Box Particle Filtering for Extended Object Tracking - MC Impulse

Box Particle Filtering for Extended Object Tracking Nikolay Petrov1 , Amadou Gning , Lyudmila Mihaylova1 and Donka Angelova§ 1

School of Computing and Communications, Lancaster University, United Kingdom  Department of Computer Science, University College London, United Kingdom §Bulgarian Academy of Sciences, Acad. G. Bonchev - 25A, Sofia, Bulgaria

Email: [email protected], [email protected], [email protected], [email protected]

Abstract—This paper focuses on real-time tracking of an extended object in the presence of clutter. This task reduces to the estimation of the object kinematic state and its extent, based on multiple measurements originated from the same object. A solution to this challenging problem is presented within the recently proposed Box Particle Filtering framework. The Box Particle Filter replaces the point samples with regions, which we call boxes. The performance of the Box Particle Filter for extended object tracking is studied over a challenging scenario with simulated cluttered radar measurements, consisting of range and bearing components. The efficiency is evaluated for different levels of clutter, number of box particles, uncertainty regions for the measurements, number of the active sensors collecting the measurements data and iterations for the contraction of the uncertainty region. Accurate estimation results are demonstrated.

within the random finite set statistics approach in [23] and in particular as a more efficient implementation of a Bernouli PF. In the present paper the Box PF addresses the problem of extended target tracking. First, a formulation of the problem is presented within the interval analysis framework. Next, this estimation problem for an extended object is solved with a Box PF. A similar example as in [24] is considered. The rest of this paper is organised as follows. The following section II gives the main idea behind the Box PF. Section III introduces elements of the interval analysis theory used in this work. Section IV presents the problem formulation within the interval analysis framework. Section V introduces the likelihood function for the Box PF in the context of extended object tracking. Section VI presents the implementation of the algorithm using an example with circular-shaped extended target. Section VII shows performance evaluation of the Box PF based on simulation results. Finally, conclusions are given in Section VIII.

I. I NTRODUCTION Tracking of extended objects such as a convoy of vehicles, formations of aircrafts, ships, or a crowd of people (considered as one formation) has been extensively studied in the recent years. Extended objects are giving rise to many measurements and need sophisticated methods to cope with them. Depending on the proximity of the object and the resolution of the sensors, the measurements obtained from the object may provide some valuable information about the size and the shape of its extent. Usually the problem is formulated as kinematic state and parameters estimation, where the parameters relate to the extent of the object of interest [1], [2], [3], [4], [5], [6], [7], [8] and the main methodology is the Bayesian framework. Various filters have been developed for extended target tracking: particle track-before-detect filters [9], cluster based approaches [10], [11], Poisson spatial models combined with particle filters [12], [13], [14], [15] and mixture Kalman filters combined with data augmentation [1]. Recently various Probabilistic Hypothesis Density (PHD) filters have been proposed [16]. As an alternative, various interval algorithms have been developed, mainly for linear systems and linear measurements [17], [18], [19]. This paper proposes a Box Particle Filter (Box PF) framework for extended object tracking. The Box Particle filter is developed in [20] and applied for localisation problems. The theoretical justification of the Box PF approach is derived in [21], [22]. The Box PF has also been applied to filtering problems

II. M AIN I DEA OF THE B OX PARTICLE F ILTERING The main idea of Box PF is to work with region-particles, instead of with point particles. This approach is suitable for dealing with various types of uncertainties, e.g. in the measurements: interval, stochastic and data association uncertainties [23]. These three types of uncertainties can be dealt with the same steps, prediction and correction, as in the generic PF. However, there are some significant differences. The prediction step is performed in a similar way as in the classical PF, however, it is with respect to box particles. When a box particle is propagated via a non-linear function, the image of it is not necessarily a box particle. Hence, a function, called inclusion function, is applied to convert the predicted region into a box particle. The measurement update step requires the calculation of the likelihood function [25]. First, the measurement noise is supposed to be bounded and a likelihood box is defined as a set containing the measurement and the noise boundaries. This generalised measurement function is calculated in a different way compared to the classical PF. The measurement update step is based on on finding the minimum boxes inside the box particles, consistent with the likelihood box. This is done with a procedure called

