Automated Model Selection based Tracking of Multiple Targets
using Particle Filtering
.
Mukesh A. Zaveri Uday B. Desai S. N. Merchant
SPA"
Lab, Electrical Engineering Dept., IIT Bombay - 400076.
Email @dress :[mazaveri,ubdesai,[email protected]
Abstract-Particle filtering is being investigated extensively
due to its important feature of target tracking based on nonlinear and non-Gaussian model. It tracks a trajectory with
a known model at a given time. It means that particle filter
tracks an arbitrary trajectory only ifthe time instant when trajectory switches from one model to another model is known
apriori. Because of this reason particle filter is not able to
track any arbitrary trajectory where transition instant from
one model to another'model is not known. Another problem with multiple trajectories tracking using particle filter is
the data association, i.e. observation to track fusion. In this
paper we propose a novel method, which overcomes both the
above problems. In the proposed me:hod an interacting multiple model based approach is used along with particle filtering, which automates the model selection process for tracking
an arbitrary trajectory. The uncertainty about the origin of an
observation is overcome by using a centroid of measurements
to evaluate weights for particles as well as to calculate likelihood of a model.
1. INTRODUCTION
The standard Kalman filter gives an optimal estimate with linear and Gaussian model assumption. For nonlinear case, one
typically uses the extended Kalman filter. In recent times,
for nonlinear and non-Gaussian models, particle filtering has
been proposed as an alternative to the extended Kalman filter [1]-[7]. Particle filtering has been extended to multiple
target tracking and different methods have been proposed for
this problem [8],[9]. In [XI the data association problem is
treated as an incomplete data problem [IO]. It treats observation to track assignment as missing data and Gibbs sampler
method is used to estimate the assignment probability. Particles are sampled from the probability density function (pdf)
representing the combined state of all the targets in [XI. In
multiple target scenario the number of state parameters varies
from target to target. Moreover the computational complexity
of this method increases exponentially as number of measurements increase, and number o f targets to he tracked increase.
Estimating the joint probability distribution of the state of all
targets makes the problem intractable in practice. Particle filter, of course, needs the knowledge of the model to track a
target; but more importantly it needs to know the time instant
when trajectory switches from one model to another model.
Now, if the target inoveinent is random, then the trajectory
formed by the target is arbitrary and there is no apriori knowledge about which model to use at a given time and when to
switch. In such a situation, particle filter suffers from the de-
generacy problem, and the pdf of a the state collapses. To
track an arbitrary trajectoty, it is incumbent to use a multiple
model based approach, namely, IMM filtering.
In this paper we propose a novel method that works with iniultiple nonlinear or non-Gaussian state space models to track
arbitrary trajectories. In the proposed inethod an interacting multiple model [II], [I21 based approach is used along
with particle filtering, which automates the model selection
process. The proposed approach is completely different than
the multiple target tracking method proposed in [8] as follows: ( I ) Our method automutes model selection to track an
arbitrary trajectory by inclusion of inultiple models instead
of tracking with on1.v one known model, ( 2 ) Instead of combined state vector representation for all targets, individual
target state model is used and track different targets independently, (3) The likelihood of an observation for a target
is treated as mixture pdf, (4) Thc calculation of importance
weight for a particle is done differently. The performance of
nearest neighbor method degrades in dense clutter, whereas
joint probabilistic data association filter and multiple hypothesis tracking methods are computationally expensive [ 131. So
in the proposed method PMHT based approach is used for
data association. PMHT algorithm [14], [IS] has been proposed to avoid the uncertainty about the origin ofan nieasurement. It uses a centroid of measurements to evaluate state
vector of a target. In the proposed approach, we have used
a centroid of the measurements to evaluate likelihood of the
model, which is required for mixing state vectors fromdifferent models in IMM based approach. The centroid is also used
to update particles. Thc proposed method is able to track multiple trajectories in the presencz o f dense clutter, and does not
require the apriori knowledge of the time when the trajectory
switches form one model to another.
2 . AUTOMATED
MODELSELECTION BASED
PARTICLE
FILTERING
In the proposed method, a target follows more than one model
and as in particle filtering, these models are assumed to be
known. But, at a given time, the model, which a target obeys,
is not known. For each model the probability density function
is approximated by a set of samples, called particles. Each
particle is assigned a weight, known as impoflance weight.
For every model, particle weights [16], [I71 are evaluated at
each time instant independently. If the trajectory does not follow any model at a given time instant its probability density
function may collapse or all importance weights may have
negligible value for respective particles. At this time instant,
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particles are initialized using mix state vector given by the
IMM filtering method and hence, it is possible to follow an
arbitrary trajectory. IMM filtering mixes the state vector from
different models using model probabilities. When a trajectory
switches from one model to another, particle weights have
marginal values if it matches a model and hence, it is reflected
in model probability. Mix state vector takes care of the likelihood of a model for a given trajectory. Model.prohability
is calcu!ated using the centroid of observations. Inclusion of
IMM based approach allows us to track an arbitrary trajectory
with different models. For observation to track association,
PMHT based approach is used which is described in [14].
