Chinese Journal of Electronics Vol.24, No.2, Apr. 2015
Bernoulli Filter for Extended Target in Clutter Using Poisson Models∗ CAI Fei, FAN Hongqi and FU Qiang (ATR Key Laboratory, National University of Defense Technology, Changsha 410073, China ) Abstract — This paper presents a joint detection and tracking filter for a single extended target in the presence of clutter measurements and missed detections. The filter is obtained by adapting the Poisson extended target model into the Bernoulli filter proposed by Mahler. The resulted filter is an optimal extended target joint detection and tracking filter. Predictor and corrector are derived follows the random set filtering framework. A particle filter implementation is presented, in which simplification methods are used to make it easy to be realized. Simulation results show that the proposed filter is effective at detection and tracking of extended target in dense clutter backgrounds. Key words — Extended target, Target detection, Target tracking, Random set.
I. Introduction Due to the increasing resolution capabilities of modern sensors, the target may occupy multiple resolution cells of the sensor, thus potentially producing more than one measurement at a given time step. Such target is denoted extended, and tracking of extended target has received increasing research attention over the past decade[1−5] . An inhomogeneous Poisson point process extended target measurement model is suggested in Ref.[6], where a Poisson distributed number of measurements is distributed around the target centroid. Using random matrices and random hypersurface as extended target models were suggested in Ref.[2] and Ref.[7], respectively. With Finite set statistics (FISST), Mahler introduced a set theoretic approach in which targets and measurements are modeled using Random finite sets (RFS). The approach allows multiple target tracking in the presence of clutter and with uncertain associations to be cast in a Bayesian framework, resulting in an optimal multi-target Bayes filter. The optimal multitarget Bayes filter is usually computationally intractable and approximations are needed such as the Probability hypothesis density (PHD) filter and the Cardinalized PHD (CPHD) filter. However, under the condition that at most a single target is present, the multi-target Bayes filter reduces to the Bernoulli
filter[8] (also known as the JoTT filter), which is both computationally tractable and theoretically optimal. Mahler[3] formulated the PHD filter for multiple extended targets tracking of the type presented in Ref.[6]. PHD and CPHD filters for tracking multiple extended targets using random matrices are proposed in Ref.[4] and Ref.[5], respectively. CPHD filter for multiple maneuvering targets is proposed in Ref.[9]. Tracking of single extended target in clutter is addressed in Ref.[10], but the target is assumed to be existing all the time. In Refs.[11] and [12], a Bernoulli particle filter for extended target named as BPF-X is proposed, which uses a simplified version of the extended target model due to Mahler[8] . The target is assumed consists of a time-vary number of measurement generating points, characterized by the same probability of detection. The number of scattering points is estimated at each time step using a empirical method and target size/shape is estimated along with target centroid. Different from Ref.[11], we use the Poisson extended target model proposed by Ref.[6] in a Bernoulli filter. As a result, only target centroid is estimated. Estimation of the scattering points number is avoid to make the resulted filter optimal. The proposed filter is a solution to the problem of joint detection and tracking of a single extended target under the presence of detection uncertainty and clutter, A practical particle filter implementation of the proposed filter is developed, in which simplification methods are used to make the proposed filter easy to be realized. The effectiveness of the proposed filter is evaluated via a simulated tracking example, which shows that it can detect and track extended target in dense clutter backgrounds.
