Group Object Structure and State Estimation with Evolving Networks and Monte Carlo Methods

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1 IEEE TRANSACTIONS ON SIGNAL PROCESSING, REGULAR PAPER, VOL. A, NO. SEPTEMBER, 1 1 Group Objec Srucure and Sae Esimaion wih Evolving Neworks and Mone Carlo Mehods Amadou Gning 1, Lyudmila Mihaylova 1, Simon Maskell, Sze Kim Pang 3 and Simon Godsill 3 1 Dep of Communicaion Sysems, Lancaser Universiy, UK QineiQ, Malvern Technology Cenre, Worcesershire, UK 3 Dep. of Engineering, Universiy of Cambridge, UK {e.gning,mila.mihaylova}@lancaser.ac.uk, s.maskell@signal.qineiq.com, {skp31,sjg }@eng.cam.ac.uk Absrac This paper proposes a echnique for moion esimaion of groups of arges based on evolving graph neworks. The main novely over alernaive group racking echniques sems from learning he nework srucure for he groups. Each node of he graph corresponds o a arge wihin he group. The uncerainy of he group srucure is esimaed joinly wih he group arge saes. New group srucure evolving models are proposed for auomaic graph srucure iniialisaion, incorporaion of new nodes, unexising nodes removal and he edge updae. Boh he sae and he graph srucure are updaed based on range and bearing measuremens. This evolving graph model is propagaed combined wih a sequenial Mone Carlo framework able o cope wih measuremen origin uncerainy. The effeciveness of he proposed approach is illusraed over scenarios for group moion esimaion in urban environmens. Resuls wih challenging scenarios wih merging, spliing and crossing of groups are presened wih high esimaion accuracy. The performance of he algorihm is also evaluaed and shown on real ground moving arge indicaor (GMTI) radar daa and in he presence of daa origin uncerainy. Keywords evolving graphs, random graphs, group arge racking, nonlinear esimaion, Mone Carlo mehods, Meropolis-Hasings sep I. INTRODUCTION During he las years group objec racking has been invesigaed in various differen applicaions including road raffic sysems, miliary surveillance and in paricular for ground moving arge indicaor (GMTI) racking [1] and roboics applicaions [] [5]. Groups of arges can be considered as formaions of eniies whose number varies over ime because arges can ener a scene, or disappear a random imes. The groups can spli, merge, o be relaively near o each oher or move largely independenly on each oher. However, i is ypical for group formaions o mainain some paerns of movemen [6] and hence he mehods for group racking differ from he mehods of This paper was submied Augus 8, revised Ocober 9, April 1 and Sepember 1. sandard muliple-arge racking. Alhough individual arges in he group can exhibi independen movemen a a cerain level, overall he group will move as one whole, synchronising he movemen of he individual eniies and avoiding collisions. In mos of he muliarge racking mehods, as opposed o groups racking mehods, racking of individual objecs is he common approach. However, here are srong moivaions o model and o sudy he behaviour of groups. One moivaion is he abiliy o saisically infer which racks are moving in formaion or are having common movemen. We may also wan o deec evens inside groups (spliing) and beween groups (merging). This informaion fis well wih a number of modern muliarge racking applicaions where one may wan o differeniae friendly objecs from enemies or o predic he inenion, desinaion and fuure manoeuvres of arges. Moreover, anoher srong moivaion for group racking is in he possibiliy of using common informaion abou he group o improve he esimaion of he objecs individual saes. For insance, in case of low deecion probabiliies and/or very noisy environmens, by modeling he arges ineracions inside groups, he deecion of sealhy arges can be faciliaed [7]. A furher moivaion is ha a user may be unable o assimilae informaion relaing o large numbers of individual objecs. Group objec esimaion makes i possible, for such a user, o deec evens or focus on paricular groups wih ineresing behaviour. Finally, an addiional moivaion is o consider common applicaions where objecs have muliple pars, each of which generaes deecions (e.g. when muliple radar scaers exis in a single exended objec). In many cases group objec racking is he only applicable approach, for insance when racking housands of arges ha may no be possible o be individually racked [8]. Rescuing people in earhquakes, floods and disaser evens also necessiae approaches where he whole group moion is moniored insead of he moion of each individual person.

2 IEEE TRANSACTIONS ON SIGNAL PROCESSING, REGULAR PAPER, VOL. A, NO. SEPTEMBER, 1 In [9] he benefi of group objec racking over individual objec racking is demonsraed over simulaed and real daa in erms of esimaion accuracy. The ineracions beween group members are modelled by repulsive forces. In hese classes of problems, group modeling offers a naural soluion. Differen models of groups of objecs have been proposed in he lieraure, such as paricle models for flocks of birds [1] [1], and leader-follower models [6]. However, esimaing he dynamic evoluion of he group srucure has no been widely sudied in he lieraure, alhough here are similariies wih mehods used in evolving nework models [13], [1]. Mehods for group objec racking also vary widely: from Kalman filering approaches, Join Probabiliy Daa Associaion (JPDA) [15], [16] o Probabiliy Hypohesis Densiy (PHD) filering [17] [19], and ohers [] []. The influence of he negaive informaion on group objec racking is considered in [5] and ground moving arge indicaor racking based on paricle filering in [1]. In [19] a coordinaed group racking model is presened, comprising a coninuousime moion of he group and a group srucure ransiion model. A Markov chain Mone Carlo (MCMC) paricle filer algorihm is proposed o approximae he poserior probabiliy densiy funcion (PDF) of he high dimensional sae. Mahler [6] oulines ha careful group arge moion models should be able o describe arge appearance and disappearance, no jus for he moion of individual arges and he degree o which arges joinly move in a coordinaed manner. Inspired by some ideas from [13], we consider he groups of objecs as evolving undireced random graphs. The novely of his paper is in he proposed approach for esimaing he group srucure joinly wih he group arge saes using a graphical represenaion. Wih his graphical represenaion, objecs are no assigned o groups bu are conneced o one anoher. This enables he cohesion of a group o be precisely modeled. The main conribuions for his work consis in: i) he developed graphical represenaion of he group srucure, ii) a second graphical model is developed for he groups which gives informaion abou muually ineracing groups and ha is also used in he daa associaion algorihm, iii) finally, arge sae esimaes (from he designed Mone Carlo mehods) wihin he same group or wihin ineracing groups are compared in order o updae he graph. The remaining pars of his paper are organised as follows. Secion II presens he evolving nework models. Secion IV formulaes he group objec racking problem joinly wih he proposed evolving nework model for he groups. Secion VI presens resuls wih simulaed daa and from real GMTI radar measuremens, wih measuremen origin uncerainy. Finally, conclusions are