Group Object Tracking with Sequential Monte Carlo Methods Based on a Parameterised Likelihood Function Nikolay Petrov1 , Lyudmila Mihaylova1 , Amadou Gning1 and Donka Angelova 2 1 Lancaster University, School of Computing and Communication Systems, UK 2 Bulgarian Academy of Sciences, Bulgaria Email: {n.petrov, mila.mihaylova, e.gning}@lancaster.ac.uk,
[email protected]
Abstract. Group objects are characterised with multiple measurements originating from different locations of the targets constituting the group objects. This paper presents a novel general Sequential Monte Carlo (SMC) approach for group object tracking applicable to various nonlinear problems. The novelty in this work is in the derivation of the likelihood function for nonlinear measurement functions, with sets of measurements belonging to a bounded spatial region. Simulation results are presented when a group of objects is surrounded by a circular region. Accurate estimation results are presented both for the group object kinematic state and extent.
1
Motivation
Group object tracking is concerned with finding the patterns of behaviour and tracks of a whole group instead of each object separately. The group of objects are generating multiple measurements which origin is unknown. Different methods are proposed in the literature for solving the problem of group object tracking, e.g., [KF09,Koc08,BFF+ 10a]. Several approaches consist in estimating the extent of the group, e.g., with random matrices as suggested in [Koc08, BFF+ 10b]. Other related works are [GS05, SH07b, SH07a, SBH06, BH09a, BH09b, NKPH10]. In general the measurement uncertainty can belong to a hypercube or to another spatial shape. In our approach, we consider the general case with a nonlinear measurement equation and measurements. The main contributions of the work is in the derived likelihood function based on a parameterised shape and in the developed Sequential Monte Carlo (SMC) filter for group objects. Then we propagate this spatial measurement uncertainty through the Bayesian estimation framework. In this work inspired by ideas from [GS05,GGMS05] and combining it with the approach from [PMGA11] we developed a novel approach for group object tracking. The remaining part of this paper is structured as follows. Section 2 formulates the problem. Section 3 derives the measurement likelihood function. Section 4 presents simulation results about the performance of the proposed apporoach and finally Section 5 summarises the results.
2
Group Object Tracking Within the Sequential Monte Carlo Framework
For the purpose of constructing a general framework, consider the state estimation of groups of objects. Each group, g = 1, ..., G, is comprised of ngT targets and the group 0 state vector is defined as X gk = x01,k , ..., x0t,k , ..., x0ng ,k , Gk , where xt,k is the state T
vector of the tth point target, t = 1, ..., ngT , at time k and Gk is a vector characterising k the group. The groups of objects give rise to a set of measurements Z k = {z m,k }M m=1 . The goal is to estimate X gk given the measurements Z 1:k = Z 1 , ..., Z k , collected up to time k. Within the sequential Bayesian framework this goal can be achieved by constructing the group state probability density function (pdf) based on the incoming sensor data. According to the Bayes’ rule the filtering pdf p(X gk |Z 1:k ) of the state vector X gk given a sequence of sensor measurements Z 1:k can be written as p(X gk |Z 1:k ) =
p(Z k |X gk )p(X gk |Z 1:k−1 ) , p(Z k |Z 1:k−1 )
(1)
where p(Z k |Z 1:k−1 ) is the normalising constant. The state predictive distribution is given by the equation Z p(X gk |Z 1:k−1 ) = p(X gk |X gk−1 )p(X gk−1 |Z 1:k−1 )dX gk−1 . (2) The evaluation of the right hand side of (1) involves integration which can be performed by the Particle Filtering (PF) approach [AMGC02] by approximating the posteg(i) rior pdf p(X gk |Z 1:k ) with a set of particles X k , i = 1, . . . , N and their correspond(i) ing weights wk . Then the posterior density function can be written as follows p(X gk |Z 1:k ) =
N X
(i)
g(i)
wk δ(X gk − X k ),
(3)
i=1
where δ(·) is the Dirac delta function, and the weights are normalised such that P (i) = 1. i wk g(i)
(i)
g(i)
Each pair {X k , wk } characterises the belief that the group g is in state X k . An estimate of the variable of interest is obtained by the weighted sum of the particles. Two major stages can be distinguished: prediction and update. During prediction, each particle is modified according to the state evolution model, including the addition of random noise. In the update stage, each particle’s weight is re-evaluated based on the new data. A resampling procedure introduces variety in the particles by eliminating those with small weights and replicating the particles with larger weights such that the approximation in (3) still holds. The residual resampling algorithm [Wv01] is applied here. The dynamics of each individual object can be described using known Markovian models p(xgt,k |xgt,k−1 ), for t = 1, . . . nT targets. The vector X gk consists of the kinematic state vectors of all objects. The same dynamics model for each individual target and for the observations is used as in [PMGA11].