II. The Proposed Filter It is assumed that at most one target can be present, and that the target can be in one of two ‘present’ or ‘absent’ modes. The target state set is denoted by X, with X = Ø denoting target absent and X = {x} denoting target with state x is presented. The FISST posterior distributions of the random
∗ Manuscript Received July 2014; Accepted Oct. 2014. This work is supported by the National Natural Science Foundation of China (No.61101186). c 2015 Chinese Institute of Electronics. DOI:10.1049/cje.2015.04.017
Bernoulli Filter for Extended Target in Clutter Using Poisson Models finite set X at time step k have the form 8 > if X = Ø < 1 − pk|k , fk|k (X) = pk|k · fk|k (x), if X = {x} > : 0, if |X| > 1
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Using the fundamental convolution formula[8] , the likelihood for the complete observation process is fk (Zk |X) X = f (W |X)κ(Zk − W )
(1)
W ⊆Zk
where fk|k (x) is the target’s track density and pk|k is the probability that it exists. 1. Motion model Target presence is modeled by a two-state Markov chain. If the target is not in the scene at time step k − 1, it may be born at the next time step with probability pB , and its born density is bk|k−1 (x). Then the Markov density has the form as follows: 8 > if X = Ø < 1 − pB , (2) fk|k−1 (X|Ø) = pB · bk|k−1 (x), if X = {x} > : 0, if |X| > 1
If a target is in the scene at time step k − 1 with state x , it is assumed that it will remain in the scene with probability pS (x ), and its motion between time steps k − 1 and k is governed by the Markov density fk|k−1 (x|x ). In this case, the Markov density has the form 8 > if X = Ø < 1 − pS (x ), fk|k−1 (X|{x }) = pS (x )fk|k−1 (x|x ), if X = {x} (3) > : 0, if |X| > 1 2. Measurement model The set of measurements obtained at time k is denoted (j) Nz,k , where Nz,k = |Zk | is the number of measureZk = {zk }j=1 ments. Zk is the union of the target generated measurements and the false alarms. The set of target generated measurements is distributed according to a Poisson model. The corresponding set likelihood is given as Y γ(x)φx (z) (4) LZk (x) = f (Zk |x) = e−γ(x) z∈Zk
where γ(x) and φz (x) denote the mean number and the spatial distribution function of the target generated measurements. Note that the target may generate no measurements with probability e−γ(x) . Thus 1 − e−γ(x) is the probability that it will generate at least one measurement. Consequently, pD (x) (1 − e−γ(x) )pD (x) is the probability that there is at least one target generated detection, where pD (x) is the probability of detecting the extended target as a whole. The false alarm process κ(Zk ) is independent of the detection and target-observation process. The number of false alarms is modeled by a Poisson distribution with known mean λC . The false alarm process is Y κ(Zk ) = e−λC λC c(z) (5) z∈Zk
where c(z) is the spatial distribution of the false alarms. The false alarms are generally uniformly distributed over the sensor field-of-view, then c(z) = 1/V , where V is the surveillance area, and (6) κ(Zk ) = e−λC (λC /V )|Zk |
X
= f (Ø|X)κ(Zk ) +
f (W |X)κ(Zk − W )
(7)
W ⊆Zk
where W is nonempty. If X = Ø, then fk (Zk |Ø)
