Autonomous Crowds Tracking with Box Particle Filtering and Convolution Particle Filtering

Autonomous systems such as Unmanned Aerial Vehicles (UAVs) need to be able to recognise and track crowds of people, e.g. for rescuing and surveillance purposes. Large groups generate multiple measurements with uncertain origin. Additionally, often the sensor noise characteristics are unknown but measurements are bounded within certain intervals. In this work we propose two solutions to the crowds tracking problem - with a box particle filtering approach and with a convolution particle filtering approach. The developed filters can cope with the measurement origin uncertainty in an elegant way, i.e. resolve the data association problem. For the box particle filter (PF) we derive a theoretical expression of the generalised likelihood function in the presence of clutter. An adaptive convolution particle filter (CPF) is also developed and the performance of the two filters is compared with the standard sequential importance resampling (SIR) PF. The pros and cons of the two filters are illustrated over a realistic scenario (representing a crowd motion in a stadium) for a large crowd of pedestrians. Accurate estimation results are achieved.


Introduction
Tracking a large number of objects requires scalable algorithms that are able to deal with large volumes of data characterised by the presence of clutter. Although groups are made up of many individual entities, they typically maintain certain patterns of motion, such as in the case of crowds of pedestrians [1]. When the number of objects in the group is huge, e.g. hundreds and thousands, it is impractical (and impossible) to track them all individually. Instead of tracking each separate component, the group can be considered as one whole entity. Large group techniques identify and track concentrations, typically the kinematic states of the group and its extent parameters [2].
The box PF [15,19,20] relies on the concept of a box particle, which occupies a small and controllable rectangular region having a non-zero volume in the state space. The box PF affords to resolve the data association problems arising from the multiple measurements originating from the crowd. This is a common case when a UAV is flying over a region and collects data, seeing the area from above. This paper has several novel contributions when compared with our previous works such as [21,22,23]. These novelties include: i ) a generalised likelihood function for the box PF is derived when the state vector consists of kinematic states and extent parameters; ii ) the likelihood of the Box PF is calculated based on optimisation, by solving a constraint satisfaction problem (CSP) with multiple measurements; iii ) the online estimation of the crowd and clutter measurement rates; iv ) an adaptive CPF is proposed. The developed CPF is able to deal with multiple measurements, including a high level of clutter. It is able to resolve the data association problem without the crowd and clutter measurement rates. The adaptive CPF can estimate both dynamic kinematic states and dynamic parameters which is a different solution from the CPF based approach for static parameters presented in [16,17,18].
Both filters have very appealing properties in solving nonlinear estimation problems. Both filters operate in the condition of uncertain and imperfect observations: fluctuating number of sensor reports.
The performance of the box PF and CPF is evaluated for two different cases. Firstly, in a fully matched case where the models used by the filter directly match that used by the simulator, and secondly, in an unmatched scenario of a realistic crowd moving through a bottleneck. Both filters are compared with the standard sequential importance resampling PF (SIR PF) [24] in terms of filter accuracy and computational complexity.
A main advantage of the box PF consists in its robustness to measurement characteristics and its ability to be implemented efficiently in a distributed way. The CPF is based on the principles of kernel based learning and can deal with problems where the likelihood is not available in an analytical form or it is difficult to calculate.
The rest of this paper is organised in the following way. Section 2 describes the state space modelling of a crowd. Section 3 is a brief overview of inference in a Bayesian framework. Section 4 presents the adaptation of the box PF for group object tracking. Section 5 introduces the CPF for crowd tracking, which is followed by a performance evaluation of the presented approaches in Section 6. Finally, conclusions are presented in Section 7.