given in Secion VII. II. EVOLVING NETWORK MODELS The evoluion of complex nework srucures has been sudied in he ligh of differen problems, such as complex neworks in communicaions, biology, social sciences, economics and Inerne (see, e.g., he surveys [13], [1]). Graph heory represens naural ways of modeling hese nework srucures. Wihin his graphical family, random graphs inroduced in he early sixy by Erdos and Renyi [6] are he firs approach aemping o model hese complex evolving neworks. A random graph of size n is simply obained by saring wih a se of n verices and by adding randomly edges beween hem. In he firs model proposed in [6], every possible edge in he graph occurs wih a chosen common probabiliy. Afer several sudies and generalisaion on random graph nework heory, recen research in neworks has been focussed on more sophisicaed evolving dynamic sysems. The main difference sems from he necessiy o coninuously change he size of he graph (e.g., due o addiion of new nodes or removal of nodes). Anoher major difference is in he probabiliies associaed o he creaion of new edges. For insance, when adding a new node, insead of using a random process wih an equal probabiliy for he generaion of new edges, a preferenial creaion of edges can be compued. The preferenial sraegy of adding edges is based on he assumpion ha a node wih a higher impac in he graph nework has a higher probabiliy o be conneced o new nodes han a second node wih less impac. For insance, for a research communiy nework, an aricle wih many ciaions has more chances o be cied han a paper wih few ciaions. The flexible approach of evolving graphs fis well o he problem of group objec racking. The closes applicaion o he group modelling ask is he World- Wide Web (WWW) nework represening a large dynamic nework where nodes and links are coninuously creaed and removed [13]. However, he nework characerising he group objec evoluion is obviously more dynamic han he WWW nework where he effecs of removed links beween nodes are ofen negligible. A significan novely in he evolving group objec nework ha we develop is ha he arges have dynamical spaial consrains. The preferenial approach is consequenly irrelevan for he group racking problem and more appropriae evolving models need o be inroduced. In his paper we exend conceps of evolving nework models o group objec nework in Secion III. A graphical represenaion models he connecions beween arges. A each ime sep new nodes are added, exising nodes are removed and he se of edges is updaed. III. AN EVOLVING NETWORK MODEL FOR GROUP MOTION ESTIMATION One of he challenges in group objec racking is in he necessiy of updaing he group srucure and mod-

3 GROUP OBJECT STRUCTURE AND STATE ESTIMATION WITH EVOLVING NETWORKS AND MONTE CARLO METHODS 3 eling he ineracions beween separae componens. For his purpose adding componens o he groups, removing ohers, spliing and merging groups are of primary imporance. In our paper, G is chosen o be an evolving undireced random graph represening boh he arges wihin he groups (nodes in he graph) and some relaions beween he group members, which is refleced by he edges beween he relaed graph nodes. Symmeric crieria as disance and velociy are used o creae an edge. However, oher crieria can be applied. G 5 x 1, x G 1 x 1, x 3 x 3 G x π 1,1 π 1, x 1 x, Evoluion model 1 x x, 1 x G 3 π x 1, x = f (x 1, x, x 3, G -1 ) 1,5 G -1 G x G 1 π 1,3 x 3 x x 3 x, x 3 x 3 G 3 A. Graphical Represenaion for he Group Objec Srucure Consider N arges consiuing he se of verices {,..., v N }. Each verex v i is associaed wih he arge sae and wih he arge sae s corresponding variance. The se of edges linking he se of verices is denoed by E. The graph srucure can hen be denoed by G = ({,..., v N }, E). One edge, in E, beween wo nodes v i and v j is denoed by (v i, v j ). In order o characerise he presence or absence of a link (edge) beween wo nodes, he disance beween hese wo considered nodes is calculaed, e.g., by he Mahalanobis disance crierion. The Mahalanobis disance is compued from he esimaed posiions and from he velociies of he separae objecs. This esimaed disance is hresholded and a decision is made abou he connecions. In his represenaion a group corresponds o a conneced componen of he graph srucure. Noe ha, wo nodes are in he same conneced componen if and only if a pah beween hem exiss. In he following secions, he groups in G are denoed {g 1,..., g ng }, where he groups g i are he conneced componens of G and n G is he number of groups in G. In [19], G represens a se of group s labels for each arge. For example, wih five arges, G = [1 1 ] means ha arges 1 and are in group 1 and arges 3, and 5 are in group. Wih he graphical represenaion, one similar group srucure is: G = ({, v,, v, v 5 }, {(, v ), (, v ), (, v 5 )}) and he groups correspond o he conneced componens of he graph G. B. Moivaions for he Group Objec Srucure Graphical Represenaion The approach proposed in his paper builds up a dynamical evoluion model insead of using ransiion probabiliies in he space of possible group srucures (e.g., see Figure 1). Algorihms of adding componens o he groups, removing ohers, spliing and merging groups by aking ino accoun geomeric disances and velociy disances beween he groups and beween he arges are proposed. In [19] he approach wih ransiion probabiliies is followed. In conras wih [19], an evoluion model G π 1, x 1, x, x 3 x 1 Fig. 1. Two approaches for modeling dynamical changes on he graph srucure. A lef: ransiion probabiliies π 1,j, j = 1,..., 5 in he space of possible group srucures (buil, for his example, from 3 exising arges wih respecive saes: x 1, x and x 3 ). A righ: an evoluion model for G according o he previous graph srucure G 1 and according o he curren saes x 1, x and x 3 of all arges. Bold (blue) ellipses denoe he curren group srucure, he ohers ellipses (ligh-green) denoe new group srucures ha may be reached in one ime sep. is designed for he group srucure by incorporaing he informaion abou closeness beween he groups and abou closeness beween arges wihin a group, in a graphical way. A each ime insan, based on he decision made abou birh and deah arges, nodes are creaed or removed inside a group. For each removed node, all is links o oher nodes are deleed, and for each new node, respecive links o neighbour nodes are added. Similarly, when an objec passes from one group o anoher, he respecive links (edges) in he considered graph disappear, and one or more links will appear in he graph of he oher group which he objec joins. A srong moivaion for such graphical represenaion is illusraed in Figure. The graphical represenaion allows an easy swich in he group srucure space: removing or adding only one edge can change he group srucure. A furher moivaion is illusraed in Figure 3 which shows wo groups g 1 and g wih he same nodes {,..., v }. These wo groups are idenical if considered as a se of indexes. When propagaing hese wo groups, using he graph represenaion, g is more likely o spli han g 1. The graph represenaion, allows, hus, o propagae more informaion han a vecor of group indexes for each arge. C. Evolving Graph Models The aim is o deermine an evoluion model G = f(g 1, X ) for he group srucure, for ime > and an iniialisaion process G = f(x ) for =.

