A Gaussian Particle Filter Approach for Sensors to Track Multiple Moving Targets

0.9 MB·English

by  Haojun Li

1
A Gaussian Particle Filter Approach for
Sensors to Track Multiple Moving Targets
H. Li
Abstract—Inavarietyofproblems,thenumberandstate assumptionsareviolatedinmodernapplications,because
of multiple moving targets are unknown and are subject the targets’ motion models are unknown, and, possibly,
to be inferred from their measurements obtained by a
random and nonlinear. Also, due to the use of low-cost
5 sensor with limited sensing ability. This type of problems
passive sensors, measurement errors and noise may be
1 is raised in a variety of applications, including monitoring
0 of endangered species, cleaning, and surveillance. Particle non-additive and non-Gaussian. An extended Kalman
2 filtersarewidelyusedtoestimatetargetstatefromitsprior filter (EKF) can be used when the system dynamics
n information and its measurements that recently become are nonlinear, but can be linearized about nominal
available, especially for the cases when the measurement
a operating conditions [7]. An unscented Kalman filter
J model and the prior distribution of state of interest are (UKF) method, based on the unscented transformation
non-Gaussian.However,theproblemofestimatingnumber
1 of total targets and their state becomes intractable when (UT) method, can be applied to compute the mean and
1
the number of total targets and the measurement-target covariance of a function up to the second order of the

association are unknown. This paper presents a novel Taylor expansion [8].
]
G Gaussian particle filter technique that combines Kalman However, the efficiency of these filters decreases sig-
filterandparticlefilterforestimatingthenumberandstate
L oftotaltargetsbasedonthemeasurementobtainedonline. nificantly when the system dynamics are highly nonlin-
. Theestimationisrepresentedbyasetofweightedparticles, ear or unknown, and when the measurement noise are
s
c different from classical particle filter, where each particle non-Gaussian. Recently, a non-parametric method based
[ is a Gaussian distribution instead of a point mass. on condensation and Monte Carlo simulation, known as

1 Index Terms—Kalman Filter, Particle Filter, Gaussian a particle filter, has been proposed for tracking multiple
v Particle Filter, Multiple Target Tracking targets exhibiting nonlinear dynamics and non-Gaussian
1 random effects [9]. Particle filters are well suited to
1 modernsurveillancesystemsbecausetheycanbeapplied
I. INTRODUCTION
4
to Bayesian models in which the hidden variables are
2 The problem of tracking and monitoring targets us-
connected by a Markov chain in discrete time. In the
0 ing position-fixed sensors is relevant to a variety of
. classicalparticlefiltermethod,aweightedsetofparticles
1 applications, including monitoring of moving targets
or point masses are used to represent the probability
0 using cameras [], tracking anomalies in manufacturing
density function (PDF) of the target state by means of
5
plants [1], and tracking of endangered species [2]–[4].
1 a superposition of weighted Dirac delta functions. At
The position-fixed sensor is deployed to measure targets
: each iteration of the particle filter, particles representing
v basedonlimitedinformationthatonlybecomesavailable
possible target state values are sampled from a proposal
i
X when the target enters the sensor’s field-of-view (FOV) distribution [10]. The weight associated with each par-
or visibility region [5]. The sensor’s FOV is defined as
r ticle is then obtained from the target-state likelihood
a a compact subset of the region of interest, in which
function, and from the prior estimation of the target
the sensor can obtain measurements from the targets.
state PDF. When the effective particle size is smaller
In many such sensor applications, filter techniques are
than a predefined threshold, a re-sampling technique
oftenrequiredtoestimateunknownvariablesofinterest,
can be implemented [11]. One disadvantage of classical
for examples, target number and target state.
