A NEW FRMULATIN F IPDAF FR TRACKING IN CLUTTER Jean Dezert NERA, 29 Av. Division Leclerc 92320 Châtillon, France fax:+33146734167 dezert@onera.fr Ning Li, X. Rong Li University of New rleans New rleans, LA 70148 fax : (504) 280-3950 nylee@uno.edu, xli@uno.edu Keywords : Tracing, IPDAF, perceivability. Abstract An improved version of Integrated Probabilistic Data Association Filter (IPDAF) based on a new concept of probability of target perceivability was introduced recently. This paper presents theoretical results of a new formulation for IPDAF. Validity and comparison of this new IPDAF its previous version is discussed through simulations results presented at the end of the paper. This provides a new basis of an integrated approach to trac initiation, confirmation, termination and maintenance. 1 Introduction The purpose of tracing is to estimate the state of a target based on a set of measurements provided by a sensor. For tracing in a clutter-free environment, the target is always assumed perceivable and the measurement is assumed to be available, unique and to arise from the target at every scan. In such a case, tracing follows conventional recursive filtering. For tracing in clutter, conventional filtering techniques cannot be used because several measurements are available at every scan and at most one measurement can arise from target when the target is perceivable by the sensor. Moreover tracing target in clutter involves trac initiation, confirmation, maintenance and termination. Trac initiation, confirmation and termination are basically decision problems whereas trac maintenance is an estimation problem compounded measurement uncertainty. We focuse our presentation here mainly on the trac maintenance problem. Trac confirmation and termination criterion will be shortly discussed at the end of this paper before simulation results analysis. Many probabilistic descriptions of decisions on trac initiation, confirmation and termination have been developed in the past [10]. A fundamental limitation of the Partly supported by NR via grant N00014-97-1-0570, NSF via grant ECS-9734285, and LEQSF via grant (1996 99)-RD-A-32. Probabilistic Data Association Filter (PDAF) [1, 3] isthe implicit strong assumption that the target is always perceivable. f course in many real situations, this is not the case. A new algorithm, called now Integrated Probabilistic Data Association Filter (IPDAF), was first developed by Colegrove and al. [4] and then by Musici and al. [13, 14] in order to remove the implicit strong assumption on target perceivability done by Bar-Shalom and Tse. Up to now the most appealing algorithm is the latest version of the IPDAF developed recently in [8, 11] which includes a more rigourous concept of target perceivability into its formalism than before. The development of a new IPDAF version presented here follows the approach of Colegrove rather than the Musici s. Simulations results show that this new IPDAF formulation performs a little bit better than the previous one [11] and has also a theoretical justification. 2 Target perceivability At, the target state of perceivability and its complement is represented by the exhaustive and exclusive events {target is perceivable} Ō {target is unperceivable} will denote both the fact that target is perceivable and the random event. When there are validated measurements at, the intersection of these events the classical data association (DA) events involved in PDAF formalism [1] θ i () {y i () comes from target} θ 0 () {none of y i () comes from target} defines a new set of existence events : E i () Ō θ i () i =1,...,m E 0() Ō θ 0 () E 0 () θ 0 () E i () θ i () i=1,...,m
