Monte Carlo-based Filter for Target Tracking with Feature Measurement∗ D. Angelova B. Vassileva Tz. Semerdjiev Central Laboratory for Parallel Processing, Bulgarian Academy of Sciences ”Acad G. Bonchev” St., Bl. 25-A, 1113 Sofia, Bulgaria E-mail:
[email protected] Abstract – Monte Carlo-based algorithm for tracking maneuvering target with a feature measurement is proposed in the paper. Amplitude Information (AI) is used as a feature for state estimation of relatively low observable target (low Signal-to-Noise Ratio (SNR)) in the presence of high rate of false alarms. Rayleigh distributed noise amplitude and Swerling 3 type target model are assumed. The stochastic filter combines the Multiple Model (MM) approach with switching models for dealing with maneuvers and probabilistic association of features and measured kinematic data. The filter performance is analyzed by simulation. Results show that the suggested algorithm can track targets with SNR down to 10 dB with acceptable percentage of lost tracks, while the filter without AI works down to 13 dB. In the case of nonmaneuvering target these limits are at lower levels. Keywords: Hybrid systems, nonlinear filtering, Monte Carlo methods, target tracking.
1 Introduction Tracking of a maneuvering object in the presence of false measurements is a difficult task due to the conflict between data association and maneuver detection. It is well known, that the Interacting Multiple Model (IMM) state estimator, combined with the Probabilistic Data Association (PDA) [2] is one of the most effective algorithms solving this nonlinear filtering problem. In the worse case of a low observable target (quantified by a low SNR), the availability of additional information can be very helpful. The presence of a feature measurement (for example the rank of a measurement [2]) increases the quality of data association procedure. A technique that uses as feature the measured target amplitude in the IMMPDA filter for track initiation and maintenance is discussed in [14]. Its effectiveness is ∗ Partially
supported by Center of Excellence BIS21 grant ICA1-200070016 and Bulgarian National Foundation for Scientific Investigations under grant No I-902/99.
ISIF © 2002
illustrated in the context of a long-range active sonar problem. A PDA-based Maximum Likelihood (ML) estimator that uses AI is derived in [13]. AI enhances the filter convergence and performance for low observable targets in passive sonar systems. The incorporation of a model for fluctuating target amplitude into the ML approach is addressed in [18]. Feature variables in the measurement vector are particularly valuable in multiple target tracking algorithms. A multiple hypothesis algorithm extended with AI is implemented and examined by simulation in [12]. The results exhibit an improved association performance and correct target type identification. Recently, a large number of publications demonstrate the ability of Monte Carlo (MC) methods for efficient nonlinear filtering [9, 10, 19]. With this numerical method, implementing a near optimal Bayesian filtering, the complicated non-linear, non-Gaussian estimation problems can be solved in real time. Simulation-based algorithms are applied and experimented in many challenging tracking problems - multiple target tracking using multiple receivers [7, 11], terrain navigation [17], ground target tracking [15], low SNR target in a pre-detection tracking situation [16]. A bootstrap filter for maneuvering target tracking which uses a feature measurement is suggested in the present paper. The objective is to explore the impact of AI on filter capability for dealing with relatively low observable target in conditions of high rate of false alarms. The proposed version of the simulation-based filter incorporates • MM approach with switching models for maneuvering target state estimation; • PDA to cope with false measurements; • AI for alleviating the low SNR problem. The resulting algorithm is studied by simulation on a surveillance radar target tracking problem. This introduction is followed by five sections. Section 2
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contains the statement of the problem. Some aspects of the implemented AI model are considered in Section 3. The features of the bootstrap filter are mentioned in section 4. In section 5 simulation results are presented and performances compared. Some concluding remarks are given in section 6.