State Space Modelling of a Crowd
The characteristics of the crowd and scene that are required to be inferred at each time step k, k = 1, 2, ..., K, are represented by an augmented state vector: where X k is the kinematic vector of the centre of the crowd, and Θ k is the parameter vector which characterises the crowd extent. Multiple measurements are received from the crowd and from clutter at each time step, thus the state vector includes λ k which is the measurement rate vector. The notation (·) T is the transpose operator. In this paper we consider the two-dimensional case, where the kinematic vector consists of the position coordinates and the velocity of the centre of the crowd and the extent of the crowd is represented by a rectangle. The resulting kinematic vector has the following form: X k = (x k ,ẋ k , y k ,ẏ k ) T (2) and the parameter vector is given by: where a k and b k represent the lengths of the sides of the rectangle in the x and y dimensions, respectively. The measurement rate vector is represented by: where λ T,k and λ C,k represents the crowd and clutter measurement rates, respectively.

Crowd Dynamics Model
The motion of the centre of the crowd is modelled by a correlated velocity model. The correlated velocity model is related to the Singer model [25] and jerk model [26] with the difference being that the velocity component is correlated in time and that the second and other higher order derivatives of position are negligible. The evolution model for the kinematic state of the target is represented mathematically by where η k represents the system dynamics noise. The state transition matrix is given by where T s is the sampling interval, ⊗ denotes the Kronecker product, I 2 denotes the 2 × 2 identity matrix, and α is the reciprocal of the velocity correlation time constant. The covariance of the system dynamics noise η k can be modelled as where σ 2 v is the variance of the velocity of the crowd centroid for a single dimension and The evolution for the crowd extent is assumed to be a random walk model, described by where the parameter noise η p,k is characterised by the standard deviation σ θ ∈ R nΘ .

Observation Model
In this paper we consider the scenario where the measurements originate from within a confined area. However, other scenarios, such as the case where the measurements only come from the border of the crowd, have a similar solution.
The total number of measurements M k , obtained at each time step from the sensor consists of the M T,k number of measurements, originating from the crowd and M C,k clutter measurements, i.e. M k = M T,k + M C,k . The number of measurements M T,k originating from the crowd is considered as a Poisson-distributed random variable with mean value of the crowd rate, λ T,k , i.e., M T,k ∼ P oisson(λ T,k ). Similarly, the number of clutter measurements is M C,k ∼ P oisson(λ C,k ). The M T,k measurements originating from the crowd are uniformly located in the area represented by the crowd. The M C,k clutter measurements are uniformly located in the region about the crowd.
Typically in point target tracking, an observation model which directly relates the states to the measurements is available, in the form given by: where ξ k represents a observation noise. However, since the crowd is an extended target 1 , there is no direct observation model. The observations can be indirectly related to the states through the sensor characteristics and the target model.
The sensor characteristics describe the relationship between the measurement point m, m = 1, ..., M k and the measurement source in a Cartesian coordinate system and is of the form: whereh(·) is the measurement function and x m k = (x m k , y m k ) T denotes the Cartesian coordinates of the measurement source in a two dimensional space. In this paper we consider the following model: where H = I 2 , and the measurement noise ξ k = (ξ 1,k , ξ 2,k ) T , is assumed (but not restricted) to be Gaussian, with a known covariance matrix R = diag(σ 2 1 , σ 2 2 ). The vector of interval measurements is are the intervals of the m-th measurement point. One way to describe these components is by representing the noise terms in equation (11) as intervals: At each time step k, the M k interval measurements are combined into an interval matrix Each measurement originates from either random clutter or the crowd but its origin is unknown. The target model describes the relationship between the states and the measurement sources for the M T,k measurements that originate from the crowd. As previously described, the measurement sources are uniformly distributed across the region which exhibits measurements, and this region is represented by the states through the following probability density: where U [x] (·) denotes the multivariate uniform probability density function (pdf) with the interval [x] as support. The support of the uniform distribution describes two independent regions which cover the area of the rectangle used to approximate the extent of the crowd:

Inference in a Bayesian Framework
Classic Bayesian inference relies on computing the posterior distribution from a prior distribution and measurements. The posterior distribution can be updated sequentially based on a prediction step, followed by an update step. The following equation describes the prediction: The measurement update is described by the following equation: The recursive relationship of equations (16) and (17) form the optimal Bayesian solution. Utilising these equations for Bayesian filtering is generally not possible since an analytical solution rarely exists. A solution for when the state space model is linear and perturbed by Gaussian noise is referred to as the Kalman filter [27]. Several techniques have been used in the more general case consisting of non-linearities and non-Gaussianity in the state space model, such as the extended Kalman filter [27], unscented Kalman filter [28] and particle filter based techniques [29] to name a few.
For further notational convenience, the marginal state is defined as follows: In this application the posterior distribution can be further factored into the following form: This factorisation implicitly states that the crowd and clutter measurement rates are independent of the kinematics and extent of the crowd. This is true for the clutter measurement rate but not necessarily valid for the crowd measurement rate. However, the variance of the prior distribution for the crowd rate is sufficient to represent the variation of the number of measurements over time.
It has been shown that a closed form recursive Bayesian solution exists for the estimation of the mean of a Poisson distribution, based on using the conjugate prior Gamma distribution [30]. The crowd and clutter measurement rates are estimated based on this concept 2 , and the focus of this paper thus lies on the calculation of the marginal posterior distribution for the states representing the kinematics and extent of the crowd, p(x k |Z 1:k , λ k ), using the novel box particle filter and convolution particle filter algorithms.

The Box Particle Filter for Crowd Tracking
This section begins with a review of the box PF in point target tracking without clutter, thus, the following subsection does not consider the extent of the target, i.e. x k = X k .

The Classic Box Particle Filter
The concept of a box particle is introduced where a box particle represents a small region with controllable size (or volume). The box PF approximates the posterior state pdf with a mixture of uniform pdfs [31,19], i.e.
For the box PF, the time update can be written as: For any transition function f , we can obtain an inclu- Thus, for all p = 1, . . . , N we can write Using interval analysis techniques, the support function 3 for the pdf terms in (21) Combining (21) and (23) gives Approximating each pdf term using one uniform pdf component may not be accurate enough. However, as for the PF, it is sufficient to approximate the first moments of the pdf. If a more accurate representation is required then each term can be approximated as a mixture of uniform pdfs as shown in [19].
Under the assumption that at time instant k, the time update pdf p(x k |z 1:k−1 ) can be represented by a mixture of N uniform pdfs with interval supports [x (p) k|k−1 ] and weights w p k−1 , the measurement update step can be performed. A probabilistic model p ξ for the measurement noise ξ k is also available. It is assumed in general that p ξ can be expressed by using a mixture of uniform pdfs. For simplicity and without loss of generality, p ξ is considered here to be a single uniform pdf, such that the box measurement [z k ] contains all realisations of (10). Then we have: p(z k |x k ) = U [z k ] (h(x k )) and according to equation (17), the measurement update can be expressed with the equation: where α k denotes the normalising constant. Each of the terms is also a constant function with a support being the following region S p ⊂ R nx , where Equation (26) represents a constraint and from its expression we can deduce that predicted supports [x  (26), we can write where | . | denotes the interval length (respectively the box volume in the multidimensional case). By combining equations (25) and (27), and keeping in mind that [x In the Sequential Importance Resampling (SIR) PF, each particle weight is updated by a factor equal to the likelihood p(z k |x (p) k|k−1 ), followed by normalisation of weights. In the Box PF this step is very similar, i.e., after contracting each box particle [x according to (28) the weights are updated by the ratio In summary, the posterior distribution is approximated by

