VARIABLE RATE PARTICLE FILTERS FOR TRACKING APPLICATIONS. Simon Godsill and Jaco Vermaak

Size: px

Start display at page:

Download "VARIABLE RATE PARTICLE FILTERS FOR TRACKING APPLICATIONS. Simon Godsill and Jaco Vermaak"

Holly Hubbard
5 years ago
Views:

1 VARIABLE RATE PARTICLE FILTERS FOR TRACKING APPLICATIONS Simo Godsill ad Jaco Vermaak Sigal Processig Group Departmet of Egieerig, Uiversity of Cambridge, U.K. ABSTRACT Here we describe recet advaces i particle filterig algorithms ad models for trackig of maoeuvrig objects i clutter. The methods develop o the basic variable dimesio particle filterig algorithms itroduced i [1], i which a ew type of dyamical model is itroduced whose state variables arrive at ukow times relative to the observatio process (hece variable rate ). Targets are assumed to follow determiistic trajectories i betwee state times, determied by a appropriate model, such as the differetial equatio model for the object. The framework allows for automatic modellig ad estimatio of the trajectories of targets usig a adaptatio of particle filterig methods [2] ito the variable dimesio settig. I this paper we itroduce more effective samplig schemes for the variable rate settig that esure future states are oly geerated as ad whe required, ew dyamical models appropriate for maoeuvrig objects, ad ew observatio models uder the assumptio of a ohomogeeous Poisso process for both targets ad clutter. Simulatios show very effective trackig performace uder challegig settigs which caot be emulated i a stadard fixed rate scheme. 1. INTRODUCTION Filterig methods for trackig of objects i oise are ow very well established, see for example [3, 4]. I straightforward settigs (close to liear/gaussia models, low clutter levels) the classical Kalma filter ad its variats ca be successfully adopted. I more challegig settigs (highly o-liear, o-gaussia models, high clutter desities ad low detectio probabilities), other umerical methods are required, ad i recet years a class of methods receivig great attetio is sequetial Mote Carlo, or particle filters [2, 5, 6]. I commo with the earlier methods the particle filter relies o a state-space represetatio of the system i terms of Markov hidde states {x } ad observatios {y }: x 1 f(x 1 x ) State evolutio desity y g(y x ) Observatio desity (1) with x f(x ) beig the iitialisatio, where deotes that the variable to the left is draw idepedetly from the probability desity o the right. The optimal ( Bayesia ) filterig recursios from time idex to 1 are the give by p(x 1 y : ) = p(x y : )f(x 1 x )dx (2) p(x 1 y :1 ) = g(y 1 x 1 )p(x 1 y : ) p(y 1 y : ) (3) The Kalma filter implemets this exactly i the liear/gaussia settig. The geeral update rule is aalytically itractable for most models of practical iterest. We therefore tur to Sequetial Mote Carlo (SMC) methods [2, 5, 6], also kow as particle filters, to provide a efficiet umerical approximatio to the update rule. These methods have gaied tremedous popularity i recet years over a wide rage of trackig applicatios. They are applicable to o-liear ad o-gaussia models, ad are able to capture multimodal distributios. The basic idea behid particle filters is simple: the target distributio is represeted by a weighted set of Mote Carlo samples. These samples are propagated ad updated usig a sequetial versio of importace samplig as ew measuremets become available. Usig the samples, estimates of the target state ca be obtaied usig stadard Mote Carlo itegratio techiques. The particle filter implemets the filterig recursios approximately by propagatio of a weighted cloud of N particles {x (i), w (i), i = 1,..., N}, i w(i) = 1. A basic filter [2, 7] implemets the followig recursio at time, ad for particles i = 1,..., N: x (i) 1 q(x 1 x (i), y :1 ), w (i) 1 w(i) g(y 1 x (i) 1 )f(x(i) 1 x(i) ) q(x (i) 1 x(i), y :1 )

