Smoothing Framework for Automatic Track Initiation in Clutter

Transcription

1 Soothing Fraework for Autoatic Track Initiation in Clutter Rajib Chakravorty Networked Sensor Technology (NeST) Laboratory Faculty of Engineering University of Technology, Sydney Broadway -27,Sydney, NSW Subhash Challa Networked Sensor Technology (NeST) Laboratory Faculty of Engineering University of Technology, Sydney Broadway -27,Sydney, NSW Abstract Autoatic track initiation in clutter within the soothing fraework is considered in this paper We derive a new soothing algorith based on integrated probabilistic data association (IPDA) filter and augented state soothing fraework We deonstrate the benefits of soothing based approaches in ulti-target tracking in clutter scenarios using tracker perforance easures like true/false track discriination statistics, track initiation/terination ties and RMS errors of confired true tracks I INTRODUCTION Soothing within the state estiation context is technically defined as a process the current easureents are used to iprove the estiates of the past states of the object of interest In the target tracking proble, this corresponds to estiating the past target states and associated tracker perforance paraeters using current easureents Forally, one can define the track estiation proble as follows Let x(t k ) denote the target state at tie k and y k y(),y(2),,y(k) denote the easureent sequence up to tie k y(i) denotes the easureent at tie i The target estiation proble can then be defined as the proble of coputing the conditional ean estiate of the target state ˆx(t k L k )Ex(t k L ) y k () and its associated error covariance P k L k E[(x(t k L ) ˆx ( t k L k ))(x(t k ) ˆx ( t k L k )) T y k L, L < and L > are for three types of estiation naely filtering, prediction and soothing respectively Soothing algoriths were shown to provide significant perforance iproveents in ters of RMS errors in several iportant tracking probles like aneuvering target tracking using IMM soothers by Helick et el [2-[2, tracking in clutter using PDA soothers by Mahalanabis et el [6 and using IMMPDA soothing for aneuvering target tracking in clutter [23 More recently, the augented state soothing fraework was used for dealing with out of sequence easureents by Challa et al [8-[9 (2) One of the very iportant assuptions ade in all these efforts is the fact that the target exists However, in reality, target existence ust first be established before using one of the above ethods Several techniques to achieve this are available in literature, like the heuristic M out of N detections ethod [9 and the Bayesian approaches like IMMPDA and IPDA IMMPDA, when used in the context of autoatic track foration [7, uses two odels - one that assues that the target is observable by the sensor with a detection probability of P D being, <P D and the other assues that the target is not observable with probability of detection P D The algorith uses a probability easure (the odel probability) for each of the odels and estiates the true target probability If that probability crosses a suitable threshold, a decision on target s presence (existence) is taken Another effective algorith to solve the autoatic track foration in clutter, referred to as Integrated Probabilistic Data Association (IPDA), is proposed by Darko and Evans [-[2 Many of its variants for use in difficult environents are proposed in [3-[4 IPDA odels target existence as a rando event satisfying Markovian properties between existence and non-existence states and provides a echanis to associate a probability easure (the target existence probability) to it Siilar to the ethod of IMMPDA, the target existence probability is estiated along with the target states and if the existence probability crosses a threshold, a decision on target s presence (existence) is taken IMMPDA and IPDA are Bayesian approaches and are aenable for treatent within the soothing fraework In this paper, we focus on IPDA based approach for target existence and develop a new algorith for autoatic track initiation in clutter within the augented state soothing fraework We investigate the effect of soothed track existence probability on tracker perforance easures, eg true/false track discriination, by coparing its perforance with the standard IPDA algorith The flow chart of the algorith is also presented in this paper The siulation results are also provided, the iproveents in true/false track statistics are found to be significant with a potential to iprove all higher layer functions of tracking systes like situation and /5/$2 25 IEEE