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contraction which remove inconsistent parts between the box particles and the likelihood box. When there is no consistency between the box state particle and the likelihood box, the box particle becomes empty with a likelihood zero. This case commonly happens when a clutter measurement is contracted with the box state particle. There are different methods for performing the contraction step [26]. The resampling step is used to introduce variety. However, the resampling step in the box PF differs from the resampling step of the generic PF. The resampling step in the Box PF can be performed by a division of box particles [23] or by other techniques.

data and equations. The CP method is also known to be simple and, most importantly, to be independent of nonlinearities. IV. P ROBLEM F ORMULATION W ITHIN THE I NTERVAL A NALYSIS F RAMEWORK The system dynamics and the sensor equations have the following general form: [xk ] = [f ]([xk−1 ], [η k ]) = [f ]([xk−1 , xk−1 ], [η k , η k ]),

z k = h(xk , wk ), (4)   T ⊂ Rnx is the unknown syswhere [xk ] = [X k ]T , [Θk ]T tem state interval vector at time step k, k = 1, 2, ..., K, where K is the maximum number of time steps and nx is the dimension of [xk ]. The notation [·] stands for an interval variable and (·)T is the transpose operator. The system function f (·) and measurements function h(·) are nonlinear in general. The interval vector [xk ] consists of the object kinematic interval state vector [X k ] ⊂ RnX and object extent, characterised by the interval parameters vector [Θk ] ⊂ RnΘ , where nΘ is the number of parameters to be estimated. The system (kinematic state and parameters) noise and measurement noise are given, respectively, by: [η k ] = ([η s,k ]T , [η p,k ]T )T and wk . The sensors are considered to collect point measurements according to equation (4). Nevertheless, to express the measurements uncertainty, intervals around these points are formed with respect to the sensor position. The interval measurements (likelihood boxes) vector is then given by [z k ] = [z k , z k ] ⊂ Rnz , for example based on the 3σ rule. Here nz denotes the dimension of the measurement vector.

III. E LEMENTS OF I NTERVAL A NALYSIS A real interval, [x] = [x, x] is defined as a closed and connected subset of the set R of real numbers. The lower bar is the minimum value of a quantity and the upper bar is the maximum value of a quantity. In a vector form, a box [x] of Rnx is defined as a Cartesian product of nx intervals: x [x] = [x1 ] × [x2 ] · · · × [xn ] = ×ni=1 [xi ]. In this paper, the operator |[.]| denotes the size |[x]| of a box [x]. The underlying concept of interval analysis is to deal with intervals of real numbers instead of dealing with real numbers. For that purpose, elementary arithmetic operations, e.g., +, −, ∗, ÷, etc., as well as operations between sets of Rn , such as ⊂, ⊃, ∩, ∪, etc., have been naturally extended to interval analysis context. In addition, a lot of research has been performed with the so called inclusion functions [26]. An inclusion function [f ] of a given function f is defined such that the image of a box [x] is a box [f ]([x]) containing f ([x]). The goal of inclusion functions is to work only with intervals, to optimise the interval enclosing the real image set and, then to decrease the pessimism when intervals are propagated. Often constraints have to be fulfilled which requires to solve the Constraint Satisfaction Problems (CSPs). A CSP often denoted H can be written: H : (f (x) = 0, x ∈ [x]).

A. Bayesian Framework According to the Bayesian framework the state vector is obtained in a recursive way based on the following equations, for the prediction  p(xk |z 1:k−1 ) = p(xk |xk−1 )p(xk−1 |z 1:k−1 )dxk−1 (5)

(1)

Equation (1) can be interpreted as follows: find the optimal box enclosure of the set of vector x belonging to a given prior domain [x] ⊂ Rn satisfying a set of m constraints f (with f a multivalued function, i.e., f = (f1 , f2 , · · · , fm )T , where the fi are real valued functions). The solution set of H is defined as: S = {x ∈ [x] | f (x) = 0}.