2 where q() is a specially desiged proposal desity, ad this is usually followed at some time steps by a resamplig operatio that selects the particles accordig to their weights ad the resets weights to 1/N. There are ow may applicatios of particle filters i the trackig area, see e.g. [8]. Most of these applicatios apply particle filterig to fairly simple (typically liear) dyamical models with more or less elaborate observatio models, icludig sometimes multiple targets ad/or radom clutter. I this paper we develop strategies based upo a variable rate state arrival process. The material takes as its basis the variable dimesio particle filters ad modellig framework itroduced i [1], i which target states are allowed to arrive at differet ad ukow rates compared with the observatio process. I this way the models are able to model parsimoiously the various turig/straight, smooth/o-smooth maoeuvres of a object. The basic particle filterig techiques required ad basic trackig models were proposed i [1], ad here we develop these further i several ways, by icorporatio of: more efficiet samplig schemes for the state process; specialised determiistic resamplig methods; ew variable rate dyamical models withi a itrisic coordiate system for the motio of maoeuvrig targets; observatio models are proposed based upo a Poisso assumptio for both targets ad clutter (i this way the data associatio problem i stadard trackig work is avoided altogether ad large umbers of clutter poits/low target detectio probabilities may readily be dealt with). We believe the ew Poisso observatio models adopted are closely related to those proposed i [9], although the derivatio ad expositio here are more straightforward. Simulatios are preseted for challegig scearios with may clutter poits ad highly maoeuvrable targets, demostratig the robustess of the variable rate filters compared with their more stadard fixed rate couterparts. 2. VARIABLE RATE MODEL The variable rate state is defied as as x k = (θ k, τ k ), where k N is the discrete state idex, τ k R deotes the state arrival time, ad θ k deotes the vector of variables ecessary to parameterise the target state (see [1]). Commoly they will iclude positio, velocity, headig, etc., variables. We will assume that the variable rate state sequece follows a Markovia process such that successive states are idepedetly draw as follows x k f(x k x k 1 ) = f θ (θ k θ k 1, τ k, τ k 1 )f τ (τ k τ k 1 ). (4) θ k θ t 1 θ t θ t1 4 θ k θ k Fig. 1. Stadard fixed rate represetatio (top) ad variable rate represetatio (bottom) The Markovia assumptio is coveiet for presetatio, but ca easily be relaxed if the models require it. See Fig. 2 for a compariso of the variable rate model with the fixed rate model. I the fixed rate model, trajectories are parameterised i terms of a state variable per time poit, eve whe a very smooth ad uvaryig maoeuvre is beig executed. The variable rate model, through a suitable choice of dyamical model, parameterises smooth straight sectios with oly a few state poits, while rapid maoeuveres require more closely spaced states. We will deote by y the vector of measuremets, with N the discrete measuremet time idex, as above. The rate of the measuremet process will typically (but ot ecessarily) be higher tha that of the state process. I our variable rate settig the observatio fuctios are specified i terms of a eighbourhood N of state idices defiig the depedece structure of the data. The data are the assumed draw idepedetly from a desity fuctio coditioal o the eighbourhood of state poits N : y g(y {x k : k N }) = g(y x N ). (5) Note that the eighbourhood is a determiistic fuctio of the time ad state sequece x 1:.... This is quite a geeral formulatio that ecompasses may

