B-spline based Extended target tracking technology in cluttered environments

In order to solve the problem of tracking single extended targets in a cluttered environment, this thesis proposes a B-spline based probabilistic data association extended target tracking method (ET-BS-PDA). Firstly, the state of the extended target is modelled using B-sample strips, secondly, all events associated with the extended target will be counted based on valid measurements, and the probability of the associated events will be computed using the full probability principle. Finally, we use probabilistic data association algorithms to update state and covariance of extended targets and verify effectiveness of extended target tracking algorithms in cluttered environments through simulations.

valid measurements in the observation region are assumed to originate from the target, while weights are assigned according to the location of each measurement originating from the target, the number of repetitions, and finally the weighted average of their respective probabilities is taken as the result and filtered for output.In this paper, a single extended target tracking filter ET-BS-PDA based on B-spline shape modelling is proposed for the cluttered environment.The algorithm uses the B-spline model to model the measurements, and then gives the detailed derivation process of the parameters such as measurement likelihood and association probability in the BS-PDA.
In the selection of evaluation metrics, the root means square error (RMSE) [16][17][18][19] and Jaccard distance are utilized, the error distance can intuitively reflect the tracking effect, the RMSE corresponds to the location of the centre of mass of the extended target, the proposed Jaccard distance corresponds to the shape contour, and the feasibility of the proposed algorithm is verified by tracking simulation experiments on cross extended targets.

Dynamic model
The spatial equation of state for a single extended target with linearized, time-discretization can be expressed as follows [2]  +1 =   *   +   #(1) where  +1 is the extended target's dynamic equation of state,   is the transfer matrix,   ~N(0,   ) is the process noise and   is the covariance matrix.Since the dynamic model of a single extended target is usually a constant velocity and constant rate model (CTRV), the target state can be represented by  = [, , , ̇, v]  , where x, y denotes the position of the target in a two-dimensional spatial coordinate system.φ denotes the orientation angle, and ̇ is the turning rate.

Spline contour function.
A spline function refers to line segment definition functions connected at nodes.We can first specify some base points in the Cartesian co-ordinate system and then use the quadratic uniform B spline function to generate the contour line of the target to be detected.Typically, () of the B spline is considered as a weighted sum of   basis functions where τ is an independent parameter of the spline and τ є R,   (τ) is the basis function of the spline, Pn is the weight, and if the nodes are equally distributed on the interval [0,   ], then we can call it a standard spline function.Further, if its basis functions and corresponding weights are defined periodically, then it becomes a periodic spline function.And it is constructed by three basis functions   (), where τ є [i,i+3] is a periodic quadratic basis function with the expressions are In addition: where τ is the wandering parameter defining the B-sample domain,    is the basis function, τє[0,n], and w is the ordered weight matrix.
Among them The c(τ) formula mentioned above only expresses a basic contour without any rotation and scaling.In order to be able to use the B-spline model better in more complex scenarios, here we further introduce the concept of scale factors, and the source of the measurement on the contour can be rewritten according to formula 1 as () =  +   *   * ()# (10) Rotation matrix of the extended target heading angle φ Scale matrix consisting of two expansion factors

Build an expansion model
Attaching the extended state consisting of the scale factor   =�  ,   �  to the state vector yields  = the extended process is given by the following equation: is the process noise and    ~(0,    ), the covariance matrix According to the above formula, the measurement model from the contour can be written as () = () + #(14) Bringing equation ( 9) into it gives () =  +   *   * () + #(15) The pseudo-measurement equation is obtained after performing the squared arithmetic operation on both ends of it as where ~(0, ), ω is the Gaussian measurement noise in the zero-mean state, and the covariance is R.

3.Probabilistic data correlation algorithm
The algorithm considers all the measurements that fall into the correlation tracking gate, and calculates separately the probability of each valid measurement originating from the target according to different cases, and different valid measurements correspond to different probability values, so we can weight the measurements according to their probability to obtain an equivalent measurement and then update the target state according to the equivalent measurement.
Assuming that the sensor receives  +1 valid measurements  +1 = � +1  � =1   at moment k+1, a valid measurement is one that falls into the relevant tracking gate, i.e., the following conditions are satisfied: where +1  denotes that the jth measurement at the k+1st moment is valid, and z k+1 j denotes the probability that the jth valid measurement at the k+1st moment originates from the association of the target, and  +1 0 denotes that there is no valid measurement at the time of k+1.Assuming that the set Z k contains all valid measurements, then we can calculate the conditional probability that the jth valid measurement originates from the target based on the set Z k : The probability of an associated event can be obtained by the following calculation procedure: Based on the Bayesian criterion,  +1  can be written in the following form: The first part of the numerator in the above equation is the likelihood function, which takes the form The second part of the numerator is the posterior conditional probability of the associated event, as follows: When the correlation parameter model is used, the probability of the associated event can be found by the following equation ： where: (33) In the above formula,  denotes the clutter density and  denotes the number of dimensions of the measurement.

