a bayesian estimation for single target tracking based on state mixture models
TRANSCRIPT
Contents lists available at SciVerse ScienceDirect
Signal Processing
Signal Processing 92 (2012) 1706–1714
0165-16
doi:10.1
$ Thi
609340
Universn Corr
E-m
liuwf@h
czhan@1 So
journal homepage: www.elsevier.com/locate/sigpro
A Bayesian estimation for single target tracking based on statemixture models$
Weifeng Liu a,n,1, Chenglin Wen a, Chongzhao Han b, Feng Lian b
a College of Automation, Hangzhou Dianzi University, 310018 Hangzhou, Zhejiang, Chinab Electronic Information Engr, Xi’an Jiaotong University, 710049 Xi’an, Shaan xi, China
a r t i c l e i n f o
Article history:
Received 10 December 2010
Received in revised form
13 November 2011
Accepted 7 January 2012Available online 17 January 2012
Keywords:
Bayesian estimation
Likelihood function
Mixture model
Single target tracking
84/$ - see front matter & 2012 Elsevier B.V. A
016/j.sigpro.2012.01.006
s work was supported in part by the NSFC (61
09, 61004087), the ZJNSF (Y1101218), and th
ity (KYS065609051).
esponding author.
ail addresses: [email protected],
du.edu.cn (W. Liu), [email protected] (C. W
mail.xjtu.edu.cn (C. Han), lianfeng1981@gma
me of the work was finished at Xi’an Jiaoton
a b s t r a c t
This paper presents a Bayesian algorithm for single target tracking using state mixture
model theory. Compared with the existing approaches, the proposed algorithm aims at
deriving the likelihood function of all measurements. Given this, an analytic Bayesian
algorithm is further proposed. Moreover, under linear Gaussian assumptions on the
dynamics and measurement model, a closed-form solution is proposed. Our study
demonstrates the effectiveness of the proposed method in single target detection and
tracking.
& 2012 Elsevier B.V. All rights reserved.
1. Introduction
Target tracking in a cluttered environment is a challengingproblem because of three aspects: the first aspect is thatclutter points (or multitarget case) confuse the target mea-surements. The second is that nonlinear system increases thedifficulty of estimation of target state. The third aspect is thetarget motion-mode uncertainty. Therefore, the first task triesto solve the following two uncertainties: target uncertaintyand measurement uncertainty. The target uncertaintydenotes that a target may be lost or false alarms may beproduced. The measurement uncertainty is due to clutterpoints or multiple targets. Various data association methodssuch as nearest neighbor filter (NNF) [1], probabilistic data
ll rights reserved.
175030, 91016020,
e Hangzhou Dianzi
en),
il.com (F. Lian).
g University.
association (PDA) [2], joint probabilistic data association(JPDA) [3] and multiple-hypothesis tracking (MHT) [4] havebeen proposed to address these two uncertainties. To reducecomputational load, the measurements found in a validationgate are used to estimate the state of a target. For the secondnonlinear aspect, some approaches are based on the lineartechniques such as extended Kalman filter, nonlinear meth-ods include particle filter and Gaussian sum approaches [5,6].The third aspect results in the multiple model approaches.The core of all these algorithms is to determine measurementlikelihood functions. Traditional approaches solve this bydeciding a target measurement and then giving the corre-sponding likelihood function. So they can be classified as adecision-before-estimation method.
Random finite set (RFS) based algorithms, where targetstates and their measurements are represented as RFSs [7],can avoid the above association step which is substituted byan implicit estimation process. They belong to a jointdecision and estimation approaches. To make them tract-able, the probability hypothesis density (PHD) filter wasproposed by Mahler [7]. Further development were given inRefs. [8–10]. The PHD filter recursively updates PHD inten-sity using the derived measurement RFS. It possesses several
W. Liu et al. / Signal Processing 92 (2012) 1706–1714 1707
prominent characteristics. First, it can deal with the uncer-tain number of targets and does not need the associationstep. Second, it is a joint decision and estimation approach.Nevertheless, the challenging problem may be its theoreti-cally abstract and incomplete.
Since the key of target tracking is to build the measure-ment likelihood function. The association based algorithmsfocus on each individual target likelihood function, whilethe RFS based algorithms emphasis on the first order oftarget RFS in the state space. To our knowledge, in the RFSframework, the first work which aims at deriving the wholemeasurement likelihood function in a cluttered conditionmay be due to Vo et al. [9]. They proposed an analyticsolution of the measurement RFS likelihood function for thesingle target system. Accordingly, in this paper we proposean analytic likelihood based on the single target statemixture models in the traditional probability framework.Research shows that the proposed likelihood has a differentexpression and is easier to understand compared with theRFS based methods.
Therefore, we first aim at deriving the measurementlikelihood function using all the measurements and furthergiving the Bayesian filter which can also avoid the explicitassociation step like the RFS approaches. In general, targetmeasurements and clutter measurements have differentdistributional characteristic. This inspires us to model themeasurements by using target state mixture models, whichis a powerful theory describing the observations coming fromdifferent random sources. Given assumptions on the mea-surements (target measurements and clutter measurements)being Poisson and independent, the mixing weights in themixture models will be known. We can thus analyticallyderive the measurement likelihood function. Based on this, aBayesian filter is further proposed. Compared with thetraditional association approaches, the proposed algorithmdoes not need the explicit association and can effectivelyreduce target lost rate. Moreover, unlike the RFS basedapproaches, the proposed algorithm is given in the familiarprobability framework and is easier to understand.