Derivation of the Box Particle Filter Posterior Distribution in Crowd Tracking
The prediction step for the crowds tracking box PF follows the same spirit as described by equations (21) to (24). However, when dealing with multiple target originated measurements and clutter measurements, the update step is required to be re-derived. When dealing with an extended target in a SIR PF, the generalised likelihood is given by [32] p where ρ = λ C,k A C represents the clutter density and A C represents the area of the region where clutter may be emitted from. We extend the generalised likelihood for the crowd tracking box PF.
A probabilistic model p ξ k for the measurement noise ξ k is available. It is assumed in general that p ξ k can be expressed by using a mixture of uniform pdfs. For simplicity and without loss of generality, p ξ k is considered here to be a single uniform pdf, such that the box measurement [z m k ] contains all realisations of (11). Then we have: . Substituting this equation and (14) into (30), we obtain The updated marginal posterior distribution for crowd tracking can then be expressed with the equation: Each of the M k product terms, , is also a constant function with a support being the following region k,m ] is the smallest box containing S p,m . Substituting (34) in (32) we have the following updated expression for the posterior distribution: The integration is approximated by a uniform distribu- , where r(x k ) represents an interval dependent on the states and measurement function. The validity of this assumption is explored in Appendix A. The posterior distribution can thus be expanded accordingly: where The posterior pdf is a weighted sum of uniform pdfs. The number of weighted uniform pdf's increases exponentially with the number of measurements, which can render the algorithm too computationally expensive for a large number of measurements. Typically, there is a large disparity between the weights of the summed uniform pdfs, since . This allows for the approximation of the posterior pdf by a single uniform pdf for each box particle. The dominating term in the uniform pdf weights is . This term is maximised when all the measurements are assumed to originate from the crowd. If the posterior pdf was approximated by this uniform pdf, the expression would be given by: The multiplication of uniform pdfs can be further simplified to obtain a single uniform pdf with a corresponding weight. This includes the intersection of the intervals of all the uniform pdfs: However, this intersection result typically does not exist or leads to a poor contraction due to the implicit assumption that the measurements originate from the crowd. A more robust approximation for the posterior pdf, which does not require explicit knowledge of the origin of a measurement is given by: where S The difference between the posterior pdf represented by equations (36) and (39) is highlighted graphically through an example in figure 1.
The algorithm for crowd tracking is summarised in Table 1.

Box Particle Filter Implementation Considerations
In general, an important step in interval based techniques used for state estimation is in interval contraction [33]. In the box PF it is required to obtain the contracted box particles by solving the CSP described by equation (33). For the crowds tracking box PF, contraction is achieved by implementing the Constraints Propagation (CP) technique. The main advantages of the CP method is its efficiency in the presence of high Predicted box particle: Eq (36) Eq (39) Contraction result for measurement 1: Contraction result for measurement 2: Contraction result for measurement 3: Posterior pdf:  (36) and (39). This example consists of 3 measurements (measurement 3 represents a clutter measurement), a single state dimension, and a single box particle.
redundancy of data and equations. The CP algorithm, which in this application is the calculation of the intersection of the box states for each particle with all the interval measurements, is illustrated in Table 2. For notational convenience, Table 2 refers directly to the supports of the uniform distributions found in the posterior distribution, for example in equation (35).
Generally, in particle filtering, there are a variety of different resampling schemes available [36]. Based on the weights, a particle is replicated a specific number of times. The box PF differs by dividing a selected box particle into smaller box-particles as many times as it was to be replicated. Several subdivision strategies exist. In this paper we subdivided based on the dimension with the largest box face.
The parameter q is introduced in equation (39). This specifies the maximum number of clutter measurements that still result in a contraction of the states that exists. These are the clutter measurements which are located close in vicinity to the crowd. The area in the measurement space where a measurement can result in a contraction of the state that exists is dependent on the size of the box particle. An estimate for q can then Table 1 The Proposed Box Particle Filter for Crowd Tracking Initialisation Use the available prior information about the target's kinematics and extent parameters states to initialise the box particles.
Repeat for K time steps, k = 1, ...K, the following steps: (1) Prediction Propagate the box particles through the state evolution model to obtain the predicted box particles. Apply interval inclusion functions as described in [34,35].