(3)

Rn x

and respectively for the update p(xk |z 1:k ) =

p(z k |xk )p(xk |z 1:k−1 ) , p(z k |z 1:k−1 )

(6)

where p(xk |z 1:k−1 ) is the prior state probability density function (pdf), p(xk |xk−1 ) is the state transition pdf, p(xk−1 |z 1:k−1 ) is the posterior state pdf at time step k − 1, p(xk |z 1:k ) is the posterior state pdf at time step k, p(z k |xk ) is the likelihood function and p(z k |z 1:k−1 ) is a normalisation factor.

(2)

Contracting H means replacing [x] by a smaller domain [x] such that S ⊆ [x] ⊆ [x]. A contractor for H is any operator that can be used to contract H. Several methods for building contractors are described in [26, Chapter 4], including Gauss elimination, the Gauss-Seidel algorithm, linear programming, etc. Each of these methods may be more suitable to some types of CSP. Although the approaches presented in this work are not limited to any particular contractor, a general and well known contraction method, the Constraints Propagation (CP) technique is used in this paper. The main advantages of the CP method is its efficiency in the presence of high redundancy of

B. Model of the Extended Targets In a two-dimensional case, where the state vector consists of position coordinates and velocity of the centre of the extent, the kinematic state vector is in the from [X k ] = ([xk ], [x˙ k ], [yk ], [y˙ k ])T = ([xk , xk ], [x˙ k , x˙ k ], [yk , yk ], [y˙ k , y˙ k ])T ,

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(7)

j j , wβ,k )T , is assumed (but not restricted) to be wjk = (wd,k Gaussian, with a known covariance matrix R = diag(σd2 , σβ2 ). The interval measurements vector is [z jk ] = ([djk ], [βkj ])T , where [djk ] is the interval range and [βkj ] is the interval bearing of the measurement point j. One way to describe these components is:

and the interval parameter vector, in its general form, is [Θk ] = ([θ1,k ], [θ2,k ], · · · , [θnΘ ,k ])T = ([θ1,k , θ1,k ], [θ2,k , θ2,k ], · · · , [θnΘ ,k , θnΘ ,k ])T ,

(8)

The motion of the interval centre of the extended target is modelled by the nearly constant velocity model [27], [28]. The evolution model for the interval state of the target is (9) [X k ] = A[X k−1 ] + Γ[η s,k ].   1 Ts for the two The state transition matrix A1 = 0 1 dimensional case is given by A = diag(A1 , A1 ), Γ =  2 T Ts /2 Ts 0 0 , Ts is the sampling interval. 0 0 Ts2 /2 Ts The system dynamics noise [η s,k ] is characterised by the standard deviation parameters σx and σy . Then the system dynamics noise is represented as a Gaussian noise process 2 2 with  4 covariance Q = diag(Q1 σx , Q1 σy ), where Q1 = 3 Ts /4 Ts /2 . The evolution for the extent is assumed Ts3 /2 Ts2 to be a random walk model, described by the equation [Θk ] = [Θk−1 ] + [η p,k ],

(10)

where λT is the mean number of measurements originating from the target and ρ is the uniform clutter density. The likelihood that each of the j th measurement is related to the pth box particle is given with the relation: (p)

j + wβ,k ,

(p)

(p)

(p)

p([z jk ]|[xk ]) = p([z jk ]|[V k ])p([V k ]|[xk ]),

(17)

(p)

where [V k ] ∈ Rnv denotes an interval in the state space that is a plausible source of measurement. In practice, these (p) visible intervals [V k ] depend on the object nature, position (p) and parameters [xk ], on the sensor state (e.g., the sensor position and angle of view), and on the level of uncertainty (p) (p) assumed. The pdf p([V k ]|[xk ]) represents the probability of an interval in the state space to be a source of measure(p) ment given the (extended object) box particle [xk ]. The pdf (p) p([z jk ]|[V k ]) gives the probability of a measurement [z jk ] to (p) belong to the measurement source interval [V k ], relevant for the box particle p. Examples for these pdfs are given in the realisation of the algorithm in the next section. VI. R EALISATION OF THE A LGORITHM

(11)