Preliminary results have been presented as a confer-ence paper [11]. This paper is a more complete version.The paper is organized as follows. Section 2 is ourproblem formulation. In Section 3, the measurementlikelihood function and Bayesian filter are investigated.Section 4 discusses a special case of linear Gaussiancondition and a closed-form solution is proposed. Twosimulations are, respectively, given in the above twosections. Section 5 concludes this paper.
2. Problem formulation
2.1. System model
The single target system model is given by
xk,1 ¼ qk�1ðxk�1;1,ok�1Þ ð1Þ
zk ¼ hkðxk,1Þþvk ð2Þ
where qk�1ð�Þ is the target state function, hkð�Þ is themeasurement function, xk,1 is the target state vector, ascompared with clutter state xk,0 (defined in the following).
zk denotes the target measurement, ok�1 is process noisevector, vk is measurement noise vector with a distribu-tional density gkð�Þ. Note that the state function (1) has nocontrol term because the tracked target belongs to a non-cooperative target in general, while for a cooperativetarget (or signal) the control term will be consideredand some tracking control schemes are then adopted[12,13]. Besides, in this paper, we only consider the singlesensor, although tracking a target under multiple sensorcondition also has a very important application such aswireless sensor [14].
2.2. Single target state mixture models
Mixture models describe the observations fzi,i¼1, . . . ,ng by (mixture models usually model the observa-tion distribution feature and thus is time independent, sothe time subscript is omitted here)
f ðzi9yÞ ¼ p1f 1ðzi9y1Þþ � � � þpmf mðzi9ymÞ ð3Þ
where f jðzi9yjÞ is the jth distribution, y¼ fp1, . . . ,pm,y1,. . . ,ymg are the mixing weights and parameters for individualdistributions, m is the number of distributions.
Let zk ¼ fzikg
nk
i ¼ 1 be measurements which originate fromtarget and clutter or false alarms at time k. We can imagea clutter state xk,0. Further, let Xk ¼ fxk,0,xk,1g be state setand PDðxk,1Þ be target probability of detection. The singletarget mixture models are thus given by
f kðzik9XkÞ ¼ pk,0ðXkÞckðz
ik9xk,0Þþpk,1ðXkÞgkðz
ik9xk,1Þ ð4Þ
where ckðzik9xk,0Þ and gkðz
ik9xk,1Þ are, respectively, clutter
and target distributions. Accordingly, if target is notdetected, the measurement mixture models can beexpressed by
f kðzik9XkÞ ¼ pk,0ckðz
ik9xk,0Þ ð5Þ
Besides, an empty model F is proposed in the followingwhen no measurement is received:
f kðF9XkÞ ¼ 1 ð6Þ
Measurement set zk belongs to a missing data which losesthe missing variable, referred to as indicative variablesei
k ¼ feik,jg
mk
j ¼ 0, where mk is the number of targets. j¼0denotes clutter and others denote targets. The indicativevariable meets with ei
k,j 2 f0;1g andPmk
j ¼ 0 eik,j ¼ 1. ei
k,j
indicates whether the ith measurement originates fromthe jth target. By adding the variables to the measure-ments, we get the complete data fzi
k,eikg
nk
i ¼ 1. The mixingweights are defined by
pk,j ¼ Pðeik,j ¼ 19XkÞ j¼ 0;1 ð7Þ
3. Single target Bayesian estimation
3.1. Likelihood function and Bayesian estimation
The above state mixture models (4)–(6) describe howmeasurements are produced. This subsection focus onhow to derive measurement likelihood function, referredto as f kðzk9XkÞ.
W. Liu et al. / Signal Processing 92 (2012) 1706–17141708
Proposition 1. Assume that the measurements fzikg
nk
i ¼ 1 are
mutually independent. The target measurements and clut-
tered are of distributions ckð�Þ and gkð�Þ, respectively. Then
the likelihood function of the measurements is
f kðzk9XkÞ ¼$0kðXkÞ
Ynk
l ¼ 1
ckðzik9xk,0Þ
þXnk
i ¼ 1
$ikðxkÞ
Ynk
l ¼ 1,lai
ckðzlk9xk,0Þgkðz
ik9xk,1Þ ð8Þ
where
$0kðXkÞ ¼ ð1�PDðxk,1ÞÞ½pk,0ðXkÞ�
nk ð9Þ
$ikðXkÞ ¼ PDðxk,1Þ½pk,0ðXkÞ�
nk�1pk,1ðXkÞ ð10Þ
The detail proof process is given in Appendix A. With thislikelihood function, we further can derive the single targetBayesian estimation, which is the key of this paper.