(3) Output
Obtain an estimate for the state of the group object as a weighted sum of all of the particles: Further, a point estimate for the state can be obtained as the midpoint of the box estimate of the state.
(4) Resampling (a) Compute the effective sample size: be determined through: The estimated clutter measurement rate is used to obtain an approximate ρ k : where the area of the clutter region is given by A CR = A S − A T , A S is the total area observed by the sensor, and A T is the area of the crowd, approximated from the estimate of the crowd at the previous time instant, k − 1. For the given crowd tracking problem, the area A CT is given by: where the notation x and x refers to the infimum and supremum of box x, respectively. The factor of 4 in equation (43) was introduced to take into account that the area A CT also includes the region inside of the crowd, where no clutter measurements are found. It is important to note that the algorithm is fairly robust to the value of q as this represents a maximum number of clutter points, and not the actual number of clutter points. Solve the CSP to contract each box particle with all of the measurements. , ,

The Convolution Particle Filter for Crowd Tracking
This paper develops an adaptive CPF algorithm for crowds tracking. The CPF approach relies on convolution kernel density estimation and regularisation of the distributions, respectively, of the state and observation variables [17,18,37]. The CPF belongs to a class of particle filters with valuable advantages: simultaneous estimation of state variables and unknown parameters and continuous approximation of the corresponding pdf. Being likelihood free filters makes them attractive for solving complex problems where the likelihood is not available in an analytical form.
The key novelty of the proposed adaptive CPF algorithm stems from: 1) its ability to deal with multiple measurements, including high level of clutter, 2) ability to resolve data association problems, without the need to estimate clutter parameters, 2) estimation of dynamically changing parameters of crowds jointly with the dynamic kinematic states.
For the purposes of crowds tracking the marginal posterior state distribution has to be calculated and can be expressed to be independent of the clutter and measurement rates, reducing the expression from equation (19) to: The CPF relies on the following representation of the conditional state density: Suppose, that we can sample from the state and measurement pdfs, p(x k |x k−1 ) and p(z m k |x k ), respectively. Then we can obtain a sample from the joint distribution {x . . , N } at time step k by k successive simulations, starting from the sample of the initial distribution p 0 (x). We can obtain the following empirical estimate of the joint density The kernel estimate p N k (x k , Z 1:k ) of the true density p(x k , Z 1:k ) is obtained by convolution of the empirical estimate (49) with an appropriate kernel where and K x h and K Z h are the Parzen-Rosenblatt kernels of appropriate dimensions and bandwidth h. According to equation (48), the estimate of the posterior conditional state density has the following form: When dealing with point targets, the measurements are modelled as points in the measurement space. However, in the application of crowd tracking, a single point in the state space translates into a region in the measurement space. The role of the kernel in the point target case can be interpreted as a conversion of the measurement point to a measurement region. Thus there is no need for the specification of a kernel in the crowd tracking CPF framework, as the densities that describe the sensor characteristics and target model can be used to obtain an approximate region in the measurement space for each predicted particle, and are thus equivalent to the kernel. The bandwidth h of the kernel varies according to the state, resulting in a variable bandwidth which adds additional flexibility to the CPF while also removing the need to specify a bandwidth parameter. In this application the kernel is approximated as a variable uniform distribution.
An advantage of the proposed CPF framework is that it implicitly resolves the data association problem. Since there are multiple measurements assumed to be independent, the weights of individual measurements are multiplied to obtain a single weight for the particle. However, clutter measurements may occur outside of the support of the adaptive uniform kernel. This would result in particles having a weight of 0 when evaluated by the kernel. To overcome this, the adaptive uniform kernel based on the crowd is added with a uniform distribution which covers the entire observation area of the sensor. The advantage to such an approach is that it removes the need for the estimation of the clutter and measurement rates when only the kinematic states and extent parameters are of interest.
The weights are updated sequentially according to For the crowd tracking problem presented, the kernel (53) is a compositional kernel comprised of a sum of two uniform pdfs: where the support SS is the entire region observed by the sensor, and the support CS is related to the location of crowd measurements given the particle state. In this paper we utilised the region, r(x k ), as described in Appendix A.
A detailed description of the CPF algorithm is given in Table 3. Table 3 The Convolution Particle Filter for Crowd Tracking I. Initialisation: k ) -measurement region sampling 3) Weights updating: f or i = 1, . . . , N Update the weights according to (53), 5) Estimating the output state: k are the normalised weights.