2 threat assessent The paper is organized as follows Following introduction, section II forulates the autoatic track forulation proble as conceptualized by IPDA The theory of augented state IPDA soother is described in section III The flow chart of the algorith is presented in section V The siulation scenario and results are presented in section VI Conclusions are drawn in section VII II PROBLEM FORMULATION Target tracking algorith starts with a priori knowledge of target dynaic odel Each target within the surveillance region is assued to follow a dynaic equation of x k Fx k Gw k (3) target state x k consists of kineatic states eg position, velocity etc F is the state transition atrix w k is the noise (called as process noise ) w k is assued norally distributed with ean zero and variance Q It is also assued that Ew k w j if k j A easureent odel is defined as y k Hx k v k (4) H is the state to easureent transition atrix v k is noise (called as easureent noise ) v k is assued to be norally distributed with ean zero and variance RItisalsoassuedthatEv k v j if k j Moreover Ew i v j for any i, j IPDA takes track existence as a rando event and finds the probability of the event to solve the proble of autoatic track aintenance IPDA odels the existence of track as a two state rando variable, E k, E k refers to the event that the track exists at tie t k E k refers to the event that the track does not exist at tie t k A target or track can also switch between these two states according to a predefined switching probability atrix which is [ Γ Γ Γ (5) Γ Γ Γ ij p(e k j E k i) i, jɛ, (6) In the rest of the text, E k will be denoted as E k and E k will be denoted as E k with the definition of Γ ij considered to be understood as defined by (6) IPDA solves the uncertainty in target existence autoatically by estiating p(e k L y k ) (7) again L,L < and L> are for filter, prediction and soothing type of estiation respectively The state is estiated with the condition that the target exists and thus the state estiation is redefined by the introduction of a conditional paraeter as p(x k L E k L,y k ) (8) III FIXED LAG AUGMENTED STATE IPDA (AS-IPDA) SMOOTHING In an augented approach for a lag of N, the target dynaic odel and easureent equation of (3) and (4) respectively will be replaced by the augented vectors as X k FX k Gw k (9) Y k HX k v k () X k [ T x k x k x k N () F I F I (2) I Y k [y k (3) H [ H (4) The noise variance atrix Q k will also be adjusted to Q Q (5) But IPDA concept suggests that along with the target s state, its existence event also needs to be augented Based on that conceptual fraework, two possible cobinations of target state and its existence are possible at any given instant of tie These are: Ck x k,e k, target exists and so does its state Ck 2 φ, E k, target does not exist and so does not its state Thus for an entire fixed lag of N, the augentation can be carried out in the following anner [ C a k Ck b Ck N d T (6) a,b,,dɛ, 2 This suggests that there can be ore than one augented hypotheses possible Fro the ipleentation point of view of IPDA, when a target goes out of existence, it reains that way for all future tie In that context, the transition atrix of (5) can be ade ore specific [ Γ Γ Γ (7)

3 All published results of IPDA follow this transition atrix Therefore not all the cobinations of C k s are valid in (6) Thus at any tie, eliinating the ipossible hypotheses, there reain N 2 perissible augented hypotheses These are Hypothesis : Hypothesis : H k [X k, E k H k [Xk, Ek,, 2,,N Hypothesis n: x k,e k x k,e k x k N,E k N φ, E k φ, E k x k,e k x k N,E k N (8) (9) φ, E k H n k [Xk n, Ek n (2) φ, E k N Except for hypothesis one, the other hypotheses assue that the target does not exist at the current tie Thus the soothing of a track is concerned only for the first hypothesis Thus the underlying Bayesian approach for developing an IPDA soothing algorith reduces to the calculation of probability density, p(x k, E k y k )p(x k E k,y k )p(e k y k ) (2) Furtherore the existence probabilities at each tie instant are readily given by, p(e k )p(e k y k ) (22) d p(e k d )p(e k y k ) p(e j k yk ) (23) j d, 2,,N The conditional state estiate of (2) and the existence probabilities of (22) and (23) together solve the IPDA soothing proble In the next section, the calculation of these estiates will be carried out IV DERIVATION OF AS-IPDA SMOOTHING In this section, one iteration of state estiate and existence probability estiates are derived separately for clarity A Conditional State Estiate The conditional state estiate p(x k E k,y k ) can be expanded through Bayes Theore : p(x k E k,y k ) p(x k E k,y k,y k ) p(y k X k, E k,y k )p(x k E k,y k ) p(y k E k,y k ) Likelihood P rediction Noralization (24) The three ters likelihood, prediction and noralization will be derived in step by step The a priori target state is assued known (either fro previous iteration or fro initialization of the tracks) with a Gaussian distribution having ean ˆX k k and co-variance P k k Likelihood To calculate likelihood, the assuptions ade in the literature are the nuber of expected validated easureents conditioning on the past easureent history at tie k is a Poisson distribution with density paraeter, P ( k y k ) P ( k ) k e k! if k k P DP Gp(E k y k ) if k > (25) (26) k is the nuber of validated easureents at the tie t k P D and P G denote the detection probability and gate probability respectively while p(e k y k ) can be obtained fro p(e k y k ) by using Markov Transition Probability as p(e k y k )Γ p(e k y k ) (27) the hypotheses that α : all validated easureents are false easureent or clutter α i :i-th validated easureent is target originated and all others are false easureents These are copleentary sets and hence the following conditional probabilities can be defined ) No validated easureent is target originated given the target exists P (α X k, E k,y k ) P DP G (28) 2) The i-th validated easureent is target originated given the target exists P (α i X k, E k,y k ) PDPG k (29)