Proposition 2. Given prior distribution pk9k�1ðXk9z1:k�1Þ
and the likelihood function (8), assume that the distribution
of clutter is stable, that is, pk9k�1ðxk,09z1:k�1Þ ¼ pkðxk,09z1:kÞ.Then, the single target Bayesian posterior distribution is
pkðxk,19z1:kÞ ¼1
Cf
($0
k ðXkÞYnk
l ¼ 1
ckðzlk9xk,0,D0Þþ
Xnk
i ¼ 1
$ikðXkÞ
�Ynk
l ¼ 1,lai
ckðzlk9xk,0,D1Þgkðz
ik9xk,1,D1Þ
)pk9k�1ðxk,19z1:k�1Þ
ð11Þ
The proof is given in Appendix B. This expression shows anexplicit form for single target Bayesian estimation in acluttered environment. Coefficients $0
kðXkÞ, $ikðXkÞ can be
calculated through the probability of detection PDðxk,1Þ,mixing weights pk,0ðXkÞ and pk,1ðXkÞ. For a known PDðxk,1Þ
and mixing weights in (4), the Bayesian filter (11) can bedirectly calculated. Otherwise, the mixing weights need to beestimated.
3.2. Derivation of mixing weights
Here three cases are, respectively, investigated as follows.
3.2.1. Case 1: clutter measurements with known intensities
In this case we propose the following proposition tocompute the mixing weights.
Proposition 3. Let the number of the target measurements
and the number of the clutter measurements be Poisson
distributions with intensities lðXkÞ and rðXkÞ, respectively.
Then the mixing weights can be given as follows:
fpk,0ðXkÞ,pk,1ðXkÞg ¼rðXkÞ
rðXkÞþlðXkÞ,
lðXkÞ
rðXkÞþlðXkÞ
� �ð12Þ
The detail proof is given in Appendix C. In this paper, atarget is assumed to generate at most one measurement.
Thus, the mixing weights follow that
fpk,0ðXkÞ,pk,1ðXkÞg ¼rðXkÞ
rðXkÞþ1,
1
rðXkÞþ1
� �ð13Þ
In assumption of a single measurement, the number oftarget measurement may be 1 or 0 due to miss detection.Nevertheless, this case of no detection have been con-sidered in the first term of RSH of Eq. (8). This equationonly deals with the case of detection.
3.2.2. Case 2: clutter measurements with unknown
intensities
Clutter measurements which consist of mc mixturedistributions are shown as follows:
ckðzkÞ ¼ p1k,0ðzk9XkÞc
1k ðXkÞþ � � � þpmc
k,0ðXkÞcmc
k ðzk9XkÞ ð14Þ
where fpjk,0ðXkÞg are the mixing weights of the clutter
measurements and fcjkðXkÞg are the corresponding clutter
distributions. Some clutter intensities may be unknown.Learning algorithms such as Markov Chain Monte Carlosampler and EM algorithm, which we adopt here, areoften proposed to derive the intensities. The detail pro-cess of the EM algorithm is proposed in Table 1, whereLthreshold is a threshold for the change of the unknownclutter number in two iterative steps. Q(t) is log-like-lihood function [15]. Thus, the mixing weights of theclutter distributions are estimated by
fp1k,0ðXkÞ, . . . ,pmc
k,0ðXkÞ,pk,1ðXkÞg ¼l1k,0ðXkÞ
9zk9, . . . ,
lmc
k,0ðXkÞ
9zk9,lk,1ðXkÞ
9zk9
( )
ð15Þ
where p1k,0ðXkÞ, . . . ,pmc
k,0ðXkÞ are the mixing weights ofclutter measurements. l1k,0, . . . ,lmc
k,0 are the correspondingnumbers of measurements of clutter distributions ineach scan.
3.2.3. Case 3: no measurement
The number of measurements is zero when no mea-surement received. The empty model (6) is proposed inthis case. This implies that posterior density pðxk,19z1:kÞ isequal to prior density pk9k�1ðxk,19z1:k�1Þ. Therefore, thepredicted state is used here.
3.3. Sequence Monte Carlo implementation
The above Bayesian equation (11) is commonly non-linear and non-Gaussian. The sequence Monte Carloalgorithm is proposed here. For simplicity, the transitionfunction is adopted as proposal distribution. Assume thatat time k�1, a set of particles fwi
k�1,sik�1g are available.
Then, in the prediction step the particles are given asfollows:
wik9k�1 ¼wi
k�1
sik9k�1 � pðxk,19s
ik�1Þ
where pð�9�Þ is transition density for target state function.In the update step, the particles are updated as follows:
wik ¼
1
Cgðzk9s
ik9k�1Þw
ik9k�1
Table 1The intensity learning process using the EM algorithm.