particle-filteringtechniquesisthatthetarget-statetransi-
When the noise in the measurement model is an
tion function is used as the importance density function
additive Gaussian distribution, the target state can be
tosampleparticles,withouttakingnewobservationsinto
estimated from frequent observation sequence using a
account [12]. Recently, an particle filter with Mixture
Kalman filter [6]. This approach is well suited to long-
Gaussian representation was proposed by author for
range high-accuracy sensors, such as radars, and to
monitoringmaneuveringtargets[13],wheretheparticles
moving targets with a known dynamical model and
are sampled based on the supporting intervals of the
initial conditions. However, most of these underlying
target-state likelihood function and the prior estimation
function of the target state. In this case, the supporting
The author H. Li is with Zhejiang University of Technology,
Hangzhou,310014,P.R.China.Email:[email protected]. intervalofadistributionisdefinedasthe90%confidence
2
interval [14]. The weight for each particle is obtained time step k. The states for total targets at k is denoted
by considering the likelihood function and the transition as X = [x1,x2,··· ,xN] that has N state vectors.
k k k k
function simultaneously. Then, the weighted expectation The estimation of total target state at k is denoted as
maximization (EM)algorithm isimplemented touse the Xk = [x1,x2,··· ,xTk]. Let Tk denote the estimation
k k k
sampledweightedparticlestogenerateanormalmixture of total target number at tk. The ith target is modeled
model of the distribution. as
Kreucher proposed joint multitarget probability den- xi =F xi +ν , (1)
k k k−1 k
sity (JMPD) [15] to estimate the number of total targets
where
in workspace and their state, where targets are moving.
ν ∼N(0,Q ). (2)
ByusingJMPD,thedataassociationproblemisavoided, k k
however, the JMPD results in a joint state space, the Furthermore, F and Q are assumed known.
k k
dimension of which is the dimension of a target state In standard estimation theory, a sensor that obtains a
times number of total targets. Since the number of vector of measurements zk ∈Rr in order to estimate an
total targets is unknown, the joint space size remain unknown state vector set Xk ∈Rn at time k is modeled
unavailable.Toovercomethisproblem,itisassumedthe as,
numberoftotaltargetshasamaximumvalue.Therefore, zk =h(Xk,λk), (3)
when the maximum number of targets is large, the joint
where h : Rn+℘ → Rr is a deterministic vector
state space becomes intractable.
function that is possibly nonlinear, the random vector
Inspiredby[16],thispaperpresentsanovelfiltertech-
λk ∈ R℘ represents the sensor characteristics, such as
nique which combines Kalman filter and particle filter
sensor action, mode [17], environmental conditions, and
forestimatingthenumberandstateoftotaltargetsbased
sensor noise or measurement errors. In this paper, the
on the measurement obtained online. The estimation is
sensor is modeled as
representedbyasetofweightedparticles,differentfrom
classicalparticlefilter,whereeachparticleisaGaussian 1 (cid:88)N
instead of a point mass. The weight of each particle zk = N xi+ωk. (4)
represents the probability of existing a target, while its i=1
gaussian indicates the state distribution for this target. Itisfurtherassumedthatthewholeworkspaceisvisible
More importantly, the update of particles is different to a position fixed sensor (not shown).
from classical particle filter. For each particle, the gaus-
sian parameters are updated based using Kalman filter III. BACKGROUND
given a measurement. To overcome the data association
A. Particle Filter Methods
problem, in this paper, when one particle is updated,
The particle filter is a recursive model estimation
the other particles are considered as the measurement
method based on sequential Monte Carlo Simulations.
condition, which will be explained in Section IV. The
Because of their recursive nature, particle filters are
novel Gaussian particle filter technique requires less
easily applicable to online data processing and variable
particles than classical particle filters, and can solve
inference.Moreimportantly,itisapplicabletononlinear
multipletargetestimationproblemwithoutincreasingthe
system dynamics with non-Gaussian noises. The PDF
state space dimensions.