This section presents the different steps of the Box PF applied to the problem of extended target tracking. The particularity here is that we have joint state and parameter estimation. For this realisation of the algorithm the parameter in the state vector is considered to be the radius of an “interval circle”, i.e. [Θk ] = [rk ] = [rk rk ]. After the initialisation of the particles, Step A describes the state prediction, where the box particles are propagated through the system dynamics’ equation and through the parameters’ equation. Step B.I takes into account that some

The range and bearing components are described respectively:  j j2 j (12) dk = xj2 k + yk + wd,k , xjk

(15)

The main difference between the probabilistic interpretation of the classical PF and the Box PF is in the type of pdf used to represent the measurements uncertainty. In Box PF the posterior state pdf can be represented as a sum of uniform pdfs [29]. To reflect the presence of clutter the total likelihood is calculated in a way similar to [13]:  M   λT (p) (p) j 1+ p([z k ]|[xk ]) , p([z k ]|[xk ]) = (16) ρ j=1

A scenario with a network of multiple low cost sensors positioned along the trajectory of movement is considered. One or more of them can be active to collect measurements at a particular time. To select the active sensor(s) at the instance of time k, the estimated state at the previous time step k − 1 is propagated through the state evolution model xk−1 ]). The sensor(s), with coordinates [ˆ xk|k−1 ] = [f ]([ˆ xk|k−1 ] is/are {xs,k , ys,k }, closest to that predicted position [ˆ selected as active during time step k. The collected measurements consist of range and bearing. In this work the performance evaluation is done using simulated measurements data. The number of measurements M obtained at each time step from an active sensor consists of N measurements originating from the target and C clutter measurements, i.e. M = N + C. The measurement equation for the measurement point j, j = 1, ..., M is of the form:

ykj

[βkj ] = βkj + [−3σβ , +3σβ ],

V. M EASUREMENT L IKELIHOOD FOR THE B OX PF IN THE CONTEXT OF E XTENDED O BJECT T RACKING

C. Observation Model

βkj = tan−1

(14)

i.e. the error in the range measurements is assumed to be dependent on the distance between the observed point and the sensor.

where the interval parameters noises [η p,k ] are characterised by σ θ ∈ RnΘ . Then again, the augmented interval state vector is [xk ] = ([X k ]T , [Θk ]T )T .

z jk = h(xk , wjk ).

[djk ] = djk (1 + [−3σd , +3σd ]),

(13)

where xjk and ykj denote the Cartesian coordinates of the actual point of the source which generates the measurement in the case of two dimensional space. The measurement noise

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measurements fall outside the visibility region of the sensor or the plausibility region for the scenario and these measurements are hence discarded automatically as unreliable. Step B.II performs contraction between the plausible measurement sources and the predicted measurement boxes. In order to facilitate the solution of the considered problem, instead of working in polar coordinates with respect to the sensor, a transformation to polar coordinates with respect to each of the box particles is performed in Step B.III. Step B.IV takes into account the visibility area of the sensor for each of the box (p) (p) particles and calculates the likelihood term p([V k ]|[xk ]). Step B.V is devoted to introducing variety in the box particles by resampling. Further we iterate the likelihood calculation and the resampling step several times in order to perform the contraction. This approach is chosen due to the sophisticated relationship between the predicted box particles and the predicted measurement boxes. Finally, in Step B.VI, the overall box estimate is represented as the weighted sum of all box particles. Note, however, that as an output the centre of the box is consider as the final state estimate.

for the range and bearing intervals with respect to the sensor. Their values can be chosen in a way which reflects a particular ˆk − x ˆk , yˆk,c = yˆk − yˆk and scenario. The notations x ˆk,c = x rˆk,c = rˆk − rˆk are, respectively, for the midpoints of the predicted x and y interval position coordinates and interval radius of the target. The indicator functions is given by:  1 if (18) and (19) hold, j (20) I = 0 otherwise. Visualisation of the indicator functions is plotted with black line in the example in Fig. 1(c)-(d). II. Contraction I (with Respect to the Sensor) For each of the N particles, p = 1, ..., N , an area of interest is defined based on their position: (p)

[dk ] where (p)

dk

(p)

dk

Initialisation

N (p) The initial particles [x1 ] are generated from the

= =

=

(p)