3
Measurement Likelihood for Group Object Tracking
In the context of group object tracking, the prediction step is generally well studied for various classes of interval and spatial representations of uncertainties. Ellipsoids, spheres and polytope families can be easily propagated when the system dynamics and sensor models are linear. The update step with the likelihood calculation is less studied or often studied with a restriction to a particular class of sets of a particular type of measurements. The aim of this paper is to derive general likelihood calculation procedures based on Monte Carlo methods without a restriction on the type of set of interest or the type of measurement available. For that purpose, the main idea of this paper is to introduce a sampling step for regions of interest in the groups regions. This sampling aims to represents the probability of a given point, in the state space, to be the origin of a measurement. This section describes in detail the newly proposed method. As an illustration, the case of a group object with circular form is considered. Without loss of generality, assume that there is only one static sensor described by its state vector xs,k . Assume that at each measurement time k the groups generate PG a matrix Z k of Mk = g=1 ngT Mt,k measurement vectors. The number of measurements Mk,t originating from the target t at time step k is considered Poisson-distributed random variable with mean value of λt , or Mt,k ∼ P oisson(λt ). The measurements originating from a target t are assumed to be conditionally independent, i.e. Mt,k
p(Z k |xt,k ) =
Y
p(z m,k |xk ).
(4)
m=1
For each target t, the generation of measurements over the observation space at time step k is modelled as a nonhomogeneos Poisson point process. Let λt (z m,k |xt,k , Gk ) be the intensity of the process. The pdf of each measurement z m , produced in a region A of the observation space (sensor field-of-view) can be expressed by p(z m |xt , G = λt (z m |xt , G)/µt (A),where µt (A) is the expected number of measurements falling in A. (For convenience, the time subscript k is omitted). We assume that µt (A) is known. It is assumed also that the measurements received over the observation volume result from the superposition of the measurements from ngT conditionally independent nonhomogeneous Poisson point processes. Since the superposition of independent Poisson Pngt processes is a Poisson process with parameter that is t=1 , the expected total number Pngt µt (A). Similarly of measurements, received in the frame k is given by µ(A) = t=1 to [GGMS05], the joint likelihood of the number ngT of measurement falling in the observation volume A is given by g
p(Z k |X gk )
=
nT Mk X Y m=1 t=1
µt (A)p(z m,k |xgt,k ).
(5)
3.1
Introduction of the Notion of the Visible Surface
The measurement likelihood p(z m,k |X gk ) of the group target can be represented with the relation Z p(z m,k |X gk ) = p(z m,k |V k )p(V k |X gk )dV k , (6) Rnv
where V k ∈ Rnv denotes a source of measurement in the state space. In practice, these visible sources V k depend on the object position, nature and parameters X gk and on the sensor state (e.g., the sensor position and angle of view). The pdf p(V k |X gk ) represents the probability of a point in the state space to be a source of measurement given the group X gk . The surface of the group with state X gk visible from the sensor with a state vector xs,k is denoted by Vk (X gk , xs,k ). We assume that the sources of the measurements at time step k, given the target state and the sensor prior state are uniformly distributed along the region Vk (X gk , xs,k ), visible from the sensor xs,k , i.e. p(V k |X gk ) = p(V k |X gk , xs,k ) = UVk (X gk ,xs,k ) (V k ),
(7)
where UVk (X gk ,xs,k ) (·) represents the uniform pdf with the support Vk (X gk , xs,k ). Inserting (7) into (6) gives Z p(z m,k |V k )UVk (X gk ,xs,k ) (V k )dV k . (8) p(z m,k |X gk ) = Rnv
3.2
Parametrisation of the Visible Surface
The likelihood (6) can be calculated using the following SMC approximation: Z g(i) p(z m,k |X k|k−1 ) = p(z m,k |V k )p(V k |X gk|k−1 )dV k Rnx
≈
Mk X
(`)
(`)
g(i)
p(z m,k |V k )p(V k |X k|k−1 ).
(9)
`=1 g(i)
(i)
After the prediction step, a set of weighted particles {(X k|k−1 , wk|k−1 )}N i=1 is g(i)
available. First, the visible area Vk (X k|k−1 , xs,k ) is determined. Then for each parg(i)
(`)
g(i)
ticle X k|k−1 the likelihood function p(V k |X k|k−1 ) is defined from the support g(i)
Vk (X k|k−1 , xs,k ), with ` = 1, ..., S, where S = QN and Q is the number of samples generated uniformly inside each of the N particles. (`) The term p(z m,k |V k ) in (9) can be easily calculated in a classical way depending on the problem, for example using a Gaussian measurement noise distribution (`) p(z m,k |V k ) = p
1 2π||R||
e−
(z m,k −z
(`) (`) T )R−1 (z m,k −z ) k k 2
.