Since target measurement cannot arise out target perceivability, events E i (),i = 1,...,m are actually impossible. Therefore, we have E i () and P {E i () Y } = P {E i () Y 1 } = P {E i ()} = 0 for i =1,...,m. nly events E 0(), E 0 () ande i () (i=1,...,m ) may have a non null probability to occur. 3 Target state estimation 3.1 Case 1 : m 0 Using the total probability theorem, one has ˆx( ) =β 0()ˆx 0( )+ β i ()ˆx i ( ) where ˆx i ( ) E[x() E i (), Y ] is the updated state estimate conditioned on the event E i () that the target is perceivable and the ith validated measurement is correct. ˆx 0( ) is state estimate conditioned on the event E 0(), that the target is unperceivable and none of the measurements are target-originated. β i () P {E i () Y } are called the integrated a posteriori data association probabilities because they tae into account (integrate) both data association and target perceivability events. Conditional state estimates are given by the classical PDAF formalism i=0 ˆx i ( ) =ˆx( 1) + K()[y i () ŷ( 1)] The gain K() is the same as in the standard Kalman filter [2] since conditioned on E i () there is no measurement origin uncertainty. For i = 0 andi=0,ifnoneofthe measurements is correct (whatever the perceivability of target is), the estimates are : ˆx 0( ) =ˆx 0 ( )=ˆx( 1) Combining these conditional estimates yields the state update equation ˆx( ) =ˆx( 1) + K() β i [y i () ŷ( 1)] i=1 The covariance P( ) of estimation error is given by P( ) =P 1 ˆx( )ˆx ( ) (1) some reason (i.e. sensor momentary down, target occasionnally hidden, etc), the best we can do is to use the predicted value of the state estimation error covariance as the updated one, that is P 0( ) =P( 1) Under hypothesis E 0 (), the covariance P 0 ( ) isgiven by [7, 8, 9] P 0 ( ) =[I+q 0 K()H()]P( 1) (2) where q 0 is a weighting factor given by [7, 8, 9] q 0 P d(p g P gg ) 1 P d P g P dp g (1 c T ) 1 P d P g 0 P g, P gg and c T are given by P g P {χ 2 n y γ} P gg P{χ 2 n y+2 γ} c T Γ γ/2(1 + n y /2) (n y /2)Γ(n y /2) c T is a ratio of incomplete Gamma function defined by Γ α (x) α 0 u x 1 e u du From equation (1) and previous expressions, we get P( ) = β 0()P( 1) (3) +β 0 ()[I + q 0 K()H()]P( 1) +(1 β 0() β 0 ())P c ( )+ P() The stochastic matrix P() has the same expression as in the standard PDAF [1] and will not be given here due to space limitation. 3.2 Case 2 : m =0 When there is no validated measurement in the gate Y = {Y() =,m =0,Y 1 }, we theoretically have using theorem of total probability ˆx( ) =P 1,0ˆx ( )+(1 P 1,0 )ˆxŌ( ) where P 1,0 is given by (see section 4.1) P 1 = i= 0,0,... β i ()[ˆx i ( )ˆx i( )+P c ( )] P c ( ) = [I K()H()]P( 1) When the target is present in the environment but is not perceivable (hypothesis E 0()) by the sensor at for P 1,0 = (1 P dp g )P 1 1 P d P g P 1 P 1 P { Y 1 } is given in section 4.2 and ˆx ( ) E[x(),m =0,Y 1 ] ˆxŌ( ) E[x() Ō,m =0,Y 1 ] (4)
But actually, when there is no measurement, there is no way to improve state estimate whatever the perceivability of the target is; thus we have and therefore ˆxŌ( ) =ˆx ( )=ˆx( 1) (5) ˆx( ) =ˆx( 1) (6) The covariance P( ) associated the estimation error is then given by P( ) =P 1,0 P ( )+(1 P 1,0 )PŌ( ) Hence we get P ( ) = [I+q 0 K()H()]P( 1) PŌ( ) = P( 1) P( ) =[I+q 0 P 1,0 K()H()]P( 1) (7) 3.3 State prediction Prediction of the state and measurement is done using classical Kalman filter equations. The predicted covariance is given by P( +1 )=F()P( )F ()+Q() (8) where P( ) isgivenby(3)or(7) depending on m. 4 Association probabilities When m 0, one has to evaluate for i = 0, 0, 1...m β i () P{E i () Y(),m,Y 1 } (9) Using Bayes rule, one gets β i () = 1 c P1 i Pi 2 P{ m,y 1 } β 0 () = 1 c P1 0P 2 0P{ m,y 1 } β 0() = 1 c P1 0 P2 0 P{Ō m,y 1 } where c is a normalization constant such that m i= 0,0,... β i() =1. for i =1...m and assuming the true measurement residual Gaussian distributed in the validation gate of volume V and false alarms uniformly and independently distributed in V µ F,wehave[1,2,8] Pi 1 Pi 2 p[y() E i (),m,y 1 ]=V m+1 e i () P {θ i(),m,y 1 }= 1 P dp g c 1 m P{ m,y 1 } P 1,m e i () Pg 1 N [(y i () ŷ( 1)); 0; S()] c 1 P d P g +(1 P d P g )ξ ξ µ F (m ) µ F (m 1) for i =0,wehave P0 1 p[y() E 0 (),m,y 1 ]=V m P0 2 P{θ 0 (),m,y 1 }= ξ c 1 (1 P d P g ) P { m, Y 1 } P 1,m for i = 0, we have P 1 0 p[y() E 0(),m,Y 1 ]=V m P 2 0 P{θ 0() Ō,m,Y 1 }=1 P{Ō m,y 1 }=1 P 1,m The conditional predicted target perceivability probability P 1,m will be explicited in the next section. Combining previous equations yields following final expressions