X
= f (Ø|Ø)κ(Zk ) +
f (W |Ø)κ(Zk − W )
W ⊆Zk
= 1 · κ(Zk ) + 0 = κ(Zk )
(8)
If X = {x}, then fk (Zk |{x})
X
= f (Ø|{x})κ(Zk ) +
f (W |{x})κ(Zk − W )
W ⊆Zk
X
= (1 − pD (x))κ(Zk ) + pD (x)
LW (x)κ(Zk − W )
W ⊆Z
= κ(Zk ) 1 −
pD (x)
X
+ pD (x)
W ⊆Zk
! κ(Zk − W ) LW (x) (9) κ(Zk )
3. Analytic solution of the proposed Filter The problem of joint detection and tracking can now be formulated in the framework of recursive Bayesian estimation as follows. Predictor Since the target state and motion models are the same, the prediction equations are identical to those of the standard Bernoulli filter. Here we present only the final results. Given pk−1|k−1 and fk−1|k−1 (x), the pk|k−1 and fk|k−1 (x) are predicted as pk|k−1
Z
= pB (1 − pk−1|k−1 ) + pk−1|k−1
pS (x )fk−1|k−1 (x )dx
and fk|k−1 (x) =
pB bk|k−1 (x)(1 − pk−1|k−1 ) pk|k−1 R pk−1|k−1 pS (x )fk|k−1 (x|x )fk−1|k−1 (x )dx + pk|k−1
Corrector Given pk|k−1 and fk|k−1 (x), the pk|k and fk|k (x) are updated according to pk|k =
1 − fk|k−1 [pD ] +
P
p−1 k|k−1 − fk|k−1 [pD ] +
W ⊆Zk
P
k −W ) fk|k−1 [pD LW ] κ(Z κ(Zk )
W ⊆Zk
k −W ) fk|k−1 [pD LW ] κ(Z κ(Zk )
and fk|k (x) = fk|k−1 (x)
P k −W ) 1 − pD (x) + pD (x) W ⊆Zk LW (x) κ(Z κ(Zk ) · P k −W ) 1 − fk|k−1 [pD ] + W ⊆Zk fk|k−1 [pD LW ] κ(Z κ(Zk )
Chinese Journal of Electronics
328 where
Z fk|k−1 [h]
h(x)fk|k−1 (x)dx
Proof The update equation of the Bayes filter formulated in the random set framework is given by fk|k (X|Z1:k ) =
fk (Zk |X) · fk|k−1 (X|Z1:k−1 ) f (Zk |Z1:k−1 )
(10)
where f (Zk |Z1:k−1 ) Z = fk (Zk |X) · fk|k−1 (X|Z1:k−1 )δX = fk (Zk |Ø) · fk|k−1 (Ø|Z1:k−1 ) Z + fk (Zk |{x}) · fk|k−1 ({x}|Z1:k−1 )dx = κ(Zk )(1 − pk|k−1 ) + κ(Zk ) · pk|k−1 1 0 X κ(Z − W ) k A · @fk|k−1 [1 − pD ] + fk|k−1 [pD LW ] κ(Zk ) W ⊂Zk 0 = κ(Zk )@1 − pk|k−1 · fk|k−1 [pD ] X
+pk|k−1
W ⊂Zk
1 κ(Zk − W ) A fk|k−1 [pD LW ] κ(Zk )
If X = Ø, then Eq.(10) becomes fk|k (Ø|Z1:k ) =
1 − pk|k−1
8 i < f (Zk |{xk }) , i wk = f (Zk |Ø) : 1,
and we can get pk|k
=
1 − fk|k−1 [pD ] +
P
p−1 k|k−1 − fk|k−1 [pD ] +
W ⊆Zk
P
k −W ) fk|k−1 [pD LW ] κ(Z κ(Zk )
W ⊆Zk
k −W ) fk|k−1 [pD LW ] κ(Z κ(Z ) k
If X = {x}, then Eq.(10) becomes
= fk|k−1 (x)
P k −W ) 1 − pD (x) + pD (x) W ⊆Zk LW (x) κ(Z κ(Zk ) · −1 P k −W ) pk|k−1 − fk|k−1 [pD ] + W ⊆Zk fk|k−1 [pD LW ] κ(Z κ(Z ) k
and we can get fk|k (x) fk|k ({x}|Z1:k ) pk|k
= fk|k−1 (x)
for Eki = 1
(11)
for Eki = 0
This operation[14] greatly reduces the computational requirement of the particle filter implementation. It results in a similar particle filter corrector as that in Ref.[8] in essence, but the derivation process and resulted expression are much simplified. From Eq.(8) and Eq.(9) we can get X LW (xi )κ(Zk − W ) f (Zk |{xik }) k = 1 − pD (xik ) + pD (xik ) f (Zk |Ø) κ(Zk ) W ⊆Zk
fk|k ({x}|Z1:k )
=