Input: x̂1k9k�1 ,pð0Þ,1
k9k�1;0, . . . ,pð0Þ,mc
k9k�1;0,pð0Þ
k9k�1;1,t¼ 0
yð0Þk ¼ fyð0Þ,1k,0 , . . . ,yð0Þ,mc
k,0 ,yð0Þk,1g //ninitialization
while 9lðtÞk,0�lðt�1Þk,0 94Lthreshold
uðtÞ,1i,0 ¼ c1k ðz
ik9yðtÞ,1k,0 Þ, y, uðtÞ,mc
i,0 ¼ cmc
kðzi
k9yðtÞ,mc
k,0 Þ;//nclutter density
uðtÞi,1 ¼ gkðzik9yðtÞk,1Þ //ntarget density
eðtÞ,i,1k,0 ¼
pðtÞk,0uðtÞi,0Pnk
i ¼ 1 pðtÞ,jk,0 uðtÞ,ji,0 þp
ðtÞk,1uðtÞi,1
,y, //nclutter indicative variable
eðtÞ,i,mc
k,0 ¼pðtÞ,mc
k,1 uðtÞ,mc
i,0Pmc
j ¼ 1 pðtÞ,jk,0 uðtÞ,ji,0 þp
ðtÞk,1uðtÞi,1
//nclutter indicative variable
eðtÞ,ik,1 ¼pðtÞ,mc
k,1 uðtÞ,1i,1Pmc
j ¼ 1 pðtÞ,jk,0 uðtÞ,ji,0 þp
ðtÞk,1uðtÞi,1
//ntarget indicative variable
//nmeasurement numbers
flðtÞ,1k,0 , . . . ,lðtÞ,mc
k,0 ,lðtÞk,1g ¼ fPnk
i ¼ 1 eðtÞ,i,1k,0 , . . . ,
Pnk
i ¼ 1 eðtÞ,i,mc
k,0 ,Pnk
i ¼ 1 ei,ðtÞk,1 g
fpðtþ1Þ,1k,0 , . . . ,pðtþ1Þ,mc
k,0 ,pðtþ1Þk,1 g //nmixing weights
¼lðtÞk,0Pmc
j ¼ 1 lðtÞ,jk,0 þ lðtÞk,0
, . . . ,lðtÞ,mc
k,0Pmc
j ¼ 1 lðtÞ,jk,0 þ lðtÞk,0
,lðtÞk,1Pmc
j ¼ 1 lðtÞ,jk,0 þ lðtÞk,0
8<:
9=;
E-step: //nlog likelihood function
Q ðtÞ ¼Pnk
i ¼ 1fPmc
j ¼ 1 eðtÞ,i,jk,0 log½cjkðz
ik9yðtÞk,0Þ�þeðtÞk,1log½gkðz
ik9yðtÞk,1Þ�Þg
M-step:
fyðtþ1Þ,1k,0 , . . . ,yðtþ1Þ,mc
k,0 ,yðtþ1Þk,1 g ¼ argmaxy1
k,0 ,...,ymck,0
,yk,1Q ðtÞ
t¼ tþ1, lðtÞk,0 ¼ 0
for j¼1 to mc
if the jth clutter intensity is unknown
lðtÞk,0 ¼ lðtÞk,0þ lðtÞ,jk,0
end while
output:flðtÞ,1k,0 , . . . ,lðtÞ,mc
k,0 ,lðtÞk,1g
−1000 −800 −600 −400 −200 0 200 400 600 8000
200
400
600
800
1000
1200
1400
1600
x(m)
y(m
)
True trajectoryThe proposed algorithmUKF−PDA
Fig. 1. Target trajectory in x–y plane (with 190 clutters in average). The
extraneous measurements with intensities rðXkÞ ¼ 5.
W. Liu et al. / Signal Processing 92 (2012) 1706–1714 1709
sik ¼ si
k9k�1
where C is the normalization constant. In re-samplingstep, we use the Gaussian disturbance to increase thediversity of the particles.
3.4. A nonlinear experiment
In this experiment, a CT movement with unknownturn rate [10] is proposed to verify the given algorithm.The surveillance region is ½0,p� � ½0;2000� rad m. Thetarget is birth at location (0,0) m. The sampling period isT¼1. The target dynamics and measurement function aregiven by
xk ¼ Fðok�1Þxk�1þGk�1wk�1
xk ¼ ½pk,x, _pk,x,pk,y, _pk,y�T
ok ¼ok�1þ$k�1
Fðok�1Þ ¼
1sin ok�1T
ok�10 �
ð1�cos ok�1ÞT
ok�1
0 cos ok�1T 0 �sin ok�1T
01�cos ok�1T
ok�11
sin ok�1T
ok�1
0 sin ok�1T 0 cos ok�1T
266666664
377777775
zk ¼ arctanpk,y
pk,x,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2
k,xþp2k,y
q� �T
þnk
The probability of detection is PDðXkÞ ¼ 0:98. lc ¼ 3�10�2
ðrad mÞ�1 is the average clutter intensity. This impliesthere are around 190 clutter points in average in each scan.The clutter points can be generated in the whole surveil-lance region as literature [16] except in the validation gate.The process noise covariance is Qk ¼ diagð½25;25�T Þm2,measurement noise covariance is Rk ¼ diagð½s2
y ,s2r �
T Þ, wheresy ¼ 2p=180 rad,sr ¼ 10 m. The initial state distribution ofthe target is x0 �N ðx; x0,B0Þ, where x0 ¼ ½0 m,60 m=s,0 m,40 m=s,0:1 rad s�T , B0 ¼ diagð½100;4,100;4,0:0004�T Þ. Theestimated state vector is ½pk,x, _pk,x,pk,y, _pk,y,ok�
T which areposition, velocity and turn rate. Each particle si is thus a five-dimension vector. About 500 particles are proposed for atarget. In order to validate the proposed algorithm, weconsider extraneous measurements as in Ref. [10] and theclutter mixture models is proposed as follows:
ckðzk9XkÞ ¼ p1k,0Uð½0,p� � ½0;2000�Þ
þp2k,0N ðzk; ½atanðpk,y=pk,xÞ,2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2
k,xþp2k,y
q�T ,DkÞ ð16Þ
where p1k,0 and p2
k,0 are, respectively, the clutter weights ofuniform distribution and Gaussian distribution. The latterGaussian distribution describes the extraneous measure-ments. Though the extraneous measurements and targetstate are dependent, Eq. (11) is effective because the stateinvolve the information of the extraneous measurements.We verify the proposed algorithm in the following two casesand compare it with the UKF-PDA filter
1.