4 IEEE TRANSACTIONS ON SIGNAL PROCESSING, REGULAR PAPER, VOL. A, NO. SEPTEMBER, 1 The vecor X = (x,1,..., x,n ) comprises he sae vecors of all he arges and f denoes he desired evoluion model. The sysem { =, G = f I (X ), (1) >, G = f NS f NI f EU (G 1, X ), shows he decomposiion of he evoluion model f according o he ime and according o hree disincive seps: edge updae, node incorporaion and node removal where denoes he composiion operaion; f I is an Iniialisaion model ha will be defined in Secion III-D; f EU is he graph Edge Updaing model ha will be defined in Secion III-E; f NI is he graph Nodes v g 1 g 1,1 v g 1, v g g,3 v g 3 (a) Spliing group (b) Merging group Fig.. One srong moivaion of using a graphical represenaion. In his example, wih 3 nodes, in (a) a simple removal of one edge can model a spliing group g 1 ino groups g 1,1 and g 1,. In (b), in conras wih (a), one new edge can model merging of wo groups g and g 3 in one new group g,3. v v v v g 1 g Fig. 3. Moivaion of using a graphical represenaion. In his example, wih nodes, graphs represen groups. These wo groups are idenical if considered as a se of indexes. A lef, he graph represening g 1 conains more edges han he one, a righ, represening g : g is more likely o spli han g 1. Incorporaion model ha will be defined in Secion III-F; f NS is he graph Nodes Suppression model ha will be defined in Secion III-G. D. Graph Iniialisaion- Model f I In his Secion, we assume ha, a ime =, he number of arges and heir respecive saes are known, given by one of he deecion echniques from [16]. Le us consider N arges consiuing he se of verices {,..., v N }. Each verex v i is associaed wih he arge sae x,i a ime =, as well as he arge sae s corresponding variance marix P,i. Model 1, given below describes he proposed edge iniialisaion mehod where E is he se of edges linking he se of verices {,..., v N }. Iniially E is he empy se { }. The Mahalanobis disance d i,k beween verices v i and v k is calculaed and we evaluae wheher i exceeds a chosen decision hreshold ε. The edge beween nodes v i and v k is denoed by (i, k). Using Model 1, Model 1.f I -The Edge Creaion Process. E = { } FOR i = 1,..., N 1 FOR k = i + 1,..., N CALCULATE d i,k IF d i,k < ε, E = E {(i, k)} he iniial graph srucure G = ({,..., v N }, E ) is hen obained. E. Edge Updaing- Model f EU The evolving graph of group of arges is more dynamic han hose sudied in he lieraure [13]. Exising edges should be updaed a each ime insan since he graph srucure is relaed wih he dynamic spaial configuraion. In a sraighforward way, Model 1 can recalculae he disance beween any pair of nodes. However, he compuaional complexiy can be reduced when some informaion abou group cenres (means, covariances and he disances beween hem) is used. For each group g we define is cenre O g = 1 n g x g k v k ϵg and is corresponding average covariance marix P g = P g k where n g is defined as he number of 1 n g v k ϵg arges in g. The cenre and covariance marix of each group can be characerised differenly, e.g., based on a mixure of Gaussian componens. Using he Mahalanobis disance crierion, an appropriae hreshold ε >> ε, and based on Model 1, a second graph G = ({v 1,..., v n G }, E ) can be inroduced wih nodes v i being characerised by heir posiion O g i. A couple of conneced nodes in he se E can be inerpreed as wo groups ha can possibly have ineracions (exchange of arges). Model summarises he edge updaing process beween neighbouring groups. The graph G will also be used in he node incorporaion process.