2 Problem definition Consider the following discrete-time target motion and sensor model xk zk
= =
f (xk−1 , mk , vk−1 ) h(xk ) + wk ,
k = 1, 2, ...
defined under the assumption that the observations at k depend only on the state at that time. The likelihood function fuses in a suitable way the likelihood of measured kinematic parameters and amplitude likelihood ratio (details are described in Sections 3 and 4). Given a prior initial distribution p(t0 , s0 ) = q0 (s©0 ), s0 ∈ S, the posterior distribution p(t , s ) = k k ª P r X(k) = sk /Z k may be calculated in the following recursive manner for k ≥ 1 and sk ∈ S [5]: Motion Update Z ? p (tk , sk ) = qk (sk /sk−1 )p(tk−1 , sk−1 )dsk−1
(1) (2)
where xk ∈ Rnx is the target continuous-valued base state vector, zk ∈ Rnz is the measurement vector, f and h are the object evolution function and measurement model function, respectively [1]. The discrete- valued modal state mk ∈ {1, 2, · · · , r} represents the target motion mode, which is in effect during the sampling period, ending at time k. The sequence of modes is modeled as a time-homogeneous, r-state, first-order Markov chain with known initial Pi = P r {m0 = m(i)} and transition probabilities pij = P r {mk = m(j)/mk−1 = m(i)} , i, j = 1, r. vk ∈ Rnx and wk ∈ Rnz are respectively process and measurement random noise sequences with known parameters. Let X(k) ≡ X(xk , mk ) denote the hybrid state of the target in the hybrid state space S. We assume (according to [5]) that the stochastic process {X(k), k ≥ 0} describing the target evolution over time is Markovian on S. Let us define the transition function qk (sk /sk−1 ) = P r {X(k) = sk /X(k − 1) = sk−1 } for k ≥ 1 and let q0 (s0 ) be the probability density function (pdf) for X(0) [5]. At discrete times k = 1, 2, · · ·, the sensor reports observations which contain the kinematic components of the measurement vector zk as well as AI ak that exceeds some detection threshold τ ¯ ¯ ¯ zk ¯ a ¯ ¯ zk = ¯ (3) ak ¯
(5)
Information Update p(tk , sk ) =
1 Lk (z(k)/sk )p? (tk , sk ) C
(6)
where C is a normalizing constant. The realization of simulation-based nonlinear tracking filters rely on these basic Bayesian equations.
3
Amplitude information at the detector output
The strength of a target return, measured at the output of the single observation detector, is referred to here as the measurement amplitude. The performance of this detector depends largely upon the noise density, determined from the threshold, and the probability of detection, determined by the threshold and the received SNR for target returns. The threshold value τ can be obtained from Z ∞ Pf a = p0 (a)da , (7) τ
where Pf a is specified probability of false alarm and p0 (a) is the pdf of the amplitude a, measured at the output of the detector if it is due to noise only. The probability of detection Pd is given by Z ∞ Pd = p1 (a)da , (8) τ
l(k) {zkaj }j=1
The set of observations Z(k) = received at k could be from the target and/or a result of noise disturbances. The tracking problem can be stated in the Bayesian framework of estimating the posterior distribution on the target state space given the cumulative observation set k Z k = {Z(j)}j=1 obtained through time k. The sensor information in the sequential Bayesian filtering consecutively updates the target distribution by using a likelihood function Lk (z(k)/sk ) = P r {Z(k) = z(k)/X(k) = sk }
(4)
where p1 (a) is the pdf of the detector output amplitude a if it is due to target and noise. The pdf p1 (a) is parameterized by the expected SNR of the returns. The amplitude a > τ likelihood ratio, required for the computation of likelihood function (4), is defined as [14] λ=
Pf a p1 (a) , a>τ Pd p0 (a)
(9)
The problem of one dominant-plus-Rayleigh fluctuating signal is considered now. This type of signal fading is known in the radar literature as the Swerling 3 case
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[4][p 152]. The one dominant-plus-Rayleigh pdf is given by [3][p 313] 3A2 9A3 exp , A≥0 4 2A0 2A20
(i)
where vk−1 is a sample drawn from the process noise pdf p(vk−1 ).