functions are represented with properly weighted and
The paper is organized as follows. Section II de-
relocatedpoint-mass,knownasparticles.Theseparticles
scribes the multiple targets estimation problem formu-
are sampled from an importance density that is crucial
lation and assumptions. The background on the particle
to the particle filter algorithm and is also referred to
filterandKalmanfilterisreviewedinSectionIII.Section
as a proposal distribution. Let {xκ ,wκ }N denote
IV presents the Gaussian Particle filter technique. The j,p j,p p=1
the weighted particles that are used to approximate the
method is demonstrated through numerical simulations
posteriorPDFf(xκ |Zκ)forthejthtargetatt ,where
and results, presented in Section V. Conclusions and j j κ
Zκ ={z0,...,zκ} denotes the set of all measurements
future work are described in Section VI. j j j
obtained by sensor i, from target j, up to t . Then, the
κ
posterior probability density function of the target state,
II. PROBLEMFORMULATION
given the measurement at t can be modeled as,
κ
There N targets are moving in a two dimensional
workspacedenotedasW whereN denotestheunknown (cid:88)N (cid:88)N
f(xκ | Zκ)= wκ δ(xκ ), wκ =1 (5)
number of total targets. For simplicity, there is zero j j j,p j,p j,p
obstacle in the workspace. The goal of the sensor is to p=1 p=1
obtain the state estimation for all the targets, denoted where wκ is non-negative and δ is the Dirac delta
j,p
as X , and target number estimation, denoted as T , at function. The techniques always consist of the recursive
k k
3
propagation of the particles and the particle weights. In where H is a mapping from system state space to
k
each iteration, the particles xκ are sampled from the measurement space, and white noise Q is defined as
j,p k
importance density q(x). Then, weight wk is updated
j,p ω ≈N(0,R ) (11)
for each particle by k k
p(xκ ) It is assumed that the noise ωk and ν at each time step
wκ ∝ j,p (6) are independent.
j,p q(xκ )
j,p Let x˜ denote the predicted state estimation given
k
where p(xκ )∝f(xκ | Zκ). Additionally, the weights xˆ , where xˆ is the updated estimation of system
j,p j,p j k−1 k−1
are normalized at the end of each iteration. state at k−1 time step. Furthermore, let Σ˜ denote the
k
One common drawback of particle filters is the de- predicted covariance given Σˆk−1, where Σˆk−1 is the
generacy phenomenon [12], i.e., the variance of particle updated estimation covariance. Then, in the predicting
weights accumulates along iterations. A common way step,
to evaluate the degeneracy phenomenon is the effective
x˜ =F xˆ +B u (12)
sample size N [11], obtained by, k k k−1 k k
e Σ˜ =F Σˆ FT +Q (13)
1 k k k−1 k k
N = (7)
e (cid:80)N (wκ )2 In the updating step, the measurement zk is used, to-
p=1 j,p
gether with above predicted state and covariance, to up-
wherewjκ,p, p=1,2,...,N arethenormalizedweights. date the state and covariance. The residual, yk between
Ingeneral,are-samplingprocedureistakenwhenN <
e measurement and predicted state is given by
N , where N is a predefined threshold, and is usually
s s
set as N. Let {xκ ,wκ }N denote the particle set yk =zk−Hkx˜k (14)
2 j,p j,p p=1
that needs to be re-sampled, and let {xκ∗,wκ∗}N
j,p j,p p=1 The innovation covariance S is given by
k
denote the particle set after re-sampling. The main idea
of this re-sampling procedure is to eliminate the parti- Sk =HkΣ˜kHTk +Rk (15)
cles having low weights by re-sampling {xκ∗,wκ∗}N
j,p j,p p=1 Then, the optimal Kalman gain is calculated as
from {xκ ,wκ }N with the probability of p(xκ∗ =
j,p j,p p=1 j,p
xκ ) = wκ . At the end of the resampling procedure, K =Σ˜ HTS−1 (16)
j,s j,s k k k k
wκ∗,p=1,2,...,N are set as 1/N.