5 GROUP OBJECT STRUCTURE AND STATE ESTIMATION WITH EVOLVING NETWORKS AND MONTE CARLO METHODS 5 Model. f EU -Edges Updaing Process. FOR i = 1,..., n G 1 APPLY Model 1 o he se of nodes in g i and updae E FOR k = i + 1,..., n G IF edge (i, k) E FOR each node in group g i, CALCULATE he disance o each node in group g k COMPARE wih ε and updae E i = n G APPLY Model 1 o he se of nodes in g i and updae E Model can be illusraed using he example from Figure. The considered graph conains 3 groups of 1 nodes. In Figure (a), by inroducing he cenre of each group, he graph G is represened: i conains 3 nodes, corresponding o he cenre of each group, and one edge beween g 1 and g. Figure (b) and (c) illusraes he updae of Model. g 3 v 7 v v 5 v 9 O g3 1 O g1 g 1 G G v 6 O g v 8 g v (a) Descripion of Model ( f EU ) v v 7 v v 7 v 5 v 5 v 9 v v 6 v 6 v 8 v 8 Fig.. Model : (a) use a second graph srucure G and, for he edge updaing process, (b) calculae disances beween nodes in he same group and (c) calculae disances beween nodes in groups ha are conneced hrough G. In each group, disances beween any couple of nodes are calculaed as shown in Figure (b). Furhermore, in Figure (c), for any couple of groups (g i, g j ) conneced in graph G (in his example, only g 1 and g are conneced). The disances beween any couple of nodes (v i, v j ), chosen respecively in groups g i and g j, are calculaed. The use of Model, in his example, avoids calculaions of disances beween nodes in g 3 and nodes in g 1 and g, respecively. F. New Node Incorporaion-Model f NI Classical approaches rely on eiher random or preferenial approaches (he mixure of he wo also exiss) in order o assign edges o he new nodes. Addiionally, in classical graph echniques, he number of new edges assigned o each new node is fixed. The approach v v (b) (c) proposed in his paper differs from he above menioned echniques. For he purposes of group racking, he Model 3. f NI - Incorporaion of new nodes. Consider group i = 1 NodeNearGroup = false DO CALCULATE d new,i IF d new,i < ε NodeNearGroup = rue FOR each node in g i, CALCULATE he disance beween v new and each node in g i COMPARE wih ε and updae E FOR k = i n G IF edge (i, k) E CALCULATE he disance beween v new and each node in g k COMPARE wih ε and updae E i = i + 1 WHILE(i = n G + 1 or NodeNearGroup = rue) disance calculaed based on he ineracion crierion should be used o creae edges wih he exising nodes and he number of edges is hen deermined by he nodes spaial configuraion. Consider a new node (verex) denoed as v new and is sae x new. Depending on he sae x new and in comparison wih he exising n G nodes, new edges have o be creaed. A simple way is o evaluae he crierion for he ineracion beween every pair (v new, v i ). In order o opimise he compuaional ime, he graph G defined in Secion III-E can be used. Model 3 shows he edge updaing process when incorporaing a new node, where d new,i is he Mahalanobis disance beween v new and O gi (d new,i = Mahalanobis-disance ((x new, P new ), ( 1 n gi k, 1 n gi v k ϵg i x gi v k ϵg i P gi k )); he fixed hreshold ε > ε inroduced in order o see wheher he new node v new is ineracing wih a node in a group g. Le us illusrae Model 3 using he example from Figure 5. The considered graph conains groups of 1 nodes. In Figure 5 (a), by inroducing he cenre of each group, he graph G is represened: i conains nodes, corresponding o he cenre of each group, and wo edges beween, respecively, g 1 and g and g 3 and g. Disances d new,i beween he new node v new and cenres of groups O i are compued. The principle of Model 3 is o calculae disances d new,i unil finding one neighbour group of node v new according o a hreshold ε or unil reaching he las index i (i = n G ). Noe ha ε is chosen such ha ε << ε so ha a new node close o one group g according he hreshold ε is far from any group ha is no conneced wih g according o he hreshold ε. For he example presened on Figure 5, g 1 and g are no neighbours of v new according o he disance crierion. In conras, g 3 saisfies he disance crierion. Then, he calculaed disances of Model 3, used o updae graph G, are illusraed. Disances beween

6 6 IEEE TRANSACTIONS ON SIGNAL PROCESSING, REGULAR PAPER, VOL. A, NO. SEPTEMBER, 1 v (a) g 3 v 7 1 O g1 g 1 v new v 9 O g3 v 6 O g v 8 g 3 g G G v Descripion of Model 3 ( f NI ) v 1 Og 3 v 7 g 3 (b) v 9 g 1 v new v 6 v 8 g g v likelihood funcion p(z X ) can be calculaed, he purpose is o compue sequenially he sae PDF for each group of objecs. The changes of he groups such as merging and spliing are aken ino accoun during he graph updae process. Addiionally, he groups movemens are assumed independen. Under he Markovian assumpion for he sae ransiion, he Bayesian predicion and filering seps can be wrien as follows: p(x, G Z 1: 1 ) = p(g X, Z 1: 1 ) p(x Z 1: 1 ) = p(g X,G 1 ) p(x X 1,G 1 )p(x 1,G 1 Z 1: 1 )dx 1 dg 1, () v 5 Fig. 5. Model 3: (a) use graph srucure G by calculaing he disance from he new node v new o he cenres of all groups. Once one group g i saisfies he disance hreshold, (b) calculae he disances beween v new and any node in g i in addiion o he disance beween v new and any node in a group conneced o g i hrough G. v new and any node in g 3 are calculaed as shown in Figure 5 (b). Furhermore, since g is conneced o group g 3, in graph G, disances beween v new and any node in g are also calculaed. The use of Model 3, in his example, avoids he calculaion of disances beween v new and nodes in g 1 and g, respecively. G. Old Node Suppression-Model f NS This is he simples graphical evoluion modeling par and consiss of removing deah arges by removing corresponding nodes and heir relaed edges. A arge in he graph will be