β i () = 1 c e i()p 1,m (10) β 0 () = 1 c b 0()P 1,m (11) β 0() = 1 c b 0()(1 P 1,m ) (12) b 0 () m 1 P d P g ξ V P d P g (13) b 0() m 1 [ ] Pd P g +(1 P d P g )ξ V P d P g (14) Remars : These new expressions for β i () are coherent Bar-Shalom derivation (taing into account Li and Guézengar correction term) for which perfect perceivability of the target was assumed (replace P 1,m by 1 into (10)-(12)). Following [5, 3, 8, 11] additional amplitude and/or recognition information can easily be taen into account in β i () by replacing e i () terms by e i ()ρ i (). ρ i () being the ratio of the pdfs of the amplitude feature/or another recognition feature of thetruetothefalsemeasurements. Since the true clutter density is never nown in most of practical applications, it must be estimated on line. The simpliest estimator ˆλ = m V is mostly used. The more appealing estimator suggested in [12] ˆλ =0 (when m =0)andwhenm 0 ˆλ = 1 V (m P d P g P 1,m ) (15)
should provide theoretically better results than the previous one. However since P 1,m is itself a function of the unnown clutter density λ (as we will see in (16) and(18)), this estimator cannot be used directly out a good model for the prediction of λ. A better issue is to use one of the three classes of theoretically solid estimators of the clutter density based on the Bayesian (conditional mean) estimation, the maximum lielihood method and the least squares method developed in [8, 12]. 4.1 Derivation of P 1,m The evaluation of β i () (i = 0,0,...m ) requires the computation of conditional predicted target perceivability probability P 1,m P { m, Y 1 }. Due to space limitation, we only give here final expression for P 1,m (see [8] for details). After elementary algebra, it can be shown (using Poisson prior for µ F ) state for the target { } can be modeled as a first-order homogeneous Marov-chain, i.e. π 11 P { 1 } π 21 P { Ō 1 } 4.3 Closed form of β i The following relations can be obtained from (10)-(14)and (16) using elementary algebra (1 φ )P 1 = 1 c 1 λv m (1 P d P g )(1 ɛ )P 1 + 1 c 1 P d P g m c = V m 1 φ P 1 1 ɛ P 1 V (1 ɛ )P 1 e i () i=1 P 1,m = (1 ɛ )P 1 1 ɛ P 1 (16) β 0 ()+ β i () = (1 φ )P 1 1 φ P 1 i=1 P { Y } P 1 Y 1 } P { { (17) ɛ P d P g m =0 P d P g (1 m λv ) m 0 (18) The derivation of unconditional predicted target perceivability probability P 1 will now be presented. 4.2 Derivation of P 1 and P P 1 and P can be obtained from Bayes rule and tae the following concise forms [8, 12, 11] where P 1 = π 11P 1 1 + π 21(1 P 1 1 ) (19) P = (1 φ )P 1 1 φ P 1 { P d P g m =0 φ P d P g (1 1 m λ i=1 e i) m 0 (20) (21) Thus, unconditional perceivability probabilities P 1 and P can be computed on-line recursively by (19) and (20) assoonasthedesign parameters π 11, π 21 and P1 0 have been set. First theoretical investigation on design of tracers for perceivability probability enhancement can be found in [10]. In this reference (and in our simulations), authors assume that the sequence of perceivability 1 P 1 β 0() = 1 φ P 1 =1 P =P{Ō Y } Rearranging expressions (10)-(14) yields the following useful concise forms for β i () β i () =E i ()/C β 0 () =B 0 ()/C β 0() =B 0()/C where the normalization constant is C =1 φ P 1 and E i (), B 0 (), B 0() are defined as E i () 1 P d P g V (1 ɛ )P 1 c 1 m e i() B 0 () 1 λv (1 P d P g )(1 ɛ )P 1 c 1 m B 0() 1 P 1 5 Trac confirmation The trac confirmation or termination can be done using different approaches. In [11] authors propose to compare the probability of perceivability some given confirmation and termination thresholds P c and P t as follows if P P c δ L = H c () (confirmation) if P P t δ L = H c () (termination) Their choices for P c and P t for tracing probability enhancement can be found in [10]. We propose here another approach based on the Sequential Probability Ratio Test
