M + g ˆ g π i fi z x ϑs i ps ( x ) ( ) The updated paamete can be obtained by taing the deivative with espect to the paametesϑ i (fo s M + ) and set the deivative to be zeo. 4 Paticle filte implementation () In this section we give a paticle implementation of the above method. 4 76

5 Step : Given a goup of paticles { x ϖ } N i which epesent the centoid state of the extended taget at time -. Step : Estimate the unnown paametes ϑ i about the taget extent by using of the measuement set Z as descibed in section 3.. The taget state x is appoximated by the pedictive state of the centoid. Step 3: Sample (. x q x Z) whee q(..) is the poposal distibution. Step 4: Compute the weights η ( Z x ) f ( x x ) ϖ (3) ϖ q( x x Z) The nomalized weights is ϖ ϖ / ϖ i (4) Step 5:The centoid state is appoximated as xˆ ϖ x (5) i 5. Simulation esults The following example is povided to validate the effectiveness of the pesented extended taget tacing method. Dynamic model: The motion of the taget centoid follows the following dynamic equation: x( + ) Φ x( ) +Γv ( ) (6) whee x ( ) [ x( ) x ( ) y( ) y ( )] T which means the position and velocity of the taget centoid at x-y plane T Φ diag{ Φ Φ } Γ diag{ Γ Γ } Φ T / Γ the sample inteval T is s. The initial T state of the taget centoid is x 0 [00 00 ] T and the covaiance matix of the pocess noise is designed as Q diag[9 4]. We assume that spatial distibution of taget extent is the mixtue of two Gauss distibution with the mean at the taget centoid and the covaiance Σ diag{ } Σ diag{ } Measuement model: We assume only the position infomation of the measuement souce is detected so the measuement equation fo pimay taget-geneated measuements is z( ) Hx( ) + w ( ) (7) whee the measuement matix is H The covaiance matix of the measuement noise is designed as R I and I is an identity matix. Fo the extaneous taget-geneated measuements they ae modeled as the mixtue of two Poisson RFS with intensity vw ( z ) ( ; ; ) λe N z s Hx Ψ s λ E E s 50 λ 60 T Ψ s HΣ s H + R. Clutte is also modeled as a Poisson RFS with intensity vw ( z ) ( ) x λw u z whee uz ( ) is the unifom distibution ove the obsevation egion. λ W is the expected numbe of clutte which is given an aveage λ W 0. To implement the paticle filte 500 paticles ae used at each time step and the tansition is used to be the poposal. The measuements ae suppoted by 00 MC uns pefomed on the same taget taectoy but with independently geneated measuements fo each tial. Fig shows the eal and estimated position fo the taget centoid in one tial by using of Pobability Data Association method (PDA) and the pesented method espectively. The RMSE cuve of the taget centoid is illustated in Fig.. RMSE y position PDA pesented method eal position of the centoid x position Fig. the eal and estimated position fo the centoid RMSE PDA pesented method Time Fig. RMSE of the taget centoid Fig.3 shows the taectoy of the taget centoid the measuements fom the taget extent based on the mixtue distibution and the estimated taget extent (illustated by two oint ellipsoids) fo seveal snapshots in one tial. In 5 77

6 Fig.4 the estimated and eal taget extents ae shown at scan 30 t. []Macus Baum and Uwe D.Hanebec Random Hypesuface Models fo Extended Obect Tacing IEE Poc.-Rada Sona Navigation50(6) extaneous meauement of component extaneous meauement of component [3]Kevin GilholmSamon Godsill Possion models fo extended taget and goup tacing Poc.of SPIE Vol R-. y position 0 00 [4] Dona Angelova and Lyudmila Mihaylova. Extended Obect Tacing Using Monte Calo Methods. IEEE TRANS on Signal PocessingJUNE007 y/s x position Fig.3 measuements fom taget extent and the estimated taget extent eal taget extent estimated taget extent x/s Fig.4 the estimated and eal taget extent at scan t30 6 Conclusions and discussions In this pape an effective appoach is developed to deal with the extended taget tacing poblem with unnown paametes about taget extent. The pesented appoach appoximates the taget extent by using of a finite mixtue model athe than a simple geometic shape. Simulation esults show its effectiveness to deal with extended taget tacing poblem. Howeve when the extended taget maes the fast otation movement and the senso only eceives small amount of measuements at each scan the pesented method doesn t wo well and multi-scan infomation is needed to achieve good pefomance. Moe effective method to deal with the fast otation movement will be done in futhe eseach. Acnowledgement The wo is sponsoed by the national ey fundamental eseach & development pogams (973) of P.R. China (007CB3006) and National Natual Science Foundation of China ( /F0309). Refeence [] K.Gilholm and D.Salmond Spatial distibution model fo tacing extended obects IEE Poc.-Rada Sona Navigation005. [5] Michael Feldman Dietich Fanen Advances on Tacing of Extended Obects and Goup Tagets using Random Matices th Intenational Confeence on Infomation Fusion. [6]Macus Baum and UweD.Hanebec Extended Obect Tacing based on Combined Set-Theoetic and Stochastic Fusion th Intenational Confeence on Infomation Fusion. [7] Koch J.W Bayesian Appoach to Extended Obect and Cluste Tacing using Random Matices IEEE TRANS on ON AESVOL.44NO.3JULY008. [8]Daniel Cla and Simon Godsill Goup Taget Tacing with the Gaussian Mixtue Pobability Hypothesis Density Filte ISSNIP 007. [9] D.Salmond and N.Godon Goup and extended obect tacing IEEE Colloquium on Taget tacing algoithms and applications999. [0] Koch J.W Bayesian appoach to extended obect and cluste tacing using andom matices IEEE Tans on AES44(3)008. [] Lundquist C Ogune U Gustafsson F Extended Taget Tacing Using Polynomials With Applications to Road-Map Estimation IEEE Tansactions on Signal Pocessing59()0. []D.Salmond and M.PaTac maintenance using measuements of taget extent IEE Poceedings of Rada Sona and Navigation.50(6)pp Dec.003. [3] B.Ristic and D.J.Salmond A study of a nonlinea filteing poblem fo tacing an extended taget ISIF 004. [4]Ba-Tuong Vo Ba-Ngu Vo and Antonio Cantoni Bayesian filteing with andom finite set obsevations IEEE on SP 00856(4) [5]R. Mahle Random sets in infomation fusion: An oveview in Random Sets: Theoy and Applications Spinge-Velag 997 pp