We propose to implement the proposed filter via a particle filter technique. In this hybrid state estimation problem, n Ns particles {Ekn , xn k , wk }n=1 are used to approximate the Bayes posterior. The binary variable Ekn ∈ {0, 1} is used to denote target existence: if Ekn = 0, then Xkn = Ø and the state vector n n n xn k is undefined; if Ek = 1, then Xk = {x k }. The weights PNs n Ns n {wk }n=1 are normalized such that n=1 wk = 1. The simplest and most widely implemented particle filter is the SIR particle filter where the target prior density is used as proposal and resampling is applied at every time step. Resampling is performed to mitigate particle degeneracy whereby all but a few particles get negligible weights. In the resampling step, particles with low weights are discarded and those with high weights are multiplied and the resulting particles have equal weights, that is wkn = 1/Ns . The prediction stage of the filter firstly predict the tarn , pB and pS (xn get existence variable Ekn using Ek−1 k−1 ). The next step is to predict the state vector for those particles n = 0 and with Ekn = 1: for newborn particles (i.e., Ek−1 n = 1), the state vector is drawn as a sample from the Ek|k−1 n =1 birth density bk|k−1 (x); for persisting particles (i.e., Ek−1 n n and Ek|k−1 = 1), the state vector is predicted as xk|k−1 ∼ n fk|k−1 (x|xn k−1 ). For those particles with Ek = 0, the state vector is undefined and no prediction is needed. The update stage of the filter incorporating measurement information is achieved via weighted resample–the weight for a particle being proportional to its likelihood[13] . Thus for the ith particle, (Eki , xik ), the resampling weight wki ∝ f (Zk |X). Since the weights are only required up to proportionality, we may divide through by f (Zk |Ø) and set:
1−p ·f [p ] P k|k−1 k|k−1 D κ(Zk −W ) +pk|k−1 W ⊆Zk fk|k−1 [pD LW ] κ(Zk )
= 1 − fk|k (Ø|Z1:k )
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(12) Notice that the summation item in Eq.(12) is computational demanding because it should be taken over all possibilities of W . We see that there are |Zk | possibilities of the value |Zk |! of |W |, and for a certain |W |, there are pos|W |!(|Zk | − |W |)! sibilities of W . So we seek to simplify it to make realization of the proposed filter possible. By substituting of Eq.(4) and Eq.(6), we have X
k −W ) 1 − pD (x) + pD (x) W ⊆Zk LW (x) κ(Z κ(Zk ) P k −W ) 1 − fk|k−1 [pD ] + W ⊆Zk fk|k−1 [pD LW ] κ(Z κ(Z ) k
III. Particle Filter Implementation
LW (x)κ(Zk − W )
W ⊆Zk
P
=
X
W ⊆Zk
e−γ(x)
Y
γ(x)φz (x)e−λC (λC /V )|Zk −W |
z∈W
= (λC /V )|Zk | e−(γ(x)+λC )
X W ⊆Zk
(V γ(x)/λC )|W |
Y z∈W
(13) φx (z)
Bernoulli Filter for Extended Target in Clutter Using Poisson Models According to Ref.[1], this equation exists: n Y
(1 + caj ) = 1 +
j=1
n X
X
ci
i=1
ak1 ak2 · · · aki (14)
{k1 ,...,ki }⊆{1,...,n}
where {k1 , . . . , ki } ⊆ {1, . . . , n} means selecting arbitrary i elements (mutually different and with no inherent order) from the set {1, . . . , n}. Using this result, Eq.(13) can be rewritten as X
LW (x)κ(Zk − W )
W ⊆Zk |Zk |
X
= (λC /V )|Zk | e−(γ(x)+λC ) X
×
(V γ(x)/λC )|W |
|W |=1
φx (zk1 ) · · · φx (zk|W | )
{k1 ,...,k|W | }⊆{1,...,|Zk |}
= (λC /V )|Zk | e−(γ(x)+λC ) 0 1 |Zk | Y ×@ (1 + (V γ(x)/λC )φx (zj )) − 1A
(15)
j=1