4 Based on the above defined paraeters, the likelihood in (24) is calculated as p(y k X k, E k,y k ) k i p(y k X k, E k,y k,y k,α i, k ) P ( k y k )P (α i X k, E k,y k ) ( ) k P( k )P (α X k, E k,y k ) ( ) k P ( k ) p(y k (i) X k, E k,y k,α i) P (α i X k, E k,y k ) ( ) k P( k )P (α X k, E k,y k ) ( ) k k P( k) ( ) k P( k ) V k P DP G P DP G is the volue of the easureent validation gate at tie t k Under Gaussian assuption of easureent noise, the likelihood of i-th validated easureent is also Gaussian and hence the likelihood ter within the suation sign is a Gaussian PDF, Fro (24), the noralization is p(y k E k,y k )δ p(y k X k, E k,y k ) p(x k E k,y k )dx k X k ( ) k P( k ) P DP G P DP G N (y k (i); HX k, R) Vk X k In (36) N (X k ; ˆX k k, P k k )dx k (36) N (y k (i); HX k, R)N (X k ; ˆX k k, P k k ) p(y k (i) X k, E k,y k,α i)p (α i X k, E k,y k ) N (y k(i); HX k, R)N (X k ; ˆX k k, P k k ) N (y k (i); H ˆX N(y k (i); H ˆX k k, S) k k, S) N (X k ; ˆX k k (i), P k k (i)) N(y k (i); H ˆX k k, S) (37) p(y k (i) X k, E k,y k,α i) (3) S HP k k H T R K P k k H T (S) ˆX k k (i) ˆXk k K(y k (i) HˆX k k ) P k k (i) (I KH)P k k Therefore (36) becoes p(y k (i) X k, E k,y k,α i) N(y k (i); HX k, R) (3) Therefore the expression for likelihood fro (3) is ( ) p(y k X k, E k,y k k ) P( k ) Vk P DP G P DP G N (y k (i); HX k, R) Prediction (32) Given the linear process and easureent equations of (25) and (26), the prediction can be directly derived fro Kalan filter theory and is given as p(x k E k,y k ) N (X k ; ˆX k k, P k k ) (33) Noralization ˆX k k F ˆX k k (34) P k k FP k k F T Q (35) p(y k E k,y k )δ ( ) k P ( k ) X k P D P G P D P G N (y k (i); H ˆX k k, S) N (X k ; ˆX k k (i), P k k (i))dx k ( ) k P ( k ) V k P D P G P D P G N (y k (i); H ˆX k k, S) (38) Now putting all the respective expressions in (24),the conditional state estiate becoes p(x k E k,y k ) δ N (X k; ˆX k k, P k k ) ) k P( k ) P DP G P DP G N (y k (i); HX k, R) ( ) k P( k )( P DP G) N (X k ; ˆX k k, P k k ) δ δ ) k P( k )P DP G N (y k (i); HX k, R)N (X k ; ˆX k k, P k k ) (39)