Performance Evaluation
In this work the performance evaluation is done using simulated measurements data. All simulations were performed on a mobile computer with Intel(R) Core(TM) i7-4702HQ CPU @ 2.20GHz with 16GB of RAM.

Test Environment
Two different crowd simulations were used to demonstrate the performance of the crowd tracking box PF and CPF.
Rectangular Group Object Simulator: A crowd with a rectangular extent located in a two dimensional plane. The centre of the crowd undergoes motion according to a correlated velocity model. The lengths of the sides of the crowd vary at each time step according to a random walk. Crowd measurements comprise of a number of points uniformly located within the confines of the crowd at each time step. In addition to the crowd measurements, clutter measurements are also present, uniformly located in a region about the crowd.
Realistic Crowd Simulator: Individuals within the crowd are represented as points moving in a two dimensional space. The dynamics of the group is determined by forces acting on those individuals: Forces of attraction towards one or more static goal points; constrained forces of repulsion between the elements of the group; constrained forces of repulsion from a set of linear contextual constraints. The net effect is that a crowd of individuals will move in a reasonably realistic manner between constraints. The simulator outputs a set of points corresponding to the positions of each individual in the crowd at each sampling step. The positions of the individuals represent the measurement sources. Additionally, clutter measurements are also present, uniformly located in a region about the crowd.

Rectangular Group Object Simulator Results
This section presents results based on the Rectangular group object simulator. The parameters are as follows: where x i represents the ground truth,x i represents the filter estimate, and N M C represents the number of Monte Carlo runs.
The first set of results illustrate how the box PF, CPF and SIR PF perform when estimating the marginal posterior distribution, p(x k |Z 1:k , λ k ), with measurement and clutter rates assumed known. Only 4 box particles are required to track the crowd. For comparison, the CPF and SIR PF were also run with 4 particles, however, this resulted in consistent filter divergence due to particle degeneracy. Instead the number of particles were selected based on achieving a similar computational expense for all algorithms. The number of Monte Carlo runs is 100. The resultant RMSE values are illustrated in Figure 2. The comparison of the computational complexity for these results are presented in Table 4. It is worth noting that the implementation of the box PF utilises the INTLAB toolbox for performing interval operations. INTLAB was initially designed and optimised for estimating rounding errors. We believe that utilising alternative methods for the interval operations could significantly reduce the computational complexity of the box PF. The box PF and CPF are able to lock on to the crowd significantly faster than the SIR PF. It is noted that the RMSE is generally higher for the box PF once all filters have locked onto the crowd. This can be attributed to the approximations made in the derivation of the marginal posterior pdf. The SIR PF is also matched in terms of the model noise and likelihood expression.
The interested reader is referred to our previous works for Box PFs for point targets where a detailed comparison is presented, with Bernoulli filters in [38] and with a Probability Hypothesis density (PHD) filter [39]. In these works it is shown that the Box PF for point target tracking requires a significantly smaller number of box particles compared with the particles needed in the Bernoulli and PHD filters, including the computational cost. The Box PF can be used is sensor network systems as it has been shown in [40] and its key advantage is that it provides accurate estimation results with a small number of particle. The Box PF can also be used in industrial applications, e.g. such as those in [41,42] and other network control systems.
The second set of results re-iterate the experiment with a significant increase in the number of particles for the        Figure 3, and the computational cost comparison for these results are presented in Table 5. Increasing the number of particles     in the CPF and SIR PF decrease the amount of time required to lock on to the crowd, however, the faster lock comes at a significantly larger computational burden. Table 6 Matlab computational time corresponding to the results in Figure 4. Time (s)   4 13.47 16 25.22 The third set of results focuses on the effect of jointly estimating the crowd and clutter measurement rates on the box PF performance. This is compared with the performance of the box PF for the ideal case where crowd and clutter measurement rates are known. The resultant RMSE values are illustrated in Figure 4. The computational cost comparison for these results are presented in Table 6. The joint estimation results in an increase in the time required to lock onto the crowd, however, this is overcome by increasing the number of box particles at the cost of an increased computational burden.