Algorithm 1 Particle filter implementation i s 1: Input: {Ek−1 , xik−1 }N i=1 , Zk 2: for i = 1 : Ns do 3: Draw u ∼ U [0, 1] i 4: if Ek−1 = 0 and u < pB then i 5: Ek|k−1 =1 i 6: else if Ek−1 = 0 and u < pB then i =0 7: Ek|k−1 8: else i i 9: Ek|k−1 = Ek−1 10: end if i i 11: if Ek−1 = 0 and Ek|k−1 = 1 then i 12: Draw xk|k−1 ∼ bk|k−1 (x) i i 13: else if Ek−1 = 1 and Ek|k−1 = 1 then i 14: Draw xk|k−1 ∼ fk|k−1 (x|xik−1 ) 15: else 16: xik|k−1 is undefined 17: end if 18: Assign the particle a weight, w ˜ki , according to Eq.(11) 19: end for P s i 20: Normalize weights: wki = w ˜ki / N ˜k , for i = 1, . . . , Ns i=1 w 21: for i = 1 : Ns do 22: Select index j i ∈ {1, . . . , Ns } with probability wki 23: 24: 25:
then X LW (x)κ(Zk − W ) κ(Zk ) W ⊆Zk 1 0 |Zk | Y = e−γ(x) @ (1 + (V γ(x)/λC )φx (zj )) − 1A
(16)
This convenient result allows the weight computation without exhaustive search all the possible W and makes real-time realization of the particle filter possible. A pseudo-code description of one iteration of the particle filter is summarised in Algorithm 1. The input to the algoi }, which rithm at time k are: the set of particles {xik−1 , Ek−1 approximates xk−1|k−1 and pk−1|k−1 ; the current set of measurements Zk . The weights of the particles are not required since they are all equal to 1/Ns after resampling. Lines 3-17 of algorithm 1 are prediction steps of the Bernoulli particle filter. Line 18 is the correction step. Lines 22-23 are resampling steps. The systematic resampling algorithm[15] is used in this study, its computational complexity is O(Ns ). We can also see that the prediction and correction of each particle can be paralleled in real time application. Using output of Algorithm 1, the pk|k is estimated by PNs
i=1
Eki
Ns
(17)
If pˆk|k exceeds a certain threshold τ ∈ (0, 1), target presence is declared. The target state is estimated then by ˆ k|k = x
PNs
i i i=1 x k Ek P Ns i i=1 Ek
i
i
j Eki = Ek|k−1 , xik = xjk|k−1
end for s Return: {Eki , xik }N i=1
IV. Simulation
j=1
pˆk|k =
329
(18)
We present a tracking example as proof-of-concept for the proposed algorithm. For comparison, performance of the BPFX is also shown. The BPF-X is implemented by setting L∗ = 1 because we are interested only in the target centroid estimate, and in this case it is adequate to use the standard point-target BPF (i.e. L∗ = 1)[11] . The reader is referred to Ref.[11] for more details on the BPF-X. An 2-D (planar) tracking scenario is considered over a surveillance area of 40×40m2 . A total of 60 time steps are used in the simulation. An extended target is introduced at time step 11 and deleted at time step 51. The target state is denoted by x = (x, x, ˙ y, y) ˙ T , where (x, y) and (x, ˙ y) ˙ represent the target Cartesian position and velocity components, respectively. Motion of the target is modeled according to the nearly-constant velocity model: 0 1 1 Ts 0 0 B0 1 0 0 C C (19) xk = B @ 0 0 1 Ts A xk−1 + wk 0 0 0 1 01 4 1 3 1 T T 0 0 4 s 2 s B 1 3 Ts2 0 0 C 2 B 2 Ts C (20) Q = σw 1 4 1 3A @ 0 T T 0 4 s 2 s 1 3 0 0 T Ts2 2 s where σw = 0.001m/s2 and the sampling interval is Ts = 1s. The mean number of target measurements per frame is fixed: γ(x) = 6. Measurements from the target are uniformly distributed within a circle that is positioned centered on the target centroid (xk , yk ) with fixed radius: R = 3m. The probability of target detection is fixed: pD (x) = 0.95. The false alarms are uniformly distributed over the sensor field-of-view, and are Poisson distributed in time with λC = 20.