Known Poisson intensities. The Poisson intensity of theuniform clutter is r1ðXkÞ ¼ 190 and the Poisson inten-sity of the Gaussian clutter is r2ðXkÞ ¼ 5. This impliesthat the mixing weights of the clutter and targetare p1k,0ðXkÞ ¼ 190=ð190þ5Þ, p2k,0ðXkÞ ¼ 5=ð190þ5Þ, pk,0
ðXkÞ ¼ ð190þ5Þ=ð190þ5þ1Þ, and pk,1ðXkÞ ¼ 1= ð190þ5þ1Þ.
2.
Unknown Poisson intensities. The Poisson intensity ofthe extraneous measurements is also r2ðXkÞ ¼ 5 but
0 10 20 30 40 50−2000
−1000
0
1000
2000
time step(s)
x(m
)ClutterTrue trackExtraneous measurementsEstimation: the proposed algorithmEstimation: UKF−PDA
0 10 20 30 40 500
500
1000
1500
2000
time step(s)
y(m
)
Fig. 2. Target trajectory in x,y directions (with 190 clutters in average). The extraneous measurements with intensities rðXkÞ ¼ 5.
0
200
400
600
RM
SE
(m)
0 0.005 0.01 0.015 0.02 0.025 0.030
0.2
0.4
0.6
0.8
1
clutter intensity (rad.m)−1
0 0.005 0.01 0.015 0.02 0.025 0.03
clutter intensity (rad.m)−1
CP
EP
(%)
The proposed algorithmUKF−PDA
Fig. 3. RMSE and CPEP for varying clutter intensities (500 MC runs). The
extraneous measurements with intensity rðXkÞ ¼ 5.
0 10 20 30 40 500
1
2
3
4
5
6
7
8
9
10
11
time step(s)
num
ber o
f ext
rane
ous
mea
sure
men
ts
True number of extraneous measurementsEstimation number of extraneous measurements
Fig. 4. Estimation of the extraneous measurement number against time
using the EM algorithm. The extraneous measurements with unknown
intensity rðXkÞ ¼ 5.
W. Liu et al. / Signal Processing 92 (2012) 1706–17141710
remains unknown. We use the EM algorithm to esti-mate the intensity at each time step.
Fig. 1 shows the true track and the estimation track in x-y
plane with a known poisson intensity. The UKF-PDAconsistently loses the track due to the extraneous mea-surements and heavy clutter. This is also seen in Fig. 2which gives track in x, y coordinates, respectively. Fig. 3further illustrates root mean square error (RMSE) andcircular position error probability (CPEP) against clutterdensity in 500 Monte Carlo runs on the same targettrajectory with independently generated measurements.The corresponding error radius is 50 m. It can be seen
from Fig. 3 that when clutter intensity is varied at0�0:03 rad m�1, the RMSE for the proposed algorithm isvaried at 27–44 m in comparison with the UKF-PDA filterbeing 58–572 m. In lost rate, the CPEP metric inFig. 3 shows that the proposed algorithm is varied at2.4%–12.4%, while the UKF-PDA algorithm is varied at10%–94.8%. These show that the proposed algorithm per-forms better than the UKF-PDA filter. A reasonable expla-nation is that the proposed algorithm retains moreinformation than UKF-PDA filter. For example, when themeasurement errors are larger in several continuoussteps, the UKF-PDA filter tends to lose target. In theunknown intensity case, Fig. 4 suggests that the number
W. Liu et al. / Signal Processing 92 (2012) 1706–1714 1711
of the unknown clutter measurements can be approxi-mated well by the proposed learning process.
4. Bayesian estimation under linear Gaussian case
Given linear Gaussian (LG) assumptions for the targetdynamic and measurement function, the system functionsof Eqs. (1) and (2) reduce to
xk,1 ¼ Fk�1xk�1;1þGk�1ok�1 ð17Þ
zk ¼Hkxk,1þvk ð18Þ
where process noise ok�1 and measurement noise vk are,respectively, zero-means and whites. Since the predictedstep is the same as the classic Kalman filter, this sectionconsiders the special case of update step of Bayesianequation (11).
4.1. Basic assumptions and the LG Bayesian estimation
Proposition 4. Under the following assumptions:
A.1
The functions of dynamic and measurement follow theLG conditions.
A.2 The probability of detection PD is state independenceand a constant and the clutter follows Poisson with
intensity l.