j,p Then, the state and covariance can be updated by
xˆ =x˜ +K y (17)
k k k k
B. Kalman Filter Methods Σˆ =(I−K H )Σ˜ (18)
k k k k
The well known Kalman filter is also a recursive
method to estimate system/target state based on a mea- IV. METHODOLOGY
surement sequence, minimizing the estimation uncer- Inthispaper,anovelGaussianparticlefiltertechnique
tainty.Themeasurementofthesystemstatewithanaddi- follows the main idea of particle filter for estimating
tiveGaussiannoisearegivenbythesensor.Then,ineach the number and state of total targets based on the
iteration, the Kalman filter consists of two precesses: i) measurement obtained online. Different from classical
it predicts the system state and their uncertainties; ii) particle filter, each particle here is a gaussian instead of
it updates the system state and uncertainties with the a point mass. The estimation for number of total targets
measurement that newly becomes available. The system andtheirstateispresentedbyasetofweightedparticles.
dynamics is given as The ith particle at time k is denoted as
xk =Fkxk−1+Bkuk+νk (8) Pki ={wki,N(xik|µik,Σik)} (19)
where subscript k and k − 1 denote the current and where w is the probability of existing a target having a
i
previous time index, while Fk is the system discrete statedistributionasN(µ,N(xi|µ,Σi)).Bythisparticle
transition matrix, and Bk and uk are the control matrix definition,thedimensionsofthesystemstateremainsthe
and control input. νk is the white noise, defined as same as the dimensions of each individual target. When
these particles are available, the estimated number of
ν ≈N(0,Q ) (9)
k k total targets can be given as T =(cid:80)Np w , where N is
i=1 i p
where Σ is the covariance. At kth time step, an mea- the particle number.
k
surementofthesystemtruestatex ismadebyasensor, Noticethattheparticlerepresentationisdifferentfrom
k
is given by classical particle filter, where each particle represents a
z =H x +ω (10) possible value of system state. The updating of each
k k k k
4
particleandtotalweightsarealsodifferentfromclassical Then, by applying Kalman procedure
particle filter. Kalman filter is used to update each
y =z −H µ (24)
particle,theweightandthedistribution.Noticethatsince k k k k
the measurement at each time step is conditioned on all Sk =HkΣkHTk +Rk (25)
the targets in the FOV, while in the classical Kalman K =Σ HTS−1 (26)
k k k k
filter method one measurement is associated with one
µ =µ +K y (27)
k k k k
target, which means data association problem is avoid.
Σ =(I−K H )Σ (28)
Therefore, the Kalman filter is modified to updated k k k k
the particles which are coupled by one measurement, Its proof can be found in the appendix.
and some approximations and assumptions are further Once each particle appearing in combination E has
needed. been updated, the weight w is for the particle combi-
c
nation E can be obtained by
1
w = Π(w )ei(1−w )1−ei ×
c i i (2π)2(cid:107)Σ−1(cid:107)
c
×exp{−(z −µ )TΣ−1(z −µ )} (29)
k c c k c
where c ∈ I , where I is the combination index, and
E E
Similar to Kalman filters and particle filters, the al- µ ) and Σ−1 is given by
c c
gorithm proposed in this paper is a recursive method.
(cid:88)
µ =H µi (30)
Assume that at time step k, the measurement zk is c k k
available, and the estimation of the system at time step Σ =H (cid:88)ΣiHT +R (31)
c k k k k
k−1isrepresentedbyaparticleset,denotedasP =
k−1
{P1 ,P2 ,...,PNp }, where N is the number of Then, insert particle {Ni(µ ,Σ )} into a set G for
k−1 k−1 k−1 p k k c
all particles. By using the target dynamic function 1, combinationc∈I ,thesetG hasaweightw .Afterall
E c c
Pk−1 canbeupdatedtoP˜k withoutusingthezk.Dueto Gc the combination of E that Π(wi)ei(1−wi)1−ei >(cid:15)
limitofFOV,onlyafewparticlesmayhavecontribution is obtained. Weights are updated by
ptoartthicelems eliaesuinremtheenFt.OLVe,twPhSiledleentoPt¯e =setP˜ikn/cPluSdindgentohtee wc = (cid:80)wwcc (32)
the compensation set. Only the particles in P are
S . Then in each group G , the weight of ith particle is
updated. Please Note the size of P is small. Without c
generality, assume that P = {P˜1,PS˜2,...,P˜s}, where updated as
S k k k wi
s the number of particles The update of each particle in wi = k ∗w (33)
c Πwi c
P is calculated separately. Without generally, we focus k
S
on updating P˜kj, Let a boolean set E =[e1,e1,...,es], The particles in all set Gc are updated from the same
where e ∈ {0,1}. For any E with e = 1 such that particle in the previous set. If two particles in is close
i j
Π(wi)ei(1 − wi)1−ei > (cid:15), where (cid:15) is a predefined enough, then they are combined as one particle, the
threshold, a particle is calculated and denoted as p˜ , weightofwhichissetasthesummationofbothweights.