For particle filter. a set of weighted particles are drawn from
the posterior pdf of the state. The pdf can be approximated
using discrete sums in place of integrals as follows:
where Y,,, = (!,I, yz, . . . , y t } is a set of measurements up to
time t and yt is a measurement available at time t. z i (1 5
i 5 N) represents i"' sample drawn from pdf at time t. Here,
A' is the total number of samples used to represent pdf and
J,,:;is the Dirac delta function. Based on this approximation,
any moment can be evaluated [ 181. It can be written as
k
N
E ( g t ( z t ) )= . r s i ( ~ t ) l J ( ~ t I Y i : t ) ~ C,=I
~t
clt(d)
(2)
The particles Z; are assumed to be independent and identically distributed. As N + CO estimation converges to its
true value [17]. Generally it is difficult to sample from the
posterior pdf. But it is easy to sample from the proposal distribution function q(:~~lY~:~).
There are various method for
sampling from the proposal function. Sequential importance
sampling,(S!S) is one of these techniques. Each particle is
weighted by an importance weight and it is given by.
Wt(Zt)
(3)
=
The proposal function should be chosen to minimize the varil
anceoftheimportanceweights [16]. Themostpopularchoice
for proposal function [ I ] is
q(2:tlx*-,,vt)
= P(4"t-l)
N
(6)
i=l
where G: =
$$ is normalized weight. The major prob-j=,",
For our algorithm, Y and X denpte the observation process
and the state process respectively. Y' is a set of all ohservation set for time 1 5 t, where t. is current time. Y ( t )
and X ( t ) represent the realization of observation process and
state process at time t, respectively. At time t., a vector of
measurements is received, Y ( t ) =
. . .U,,,,,), where
rnt represents the number of measurements received. Sim:.
Here. N,is the total
ilarly, X ( t ) = ( ~ ~ ( 1. .),T;~(A'*)).
number of targets at time instant t and Z:,,(.S) (1 5 s 5 N,)
represents the combined state estimate for target
the state estimate of target ,s due to model ni. at time t, where
1 5 m 5 M. M is the total number of models used to track
a particular target.
To overcome the uncertainty about the measurement origin,
'
h is a set of a11 its rean assignment process K is used and l
alization for time 1 t. .Its realization at time t is denoted
by, K ( t ) = ( k l ( t ) ,. . . , k , , > , ( t ) ) where K ( t ) is an assign-
<
ment vector and each element of vector k j ( f ) = .s indicates
that target s produces measurement j at time t . The measurement to track assignment probability lI at time t is given by,
n(t)= ( n l ( l ) ,.. . > n , ( N , ) )Here,
.
n1(.s)indicatestheprobability that a measurement originates from the target s. This
probability is independent of the measurement, i.e.,
n*(.s)=p(lc,(t) = s ) ,
v j = 1: . . . ; 1 n ,
(7)
It is assumed that one measurement originates from one target
or clutter, which leads to following constraint on assignment
N
probabilities, C;L1nf(s) = 1.
Each element of assignment vector K ( t ) is assumed to he independent ofedch other.
With this problem formulation, the proposed algorithm, automated model selection based tracking using particle filter is
described as.follows:
(4)
The problem with the above choice is that the most recent
measurement is not incorporated but it is vely easy tu implement. This simp1,ifies the evaluation of weights ~ t ( : c t ) and
written as
. .
U J t ( Z t )= P(?/t/zt)
(5)
and it can he shown that, expectation in (2) can he written as,
E(.qt(rt)) =~. x 9 t ( ~ : ) G t ( d )
resampling is performed to eliminate the particles with low
weights and multiply particles with high weights. There are
number of resampling methods: sampling importance resanipling (SIR), residual iesampling and minimum variance sampling. In. our proposed method. residual resampling method
is used because it is coinputationally less expensive and the
variance is small than that given by SIR method.
'.
lem with the above technique is that the variance of the i,m,portance weights increases over time. It results into degeneracy phenomenon. TO;overcome this degeneracy,problem a
1. Initialize particles for each m o d e h (1 5 111 5 M ) , by
drawing samples z' ( i = 1 , . . . , N ) from the priorp,,,(z") for
each target s (1 5 s 5 N,).
Initialize model probability and
transition probability. Here, Nl represents the total number
of targets at time t.
2. F o r t i m e t = l , Z , . . .
(a) Update particle weights:
For each target and for each filter model 7% initialize assignment probabilities n t ( s )and repeat the following steps (i)-(v)
during each iteration, till error converges to a fixed threshold
value, i.e. ~ ~ P ( J ~ - ~ ) ( . s ) - P ( P ) (,s)I/ < E . Initially?"' = 3"'
where :c'" is predicted state vector for model ni at previous
time.
i. Evaluate the likelihood for an observation falling inside
the validation region formed using predicted state.vector with
.
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.
..