removed if he measuremens do no conain any informaion abou i afer a cerain period of ime. IV. PROBLEM FORMULATION Consider he problem of racking he moion of groups of arges. Each arge i is characerised by is sae vecor x,i = (x,i, ẋ,i, y,i, ẏ,i ) (comprising he posiions x,i, y,i and velociies ẋ,i, ẏ,i in x and y direcions respecively); denoes he ranspose operaion. Targes which are close o one anoher end o form a group. The Mahalanobis disance d i,k is chosen as a crierion of closeness beween he arges wihin a group. A each ime insan, he se of objecs racked in a group g can be modeled by a Random Finie Se (RFS, see [6]) ha incorporaes he sae vecors of he group members, X g = { } x g,1, x g,,..., x g,n g (ng is he random size of group g). Knowing he group srucure G = {g 1,..., g ng } (n G is he number of groups), he join sae for he all he arges in he n G groups has he expression X = {X g 1,..., X g n G }. A ime a measuremen vecor z is received which can be described as a funcion of he sae X = {X g 1,..., X g n G }. Assuming ha he measuremen v 5 p(x, G Z 1: ) = p(z X, G ) p(x, G Z 1: 1 ) p(z Z 1: 1 ) where Z 1: is he se of measuremens up o ime and z is he curren vecor of measuremens. The ransiion PDF p(g X,G 1 ) of he group srucure can be calculaed knowing he predicion of he arge sae and group srucure in he previous ime insan, and using he graph evoluion model inroduced in Secion III-C. The ransiion PDF p(x X 1,G 1 ) of he sae of all arges is calculaed knowing he previous ime arge saes and group srucure PDF p(x 1,G 1 Z 1: 1 ). Wih he assumpion of independence beween group moions, he PDF p(x X 1,G 1 ) can be decomposed in he following supplemenary equaion p(x X 1,G 1 ) = p(x gi X g i 1 ), () g i G 1, (3) where p(x g i X g i 1 ) is he ransiion densiy of he se of arges from he group g i. In order o perform he correcion sep, he likelihood funcion p(z X, G ) of he whole sae has o be evaluaed by means of a daa associaion approach. In his paper, he JPDA algorihm [16] is used o resolve he measuremen origin uncerainy. In addiion, in he gaing process in he JPDA algorihm is enhanced by using oher informaion abou he graph srucure, such as he disance beween groups. Figure 5 shows an example where groups g 3 and g can be considered separaely from groups g 1 and g. Noe ha he graph G esimaed a each ime insan is applied in he edge updaing process and in he nodes incorporaion seps which leads o reducion of he daa associaion compuaions. A each ime sep, he graph G can also be used in he gaing process. Indeed groups of he same graph G s conneced componen can be gahered in separae daa associaion process: he graph G offers a sraighforward mehod of clusering he arges for he daa associaion process. Denoe by {g 1,..., g n }, he se of n G G conneced componens in graph G. Any conneced componen g i can model a se of groups ha are close enough

7 GROUP OBJECT STRUCTURE AND STATE ESTIMATION WITH EVOLVING NETWORKS AND MONTE CARLO METHODS 7 o be reaed in he same daa associaion algorihm. Under he independence assumpion beween he g i, he following equaion (5) can be wrien p(z X, G ) = p(z X, G, G ) = i=1,...,n G p(zg i X g i 1 ), (5) where X g i 1 is he se of arges saes belonging o he groups in g i. The vecor zg i comprises he subse of measuremens relaed wih he group in g i. For example, z g i can be chosen by gaing measuremens using he se of arges sae X g i A. Model of Individual Targes 1. The nearly consan velociy model [7], [8] is used for he updae of each node of he graph, i.e., for modelling he moion of each arge wihin a group. In wo dimensions, he sae of he ih arge is given by: x,i = Ax 1,i + Γη 1, (6) ( ) 1 T where A = diag(a 1, A 1 ), A 1 =, Γ = 1 ( ) T/ 1, T is he sampling inerval T/ 1 and η 1 is he sysem dynamics noise. In order o cover a wide range of moions, he velociy should be approximaely consan over sraigh line rajecories and he velociy change should be abrup a each urn (especially for he direcion of he velociy). Then, he sysem dynamics noise η 1 is represened as a sum of wo Gaussian componens p(η 1 ) = αn (, Q 1 ) + (1 α)n (, Q ), (7) Q 1 =diag(σ, σ1), Q = diag(σ, σ); σ is a sandard deviaion assumed common and consan for x and y; σ 1 σ are sandard deviaions allowing o model respecively smooh and abrup changes in he velociy. The fixed coefficien α has values in he inerval [, 1]. In addiion, o model he ineracion beween objecs in each group, he average velociy of group objecs is used in (6) insead of he velociy of each group componen. For each group g, in he group srucure G and for each x g,i X g = { x g,1, x g,,..., x g,n g } we have he following equaion x g n,i = xg 1,i + g (Bx g 1,j ) + Γη 1, (8) j=1 ( where and B = diag(b 1, B 1 ) wih B 1 = T ) n g. More sophisicaed models can be considered o model arges ineracions in each group such as he developed in [19]. B. Observaion Model Range and bearing observaions from a nework of low cos sensors posiioned along he road are considered as measuremens. The measuremen vecor z,i for he ih arge conains he range r,i o each arge and he bearing β,i. The measuremen equaion is of he form: z,i = h(x,i ) + w,i, (9) where h is he nonlinear funcion ( h(x,i ) = x,i + y,i, y ) an 1,i x,i (1) and he measuremen noise w,i is supposed o be Gaussian, wih a known covariance marix R. C. Paricle Filering Algorihms for Group Moion Esimaion In his paper, wo approaches are proposed. The impac of incorporaing or no he group srucure in he sae is sudied, also from he poin of view of is compuaional complexiy. One way of considering he group srucure is o propagae, a each ime sep, a deerminisic group srucure using he previous group srucure G 1 and he curren esimae of all he arge saes denoed by X, i.e., G = f(g 1, X ). Alhough he complexiy of such an approach is reduced, i does no provide informaion abou he group srucure uncerainy. This group srucure evoluion model has been inroduced in [9]. In conras, by considering an augmened sae (insead of X, he sae is now (X, G )), he group srucure uncerainy can be accouned for in a beer way. This group srucure evoluion model wih an augmened sae has been sudied in [3]. In he nex wo subsecions, paricle filering algorihms are presened combined wih hese wo approaches. 