j
Then, the modified Kalman filter is used to give the The distance between µi and µj is defined as Mahal-
updatedgaussianparametersofallparticleswithe =1. distance
i
According to sensor model 4, the measurement is given (µi−µj)TM(µi−µj) (34)
by
and its covariance is updated as
z = (cid:80)si=1µikei+ 1 (cid:88)s e Σ +ω (20) Σ=(cid:88)Σi (35)
k (cid:80)s e ((cid:80)s e )2 i i k k
i=1 i i=1 i i=1,i(cid:54)=j
V. SIMULATIONANDRESULTS
Compare the above function to (10), we have following
As shown in figure (1), N targets, represented by
setting
blue dots, are moving in the 2 dimensional workspace.
1 The whole workspace is visible to a position fixed
H =I× (21)
k ((cid:80)s e ) sensor(notshown)andtheworkspaceisdiscretizedinto
i=1 i
(cid:80)s µie 12×12 cells. Each cell represents a 1×1 rectangular
zk =zk− i=(cid:80)1,si(cid:54)=je k i (22) area. The ith cell, denoted as Ci,i ∈ C, is defined by
i=1 i [xul,yul,xdr,ydr], where C is the cell index set and
s i i i i
1 (cid:88) (xul,yul) and (xdr,ydr) are up left and down right cor-
Rk = ((cid:80)si=1ei)2 i=1,i(cid:54)=jeiΣi+ωk (23) neri cooirdinates oif thei ith rectangular area respectively.
5
Only M cells can be measured at each time step k, same cell, then the detection probability is p (T) =
d
and they don’t have to be adjacent. The goal of the p(1+SNR)/(1+T×SNR) and the ith sensor measurement at
d
sensor is to estimate the target states and target number time k can be evaluated by
at time k. In this paper, information value function (cid:40)
p (T) zk =1
based α divergence is used to select the best M cells p(zk|Xk,Tk,λk ,λk)= d i
to measure at each step [18]. The estimation of target i a,i c 1−pd(T) zik =0
statesandtargetnumberattimek isrepresentedbyjoint
N
(cid:88)
multitarget probability density(JMPD) and it is updated T = (xk ≥xul)∩(xk <xdr)
j c j c
after obtaining new measurements.
j=1
∩(yk ≥yul)∩(xk <xdr), c=λk (39)
j c j c a,i
&! p (T)=p(1+λkc)/(1+Tλkc) (40)
d d
! ! ! ! ! ! ! ! ! ! ! !
where xk,yk are two position components of xk ∈ Xk
! ! ! ! ! ! ! ! ! ! ! ! j j j
andTk isthetargetnumber.Additionally,operators”≥”
! ! ! ! ! ! ! ! ! ! ! !
and ”<” return either 1 if true or 0 if false, while ”∩” is
! ! ! ! ! ! ! ! ! ! ! !
the Boolean operator ”and”. For example, as shown in
! ! ! ! ! ! ! ! ! ! ! !
figure[1],whenc=k,T =2,similarly,whenc=j(i),
! ! ! ! !$! ! ! ! ! ! ! ! T =1(0).