(SPRT). To perform this test, one has to choose the probability of decision error of first and second ind defined as P F = P (δ W ()=H c () H c ()) P M = P (δ W ()=H c () H c ()) where δ W () corresponds here to the decision taen by the following decision rule (SPRT) at if SPR() A δ W ()=H c () (confirmation) if SPR() B δ W ()=H c () (termination) SPR()= P{Y, } P{Y,Ō} = (1 φ )P 1 1 P 1 (22) The decision δ W is postponed to next scan whenever B< SPR() <A. The SPRT bounds A and B are given by A = 1 P M P F and B = P M 1 P F Actually, both criteria presented here concern only instantaneous decision. However during tracing process, a trac can either be declared confirmed during several scans, terminated or being a tentative trac because of the possible fluctuations of P computed by the IPDAF. This phenomenon generates difficulties to evaluate IPDAF performance in Monte Carlo simulations because we could average some tracs which correspond most of the only to tentative tracs based on false measurements. To overcome this problem, two other procedures have been used in our simulations for trac confirmation decision. These procedures are both based on the percentage of instantaneous confirmation decision over the of interest. Thus in the following, a trac will be deemed confirmed by the procedure 1 or by procedure 2 whenever at least 80% of, one has δ L = H c () orδ W =H c () respectively. 6 Simulation results The purpose of our simulation was to compare the performance of this new IPDAF version the previous one [8, 11]. The two-dimensional scenario of [14, 11] was simulated. 500 independent runs were used in the simulation; each 21 scans. The number of false measurements satisfies a Poisson distribution density λ =0.0001/scan/m 2. A single target is moving constant velocity the following dynamic model x( +1)=Fx()+v() where x() =[xẋyẏ] is the target state vector at and F is the following transition matrix [ ] [ ] F1 1 T F = F F 1 = 1 0 1 where T =1sis the sampling period. The process noise v() is a zero-mean white Gaussian noise nown covariance Q() [ ] Q1 Q=0.75 Q 1 Q 1 = [ ] T 4 /4 T 3 /2 T 3 /2 T 2 The sensor introduced independent errors in x and y root mean square (RMS) value r =5m. For each run, the true initial state for the target was x(0) = [ 50m 35m/s 0m 0m/s]. To simplify simulations, initial state estimate ˆx(0 0) for each run follows N (x(0), P(0 0)) P(0 0) given by [1, 2] P(0 0) = [ ] P1 P 1 P 1 = [ r 2 ] r 2 /T r 2 /T 2r 2 /T The uniform initial predicted probability of perceivability P1 0 0 =0.5was used. To get a fair comparison between results, the true value of clutter density λ instead of ˆλ wasusedinbothipdafimplementedversions. Both IPDAF versions used the same set of design parameters [10] (P g =0.99, P d =0.9, π 11 =0.988, π 21 = 0). Two cases were simulated for performing confirmation procedures (case 1 P t =0.0983, P c =0.9, P M = P F =0.1 and case 2 same P t and P c but P M = P F =0.05 for procedure 2). Figure 1 and 3 shows the percentage of confirmed tracs for both IPDAF versions based on procedure 1 and 2 for case 1 and case 2 respectively. Figures 2 and 4 shows the corresponding RMS position errors. Results obtained by procedure 1 and 2 for each IPDAF implementation for case 1 are very close and only 2 curves appear actually on figures 1 and 2. The comparison of RMS velocity errors are not given here due to space limitation and because there is no significant difference between plots. Results indicates that this new version of IPDAF gives on average as good performance as its previous version concerning the percentage of confirmed tracs. Concerning the RMS position errors, this new IPDAF version performs a little bit better than the previous one. 7 Conclusions A new formulation of IPDAF based on the recently developed probability of perceivability has been presented theoretical justification. Specially, this filter formulation is fully coherent and intuitively appealing the PDAF formulation as soon as the probability of perceivability becomes unitary. A new approach for trac confirmation and termination based on sequential probability ratio test has also been given. Performance of this new IPDAF is quite comparable to the previous version of IPDAF concerning the percentage of confirmed tracs but is a little bit better on average in terms of RMS estimation errors. An extension of this approach to mutitarget tracing can be found in [6].