(10)
Resampling: On receipt of the new measurement set l(k) Z(k) = {zkaj }j=1 resampling with replacement is per-
where A0 is the peak of the pdf, or most probable value of A. This pdf describes return from a target consisting of one large dominant reflector plus many small random scatters. The optimum detector for amplitude modulated narrowband signal in the presence of Gaussian noise is a quadrature receiver or a bandpass matched filter for each discrete range and bearing cell. Assuming a normalized background noise, the probability density of the output signal envelope a for noise only and target present are as follows [3][pp 304–315]
formed, where each prior sample sk is drawn with a probability determined by the normalized weight:
p(A) =
2
p0 (a) = exp (− a2 ) , ³ 9 1+ p1 (a) = p0 (a) (A2 +3) 2 0
A20 a2 2(A20 +3)
´
A 2 a2
exp 2(A02 +3)
∗(i)
∗(i)
Lk (z(k)/sk ) , for i = 1, N , qi = PN ∗(ii) ) ii=1 Lk (z(k)/sk where the likelihood function accounts for all possible measurement-to-target association hypotheses at the current scan k [2]. The likelihood of kinematic measurement is multiplied by its respective amplitude likelihood ratio λ (9) l(k)
.
∗(i)
Lk (z(k)/sk ) =
0
(11) According to (7), (8) and (11) the probability of detection is given by ³ ´ ln Pf a 3 ln Pf a A2 ) . Pd = A23+3 1 + 30 − 2(1+3/A exp (− (A2 +3) 2) 0 0 0 (12) ¯ ¯ denotes It is important to know that A20 = 3R/4, where R the average SNR.
4 Bootstrap filter
where1 ej (zkj ) = N [zkj ; h(x∗i k ), Rk ] and V (k) is the volume of a validation region G, set up around the predicted ∗(i) ∗(i) measurement set of samples {zk = h(xk ) : i = 1, N }. Rk is the measurement error covariance. Output: The posterior mean E[xk /Z k ] and associated covariance V [xk /Z k ] of the target base-state are evaluated approximately by the set of samples
The simulation-based filter replaces the evolving pdfs in equations (5) and (6) ¡ ¢ ¡ ¢ ¡ ¢ k−1 ∗ k−1 k p tk−1 , sk−1 /Z
→p
tk , sk /Z
i=1
1 X (i) (i) (x − x bk )(xk − x bk )T N − 1 i=1 k
(15)
N
Vkx =
An estimate of the modal-state posterior probability pbj = P r{mk = j/Z k }, j = 1, r can be easily obtained as2 [6]
i=1
where sk = {xk , mk } contains the components of hybrid state vector. The consecutive steps of the filter can be written as follows: Initialization: k = 0; (i) For i = 1, N draw the base states x0 from the prior initial (i) distribution p(t0 , x0 ) and draw the initial modal states m0 according to the initial probabilities of Markov chain. Thus (i) the initial hybrid o n states s0 are generated. (i)
(14)
→ p tk , sk /Z
n oN n oN n oN (i) ∗(i) (i) → sk → sk sk−1
Starting from sk−1 : i = 1, N
N 1 X (i) x N i=1 k
x bk =
by predicting and updating their respective sets of random samples
i=1
Pf a X 1 − Pd + ej (zkj )λj (ajk ) (13) V (k) l(k) j=1
carry out:
Sampling: For each i = 1, N first realize the modal(i) ∗(i) state sampling mk−1 → mk according to the finite state Markov chain with known parameters. Second, obtain the ∗(i) ∗(i) (i) ∗(i) (i) base-state samples xk as xk = f (xk−1 , mk , vk−1 ),
pbj =
1 (i) (i) |s : mk = j, N k
i ∈ {1, 2, · · · , N }|
The suboptimality of the simulation-based procedure is due to the finite number of samples used. The main advantage of this algorithm in this particular application is that measurement association and target behaviour hypotheses from previous time steps are implicitly included in the sample set.
5
Monte Carlo Simulation
The problem of tracking a maneuvering target by a surveillance radar in the presence of false alarms is examined.
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1 N [a; a, A] 2 |.|
denotes the Gaussian pdf with mean a and covariance A. denotes the cardinality of a set.