3 Bayesia models. Typically the likelihood is calculated as g(y ˆθ (x N )), where ˆθ (x N ) is a determiistic fuctio of x N. We will propose specific examples of both dyamical models ad observatio fuctios i a later sectio. 3. VARIABLE RATE STATE ESTIMATION The objective will be to estimate recursively the sequece of variable rate state poits as the measuremets become available. All the iformatio cocerig the variable rate state sequece is cotaied i its coditioal probability distributio. I keepig with the stadard filterig omeclature this coditioal distributio will be referred to as the variable rate filterig distributio, defied as y 1: ), (6) where y 1: = (y 1 y ) is the sequece of available measuremets up to time idex ad x 1:k = (x 1 x k ) deotes a sequece of k hidde state variables; k deotes the idex of the state variable havig largest time idex τ k withi all eighbourhoods N 1 through N, i.e. k = {k N 1: : τ k > τ l, l N 1:, l k} (7) where N 1: = k=1 N k. Note that k is a radom variable that depeds determiistically o the sequece of x k values. The variable rate filterig distributio has variable dimesio support sice k itself is a radom variable (to be estimated alog with the hidde state sequece). Hece the problem ca be cosidered as oe of dyamical model ucertaity as well as parameter estimatio. For a recursive iferece procedure we require a update rule of the form y 1: ) y p(x 1:k y 1: ), (8) i.e. oce a ew measuremet is received, the variable rate filterig distributio at the previous time step is icremetally updated to yield the ew variable rate filterig distributio at the curret time step. Usig Bayes theorem ad the modellig assumptios of the variable rate model, the ew variable rate filterig distributio ca be related to that at the previous time step by y 1: ) g(y x N ) p(y 1: x 1:k ) p(y 1: x 1:k ) f(x k x 1:k k )p(x 1:k y 1: ). The first term i the above expressio is the observatio likelihood, whereas the third term deotes a repeated applicatio of the variable rate state evolutio model i (4) to complete the local eighbourhood for the ew measuremet y, defied as f(x k:kl x k 1 ) = kl l=k f(x l x l 1 ). (9) It is worthwhile to ote that the secod term, the ratio of likelihoods, will be uity i may cases, sice the additio of states x k will ot alter the eighbourhood structures for data poits y 1: ad hece 1:k the ratio of likelihoods is oe. I practice this ca be esured for all of the models of iterest here by a suitable defiitio of the eighbourhood structure. We will assume i all future calculatios that eighbourhoods are costructed i this way so that future states beyod k caot alter the eighbourhood structure for y, ad the update equatio the simplifies to y 1: ) g(y x N )f(x k x 1:k k )p(x 1:k y 1: ). This sequetial update rule is similar i form to that for stadard state-space models, comprisig a likelihood ad a dyamic compoet. There is, however, a very importat differece: owig to the form of the variable rate model the umber of state poits required to represet ay sectio of the target trajectory is a ukow radom variable. This will play a importat role i the developmet of efficiet umerical techiques to implemet the update. The basic form of the particle filter required for this task is as i [1], summarised here for coveiece. Assumig that we have a set of weighted samples approximately distributed accordig to the variable rate filterig distributio at the previous time step, i.e. {x (i), w (i) 1:k }N i=1 y 1: ), the particle filter update step proceeds as below whe a ew measuremet becomes available, for each i = 1,..., N: x (i) k 1:k w (i) q(x k x(i), y 1:k k ) g(y x (i) w (i) N )f(x (i) x (i) ) k 1:k k q(x (i) x (i), y k 1:k k ) ad as before resamplig may be carried out as ad whe required, resettig the weights to 1/N. q() is ay appropriate proposal fuctio which ca sequetially,