5 Using (37), (39) can be reduced to p(x k E k,y k ) ( ) k P( k )( P DP G) N (X k ; ˆX k k, P k k ) ( ) k P( k )P DP G δ N (X k; ˆX k k (i), P k k (i)) δ β k ()N (X k ; ˆX k k, P k k ) i N (y k (i); H ˆX k k, S) β k (i)n (X k ; ˆX k k (i), P k k (i)) β k (i) N(X k ; ˆX k k (i), P k k (i)) (4) β k () ( ) k P( k )( P DP G) (4) δ Vk β k (i) ( ) k P( k )P DP G δ Vk N (y k(i); H ˆX k k, S) (42) and taking ˆX k k () ˆX k k (43) P k k () P k k (44) Fro (4), the estiates of state and covariance are derived as ˆX k k β k (i) ˆX k k (i) (45) i k P k k β k (i)p k k (i) i i β k (i) ˆX k k (i) ˆX k k (i) T ˆX k k ˆX T k k (46) Expressions in (45) and (46) give the state estiate and its co-variance atrix conditioned on the target existence B Existence Probability Estiate Soothing of existence probability requires two steps, first, calculation of the probabilities of the augented existence hypotheses second, fro there calculation of existence probabilities at each tie instant using (22) and (23) Here the derivation of the probabilities is shown in details Hypothesis : p(e k y k )p(e k y k,y k ) p(y k E k,y k )p(e k y k ) δ [p(e k,,e k N,E K N y k ) p(e k,,e k N, E K N y k ) δ [p(e k E k )p(e k N E k N )p(e k N y k ) p(e k E k )p(e k N E k N )p(e k N y k ) δ (Γ)N p(e k N y k ) (47) Hypothesis : p(e k yk ) p(y k E k,y k )p(e k y k ) p(e k y k,y k ) N p(y k E k,yk )p(e k yk ) p(y k E n k,yk )p(e n k yk ) (48) p(y k E k,yk )p(e k yk ) p(y k E k,y k )p(e k,,e k,e k,,e k N y k ) ) k P( k ) [ p(e k,,e k,e k,,e k N,E k N y k ) p(e k,,e k,e k,,e k N, E k N y k ) ) k P( k )(Γ ) Γ (Γ ) N p(e k N y k ) (49) Hypothesis : n p(e n k yk )p(e n k y k,y k ) p(y k E n k,yk )p(e n k yk ) p(y k E k,y k )p(e k,,e k N y k ) ) k P( k ) [ p(e k,,e k N,E k N y k ) p(e k,,e k N, E k N y k ) ) k P( k ) [ Γ N Γp(E k N y k ) Γ N ( p(e k N y k )) Both in Hypothesis and Hypothesis n, the target does not exist at the current tie t k and so by definition P (α E k,y k ) P (α i E k,y k ) Therefore the likelihood, p(y k E k,y k ) for i, 2,, k Vk (5) ) k (5) and is used in the derivation of (49) and (5) Thus (47) through (5) give the probabilities of the augented existence hypotheses Fro these expressions, track existence probability at each tie step (of the entire lag of N) can be obtained by using (22) and (23)

6 V ALGORITHM FLOW CHART In this section the proposed soothing algorith is converted into a flow chart for the direct ipleentation Start Track Initialization using first two scans of easureent and through velocity differencing ethod clutters per scan The clutters are also uniforly distributed in the surveillance region The tracks are initiated by two-point differencing ethod assuing the axiu velocity range of 5unit/sec The easureent validation gate threshold is 9, which ensures a gating probability of P G 99 Probability of detection is assued to be 9 If the existence probability of a track equals or goes above 9, ie K3 K<Finishing Tie No End p(e k y k ) > 9 Yes Yes NValid Track No at K i i<n If i th Track Duration > Fixed Lag If Track Duration Fixed Lag Prepare the Augented State Vector and Covariance Matrices Augented State IPDA soother No No No Standard IPDA Filter the track is oved fro tentative to confired while if p(e k y k ) < 5 the track is terinated Chi-square test is carried out on detected tracks If the statistical distance between two detected tracks is less than 5, the tracks are assued to be identical and are erged For the coparison between soother and filter Monte carlo siulation runs are carried out Existence Probability Calculation Decide about the validity of the Track A Terination Tie Detection ii KK, New Track initialization using unused Measureent Fig Flow Chart of AS-IPDA soothing VI SIMULATION RESULT Siulation is carried out to investigate the perforance of the proposed AS-IPDA soother with standard IPDA filter The siulation scenario consists of non-aneuvering targets oving in an one diensional surveillance region The target state is assued to be consisting of position and velocity The state transition atrix is defined as [ T F (52) the sapling interval T isassuedtobe The sensor receives the position of each target Hence the state to easureent conversion atrix is defined as H [ (53) The syste noise has a variance of and the easureent noise variance is 25 The nuber of clutter is generated according to a poisson distribution with an average of 5 In this siulation scenario the siulation is carried out for 4 scans while the single target is dropped at 3th scan The rest scans consist of only with the clutter Two different switching atrices are used for the coparison purpose - one with Γ 98 and the other with Γ 9, by definition, Γ Γ The results for these two cases are shown in Table-I and Table-II TABLE I FIRST CASE : Γ 98, ACTUAL TERMINATION TIME 3 Filter Detection Soother detection Lag Lag 2 Lag 3 Lag TABLE II SECOND CASE : Γ 9, ACTUAL TERMINATION TIME 3 Filter Detection Soother detection Lag Lag 2 Lag 3 Lag