1) Deerminisic Updae of he Graph Srucure: We denoe by N p he number of paricles and L is he curren index of a paricle. Having in mind equaions ()-(5), he implemened algorihm is described as Algorihm 1, where he proposal PDF is of he form: X g i,(l) : 1, z : 1) = p(x g i ) (where X g i,(l) 1 ) is he ransiion PDF, for he arge s sae in he group g i, under he assumpion ha he ineracion beween arges is wih respec o he group srucure G 1 ). In order o sample from his ransiion PDF, a nearly consan velociy model (6) q gi (X gi p(x g i is used for each componen X g i,(l) 1 X (L) 1 o obain Xg i,(l). X g i,(l) 1 of he paricle JPDA Combined wih he Esimaed Group Srucure In sep of Algorihm 1, he daa associaion problem is resolved by he JPDA algorihm [16]. The graph srucure is used in he firs sep of he JPDA algorihm. Informaion conained in he graph srucure is used o

8 8 IEEE TRANSACTIONS ON SIGNAL PROCESSING, REGULAR PAPER, VOL. A, NO. SEPTEMBER, 1 Algorihm 1. The paricle filer wih deerminisic updae of he group srucure 1. Predicion sep (Eq. ()) FOR all g i G 1 FOR L = 1... N p, DRAW samples from he proposal PDF X g i,(l) q gi (X g i Xg i,(l) : 1, z : 1) CALCULATE he average prediced sae vecor X 1 ESTIMATE G using G = f(g 1, X 1 ). Updaing sep (Eq. (3)) FOR L = 1... N p CALCULATE he likelihood funcion according o: equaion (5) and using a JPDA algorihm [16] UPDATE and NORMALISE he weighs CALCULATE he esimae X of he curren sae vecor X UPDATE G using G = f(g 1, X ) 3. Resampling Perform he resampling sep if ˆN eff < N hr cluser he daa associaion problem ino disinc subproblems (Eq. (5)). This clusering sage helps reducing he compuaion ime during he gaing process (his gaing process, in urn, is imporan for reducing he number of daa associaions hypoheses). The weigh updae is hen performed by muliplying he likelihood by he previous ime weighs (Eq. (3)) Finally, in sep 3 for each arge we esimae he corresponding efficien componens in he paricles X (L) and resample if he number of efficien paricles ˆN eff is less han a hreshold N hr : ˆNeff < N hr [31]. ) Augmened Sae for a Graph Srucure Uncerainy Esimaion: In his Secion we presen a paricle filering echnique wih a Meropolis-Hasings (MH) sep for group objec moion esimaion. Due o he augmened sae wih he graph srucure, each paricle conains he arges sae and he group srucure. In general, he MH seps are known o allow using less number of paricles han he classical paricle filer. We are, hen inroducing hese MH seps in order o reduce he size of he paricle cloud. Having in mind ()-(5), he implemened evolving group model is described as Algorihm, where he samples X g i,(l) are drawn from he proposal PDF (X gi,(l) X gi,(l) : 1, z : 1) = p(x g i,(l) X g i,(l) 1 ). The samples G (L) for he graph srucure are drawn from he PDF Q(G X :, G 1 ) = p(g X, G 1 ). To sample from he proposal PDF q gi, a nearly consan velociy model (6)-(7) is used for each componen q (L) g i X g i,(l) 1 of a paricle X (L) 1 o obain Xg i,(l). The ineracions wihin each group are modeled based on he mean velociy of group componens (from he consan velociy model insead of he velociy of each group componen). To sample from he proposal PDF Q, he group sruc- Algorihm. Paricle filering wih a sae augmened by he group srucure 1. Predicion sep FOR L = 1... N p FOR all g (L) i G (L) 1 DRAW a sample X g i,(l) PDF q (L) g i : X g i,(l) q (L) g i from he proposal (X g i,(l) X g i,(l) : 1, z : 1) DRAW a sample G (L) from a proposal PDF Q G (L) Q(G X (L) :, G(L) 1 ). Updaing sep FOR L = 1... N p CALCULATE he likelihood funcion according o: equaion (5) and using a JPDA algorihm [16] Run he Meropolis-Hasings algorihm wih m seps (see Algorihm 3) UPDATE and NORMALISE he weighs CALCULATE he esimae X of he curren sae vecor X 3. Resampling Perform he resampling sep if ˆN eff < N hr ure evoluion model G = f(g 1, X ) inroduced in Secion III-C, is used. In sep of Algorihm, he likelihood is calculaed by assuming independence beween clusers of measuremens corresponding o each group. The MH sep is described in Algorihm 3 and is ieraed m ime seps (m being chosen beforehand). The MH algorihm is inroduced o sample from he join PDF p(x, G Z 1: ). In sep he likelihood and he weigh updae is performed, similarly o Algorihm 1, using he JPDA algorihm. Finally, in sep 3, for each arge we esimae he corresponding efficien componens in he paricles X (L) and resample if he effecive number of samples ˆN eff is less han a hreshold N hr [31]. Algorihm 3. Meropolis-Hasings sep wih he group srucure FOR L = 1... N p FOR all g (L) i G (L) 1 DRAW a new sample X g i (prop) PDF q g (L) i, X g i,(l) X g i (prop) using he proposal : 1 and z : 1(see Algorihm ): q g (L) i : 1, z : 1) DRAW a new sample G prop using a he proposal PDF Q, X (prop) and G (L) 1 (see Algorihm ): G prop Q(G X (prop) (X g i,(l), G (L) 1 ) X g i,(l) CALCULATE he likelihood for X (prop) CALCULATE he accepance raio ρ = min(1, p(z X (prop) ) UPDATE (X (L) p(z X (L) ), G (L) ) ) and is likelihood Noe ha, he paricle filer presened in subsecion IV-C1 can be implemened wih a MH sep, and respecively, he paricle filer proposed in subsecion IV-C can be implemened wihou he MH move sep.