A.3 The target produces one measurement at most. A.4 The process noise and measurement noise are inde-pendent, i.e., covðok,vkÞ ¼ 0 and with covariances
covðokÞ ¼Qk and covðvkÞ ¼ Rk, respectively. If the
prior distribution of the target state follows Gaussian
mixture
pk9k�1ðxk,19z1:k�1Þ ¼XJk9k�1
j ¼ 1
pk9k�1,jN ðxk,1;mk9k�1,j,Pk9k�1,jÞ
ð19Þ
Then, the LG Bayesian estimation is given by
pkðxk,19z1:kÞ ¼Xnk
i ¼ 0
XJk9k�1
j ¼ 1
pik,jqk,jðz
ikÞN ðxk,1;mi
k,j,Pik,jÞ ð20Þ
The total number of the Gaussian terms is Jk ¼ ðnkþ1ÞJk9k�1.The posterior distribution is composed of two parts.
�
Miss detection terms: for i¼ 0,j¼ 1;2, . . . ,Jk9k�1p0k,j ¼ n
0kpk9k�1,j ð21Þ
m0k,j ¼ mk9k�1,j ð22Þ
qk,jðz0kÞ ¼ 1 ð23Þ
P0k,j ¼ Pk9k�1,j ð24Þ
�
Detection terms: i¼ 1, . . . ,nk,j¼ 1, . . . ,Jk9k�1pik,j ¼ ak,ipk9k�1,jqjðz
ikÞ ð25Þ
qk,jðzikÞ ¼N ðzi
k;Hkmk9k�1,j,Rk,jþHkPk9k�1,jHTk Þ ð26Þ
mik,j ¼ mk9k�1,jþKk,jðz
ik�Hkmk9k�1,jÞ ð27Þ
Pik,j ¼ ðI�Kk,jHkÞPk9k�1,j ð28Þ
Kk,j ¼ Pk9k�1,jHTk ðHPk9k�1,jH
TkþRkÞ
�1ð29Þ
The detail proof is given in Appendix D.
4.2. Target detection and state estimation
The IPDA filter [17,18] uses the probability of a trackexistence to judge whether a target survives. Bethel et al.[19] proposed an analogous process to detect whether atarget exists. Here we propose a surviving variable (SV) ws
k
which is the weighted sum of the detection terms to judgethe track existence,
wsk ¼
Xnk
i ¼ 1
XJk9k�1
j ¼ 1
pik,j ð30Þ
If the SV exceeds a certain threshold tb, then a target trackmay exist. For state estimation, we consider two rules ofMAP and MSE. Under the MAP criterion
x̂MAPk,1 ¼ argmax
mik,j
fpik,jg
i ¼ 0,...,nk
j ¼ 1,...,Jk9k�1ð31Þ
under the MSE criterion
x̂MSEk,1 ¼
Xnk
i ¼ 0
XJk9k�1
j ¼ 1
pik,jm
ik,j ð32Þ
Besides, if no limitation is adopted, the number of thecomponent terms will become explosive. The managementtechniques of the Gaussian terms are proposed. Theyinclude term combination of the approximate terms andterm deletion of the terms whose weights are less thancertain threshold.
4.3. A linear Gaussian experiment
Here a nearly constant velocity model is considered. Thesurveillance region is ½�1000;1000� � ½�1000;1000�m2.The target birth takes place at (0,0) m. The sampling timeis 1 s. The probability of detection is PD¼0.98, clutteraverage intensity is l¼ 5� 10�5 m�2, which gives anaverage of 200 clutter points per scan. Process noisecovariance Qk ¼ diagð½25;25�T Þ, measurement covarianceRk ¼ diagð½100;100�T Þ. The target initial state followsx0 �N ðx; x0,B0Þ, where x0 ¼ ½0 m,�3 m=s,0 m,3 m=s�T ,B0 ¼ diagð½100;25,100;25�T Þ. The deletion threshold istD ¼ 1:0� 10�5 and the combination threshold is tC ¼ 5 m.
4.3.1. Tracking performance
We compared the proposed algorithm with the PDA filter.Fig. 5 shows the RMSEs and the CPEP of these two algorithmsagainst clutter average intensity. We gradually increase theclutter intensity from 0 to 5:0� 10�5 m�2. It can be seenfrom Fig. 5 that the RMSEs of these two algorithms also
0
10
20
30
RM
SE
(m)
0 1 2 3 4 50
10
20
30
clutter intensity (× 10−5m−2)
0 1 2 3 4 5
clutter intensity (× 10−5m−2)
CP
EP
(%)
The proposed algorithmPDA filter
Fig. 5. RMSE and CPEP for varying clutter intensities (500 MC runs).
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
λ=3.75× 10−5
λ=2.5× 10−5
λ=1.25× 10−5
Fig. 6. The surviving variable (SV) against time (500 MC runs).
W. Liu et al. / Signal Processing 92 (2012) 1706–17141712
increase, but the proposed algorithm shows somethingadvantage in precision than the PDA filter. The reason is thatthere always exists the case where the target measurementdoes not lie in the validate gate in each run. Nevertheless, therunning speed of the proposed algorithm is slower thanthe PDA filter. Besides, the CPEP metrics are shown in Fig. 5.The plot demonstrates that with the increase of the clutteraverage intensity from 0 to 5� 10�5 m�2, the CPEPs for theproposed algorithm vary from 0% to 0.6%, while the CPEPs forthe PDA filter vary from 0% to 10%. The reason is that theproposed algorithm saves more history (prior) informationusing multiple Gaussian terms. Correspondingly, thePDA filter uses one Gaussian term holding all the priorinformation.