7 B Target state estiation 5 45 The siulation is carried out assuing the switching probability atrix paraeter Γ 98 The filter and soother are copared in ters of position estiation error and velocity estiation error A lag of three is used for the soother Result is shown in figure 2 and figure 3 Confired True Track RMS positon error RMS error of AS IPDA soother RMS error of IPDA filter 5 AS IPDA soother IPDA filter Tie RMS error 8 6 Fig 4 True Tracks detected Tie Fig 2 RMS error coparison of position with Monte Carlo Run with Detection Probability 9 D False Track Detection RMS error RMS Velocity error RMS error of AS IPDA soother RMS error of IPDA filter Tie In this siulation the false tracks are defined as those tracks that are confired, but are not true tracks (the false tracks are confired tracks that fall outside the gate fored around true target s state) The siulation is carried out with a lag of three for soother Two different detection probabilities are used, ie P D 9 and P D 75 The nuber of false tracks in each run are copared and the result is shown is figure 5 and figure 6 Fig 3 RMS error coparison of velocity with Monte Carlo Run with Detection Probability 9 8 False Track detected in Monte Carlo Runs with Detection Probability 9 False track detected by AS IPDA soother False Track detected by IPDA filter 7 6 C True Track Detection For the siulation of the soothing perforance in ters of true track detection, the scenario is set as a 5 run of the siulation for a particular fixed lag A single target reappears at the beginning of each run The soother uses a fixed lag of three At each tie instant a gate is fored around the target s state and a chi-square test is perfored for each detected track at that tie instant The confired tracks that are within the validation gate are considered as true tracks Tie average of nuber of confired true track is 479 for soother and 476 for filter Thus the percentage of lost true track is 42% and 48% for soother and filter respectively Detected true tracks against tie is shown in the figure 4 No of false track detected Monte Carlo Run Fig 5 Nuber of false track with Detection Probability 9

8 No of false track detected False Track detected in Monte Carlo Runs with Detection Probability 75 Monte Carlo Run False track detected by AS IPDA soother False Track detected by IPDA filter Fig 6 Nuber of false track with Detection Probability 75 VII CONCLUSION In this paper, augented state IPDA algorith for soothing is derived The new algorith can be used in a siilar anner as standard IPDA filter with the only difference being is the augentation of the state vectors The siulation result also shows that, besides an iproved state estiation in clutter, a better estiate of track aintenance paraeters is possible due to the introduction of the autoatic track foration approach into soothing Specially the iproveent as found in paraeter like track terination tie and false track detection can be of very high iportance for applications like Situation awareness or any other strategic decision taking area ACKNOWLEDGEMENT The authors would like to thank Centre for Autonoous Syste (CAS), UTS node for its active contribution to the research REFERENCES [ RE Kalan, A new approach to linear filtering and prediction probles, Trans ASME, Journal of Engineering, vol 82, March 96, Pages:34-45 [2 RE Kalan, RS Bucy, New results in linear filtering and prediction theory,trans ASME, Journal of Engineering, vol 83, March 96, Pages 95-8 [3, Yaakov Bar-Shalo, Xiao-Rong Li, Estiation and Tracking : Principle and Software, Boston : Artech House, c993 [4 Bar-Shalo Y and Tse, E, Tracking in a Cluttered Environent with Probabilistic Data Association, Autoatica, Vol, 45-46,975 [5 Bar-Shalo Y, Tracking Methods in a Multitarget Environent,IEEE Transactions on Autoatic Control,Issue: 4, Aug 978 Pages: [6 Yaakov Bar-Shalo,Xiao-Rong Li, Multitarget-Multisensor Tracking : Principle and Techniques,Storrs, CT,995 [7 Bar-Shalo, Y, Chang, KC, Blo, HAP, Autoatic track foration in clutter with a recursive algorith, Proceedings of the 28th IEEE Conference on Decision and Control, vol 2, 3-5 Dec 989, Pages:42-48 [8 Yaakov Bar-Shalo,Edison Tse, Tracking in a cluttered environent with probabilistic data association,autoatica, Volue, Issue 5, Septeber 975, Pages [9 FR Castella, Sliding Window Detection Probabilities, IEEE Transactions of Aerospace and Electronic Systes, AES-2, Noveber 976, Pages:85-89 [ D Musicki, REvans and SStankovic, Integrated Probability Data Association, IEEE Transactions on Autoatic Control, Vol 39, No6, June 994 [ D Musicki, REvans, Tracking in clutter using probabilistic data association,international Conference on Radar 92, 2-3 Oct 992, Pages:82-85 [2 D Musicki, REvans, S Stankovic, Integrated Probabilistic Data Association (IPDA), Proceedings of the 3st IEEE Conference on Decision and Control, 992, 6-8 Dec 992, Vol 4 Pages: vol4 [3 Musicki, D, Evans, R, Joint integrated probabilistic data association: JIPDA,IEEE Transactions on Aerospace and Electronic Systes, vol 4, Issue 3, July 24, Pages:93-99 [4 Musicki, D, Evans, R, Joint Integrated Probabilistic Data Association - 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