1 P r{a > ak > τ } = Pd
Simulation Model. Consider the following targetmeasurement model [1]: xk zk
= =
1 0 F (ω) = 0 0
F (ω)xk−1 + Gvk−1 h(xk ) + wk sin ωT ω
cos ωT 1−cos ωT ω
sin ωT T2 2
T G= 0 0
(16) (17)
ωT 0 − 1−cos ω 0 − sin ωT , sin ωT 1 ω 0 cos ωT 0 0 , T2 2
T
where ω and T are target turn rate and sampling interval, re˙ η, η) spectively. The state space vector x = (ξ, ξ, ˙ T contains target positions and velocities in horizontal (Oxy) Cartesian coordinate frame. The distance to the target D and bearing β, measured by the radar, are kinematic components of the measurement vector z = (D, β)T . Thus the form of the measurement function ¶T µp ξ ξ 2 + η 2 , arctan h(x) = η in the measurement equation (17) is completely specified. The set of models, describing multiple model configuration, includes one nearly constant velocity model (ω = 0) and two nearly coordinated turn models with known mean values ±ω for left and right turns, respectively [1]. An underlying Markovian chain with known initial and transition probabilities controls the model switching. The 2−D radar parameters are specified in the Table 1. Table 1. Sensor Spesifications Measurement accuracy: Maximum Range - σD = 100.0m Range -100 km Bearing - σβ = 0.15 deg Bearing-5 ÷ 85
Sampl. interval T = 5s
The radar produces measurements from the target with detection probability Pd ≤ 1. False measurements are modeled independently from scan to scan, with known expected number of false alarms per unit volume λf a (determined by Pf a ), uniformly distributed in the region G. The amplitude measurements ak are generated as random numbers using cumulative distribution functions in the cases of pdf’s p0 (a) and p1 (a) respectively. Since the cumulative distribution function is uniformly distributed between 0 and 1, amplitude measurements ak can be generated by solving Z ∞ 1 p0 (a)da = 1 − ξ (18) P r{a > ak > τ } = Pf a ak for noise only
Z
∞
p1 (a)da = 1 − ξ
(19)
ak
for target and noise where ξ are random numbers uniformly distributed between 0 and 1. Measures of performance: Root-Mean Squared Errors (RMSE): position RMSE (both coordinates combined) and speed (magnitude of the velocity vector) RMSE; average probability of correct mode identification; average time per update, number of tracks lost. A loss of a track is established, when the absolute position error exceeds only once a threshold (th = 15σD ), even if it returns to admissible values on later. Filter Design Parameters. The parameters of base state vector initial distribution x0 ∼ N [x0 ; m0 , P0 ] are selected as follows: P0 = diag{1502 m, 20.02 m/s, 1502 m, 20.02 m/s}; m0 contains the exact initial target parameters for each scenario. The process noise standard deviations σνj for each mode j = 1, 2, 3 in the multiple model configuration (r = 3) are as follows: σν1 = 2.2m/s2 and σν2,3 = 5.8m/s2 ; Initial and transition mode probabilities of the underlying Markov chain are: P1 = 0.6, P2 = 0.2, P3 = 0.2; p11 = 0.7, p12 = 0.15, p13 = 0.15; p21 = 0.15, p22 = 0.8, p23 = 0.05; p31 = 0.15, p32 = 0.05, p33 = 0.8. The size of the sample set is N = 3200. Simulation Experiments. The filter performance is examined over two test scenarios. A constant target SNR is used in the first scenario, averaged over the target range interval. In the second, more realistic scenario, the SNR is calculated at each scan as a function of range D. The results presented below are based on 100 Monte Carlo runs. Scenario 1. The test trajectory is depicted in Fig. 1. The target speed is constant, equal to 200 m/s. The duration of nonmaneuvering phases is 15 and 12 sampling intervals. The object accomplishes a coordinated turn maneuver with normal acceleration 30 m/s2 (turn rate ≈ 8.6 deg /s) and duration of 4 sampling intervals. The same mean turn rate values of ±8.6 deg /s are assigned to the filter left and right turn models. The performances of the filter with AI (denoted as BFAI) and without AI (BF) are examined over several combinations of Pf a and target SNR. The selected set of Pf a : 0.001; 0.01; 0.03; 0.1 is related to the following set of clutter densities λf a : 5.5e − 6; 5.5e − 5; 1.7e − 4; 5.5e − 4. This √ corresponds to the √ range and bearing cells of resolution 12σD m and 12σβ deg. The SNR set includes: 18.8, 15.8, 13.6, 12 and 10 dB. Plots of the BFAI position and speed RMS errors for all selected SNR values and λf a = 1.7e−4 are presented in Figs. 2 and 3. The respective values of the posterior nonmaneuvering mode probability are given in Fig 4. Comparative