The Realistic Crowd Simulator Results
In the realistic crowd simulator the crowd moves through a corridor which consists of a bottleneck. The crowd is initialised at the entrance of the bottleneck. This is illustrated in Figure 5. In this section a comparison between the box PF and CPF is presented to illustrate the filters operation on the realistic crowd simulator. The SIR PF is not included since it is incapable of operating without knowledge of the crowd and clutter measurements which are not available in a realistic situation. The parameters for the simulations are as follows: •         the CPF struggles to lock on to the target. Once locked, and after the crowd has passed through the bottleneck, the RMSE for the length corresponding to side a is increased. This is due to several crowd entities spreading out further away from the majority of the crowd and thus being mistaken as a clutter measurements.

Conclusions
This paper proposes a box PF and CPF framework for tracking a large crowd of entities. A theoretical derivation for the generalised likelihood function for the box PF is presented when the state vector consists of kinematic states and extent parameters. The likelihood is calculated based on optimisation, by solving a constraint satisfaction problem (CSP) with multiple measurements. An adaptive CPF is proposed able to deal with multiple measurements, including a high level of clutter. It is able to resolve the data association problem without the need to estimate the clutter parameters.
The filters adaptively track the envelop of a crowd. Both filters resolve the data association problem in an efficient way. These are two different types of filters -the Box PF works with box particles, whereas the CPF represents the probabilistic distributions with point samples. The Box PF and the CPF are compared with the generic SIR PF. The filters are both robust to sensor error characteristics. The experiments show that the Box PF is also robust to initialisation errors. The Box PF requires a significantly smaller number of (box) particles than the SIR PF and the CPF.

A Likelihood Derivation
In Section 4.2 the following approximation is presented: (A.1) In this Appendix a detailed description supporting this approximation is presented.
In order to evaluate the integral, it is required to transform the domain of the uniform distribution relating a measurement to a measurement source. The explicit expression for the pdf of this distribution is given by: It is worth noting that after the transformation, the expression in (A.3) is not necessarily uniform.

A.1 The Linear Case
For the linear observation model, given in (12), the expression in (A.3) remains uniform: g (x m k ) = The range of the uniform distribution is dependent on the noise characteristics of the sensor. The range of the second uniform distribution, U q(x k ) (x m k ), in (A.1) is dependent on the extent parameters of the target. The integral in (A.4) only exists when the two uniform distributions overlap. The overlapping region is defined by: (A.6) The approximation in (A.1) is based upon the assumption that the length of the extent is significantly larger than the sensor noise characteristics. For instance, the case when the extent tends towards an infinite length with fixed sensor noise is also equivalent to an extent with a fixed length size, and with a sensor noise tending towards zero. In this case the uniform distribution in (A.5) tends towards the Dirac delta function, i.e. Consequently resulting in equivalence in (A.1): In reality, the extent is not infinite, however in general, it is considered significantly larger than the range of the sensor noise. This is the motivating factor for the result in (A.1).

A.2 The Non-linear Case
A toy example is presented to illustrate the effect of a non-linear relationship between the sensor and a measurement source. Considering a single dimension with the following relationship,

B Crowd and Clutter Measurement Rate Estimation
The clutter rate λ C and the crowd rate λ T can be updated based on the assumption that they can be drawn from the Gamma distribution, similarly to [30]. For λ C we have p(λ C |Z k ) = GAM(λ C ; α C k|k , β C k|k ) × L(α C k|k−1 , β C k|k−1 , M C,k ) (B.1) and the updated parameters of the Gamma distribution for the clutter measurement rate are: α C k|k = α C k|k−1 + M C,k , β C k|k = β C k|k−1 + 1.