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Chinese Journal of Electronics
The particle filter parameters are set as follows, the probability of target “birth” pB is 0.05, the survival probability pS is 0.95, the initial target exist probability is p0|0 = 0.1, the a priori and birth densities of the position component are both uniformly distributed in the surveillance area, while those of the velocity component are uniformly distributed in (0, 1)m/s, 4000 particles are used. The results of a single run are shown in Fig.1, in which the measurements and position estimation results are plotted in x and y coordinates versus time. We see that the proposed filter is able to track the extended target in dense clutter backgrounds, though momentarily confused at the beginning.
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ble. The BPF-X (L∗ = 1) has a comparatively small execution time (0.03 s on the average).
Fig. 2. Monte Carlo results. (a) Average probability of existence; (b) Position RMSE
Fig. 1. A single run of the proposed algorithm. Measurements (‘×’) and false alarms (‘+’), filter estimates (gray, dashed) and true tracks (dark, solid)
Monte Carlo results using 1000 runs are shown in Fig.2. Fig.2(a) presents the averaged estimation results of the existence probability pk|k of the two filters. Setting threshold τ = 0.5 for the proposed filter, we can see that the target can be detected with one time step delay after the target’s appearance at time step 11. The pk|k reliably stays at a high value, until, again with one time step delay falls and stays at a low value after the target disappeared at time step 51. Fig.2(b) shows the estimation performance of position in terms of Root mean square error (RMSE). The RMSE of the position estimation is defined as v u M u 1 X 2 ((ˆ xm ykm − yk )2 ) (21) RMSEk = t k − xk ) + (ˆ M m=1 where x ˆm ˆkm are the estimated target position at time k and y k of the mth run and M is the number of Monte Carlo runs. We see that the position estimation converges to the truth as time step increases. From Fig.2 we also see that the BPFX (L∗ = 1) has larger target detection and estimation convergence delays compared with the proposed filter, and the BPF-X has worse estimation performance than the proposed filter after convergence. The simulation is performed in the Matlab2009b/Slackware14.1 environment on a 3.2GHz Intel I5 computer with 4GB of RAM. The execution time of each time step (prediction, correction, resample and estimation) of the proposed filter is 0.19s on the average, thus we can conclude that real-time realization of the proposed filter is possi-
Now we evaluate the detection performance of the proposed filter in the detection terminologies. We estimate the probability of false alarm pF A using frames 1 to 10 of the 1000 Monte Carlo runs, where no target is present. More explicitly,
pF A =
« 10 „ M X X 1 1 pˆm k|k ≷ τ M × 10 m=1 0
(22)
k=1
where pˆm k|k is the estimated probability of existence at time k of the mth run. Similarly, probability of detection pDf inal is computed when the target is present. Setting τ = 0.02 for the proposed filter and τ = 0.1 for the BPF-X, the detection performances of them are shown in Table 1. We see that pDf inal of the proposed filter is higher than the BPF-X under lower pF A . Thus we can conclude that detection performance of the proposed filter is better than that of the BPF-X. We also see that in the convergence region (frame 51 to 70), the pDf inal is greater than 0.99. Table 1. Detection performance Parameters Proposed BPF-X (L∗ = 1) pF A 0.0146 0.0216 pDf inal (k=11:70) 0.9862 0.9694 pDf inal (k=51:70) 0.9996 0.9993
Performances of the proposed filter under different λC s are shown in Fig.3, where λC is set as 20, 40, 60, 80 and 100, respectively, with other parameters unchanged. 200 Monte Carlo runs are performed for each λC . We see that as λC increases, the detection and estimation performances (in terms of convergence delay and the estimation after convergence) both deteriorates. However, for λC as large as 100, the target can still be effectively detected and estimated.
Bernoulli Filter for Extended Target in Clutter Using Poisson Models
Fig. 3. Monte Carlo results for different λC s. (a) Average probability of existence; (b) Position RMSE
V. Conclusions In this paper we present a joint detection and tracking filter for single extended target in the clutter background. The Poisson extended target measurement model is adapted into the Bernoulli filter. A particle filter implementation of the proposed filter is proposed, in which simplifications are used which have greatly reduced the computation load. Simulation results show the effectiveness of the proposed method. The mean number of measurements and the target extension are assumed known in this study, future works should relax these assumptions by estimating them adaptively.