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4
7.5
x 10
110
10 dB
λfa = 1.7 e−4
100 7
12 dB
90 6.5
RMSE [m/s]
y [m]
80
6
70
60
START
5.5
15.83 dB
50 5 40
18.84 dB 4.5 4.5
5
5.5
6
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7
x [m]
30
7.5
0
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10
15
20
Fig. 1: target trajectory
30
Fig. 3: BFAI - speed RMSE 0.9
600
18.84 bB
550
λ = 1.7 e−4 fa
0.8
10 dB
15.83 bB
12 bB
500
0.7
450
10 bB
0.6
RMSE [m]
25
Time, k, in scans
4
x 10
12 dB
400
0.5 350 0.4 300
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0.3
250 0.2
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150
100
0.1
0 0
5
10
15
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30
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5
10
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20
25
30
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Time, k, in scans
Fig. 2: BFAI - position RMSE
Fig. 4: BFAI - mode 1 posterior probability
results, including position and speed RMSE for BFAI and BF are shown in Figs. 5 and 6. The obtained percentage of lost tracks for SN R = 13.62dB is presented in Table 2. There is no loss of tracks for SN R = 18.84dB. At 10 dB SNR the BFAI track loss is 25%, while BF loses all tracks.
±7.64 deg /s, corresponding to 40m/s2 are assigned to the filter left and right turn models. The average SNR is computed according to the equation [8]
BF BFAI
¯ = Ωσave , R D4
(20)
where Ω = 210 dB is the generalized radar parameter, D [m] is the target range and σave [m2 ] is the average Radar-Cross-Section (RCS) of the target. Aircraft with σave = 1.2 m2 is selected in the simulations. The results (in terms of RMSE and averaged posterior probabilities) are shown in Figs. 8, 9 and 10. The following inferences can be drawn from the results obtained:
Table 2. Percentage of Tracks Lost SNR = 13.62 dB Pf a = 0.001 Pf a = 0.01 Pf a = 0.10 Pd = 0.67 Pd = 0.8 Pd = 0.92 40.0 13.0 10.0 30.0 8.0 2.0
Scenario 2. The scenario considered (Fig.7) includes three coordinated turn maneuvers with normal accelerations 40, 20 and 40 m/s2 . The target speed is constant, equal to 300 m/s. The mean turn rate values of
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• the bootstrap algorithm is very sensitive to the low Pd levels. The Pd values below 0.75 lead to a degradation in the filter performance. But the filter operation is rather reliable at high levels of Pf a . Therefore, the
210
450
λfa = 1.7 e−4
200
SNR = 13.62 dB
400
BF
λ = 5.5 e−4 fa
190
BFAI 180
RMSE [m]
RMSE [m]
350
300
250
170
160
150
140
200
BFAI 130 150 120
100
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5
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20
25
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30
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50
60
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60
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60
70
Time, k, in scans
Time, k, in scans
Fig.8: BFAI - position RMSE
Fig. 5: BF & BFAI comparison-position RMSE
90
100
λ = 1.7 e−4 fa
BF SNR = 13.62 dB
90
80
λ = 5.5 e−4 fa
80
BFAI
70
RMSE [m/s]
RMSE [m/s]
70
60
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50 40
BFAI
40
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0
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30
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Time, k, in scans
Fig. 9: BFAI - speed RMSE
Fig. 6: BF & BFAI comparison-speed RMSE
4
7
x 10
1
λfa = 1.7 e−4
0.9 6 0.8 5
0.7 START 0.6
y [m]
4 0.5 3 0.4
0.3
2
0.2 1 0.1
0
0
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6
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x 10
Fig. 7: target trajectory
0
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30 40 Time, k, in scans
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Fig. 10: BFAI - mode probabilities
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detection threshold can be lowered without significant degradation in the tracking performance; • AI measurements improve the estimation accuracy during maneuvering intervals. They provide superior convergence and more precise mode identification. The filter with AI is more robust against track loss at low SNR levels.