4 geerate ew state poits coditioal o previous oes. Note that the proposal mechaism ivolves repeatedly icreasig the umber of state poits util the eighbourhood for time is complete, i.e. the observatio likelihood ca be evaluated. I the examples here, we set q() equal to the dyamical model for the variable rate states, i.e. (9). The resultig set of weighted samples will the be approximately distributed accordig to the ew variable rate filterig distributio, i.e. {x (i), w (i) 1:k } N i=1 y 1: ). I this paper we preset two modificatios to the basic algorithm above: Determiistic resamplig. I order to retai greater diversity i the particle populatio, particles are determiistically replicated at each time step (see [6] for geeral discussio of this issue). Specifically, if a particle has weight w greater greater tha 1/N, the we assig a multiplicity m = Nw to that particle, ad a weight w /m. If a particle has weight w less tha 1/N, we preserve it i the particle set with a multiplicity of 1, ad retaiig its ow weight w. This procedure preserves exactly the same represetatio of the filterig desity, but maitais some lower weight particles that would ormally be resampled out. Of course, such a procedure ca lead to (at most) a doublig of the umber of particles at each time step, ad so a further pruig stage is icluded that preserves oly the N highest weighted particles i the filter followig the update to the ext time step. State regeeratio. This addresses a iefficiecy i the basic filter, sice state times τ k, k N are ofte well i the future (i.e. sigificatly greater tha ). I the basic filter above these state poits are ever regeerated ad ca attai very low weights at some poit i the future if they are ot cosistet with ew data poits. We address this by allowig for regeeratio of these future state poits at each time. There are several correct ways to achieve this, but we have settled o oe effective procedure. I essece we attempt to regeerate ay part of the state that does ot affect the curret (or previous) likelihood fuctios g(y x N ). Typically this will ivolve regeeratig oe or more future τ k values, coditioal upo the sequece of values ˆθ 1: used i likelihood computatio. To see how this works, cosider augmetig the variable rate state at time with the value of ˆθ (x N ), i.e. the state becomes {x N, ˆθ (x N )}. Now, suppose that some parts of x N are redudat i the sese that they ca be modified without chagig the values of ˆθ 1:. We ca the resample these elemets accordig to p(x N ˆθ 1:, x N1:, y 1: ) = p(x N ˆθ 1:, x N1: ) Fig. 2. Itrisic coordiate system (a degeerate distributio i geeral) ad the particle weights remai uchaged as before, sice this ca be regarded as a very simple Markov chai trasitio kerel of the type described i [1]. Furthermore, sice high weighted particles have a multiplicity greater tha 1 (see above), some degree of stratificatio is ofte possible, sice differet particles ca be draw from differet regios of the probability distributio. 4. VARIABLE RATE TRACKING MODELS A ew dyamical model appropriate for maoeuvrig objects uder the variable rate model is proposed, based o a itrisic coordiate system. I a itrisic coordiate system applied forces ca be represeted relative to the headig of the object, rather tha relative to the more stadard Cartesia or polar fixed coordiate frame. This we postulate is a more realistic represetatio of the thrusts applied whe turig a vehicle. Distace travelled alog the path of motio is deoted s, while agle of the path relative to horizotal is deoted ψ. Acceleratios tagetial to ad perpedicular to the motio are the give i Fig. 2.I the variable rate model we ow assume that a piecewise costat thrust, relative to the directio of headig, is applied betwee ay two times τ k ad τ k1, with tagetial compoet T T ad perpedicular compoet T P. A resistace term λ ds dt, assumed to apply i the opposite directio to the headig adds some dampig to the system. Resolvig forces tagetially ad perpedicularly: T T = λ ds dt s md2 dt 2 T P = m ds dt dψ dt These equatios are readily itegrated to give equatios for s(t) ad ψ(t) durig ay time period with fixed thrusts T T ad T P, from which the cartesia positio x(t) ca be obtaied by umerical itegratio for ay time t betwee τ k ad τ k1.

5 object trackig. The model for the measuremets is the: { N(x, σ 2 y,i y), target measuremet U R, clutter measuremet where U R is the uiform distributio over the regio of surveillace R. Uder these assumptios it ca be show from the properties of Poisso poit processes that g(y x ) Π N i=1 λ yc(y,i ), λ yc (y) = λ T N(y x, σ 2 yi) λ C U R Fig. 3. Example trajectories simulated from the itrisic coordiate model We thus have the variable rate state variables as follows: θ k = [T T,k, T P,k, v(τ k ), ψ(τ k ), x(τ k )] from which the positio, speed ad agle at ay time betwee τ k ad τ k1 may be obtaied determiistically. This implies that the variable rate eighbourhood structure for this model eed cotai oly oe elemet, i.e. N t = {k; τ k t, τ k1 > t}. To complete the dyamical model we specify the distributio of thrusts i the time iterval τ k to τ k1 : T T,k N(µ T, σ 2 T ), T P,k N(, σ 2 P ) ad the distributio of time poits: τ k1 τ k G(α τ, β τ ) where N is the ormal ad G is the gamma distributio. Some example trajectories from the model are show i Fig. 3, showig the clear ability of the model to geerate elaborate maoeuvres. May possible observatio models are available for this dyamical model, ad we have explored several. I order to test a challegig o-gaussia case, we adopt here a Poisso target model with idepedet Poisso clutter observatios. The umber of measuremets from the target at each time poit are assumed draw radomly from a Poisso distributio havig mea λ T. Each such target measuremet spatially the has a Gaussia distributio cetered o the true positio. Radom clutter measuremets are also icluded, havig mea umber λ C ad uiform spatial distributio. See [9] for a very similar model for exteded where N is the total umber of measuremets at time idex. This model thus avoids ay explicit treatmet of the data associatio problem iheret i may trackig scearios, but ca potetially hadle very high clutter desities. 5. RESULTS We have experimeted successfully i various differet scearios. To demostrate this i oe represetative case, see the followig example. Here we have a surveillace regio of size, clutter parameter λ C = 2 (i.e. a average of 2 clutter poits per time poit) ad target mea λ T =.7, i.e. the target is ofte uobserved, ad with observatio parameter σ y = 4. A maoeuvrig target is geerated usig the itrisic dyamic model with m = 5, λ =.3, µ T = 5, σ T = 5 ad σ P = 1. The data ad results are summarised i Fig. 4. Note that the target appears almost buried i clutter, but the variable rate filter is able to track its maoeuvres successfully. A more detailed aalysis shows that the filter is able to maitai may feasible hypotheses whe the data is ambiguous. By cotrast the correspodig fixed rate filter is uable to follow the trajectory ad loses track. Although it has the same itrisic model for the dyamics, it is uable to track the sustaied turig maoeuvres from the variable rate model. 6. ACKNOWLEDGEMENTS The authors ackowledge the suggestio of a Poisso observatio model for exteded targets by Kevi Gilholm ad David Salmod, ad helpful discussio with Simo Maskell, all at QietiQ UK. The work of both authors was partially supported by QietiQ Cotract CU uder the Weapo, Platform ad Effectors domai of the MOD UK research programme, ad that of Godsill by the UK Govermet s Data ad Iformatio Fusio DTC.