4.3.2. Target detection
In order to show the problem, consider two targets thatappear from the zero point but survive in different timeintervals (5, 20] s and (25, 45] s, respectively. The SV ws
k isused to judge whether a target track exists. Fig. 6 shows thatthe SV ws
k against time through 500 MC runs. It suggests thatin target surviving time the corresponding SV exceeds 0.9and remains stable. In the time of target disappearing, thenumber is below 0.9. Hence, we here use tu ¼ 0:9 as athreshold to judge whether a target track exists. It can beseen from Fig. 6 that even in the heavy clutter intensity theSV still remains stable. In addition, in time intervals (0,5] s,(20, 25] s, and (45, 50] s, where no target exists, the SV showsan increasing tendency against time. This is because a newtarget birth model is added at the zero point in each time.Thus, the number of the birth model is also increased againsttime. The SV therefore increases with time due to clutter.Nevertheless, the SV is effective for it is still below thethreshold 0.9.
4.4. Discussion: possible applications in shape or feature
detection, recognition and tracking
In previous discussion, a basic assumption is that thetarget is with a point. This is true in most radar and
infrared tracking when target is scores of or even hundredof kilometers away from the sensors. For a near distance,the target will have a shape in appearance. This showsthat the point tracking and the shape tracking does nothave a strict boundary. In this subsection, we discuss thepossible applications of the proposed algorithm in shapeor features detection, recognition and tracking.
4.4.1. Shape or feature detection and recognition
Shape or feature detection and recognition are widelyused in image processing and some image processingapproach was also used in radar tracking [20]. If the targetof interest and the background have a distinct distributioncharacteristics and the target can be described to be apoint in parametric space or feature space, the proposedalgorithm can be used directly. For example, the houghtransfer approach models a shaped target to a point inparameter space. The detection and recognition of ashaped target can be decided by the coefficients ofposterior distribution (11), which can be rewritten as
pkðxk,19z1:kÞ ¼Xmk
j ¼ 1
pk,jpk,jðxk,19z1:kÞ ð33Þ
where mk is the number of all possible shape or featuredistributions fpk,jðxk,19z1:kÞg. Under the MAP criterion, thedistribution that has the largest mixing weight indicatesthe feature of target of interest. This is in fact the decisionor selection of a feature from a group of features.
4.4.2. Shape or feature tracking
The tracking process can be seen as a continuousdetection and recognition for a target. The proposed recur-sive Bayesian estimation needs the state transfer function toobtain the prior state distribution pkþ19kðxk,19z1:k�1Þ. In theshape or feature space, a shape or feature transfer functionis required. Besides, since division of the event space fEi
kg
become intractable for multiple shapes or features, directextension of the likelihood function is a difficult thing. Someoptimal algorithms such as the EM algorithm and Markov
W. Liu et al. / Signal Processing 92 (2012) 1706–1714 1713
Chain Monte Carlo (MCMC) sampling are adopted before thederivation of the likelihood function.
5. Conclusion
In this paper a Bayesian tracking algorithm for singletarget is proposed based on the state mixture models.Traditional tracking algorithms such as PDAF, JPDA andMHT, estimate the target state through detection-before-estimation or decision-before-estimation, where the deci-sion process is finished by the association step. In ouralgorithm this decision is completed by the selection ofthe weight of posterior distribution which has the largestvalue in a group of state distributions.
The proposed algorithm belongs to a joint decision andestimation algorithm and without the explicit associationstep. The main reason is that the measurement likelihoodfunction can be derived through the state mixture modelsunder the assumption of independence of all the mea-surements. Based on this, we further propose an explicitBayesian estimation algorithm. Moreover, under LG con-dition, the recursive equations with a close form areproposed. The proposed algorithm can deal with the caseof target birth by importing the target birth model in theprior distribution. Additionally, we propose a survivingvariable to judge whether a target track exists. Experi-ments show that the proposed algorithm performs betterthan the traditional approaches.
Acknowledgment
The authors would like to express their gratitude toProfessor Shaosheng Zhou at the Hangzhou Dianzi Uni-versity for his valuable suggestion.
Appendix A. Proof of Proposition 1
According to the law of total probability, the likelihoodfunction can be expressed as
f kðzk9XkÞ ¼ f kðzk9Xk,D0ÞPðD09XkÞþ f kðzk9Xk,D1ÞPðD19XkÞ
¼ ð1�PDðxk,1ÞÞf kðzk9Xk,D0ÞþPDðxk,1Þf kðzk9Xk,D1Þ ð34Þ
where D0 denotes the event that the target is not detectedand D1 denotes target is detected. Here, assume thatmeasurements fzi
kgnk
i ¼ 1 are mutually independent. Then,the detected term f kðzk9Xk,D1Þ and the undetected termf kðzk9xk,D0Þ can be, respectively, defined by
f kðzk9Xk,D1Þ ¼Xnk
i ¼ 1
f kðzk9Xk,D1,EikÞPðE
ik9Xk,D1Þ
¼Xnk
i ¼ 1
pk,1ðXkÞ½pk,0ðXkÞ�nk�1
Ynk
l ¼ 1,lai
ckðzlk9xk,0Þgkðz
ik9xk,1Þ
ð35Þ
where Eik denotes the event that the ith measurement
comes from the target, PðEik9Xk,D1Þ is the probability of a
partition of the measurements and its probability can becalculated through coefficient terms pk,1ðXkÞ½pk,0ðXkÞ�
nk�1.Similarly, the undetected term f kðzk9xk,D0Þ is given by
f kðzk9Xk,D0Þ ¼ pnk
k,0ðXkÞYnk
l ¼ 1
ckðzlk9xk,0Þ ð36Þ
Substituting Eqs. (35) and (36) into (34), this completesthe proof of the likelihood function.