,
Decision, Identification and Estimation iylmage Processing /a33
respect to model n ~ .i.e. p,,,tyt,/i"'(p)).
Here, z$"(') represents state vector of model in at iteration p .
ii. Calculate the assignment weights for each measurement
j = 1 , . . , ,nit fur each targetr = I > . . ,iV,.
ii. Combined state update for target
.s:
"1
iii. Mix state initialization for model
111:
where p ( g ~ , l i ( J ' ) ( nis) )a mixture probability of an observation given combined state estimate x of a target n, and it is
given by
where
/iil'"
cy=l
Here, 1 ~is' a ~model probability which is described later
iii. Calculate the assignment probabilities for target s
= <i,,,p:/&!!llt
and &!!ll; =
<z,,,lij
If
w{ equals zero for model ni then initialize the particles with mix state x:rrbotherwise go to next step.
iv. Predict particles:.
For each model m ( l 5 771 5 A4); draw a process noise sample from a pdfp(v,.) and propagate particle L' by
WE
z:+~= f(x:)
iv. Calculate the centroid of measurements
v. Using an observation centroid and state vector o f a model
particles weights are calculated
,,if
JP+C
f -
f
rn
(1 c " L ~ p l ( 1 4
Jt
t
E,:=?
If
wi equals zero then go to next model otherwise obtain estimation E(:?;")for model 111 for next iteration using
(6).
vi. Particle weights are updated using an observation centroid and state vector of a model (i"'= :E"*) which is evaluated at the end of iteration by,
..
.
111;
= rt(s)p,,,(?/;mIxy)
Normalize the weights.
Perform Residual Resampling to obtain N particles distributed according to p,,,(zFlY1:,).
Obtain estimation E(zi") for inodel 712. (in our case mean
of a state) using (2).
(b) Propagate particles: For each targets (1 5 s 5 A i ) ,
i. Update model probability i n = 1 , .. .,M :
If C"' is negligible value (as it is calculated using centroid
only). initialize it with a equal probability value with a s s u m 6
tion that the centroid has equal likelihood with each model TI.
'Note: Likelihood of a model is calculated during f i s t iteralion only for
given modcl and target.
(1 5 i 5 N )
v. Obtain predicted state E(s;;,) for model rri. using (2)
vi. Combinedstate prediction n : , + l ( . ~ ) for target R.
3.
evaluate the likelihood of model for
Using a centroid y;'l"
+.I,:
SIMULATION
RESULTS
Two hundred particles are used for all simulations. The cuvariance value for a process noise and an observation noise is
assumed to be 5.0 for both z and 11 positions:Nonlinear trajectories are formed using sine wave with different frequencies and amplitudes. For simulation mixed trajectories are
generated using a CV model and a nonlinear model for various clips. It is assumed that each trajectory follows a CV
model and one nonlinear model. The parameters for nonlinear model for each trajectory are chosen differently for various clips. Tracking of the trajectories in ir91 (0.0l"h d u i ter) and ir81 (0.03% clutter) clips using interacting multiple
model based approach along with particle filtering is shown
in Figures I-(a) and I-(b) rtspectively. In Figures I 4 a ) and
l-(b), true trajectories are represented by solid line and estimated trajectories are depicted by dotted line. 0.03% clutter
level implies that 0.03% percentage oftotal numbers of pixels
in an image are noisy one. Table A and Table B depict mean
error in'position for each trajectory in respective clips with
different clutter level.
It is important to note that in the proposed method during
tracking the time instant when transition from one model to
another model takes place is not known apriori and it is random in nature. IMM filtering allows us to use mix state vector for tracking arbitrary trajectory. The particle filtering preserves nonlinearity of the model and allows us to use nonGaussian models fora process noise and an observation noise.
Figures 2-(a) and 2-(b) depict evaluation of model probabilities for different models for trajectory I in i t 9 1 and ir81 sequence respectively. Tracked trajectory' for target I shown in
Figure I for ir91 and ir81 clips can be compared with these
model probability plots.
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TENCON 20031834
Tablc A: Mean Error in Position
Clips with clutter
ir73
it90
0.01’% 0.02% 0.01% 0.02%
I
2.8669 3.2265 2.1640 2.2579
2
2.3483 2.9006 2.2403 2.3317
3
2.4089
fails 2.1929 2.7720
4
1.6615 2.9151 1.4184 1.6138
ir90
’
ir84
0.03%
1
0
.
0
3
%
0.01%
10.02%
__
(v=40) (V=42)
Traj
I
’
I
rect evaluation of the centroid of measurements and consequently, it prevents the necessary change in the model probability. But in real world application the movement of a target
during transition time instant will not be very large and true
measurement will fall within a smaller validation gate fonned
around the combined predicted target position.
4. CONCLUSION
From simulation results i f is concluded that our proposed
multiple model based particle filtering method allows us to
track any arbitrary trajectory which. follows more than one
model with the random switch over amongst these models i n
the presence of clutter.
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1 2.6512
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I
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