6 True trajectory Raw data y t Particle filter output Fixed rate particle filter 7. REFERENCES [1] S. J. Godsill ad J. Vermaak, Models ad algorithms for trackig usig tras-dimesioal sequetial Mote Carlo, i Proc. IEEE Iteratioal Coferece o Acoustics, Speech ad Sigal Processig, 4. [2] A. Doucet, S. Godsill, ad C. Adrieu, O sequetial Mote Carlo samplig methods for Bayesia filterig, Statistics ad Computig, vol. 1, pp ,. [3] Y. Bar-Shalom ad W. D. Blair, Multitarget- Multisesor Trackig: Applicatios ad Advaces, vol. III, Archtech House, Norwood, MS,. [4] Y. Bar-Shalom ad T. E. Fortma, Trackig ad Data Associatio, Academic Press, [5] N.J. Gordo, D.J. Salmod, ad A.F.M. Smith, Novel approach to o-liear/o-gaussia Bayesia state estimatio, IEE Proceedigs-F, vol. 14, o. 2, pp , [6] A. Doucet, N. de Freitas, ad N. Gordo, Eds., Sequetial Mote Carlo i Practice, Spriger-Verlag, New York, 1. [7] J. Liu ad R. Che, Sequetial Mote Carlo Methods for Dyamic Systems, Joural of the America Statistical Associatio, vol. 93, o. 443, pp , [8] B. Ristic, S. Arulampalam, ad N. Gordo, Beyod the Kalma Filter: Particle Filters for Trackig Applicatios, Artech House Books, 4. [9] K. Gilholm ad D. J. Salmod, Exteded object ad group trackig, i Symposium o Target Trackig ad Sesor Data Fusio for Military Observatio Systems, 3. [1] S N MacEacher, M Clyde, ad J S Liu, Sequetial importace samplig for oparametric Bayes models: the ext geeratio, Ca. J. Stats., vol. 27, pp , Fig. 4. Trackig example - Poisso likelihood, 4 particles. Top - true trajectory, secod dow - raw data, third dow - variable rate particle filter (all particles show), bottom - fixed rate particle filter (all particles show)

Standard Bayesian Approach to Quantized Measurements and Imprecise Likelihoods

Standard Bayesian Approach to Quantized Measurements and Imprecise Likelihoods Stadard Bayesia Approach to Quatized Measuremets ad Imprecise Likelihoods Lawrece D. Stoe Metro Ic. Resto, VA USA Stoe@metsci.com Abstract I this paper we show that the stadard defiitio of likelihood fuctio