(p)

[dk , dk ],

 (p) (p) (p) [(xk ] − xs,k )2 + ([yk ] − ys,k )2 − [rk ]),  (p) (p) sup( ([xk ] − xs,k )2 + ([yk ] − ys,k )2 ), inf (

(p)

(p)

the interval radius of the particle p is denoted [rk ], and [xk ] (p) (p) and [yk ] are the interval coordinates [Xk ] of the particle (p) (p) [xk ]. The distance interval [dk ] relates to the potentially visible area from the sensor {xs,k , ys,k }. In angular direction the interval is already defined by the measurements that fall within the indicator function defined in Step B.I. This area for one of the particles is plotted with red line in Fig. 1(e)(g). Only the portion of the measurements that falls within (p) the interval [dk ] is considered in the further steps. This contraction is realised using an intersection.

p=1

ˆ 0 , and their predicted starting position of the extended target x

N (p) = 1/N . weights are initialised as [w1 ] p=1

Step A. State Prediction The persistent box particles at time step k−1 are propagated (p) (p) through the evolution model (3), [xk ] = [f ]([xk−1 ]). An inclusion function is applied as part of the prediction step. The (p) predicted state [xk ] of one of the box particles, represented as an interval circle, is plotted in Fig. 1(a)-(d),(i) with black colour.

(p),j

[Dk

(p)

] = [dk ] ∩ [djk ]

(21)

As it is discussed in [23] the ratio of the area covered by the intervals after and before the contraction gives a term for updating the weights:

Step B. Measurement Update (p)

p([z jk ]|[V k ]) =

I. Gating The estimated state vector at time step k − 1, [ˆ xk−1 ], is xk−1 ]) propagated through the evolution model [ˆ xk ] = [f ]([ˆ to obtain a predicted position for the extended object at time step k. Based on that position, an area of interest for the measurements is defined:

S{[D(p),j ][β j ]} k

k

S{[d(p) ][β j ]} k

k

(p),j

=

|[Dk

]|

(p) |[dk ]|

,

(22)

where S{[a],[b]} denotes the surface area of the shape defined by the intervals [a] and [b], and |[·]| gives the length of the interval [·].

kd ID < [djk ] < kd ID ,

(18)

III. Transformation of the Coordinates

Iβ − ( Iβ + kβ ) < [βkj ] < Iβ + ( Iβ + kβ ),

(19)

After undergoing the first contraction the measurements are transformed from polar coordinates with respect to the sensor to Cartesian coordinates and then again to polar coordinates, but with respect to the centre of the extended object box particle. The whole transformation can be written as:

where ID , Iβ and Iβ are defined as ˆk,c )2 + (ys,k − yˆk,c )2 , ID = (xs,k − x (y −ˆ yk,c ) Iβ = tan−1 (xs,k xk,c ) , s,k −ˆ Iβ = sin−1 (

(p),j

rˆk,c ID ).

[rk

(p),j [αk ]

The parameters kd , kd and kβ correspond to gating parameters

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(p),j

] = [fr ]([Dk =

], [βkj ]),

(p),j [fα ]([Dk ], [βkj ]),

(23) (24)

where [fr ] and [fα ] denote inclusion functions [29]. An example for this transformation can be seen in Fig. 1(f). It can be seen that the inclusion function uses an approximation to represent the transformed boxes.

VI. State Estimate At the final step an estimate is obtained as weighted sum of all of the box particles: [ˆ xk ] =

IV. Contraction II (with Respect to the Particles)

(p)

(p)

(p)

where (p) Rk (p)

Rk

 (p) (p) = + (|[xk ]|/2)2 + (|[yk ]|/2)2 ,  (p) (p) (p) = rk − (|[xk ]|/2)2 + (|[yk ]|/2)2 .