Data Processing of Small Targets, SPIE, Vol.5913, 2005. [7] M. Baum and U.D. Hanebeck, “Random hypersurface models for extended object tracking”, Proc. IEEE Int. Symp. Signal Process. Inf. Technol. (ISSPIT), Ajman, United Arab Emirates, pp.178–183, 2009. [8] R.P. Mahler, Statistical Multisource-Multitarget Information Fusion, Artech House Boston, Norwood, MA, 2007. [9] Y. Fu, J. Long and W. Yang, “Maneuvering multi-target tracking using the multi-model cardinalized probability hypothesis density filter”, Chinese Journal of Electronics, Vol.22, No.3, pp.634–640, 2013. [10] A. Swain and D. Clark, “The single-group phd filter: An analytic solution”, Proc. 14th Int. Conf. on Information Fusion, Chicago, USA, 2011. [11] B. Ristic and J. Sherrah, “Bernoulli filter for joint detection and tracking of an extended object in clutter”, IET Radar Sonar Navig., Vol.7, No.1, pp.26–35, 2013. [12] B. Ristic, B.-T. Vo, B.-N. Vo and A. Farina, “A tutorial on bernoulli filters: Theory, implementation and applications”, IEEE Transactions on Signal Processing, Vol.61, No.13, pp.3406–3430, 2013. [13] N. Gordon, D. Salmond and A. Smith, “Novel approach to nonlinear/non-gaussian bayesian state estimation”, IEE Proceedings F-Radar and Signal Processing, Vol.140, No.2, pp.107– 113, 1993. [14] D. Salmond and H. Birch, “A particle filter for track-beforedetect”, Proceedings of the American Control Conference, Arlington, VA, 2001. [15] G. Kitagawa, “Monte carlo filter and smoother for non-gaussian non-linear state space models”, J. Comput. Graph. Statist., Vol.5, No.1, pp.1–25, 1996. CAI Fei received the B.S. and M.S. degrees from National University of Defense Technology (NUDT) in 2008 and 2011, respectively. He is currently working toward the Ph.D. degree at the same university. His research interests include radar signal processing, target detection and tracking. (Email:
[email protected])
FAN Hongqi received the B.S. degree in mechanical engineering and automation from Tsinghua University in 2001, and the Ph.D. degree in information and communication engineering from NUDT in 2008. He is currently an associate professor at NUDT. His research interests include radar signal processing, target tracking and information fusion, and multi-agent systems. (Email: fan-
References [1] K. Gilholm and D. Salmond, “Spatial distribution model for tracking extended objects”, IEE Proceedings-Radar, Sonar and Navigation, Vol.152, No.5, pp.364–371, 2005. [2] J.W. Koch, “Bayesian approach to extended object and cluster tracking using random matrices”, IEEE Transactions on Aerospace and Electronic Systems, Vol.44, No.3, pp.1042–1059, 2008. [3] R. Mahler, “Phd filters for nonstandard targets, I: Extended targets”, 12th International Conference on Information Fusion, Seattle, WA, pp.915–921, 2009. [4] K. Granstr¨ om and U. Orguner, “A phd filter for tracking multiple extended targets using random matrices”, IEEE Transactions on Signal Processing, Vol.60, No.11, pp.5657–5671, 2012. [5] C. Lundquist, K. Granstr¨ om and U. Orguner, “An extended target cphd filter and a gamma gaussian inverse wishart implementation”, IEEE Journal of Selected Topics in Signal Processing, Vol.7, No.3, pp.472–483, 2013. [6] K. Gilholm, S. Godsill, S. Maskell and D. Salmond, “Poisson models for extended target and group tracking”, Signal and
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[email protected]) FU Qiang received the B.S. and Ph.D. degrees from NUDT in 1983 and 2004, respectively. He is currently a professor, Ph.D. supervisor at NUDT. His research interests include automatic target recognition, precision guidance, and radar signal and data processing. (Email:
[email protected])