4.BS-PDA algorithm
Assuming that at the kth moment, the extended target is modeled with the B spline model, the target state vector is represented as where    represents the state of mass motion of the detection target, (which includes mass position, velocity, acceleration, etc.)

4.1Prediction step
The predicted state and predicted covariance at moment k+1 are denoted as  ,+1 =  , +  , *  , #(35)  ,+1| =  , *  , * ( , )  +   #(36) Assuming that the measurement noise  is a Gaussian white noise with mean 0 and covariance R, and the scaling factor   obeys a Gaussian distribution with mean   and covariance   2 , the mean matrix    and covariance matrix    of the augmented state vector are denoted as where   and   denote the mean and covariance of the target center-of-mass motion, respectively.

4.2.To calculate the extended target association probability
The association probability  ,+1 ,  of the association event  +1 ,  at moment k+1 is expressed in the model as The measurement likelihood function is where  � +1 ,,  ; 0;  +1 ,,  � can be obtained from the measurement equation ℎ * � , ,    ,    �, where    2 more shows the tracking results of the two algorithms for the extended target, from which we can see that there is not much difference between the two algorithms in the detection and trend estimation of the target center-of-mass position.Then, Figure 3 gives a zoomed-in view of the extended target shape estimation results, and the results show that the shape estimation results representing the ET-BS-PDA algorithm are more consistent with the true shape of the target than the comparison algorithm.In order to be able to visualize the effect of both algorithms on the extended target state estimation, the extended target center-of-mass estimation error plot and the proposed Jaccard distance plot are given below.5, it can be seen that the red solid line pseudo Jaccard distance of the proposed algorithm for extended targets is approximately 0.22m, while the comparison algorithm is approximately 0.24m.

6.Conclusion:
In this paper, the B-spline modelling method for extended targets and the association algorithm for probabilistic data combined with it are derived in detail for tracking a single extended target in a cluttered environment.However, the ET-BS-PDA algorithm slightly outperforms the conventional extended target algorithm in estimating the centre-of-mass position, and the shape contour estimation is more accurate.On this basis, the modeling method of a single extended target based on B-spline used in this article will also be applied to multiple extended targets, laying the foundation and providing direction for subsequent research on multi extended target tracking technology in cluttered environments.

Figure 1 .
Figure 1.Illustration of the unweighted periodic basis functions in different colors over the space of τ.A matrix form of the B spline function is used to describe the profile of the extended target in the following form: () =  *    #(5)

5 .
Simulation analysisUsing the ET-BS-PDA algorithm and RM-PDA algorithm proposed in this paper to track the extended target and uniform linear motion, and then compare the tracking effect of the two algorithms on the extended target with shape.The simulation scenario is set in the observation region with clutter environment, the area of the region is [-250,50]m × [-300,150]m, an extended target with uniform linear motion is present in the observation area and is observed and detected for 20 seconds, sampled every second with probability of detection P D =1,00, gate probability P G =0.99.The initial motion state in the two-dimensional observation scenario is  0 = [50m 50m/s 50m 40m/s].

Figure 2 .
Figure 2. Tracking Rendering Figure 3. Tracking Enlargement Figure2more shows the tracking results of the two algorithms for the extended target, from which we can see that there is not much difference between the two algorithms in the detection and trend estimation of the target center-of-mass position.Then, Figure3gives a zoomed-in view of the extended target shape estimation results, and the results show that the shape estimation results representing the ET-BS-PDA algorithm are more consistent with the true shape of the target than the comparison algorithm.In order to be able to visualize the effect of both algorithms on the extended target state estimation, the extended target center-of-mass estimation error plot and the proposed Jaccard distance plot are given below.

Figure 4 .Figure 5 .
Figure 4. RMSE error diagram for extended target centroid estimation Figure 5. Extended Target Quasi Jaccard Distance Graph Figure 4 gives the RMSE error plots of the extended target center-of-mass position estimation of the two algorithms.The red dashed line in the figure represents the error in the extended target center-of-mass estimation of the ET-BS-PDA algorithm, in contrast to the RM-PDA algorithm represented by the blue dashed line, where the errors are roughly between 0.09 m and 0.13 m, and compared with the RM-PDA algorithm, the ET-BS-PDA algorithm studied in this paper is slightly better for center-of-mass position estimation.From Figure 5, it can be seen that the red solid line pseudo Jaccard distance of the proposed algorithm for extended targets is approximately 0.22m, while the comparison algorithm is approximately 0.24m.