f kðzk9XkÞ ¼$0k ðXkÞ
Ynk
l ¼ 1
ckðzik9xk,0Þþ
Xnk
i ¼ 1
$ikðxkÞ
�Ynk
l ¼ 1,lai
ckðzlk9xk,0Þgkðz
ik9xk,1Þ
Appendix B. Proof of Proposition 2
Single target Bayesian filter is defined as follows:
pkðXk9z1:kÞ ¼1
Cff kðzk9XkÞpk9k�1ðXk9z1:k�1Þ ð37Þ
where Cf is a normalization constant. Assume that thetarget state and the clutter state are independent condi-tional on z1:k and z1:k�1, respectively.
pkðXk9z1:kÞ ¼ pkðxk,09z1:kÞpkðxk,19z1:kÞ ð38Þ
pk9k�1ðXk9z1:k�1Þ ¼ pk9k�1ðxk,09z1:k�1Þpk9k�1ðxk,19z1:k�1Þ ð39Þ
Given the noise stable condition pkðxk,09z1:kÞ ¼ pk9k�1
ðxk,09z1:k�1Þ and substituting (38) and (39) into (37).Hence, the Bayesian filter (37) is reduced to
pkðxk,19z1:kÞ ¼1
Cff kðzk9XkÞpk9k�1ðxk,19z1:k�1Þ ð40Þ
Furthermore, it follows from Eq. (34) that
pkðxk,19z1:kÞ ¼1
Cff$0
kðXkÞYnk
l ¼ 1
ckðzlk9xk,0,D0Þþ
Xnk
i ¼ 1
$ikðXkÞ
�Ynk
l ¼ 1,lai
ckðzlk9xk,0,D1Þgkðz
ik9xk,1,D1Þgpk9k�1ðxk,19z1:k�1Þ
Appendix C. Proof of Proposition 3
Let the target measurement be zt and the clutteredmeasurement be zc. Assume they are Poisson with inten-sities lðXkÞ and rðXkÞ, respectively. Here Xk ¼ fxk,0,xk,1g isthe state set of target and clutter. That is
Pð9zt99XkÞ ¼ PoissonðlðXkÞÞ
Pð9zc99XkÞ ¼ PoissonðrðXkÞÞ
where 9 � 9 is the number of the measurements. Let9z9¼ 9zt9þ9zc9, then 9zt9 and 9zc9 conditional on 9z9 followthe following binomial distribution:
Pð9zt999z9,XkÞ ¼ Bð9z9,ptkðXkÞÞ
Pð9zc999z9,XkÞ ¼ Bð9z9,pckðXkÞÞ
where parameters
ptkðXkÞ ¼
lðXkÞ
lðXkÞþrðXkÞ
W. Liu et al. / Signal Processing 92 (2012) 1706–17141714
pckðXkÞ ¼
rðXkÞ
lðXkÞþrðXkÞ
Parameters ptkðXkÞ,p
ckðXkÞ denote success probability in
each trial probability. In other words, when we receive ameasurements in a single target and cluttered environ-ment. It belongs to correct measurement or incorrectmeasurement with probabilities pt
kðXkÞ and pckðXkÞ, respec-
tively. This shows
pk,0ðXkÞ ¼ ptkðXkÞ
pk,1ðXkÞ ¼ pckðXkÞ
This completes the proof of Eq. (12).
Appendix D. Proof of Proposition 4
The mixing weights were given by (12) with assump-tions A.1–A.4
ðpk,0ðXkÞ,pk,1ðXkÞÞ ¼l
lþ1,
1
lþ1
� �ð41Þ
Thus, the coefficients $0k ,$i
k in (8) follow that
$0k ¼ð1�PDÞl
nk
ðlþ1Þnkð42Þ
$ik ¼
PDlnk�1
ðlþ1Þnkð43Þ
Then, substitute these coefficients to the likelihoodfunction in Proposition 1 and normalize it. We get
pkðzk9xk,1Þ ¼ Cf a0kþ
Xnk
i ¼ 1
aikN ðzi
k;Hxk,1,RkÞ
!ð44Þ
where
a0k ¼ 1�PD, ai
k ¼PD
lð45Þ
Cf ¼lnk
ðlþ1Þnkð46Þ
With target prior distribution pk9k�1ðxk,19z1:k�1Þ followingGaussian mixture, according to Bayesian equation (11),the posterior distribution (20) is thus derived. Notethat in this proof, the following Gaussian equations areadopted [9]:
N ðz;Hxþb,RÞN ðx;m,PÞ ¼ qðzÞN ðx, ~m, ~PÞ
qðzÞ ¼N ðz;Hmþb,RþHPHTÞ
where~m ¼ mþKðz�Hm�bÞ, ~P ¼ ðI�KHÞP, K ¼ PHT
ðRþPHPTÞ�1.
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