9 GROUP OBJECT STRUCTURE AND STATE ESTIMATION WITH EVOLVING NETWORKS AND MONTE CARLO METHODS 9 Hence, we have four differen ypes of algorihms. For conciseness, only wo filers are presened in his paper. The nex secion conains simulaion resuls and a comparison is made beween he wo presened paricle filers. 3 5 Posiions (group 3) V. SIMULATION RESULTS A. Scenario and Models Posiions (group 1) 3 y coordinae, [m] y coordinae, [m] x coordinae, [m] Fig. 8. Acual rajecory of group x coordinae, [m] 3 5 Posiions (group ) Fig. 6. Acual rajecory of group 1 The proposed echniques have been esed over a scenario in urban environmen for ground moving objec racking. The movemen of four groups (see Figures 6-9), each of hem comprising wo ground arges, is simulaed over a period of 8s. All simulaions and calculaions have been done using a GHz Processor and Malab sofware. The wo filers provide oupus in every ime second. The scenario is he following: a he beginning, groups 1 and form he same eniy and spli laer in wo groups during heir moion. In conras, groups 3 and are wo differen eniies a he beginning bu merge ino one group during he moion. In addiion, group 1, during he ime evoluion, passes near groups 3 and. Figure 1 shows his evoluion of he group y coordinae, [m] x coordinae, [m] Fig. 9. Acual rajecory of group 3.9 Group srucure size Posiions (group ) number of groups y coordinae, [m] x coordinae, [m] Fig. 7. Acual rajecory of group ime, [s] Fig. 1. Evoluion of he group srucure in ime. From s o s, hree groups are evolving: (1 + ), 3 and. Then from ime insan s o s, four groups are evolving: 1,, 3 and. Finally, from ime insan s o he end, hree groups are evolving: 1, and (3 + ).

10 1 IEEE TRANSACTIONS ON SIGNAL PROCESSING, REGULAR PAPER, VOL. A, NO. SEPTEMBER, 1 srucure wih wo changes due o, respecively, spliing and merging of groups. The scenario is challenging since he filer should be able o deal wih spliing and merging of groups and also should be able o avoid ineracions wih crossing groups coming from he opposie direcion. B. Resuls 1) Deerminisic Group Srucure Updae: The paricle filer (PF) wih sequenial imporance re-sampling (SIR) seps, described as Algorihm 1 in secion IV-C1, has been applied o he previous scenario. Figures show he performance of he filer for all he 8 arges, 3 paricles and N hr = 3. The coefficien α for he Gaussian sum in he consan velociy model has been chosen o be equal o.7 and he sampling inerval is T = 1s. The Mahalanobis disance hreshold for deermining wheher wo arges are in he same group or no, has been chosen equal o respecively 55m for he posiion and 15m/s for he velociy. These hreshold values are very sensiive o he elemens of he esimaed covariance marix for each arge. A suiable choice of hese parameers is necessary o avoid gaherings of wo arges, wih big difference in heir speed or posiion, in he same group. The Mahalanobis disance hreshold for he group cenre has been chosen imes bigger han he previous hresholds. 3 Esimaed rajecories error, [m] error, [m] ime, [s] error y 6 Fig. 1. error, [m/s] error, [m/s] 1 error x ime, [s] 1 Posiion esimaion error for eigh arges ime, [s] error vy 1 1 Fig. 13. error vx ime, [s] Velociy esimaion error for eigh arges. y coordinae, [m] x coordinae, [m] Fig. 11. Esimaed rajecories for he 8 arges from a single run. The circles represen he sensor places. groups srucures 1 g 1 = {1,, 7, 8}; g = {3, }; g 3 = {5, 6} g 1 = {1, }; g = {3,, 5, 6};g 3 = {7, 8} 3 g 1 = {1, }; g = {3, };g 3 = {5, 6}; g = {7, 8} TABLE I LISTING OF THREE GROUP STRUCTURES CORRESPONDING TO THE ACTUAL SIMULATED GROUP STRUCTURE EVOLUTION. Figures 1 and 13 show he posiion mean errors from 5 Mone Carlo runs. The developed approach provides accurae esimaes of he posiions of he separae vehicles neverheless a very simple firs degree evoluion model (consan velociy model) is used. Consequenly, abrup changes of velociy during he ime evoluion correspond o he spikes appearing on he Figures 16 and 17. Figure 1 shows a comparison beween he group srucure evoluion esimaed using he PF and he group srucure evoluion esimaed using he simulaed rajecory. One can conclude ha he group srucure is well capured by he inroduced graph evoluion model. In addiion, i is eviden ha he changes of he group srucure are no deeced a he same ime insan due o he errors. The group srucure esimaed using he PF is also changing and incorporaes five supplemenary groups srucures ( o 8) differen o he hree ones presened in Table I. These supplemenary group srucures occur essenially when, due o esimaion errors and during a shor ime, one group is abnormally spli. Furhermore, he group crossing simulaed in his scenario did no change he esimaed group srucure. ) Augmened Sae wih Group Srucure: The SIR PF wih he MH sep has been applied o his scenario. Figures show he performance of he filer for all he 8 arges. In his experimen 1 paricles have

11 GROUP OBJECT STRUCTURE AND STATE ESTIMATION WITH EVOLVING NETWORKS AND MONTE CARLO METHODS 11 Group srucure evoluion in ime ime (s) ime (s) Fig. 1. Comparison beween he group srucure evoluion esimaed using he simulaed rajecory (a he op) and using he PF (a he boom) error, [m] error, [m] error x ime, [s] error y ime, [s] Fig. 16. Posiion esimaion error for eigh arges. been used wih m = 1 ieraion of he MH algorihm. The coefficien α for he Gaussian sum in he consan velociy model has been chosen o be equal o.7. 3 Esimaed rajecories error, [m/s] 1 1 error vx y coordinae, [m] error, [m/s] ime, [s] error vy x coordinae, [m] Fig. 15. Esimaed rajecories for he 8 arges. he circles represen he sensor places. Figures 16 and 17 show he posiion and velociy mean errors for 5 Mone Carlo runs. The developed approach provides accurae esimaes of he posiions of he separae vehicles. Addiionally, Figure 18 presens he group srucures esimaed by he PF and he acual group srucures of he simulaed rajecories. The nine more relevan groups are labeled from 1 o 9 and a probabiliy is calculaed for each group a each ime. The group srucures esimaed by he PF give weighs o five supplemenary group srucures ( o 9). These supplemenary group srucures occur essenially when, due o esimaion errors and during a shor ime, one group is abnormally spli (especially when one edge is removed). Groups o 6 differ from group 3, slighly, wih one less edge and groups 7 o 9 differ Fig ime, [s] Velociy esimaion error for eigh arges. from group, slighly, wih one less edge. One can conclude ha he group srucure is well capured by he inroduced graph evoluion model. 3) Comparison Beween he Two Approaches: As expeced, he PF wih a sae augmened by he group srucure and incorporaing an MH move sep has shown slighly beer accuracy ha he PF wih deerminisic graph updae. However, he compuaional complexiy is increased subsanially (approximaely 3 imes more) compared wih he PF wihou he MH sep and wih deerminisic updae of he graph srucure. The compuaional ime for on ime sep (1s) are in average, respecively, ms for he deerminisic updae and 5ms for he second approach. Boh approaches saisfy he real ime consrain for hese simulaions wih Malab. The approach wih he augmened sae can model well he group srucure uncerainy and hence, gives more robus performance.