! ! ! ! ! ! ! ! ! ! ! ! A snapshot of simulations is shown in Fig. 3, where
! ! ! ! ! ! ! ! ! ! ! ! magenta squares represent positive measurement return
! ! ! ! ! ! ! ! ! ! ! ! and blue dots represent the true targets’ positions. The
"!
! ! ! ! ! ! ! ! ! ! ! ! simulation results are summarized in Fig. 2, where the
! ! ! ! ! ! #!! ! ! ! ! ! black curve represents the target state estimation error
andtheredcurverepresentsthetargetnumberestimation
! ! ! ! ! ! ! ! ! ! ! ! %!
error. As shown in Fig. 2, both errors decreases as more
!
measurements become available.
Fig.1. Theworkspacecontiansthreepointtargets.
The target time-discrete state transition function can
be written as
xk+1 =Fxk+wk (36)
i i i
where
 
1 τ 0 0
0 1 0 0
F=  (37)
0 0 1 τ
0 0 0 1
and wk is 0 mean Gaussian noise with covariance
i
Q=diag(20,0.2,20,0.2), and τ is the time step length,
and i∈{1,2,··· ,Tk} [15].
It is further assumed that i) the sensor can measure
any cell at time k; ii) the sensor can only measure up to
Fig.2. SimulationResult
M cells at time k. The sensor condition λk represents
c
the signal to noise ratio SNR, currently, it has only one
possible value, fixed and known. The measurement zk
i VI. CONCLUSIONANDFUTUREWORK
is a discrete variable, then joint PMF can be written as
A Gaussian particle filter that combines Kalman fil-
f(zk,Xk,Tk,λk)=f(zk|Xk,Tk,λk)f(Xk,Tk)f(λk) ter and particle filter is presented in this paper for
(38) estimating the number and state of total targets based
When measuring a cell, the imager sensor will give on the measurement obtained online. The estimation is
a Raleigh return, either a 0 (no detection) or a 1 representedbyasetofweightedparticles,differentfrom
(detection)governedbydetectingprobability,denotedas classicalparticlefilter,whereeachparticleisaGaussian
pd,andfalsealarmprobability,denotedaspf.According instead of a point mass. The weight of each particle
to standard model for threshold detection of Rayleigh represents the probability of existing a target, while its
returns, p = p(1+SNR). When T targets are in the Gaussian indicates the state distribution for this target.
f d
6

Fig.3. Snapshotofsimulations
This approach is efficient for the problem of estimating By setting ∂ =0, therefore
∂Kj
number of total targets and their state. k
1
Kj = Σj( I)T
k k (cid:80)s e
i=1 i
VII. APPENDIX 1 (cid:88)s 1
×(R + I Σi( I)T)−1
k (cid:80)s e k (cid:80)s e
Without losing generality, PS = {P˜k1,P˜k2,...,P˜ks}, i=1 i i=1 i=1 i
Eits=µjk[ea1n,de2Σ,.jk.,.g,eivse],nfzokr aannyd pPaSrt.icle such that ej = 1, = (cid:80)si=11eiΣjk(Rk+((cid:80)si=11ei)2 (cid:88)i=s1Σik)−1(.44)
(cid:80)s xie
y =z − i=1 k i) (41) Then,
k k (cid:80)si=1ei µjk =µ˜jk+Kkjyk (45)
Σjk = COV(xjk−xˆjk) Σj = Σ˜j − 1 Σ˜j
= COV(xjk−(x˜jk+Kkjyk)) k k (cid:80)si=1ei k
(cid:16) (cid:80)s xie 1 (cid:88)s 1
= COV xjk−(x˜jk+Kkj( (cid:80)i=si=11ekii) ×(Rk+(cid:80)si=1eiI i=1Σik((cid:80)si=1eiI)T)−1
+νk− (cid:80)(cid:80)si=s1x˜eikei)(cid:17) ×Σ˜jk (46)
i=1 i
(cid:16) 1
= COV (I− IKj)(xj −x˜j)
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