(10)
where R is a known measurement noise covariance. (`) g(i) A simple expression of p(V k |X k|k−1 ), in the case of circular group shape, can be (`)
g(i)
p(V k |X k|k−1 ) = UC(x(i) ,y(i) ,r(i) c,k
(i)
(i)
c,k
k|k−1
q (`) (i) 2 (`) (i) 2 (x − x ) + (y − y ) , k c,k k c,k )
(11)
(i)
where C(xc,k , yc,k , rk|k−1 ) is a circle, representing the ith particle by its two coordi(i)
(i)
(i)
nates for the center xc,k and yc,k , and the radius rk|k−1 . The total number of samples (`)
(`)
(`)
V k = {xk , yk } is given by S. An effective algorithm for generating uniform points inside circles/elipsoids is proposed in [DM01].
4
Performance Evaluation
An example with a group that consists of 50 targets of 200 is considered for the duration y x time steps. We are estimating the vector Gk = xg,k , yg,k , vg,k , vg,k , rg,k
0
character-
y x ising the group. This vector comprises position coordinates xg,k , yg,k , speed vg,k , vg,k and an extent rg,k . The initial coordinates of the targets are uniformly distributed in a square 50m × 50m. A constant velocity model is considered for the target evolution model. The initial position is defined by the mass center of the measurements with added Gaussian noise with standard deviation of 5m for both coordinates xg,k=0 and y x yg,k=0 . The velocity is vg,k=0 = N (100, 22 )m/s and vg,k=0 = N (100, 22 )m/s, respectively. The system dynamic noise is σx = 0.1m/s2 and σy = 0.1m/s2 , respectively. Measurements consisting of range and bearing are considered. They are collected by sensors randomly positioned in proximity to the group center defined by N (140, 202 )m. The number of the measurements originating from a target t is Poisson distributed random variable with λt = 5. The number of particles that used for the estimation is 100. For each of these particles 20 uniformly distributed samples are generated. Fig. 1 a) and b) and Fig. 2 a) and b) represent different plots of the same time step for the scenario described above. Visualisation of the sampling can be seen with blue colour in Fig. 1 a) and b), while the particles are plotted with the green circles in Fig. 1 b). The samples with their weights are shown in Fig. 2 a). In Fig. 2 b) the real targets can be seen as small red double circles, the measurements are represented as red dots, the estimated group extent is given as a big red circle. In order to judge the efficiency of the proposed algorithm we have calculated two measures. Both are shown on Fig. 3. The first measure represents the ratio between two estimated circle extents. We consider the mass center of all point targets and with respect to this center a circle is defined surrounding all targets. Next, based on the proposed algorithm the group extent is estimated. We form the ratio between these two extents (estimated over real). This is shown in Fig. 3 (dashed line). This plot indicates that the group size is overestimated which means that we have a bigger size of extent than the real one. The second figure characterises the percentage of targets included in the estimated group extent.
4
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x 10
x 10
2.025
2.025
2.02
2.02
2.015
2.015
2.01
2.01
2.005
2.005
2
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1.995
1.995
1.99
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2.02
4
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x 10
x 10
(a) Samples - blue dots, estimated extend big red circle, targets - small red double circles, measurements - red dots
(b) Particles - green circles
Fig. 1: Visualisation of the considered simulation scenario
4
x 10 2.006 50
2.004
Particle weights
40
2.002
30
2 1.998
20
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1.994 0 2.04
1.992 2.02
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Y coordinate, m
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2.02 1.988
2.01 4
x 10
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1.995
X coordinate, m
(a) Samples with their weights - blue dots
2
2.005 4
x 10
(b) Targets - small red double circles, measurements - red dots, estimated group extent - big red circle
Fig. 2: Visualisation of the considered simulation scenario
2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0
0
50
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200
Fig. 3: Performance evaluation averaged over 30 runs: green dashed line - extent ratio, solid blue line - percentage of targets inside the group extent
5
Conclusions
This paper presents a Sequential Monte Carlo approach for group object tracking. The novelty of this work consists in the proposed algorithm for the likelihood estimation. The performance of the algorithm is validated over an example with a group having a large number of components. We show that the algorithm successfully evaluates the range of the group extend based on measurements with an unknown origin. Acknowledgements. We acknowledge the support from the [European Community’s] Seventh Framework Programme [FP7/2007-2013] under grant agreement No 238710 (Monte Carlo based Innovative Management and Processing for an Unrivalled Leap in Sensor Exploitation), the EU COST action TU0702 and the Bulgarian National Science Fund, grant DTK 02/28/2009.
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