VII. P ERFORMANCE E VALUATION We evaluate the performance based on four criterions: rootmean-square error (RMSE), Volume of the estimated interval circle, Inclusion within the predicted box estimation and Inclusion within all of the box particles. The RMSE refers to the error between the real surface of the target and the curve in the middle of the estimated box shape. The Volume criterion [23] gives the volume within the estimated box circle. The Inclusion criterions give the percentage of the target’s surface that is within the box estimate or the hull of all of the particles, respectively. The varying parameters are initialised as follows. The number of the box particles is N = 20, 2 contraction iterations, 5 active sensors, clutter density ρ = 10−4 . The measurements uncertainty are σd = 0.005 and σβ = 0.3◦ , respectively for the range (multiplicative with the distance between the sensor and the object) and for the bearing. The intervals are formed according to the 3σ rule. The fixed parameters are the standard deviations for the x and y coordinates of the speed for the actual movement σx = σy = 0.1 and for the ˜y = 0.2 The standard deviation for random filter σ ˜x = σ walk change of the radius is σθ = 2. The mean number of measurements per scan for each active sensors is λT = 5. The parameters for the indicator function are chosen kdmin = 0.2, kdmax = 1.5 and kβ = π/8. The graphs on Fig. 2(a) show that stable results with high accuracy in the position and converging Volume are produced with 10 or more box particles. The percentage of the target’s contour that falls within the estimated box circle is 90% or above. For the area covered by all of the particles, that percentage is above 99%. As Fig. 2(b) shows the contraction step is necessary. For two or more iterations the estimation is converging. Fig. 2(c) and Fig. 2(d) show how the different values of the uncertainty in the measurements reflect on the estimation. Employing more than one sensors is beneficial for the accuracy of the estimation as seen on Fig 2(e). Fig 2(f) demonstrates that the algorithm manages to cope with clutter with density ρ = 5 × 10−4 or less.

(26) (27)

(p)

(p)

] = [Rk ] ∩ [r(p),j ]

(p),j [Ak ]

=

(p) [Ak ]

∩ [α

(p),j

(28)

]

(29)

Again a weighting term is calculated as the ratio of the area covered by the intervals after and before the contraction: p([V

(p) (p) k ]|[xk ])

=

S[R(p),j ],[A(p),j ] k

k

S[R(p) ],[A(p) ] k

(32)

The centre of the estimated interval circle [ˆ xk ] gives the final estimate of the target state. Nevertheless, the intervals for the position coordinates and for the radius determine a confidence interval.

The angular interval [Ak ] is defined by the tangent from the sensor to the outer border of the particle. The visual (p) (p) representation of the area defined by [Rk ] and [Ak ] is shown on Fig. 1(g)-(h). (p),j The transformed and contracted once measurements [rk ] (p),j and [αk ] are contracted again by an intersection with the visibility interval: (p),j

(p)

(25)

(p) rk

[Rk

(p)

wk [xk ]

p=1

For each box particle an interval of visibility from the sensor is defined in polar coordinates with respect to the centre of that (p) (p) box particle described by distance [Rk ] and angle [Ak ]. The (p) distance interval [Rk ] is defined as: [Rk ] = [Rk , Rk ],

N

.

(30)

k

Substituting (22) and (30) in (17) let us calculate the measurements likelihood function (16) and hence update the particle weights: (p) (p) (p) wk = wk−1 p([z k ]|[xk ]). (31) V. Resampling The resampling strategy for the box-particles is not as straight forward as in the classical PF algorithms. In addition to drawing N duplicated particles with equal weights (p) (p) {wk = N1 , [xk ]}N p=1 , different strategies for changing the position and the size of the uncertainty interval region can be applied here. As an example, with equal probability one of the intervals for x, y or the radius can be chosen and divided into several equi-sized subintervals. Either of these intervals can be selected with an equal probability. The newly sampled box particle has a reduced uncertainty region for one of its dimensions. The uncertainty regions can also be moved or changed in other ways. In the current work we do not evaluate the performance of the different resampling techniques.

VIII. C ONCLUSIONS This paper presents a solution to the extended object tracking problem with the recently proposed Box Particle Filtering

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(b) Based on the predicted position of the center of the extended target, {ˆ xk,c , yˆk,c }, as well as on the sensor position, an area of interest, or indicator function is formed for the measurements (20).