6 Concluding remarks PDA augmented with a feature measurement is incorporated into stochastic sampling MM filter for tracking relatively low observable maneuvering target in clutter. AI is used as the feature with appropriate probabilistic models for the target and clutter return amplitudes. Simulation results show that AI increases the filter accuracy and reliability. The algorithm can track targets with SNR down to 10 dB with acceptable percentage of lost tracks, while the filter without AI works down to 13 dB. The results confirm the fact that simulation-based techniques fuse different types of measured data in an easy and effective way. Acknowledgments We would like to thank Rudolph van der Merwe, Arnaud Doucet and Nando de Freitas for providing the software in the form of MATLAB code through the Internet.
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[8] W. Blair, G. Watson, T. Kirubarajan and Y. BarShalom, Benchmark for Radar Allocation and Tracking in ECM, IEEE Trans. on AES, 34, pp 1097–1114, October 1998. [9] A. Doucet, S. Godsill and C. Andrieu, On Sequential Monte Carlo Sampling Methods for Bayesian Filtering, Statistics and Compuing, 1999. [10] A. Doucet, A. Logothetis and V. Krishnamurthy, Stochastic Sampling Algorithms for State Estimation of Jump Markov Linear Systems, IEEE Trans. on AC, 45, pp 188–202, January 2000. [11] C. Hue, J. Le Cadre and P. Perez, The(MR)MTPF: particle filters to track multiple targets using multiple receivers, Proc. of the FUSION 2001 Conference, Montreal, Canada, 2001, pp FrC3-33–FrC3-40. [12] V. Jilkov and Tz. Semerdjiev, Multiple Hypothesis Tracking Using Amplitude Feature Measurements, Comptes rendus de l’Academie bulgare des Sciences, 49, No 11-12, pp 41–44, 1996. [13] T. Kirubarajan and Y. Bar-Shalom, Low Observable Target Motion Analyzis Using Amplitude Information, IEEE Trans. on AES, 32, pp 1367–1383, October 1996. [14] D. Lerro and Y. Bar-Shalom, Interacting Multiple Model Tracking with Target Amplitude Feature, IEEE Trans. on AES, 29, pp 494–508, April 1993. [15] M. Mallick, T. Kirubarajan and A. Sanjeev, Comparison of Nonlinear Filtering Algorithms in Ground Moving Target Indicator Tracking, Proc. of the FUSION 2001 Conference, Montreal, Canada, 2001, pp FrA3-11–FrA3-17. [16] S. Musick, J. Greenewald, C. Kreucher and K. Kastella, Comparison of Particle Method and Finite Difference Nonlinear Filters for Low SNR Target Tracking, Proc. of the FUSION 2001 Conference, Montreal, Canada, 2001, pp FrA3-3–FrA3-10. [17] C. Musso and N. Oudjane, Recent Particle Filter Applied to Terrain Navigation, Proc. of the FUSION 2000 Conference, Paris, France, 2000, pp WeB5-26– WeB5-33. [18] S. Tonissen and Y. Bar-Shalom, Maximum Likelihood Track-Before-Detect With Fluctuating Target Amplitude, IEEE Trans. on AES, 33, pp 796–809, July 1998. [19] Van der Merve R., A. Doucet, N. de Freitas and E. Wan, The Unscented Particle Filter, Technical Report, CUED/F-INFENG/IR 380, Cambridge University Engineering Department, pp 1–44, 2000.
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