12 1 IEEE TRANSACTIONS ON SIGNAL PROCESSING, REGULAR PAPER, VOL. A, NO. SEPTEMBER, ime (s) ime (s) Fig. 18. Group srucure evoluion in he simulaed rajecory (boom) and esimaed by he PF (op). The nine more relevan groups are labeled from 1 o 9 and a probabiliy is calculaed, for each group and a each ime sep, for he PF. Neverheless, ha he applicaion we consider is for groups of ground arges, he proposed echniques are quie general and can be applied o oher sysems, such as aircrafs or robos. In he nex secion, due o is robus performances and he possibiliy o provide group srucure uncerainy, only he approach described on his secion is applied o a real GMTI daa se. VI. RESULTS ON REAL DATA 9% 75% 6% 5% 3% 15% This secion presens resuls for he approach proposed in secion V-B. The validaion is performed over real GMTI radar daa shown in Figure VI provided o us by QineiQ, UK. Two groups of arges are moving on he ground by crossing heir pahs which consiues an addiional ambiguiy for he group racking algorihm. The GMTI measuremens are obained by an embedded radar on a moving airborn plaform. There is a measuremen origin uncerainy which requires he soluion of he daa associaion problem. As seen from Figure VI, here is cluer noise in he measured bearing angles and measured disances o he arges. The developed approach provides good esimaion accuracy of each vehicle rajecory posiions (see Figure ). Figure 1 shows addiionally ha he esimaed x coordinaes of he groups are close o x coordinaes calculaed from he measuremens. The proposed algorihm is able o cope wih he crossed rajecories of he groups. Trajecories X coordinae, [m] Fig.. Esimaed rajecories for he groups. The arrows show he direcions of he movemen measuremens arge 1 arge arge 3 arge 6 8 Time, [s] Time, [s] Fig. 19. Measured bearing and measured range, resp. for wo groups Time, [s] Fig. 1. This Figure shows he esimaed x coordinaes for he groups joinly wih he x coordinaes are calculaed from he measuremens (convered from range and bearing).

13 GROUP OBJECT STRUCTURE AND STATE ESTIMATION WITH EVOLVING NETWORKS AND MONTE CARLO METHODS 13 Figure presens he group srucures esimaed by he paricle filer. In he real scenario vehicles 1 and are forming group 1 and vehicles 3 and are forming he second group. To plo he Figure, only four relevan group srucures appearing during he esimaion process are labeled from 1 o (respecively G 1 : {g 1 = (1, ), g = (3, )}, (11) G : {g 1 = (1), g = (), g 3 = (3, )}, (1) G 3 : {g 1 = (1, ), g = (3), g 3 = ()}, (13) G : {g 1 = (1), g = (), g 3 = (3), g = ()} (1) and a probabiliy is calculaed for each group a each ime. From Figure one can conclude ha he group srucure is well esimaed by he inroduced graph evoluion model. In addiion, we can deduce precious informaion abou he group srucures uncerainy during he ime evoluion Time (s) 9% 75% 6% 5% 3% 15% Fig.. Group srucure evoluion esimaed by he PF. The more relevan group srucures are labeled from 1 o (see (11)-(1)) and a probabiliy is calculaed, for each group and a each ime sep. VII. CONCLUSIONS This paper presens Mone Carlo echniques for group objec srucure and sae esimaion. Evoluionary graph nework-ype models for he group srucure are proposed. The graph srucure can be deerminisically esimaed or in a probabilisic way wih a graph joinly updaed wih he samples of he paricle filer. The core idea is o mainain he srucure of a graph in which conneced componens correspond o groups of arges. The effeciveness of he proposed echniques is invesigaed and validaed over a challenging urban environmen scenario wih spliing, merging and crossing of groups. The performance of he approach is also validaed over real ground moving arge indicaor daa ses. The proposed approaches successfully esimae he arges saes and he group srucure graph wih reliable performance and accurae racking. Acknowledgemens. We are hankful he suppor from he EPSRC projec EP/E753/1, he Tracking Cluser projec DIFDTC/CSIPC1 wih he UK MOD Daa and Informaion Fusion Defence Technology Cenre. We acknowledge he suppor from he [European Communiy s] Sevenh Framework Programme [FP7/7-13] under gran agreemen No 3871 (Mone Carlo based Innovaive Managemen and Processing for an Unrivalled Leap in Sensor Exploiaion) and he EU COST acion TU7. We appreciae also he help from QineiQ, UK, including for providing us wih he GMTI radar daa, using RAISIN, QineiQ s GMTI Processor and he QineiQ PodSAR radar. We are hankful he Associae Edior and anonymous reviewers for heir consrucive suggesions o improve his paper. REFERENCES [1] M. Ulmke and W. Koch, Road-map assised ground moving arge racking, IEEE Trans. AES, vol., no., pp , 6. [] A. Jadbabaie, J. Lin, and A. Morse, Coordinaion of groups of mobile auonomous agens using neares neighbor rules, IEEE Transacions on Auomaic Conrol, vol. 8, no. 6, pp , 3. [3] J. Marshall, M. 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