  

  

(a) The currently active sensor with coordinates {xs , ys } is plotted as a small black circle. In a circular area around it are the cluttered measurements z jk , for j = 1, ..., M with their uncertainty intervals plotted in green [z jk ]. The prior of the target state [ˆ xk ] is given as an “interval circle” in black color.



 

















  









(c) Only the measurements that fall within that region are considered further during the current time step. The actual position of the of target is plotted in cyan.

















  











 













  









(p)

(d) One of the box particles [xk ] is shown with brown dashed line, representing an interval circle.















  









(e) For each of the box particles an interval of (p) interest (in red), defined by [βkj ] and [dk ], is formed based on the sensor position, the box particle position and the bearing measurements. The likelihood boxes are contracted based on the distance to the sensor (21).









  





(p),j

(f) The contracted measurements [Dk ] and [βkj ] after the first contraction step are transformed from a polar coordinate system with respect to the sensor to polar coordinates with respect to the centre of the box particle (23) and (24).

 











  

  

  











 



  







  





(g) Another interval of interest is formed for each (p) (p) box particle [Rk ], [Ak ]. It relates to the area of the box particle visible from the sensor and is plotted with blue continuous line. Fig. 1.

 







  













  





(h) The contracted once and transformed measure- (i) The prior of the target state [ˆ xk ] is given with ments are contracted again with the region defined black dashed lines, the box estimate is in red and (p) (p) the real position of the object is in cyan. by [Rk ], [Ak ] for a second contraction step.

Visualisation of the algorithm to complement the description in Section VI

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approach. Its performance is validated and evaluated over simulated data. Promising results are reported. Future work will be focused on the comparison of the proposed Box PF with other algorithms such as the PHD filter and also evaluation using real data.

[19] M. Baum and U. Hanebeck, “Random hypersurface models for extended object tracking,” in Proc. of the IEEE International Symp. on Signal Processing and Information Technology (ISSPIT), 2009, pp. 178 –183. [20] F. Abdallah, A. Gning, and P. Bonnifait, “Box particle filtering for nonlinear state estimation using interval analysis,” Automatica, vol. 44, no. 3, pp. 807–815, 2008. [21] A. Gning, L. Mihaylova, and F. Abdallah, “Mixture of uniform probability density functions for nonlinear state estimation using interval analysis,” in Proc. of 13th International Conference on Information Fusion. ISIF, Edinburgh, UK, 2010. [22] A. Gning, L. Mihaylova, F. Abdallah, and B. Ristic, “Particle filtering combined with interval methods for tracking applications,” in Integrated Tracking, Classification, and Sensor Management: Theory and Applications, M. Mallick, V. Krishnamurthy, and B.-N. Vo, Eds. John Wiley & Sons, 2012. [23] A. Gning, B. Ristic, and L. Mihaylova, “Bernouli/ box-particle filters for detection and tracking in the presence of triple measurement uncertainty,” IEEE Transactions on Signal Processing, vol. 60, no. 5, pp. 2138 – 2151, 2012. [24] N. Petrov, L. Mihaylova, A. Gning, and D. Angelova, “A novel sequential Monte Carlo approach for extended object tracking based on border parameterisation,” in Proceedings of the 14th International Conference on Information Fusion, Chicago, USA, 2011, pp. 306–313. [25] B. Ristic, “Bayesian estimation with imprecise likelihoods: Random set approach,” Signal Processing Letters, IEEE, vol. 18, no. 7, pp. 395 –398, july 2011. [26] L. Jaulin, M. Kieffer, O. Didrit, and E. Walter, Applied Interval Analysis. Springer-Verlag, 2001. [27] X. R. Li and V. Jilkov, “A survey of maneuveuvering target tracking. Part I: Dynamic models,” IEEE Trans. on Aerosp. and Electr. Systems, vol. 39, no. 4, pp. 1333–1364, 2003. [28] Y. Bar-Shalom and X. Li, Estimation and Tracking: Principles, Techniques and Software. Artech House, 1993. [29] A. Gning, B. Ristic, L. Mihaylova, and F. Abdallah, “Introduction to box particle filtering,” IEEE Signal Processing Magazine, 2012, in press.

Acknowledgements. The authors acknowledges the support of 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).

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