0 2 4 6 8 10 12 14 16 18 20 No. of deemed confirmed tracs out of 500 runs 100 90 80 70 60 50 40 30 20 10 0 Figure 1: Percentage of confirmed tracs (case 1) m 9 8.5 8 7.5 7 6.5 6 5.5 5 4.5 0 5 10 15 20 25 Figure 2: RMS position errors (case 1) No. of deemed confirmed tracs out of 500 runs 100 90 80 70 60 50 40 30 20 10 0 0 2 4 6 8 10 12 14 16 18 20 Figure 3: Percentage of confirmed tracs (case 2) m 8.5 8 7.5 7 6.5 6 5.5 5 4.5 0 5 10 15 20 25 Figure 4: RMS position errors (case 2) References [1] Bar-Shalom Y., Fortmann T.E., Tracing and Data Association, Academic Press, (1988). [2] Bar-Shalom Y., Li X.R., Estimation and Tracing: Principles,Techniques, and Software, Artech House, (1993). [3] Bar-Shalom Y., Li X.R., Multitarget-Multisensor Tracing: Principles and Techniques, YBS Publishing, (1995). [4] Colegrove S.B., Ayliffe J.K., An extension of Probabilistic Data Association to Include Trac Initiation and Termination, 20th IREE International Convention,Melbourne, pp. 853-856, (1985). [5] Dezert J., Autonomous Navigation Uncertain Reference Points Using the PDAF, in Multitarget- Multisensor Tracing : Applications and Advances, Y. Bar-Shalom Editor, Artech House, vol. 2, (1992). [6] Dezert J., Li N., Li X.R., Theoretical Development of an Integrated JPDAF for Multitarget Tracing in Clutter, Proc. Worshop ISIS-GDR/NUWC, ENST, Paris, (1998). [7] Guézengar Y., Radar Tracing in Cluttered Environment Applied to Adaptive Phased Array Radar, Ph.D. Dissertation, Nantes Univ, France, (1994). [8] Li X.R., Li N., Intelligent PDAF : Refinement of IPDAF for Tracing in Clutter, Proc. 29th Southeastern Symp. on Syst. Theory, pp. 133-137, (1997). [9] Li X.R., Tracing in Clutter Strongest Neighbor Measurements - Part 1 : Theoretical Analysis, IEEE Trans. AC,vol. 43, pp. 1560-1578, (1998). [10] Li N., Li X.R., Theoretical Design of Tracers for Tracing Probability Enhancement, Proc.ofSPIE Conf., rlando,vol. 3373, (1998). [11] Li N., Li X.R., Target Perceivability : An Integrated Approach to Tracer Analysis and Design, Proc. of Fusion 98 Intern. Conf., Las Vegas, pp. 174-181, (1998). [12] Li X.R., Li N., Zhu Y., Integrated Real-Time Estimation of Clutter Density for Tracing, submitted to IEEE Trans. on Signal Processing., (1999). [13] Musici D., Evans R.J., Stanovic S., Integrated Probabilistic Data Association (IPDA), Proc. 31st IEEE CDC, Tucson, AZ, (1992). [14] Musici D., Evans R.J., Stanovic S., Integrated Probabilistic Data Association, IEEE Trans. AC,vol. 39, pp. 1237-1241, (1994).