Extended object tracking when irregular star convex mutations

In this paper, for extended object tracking in the case of mutation of irregular stars convex shape, using Random Hypersurface Model (RHM ) to model the object, then the target shape is expressed as parameter by radial function. Considering that using circle as a prior shape information requires a long filter consumption time, the RHM-CC-IOU-UKF algorithm proposed in this paper uses the center contour method to correct the prior shape information of the object, and then uses the Intersection over Union (IOU) method to improve the object tracking accuracy in the mutation case. The object estimation shape is updated when combined with a simple filtering algorithm. Eventually, the effectiveness of this algorithm is demonstrated by simulation experiments in two scenarios.


Introduction
Object tracking has always been a hot topic that people pay attention to.Due to the limited sensor related technology, the traditional target tracking algorithm corresponds to the measurement one by one, so more attention is paid to tracking of point objects.In recent years, sensor can detect multiple measurement sources on extended targets [1] , thus producing multiple measurements.Different from point object, the extended object can not only measure the kinematic state (such as position, speed, acceleration, etc.), but also obtain the extended state of the object (such as shape, size, direction, etc.)The core of tracking is modeling the object shape, then use measurement values to eliminate certain errors, so as to improve tracking accuracy.In 2008, Koch first proposed the Random Matrix Model (RMM) method [2] , by combining two-dimensional random matrix with the extended object shape, RMM first models target shape as an ellipse, then describes extended state of using the inverse Wishart distributions [3], Gaussian or gamma distribution [4] track kinematic state.RMM is the earliest model to study the extended target, this method is simple and easy to implement.However, this model is only appropriate for tracking simple shaped targets.In [5] considered the problems of inaccurate tracking and accuracy degradation when tracking complex problems with simple models, and proposed a multiellipse model, which can track some non-elliptical objects.Although it is a great improvement over the random matrix model, it is still relatively rough when tracking some irregularly shaped objects.In order to solve the detailed information of irregular object, the random hypersurface model (RHM) mentioned in [6][7][8] , the idea of curve fitting to represent the object shape as a radial function with parameters, then is modeled by a certain order of Fourier coefficient expansion.The new algorithm RHM-CC-IOU-UKF proposed in this paper, RHM is used to model star convex shape, combined with central contour algorithm to set the appropriate prior shape for the object, to improve the accuracy of real-time tracking and lessen the effect of sudden change in object shape on the tracking effect, RHM-CC-IOU-UKF calculate the IOU method [9] of each time step, take appropriate value for judgment, and then perform the second update by means of a nonlinear filter [10].Finally, the effect of motion parameter and shape estimation of the extended object is analyzed by using RMSE and the quasi-Jaccard distance.

Random hypersurface model
Random hypersurface model is a powerful modeling method, which can not only describe common shape objects such as ellipses, circles, rectangles, etc., but also can describe irregular shapes such as star convex shapes [11] in figure 1.
Star convex object parameter vector

, y
x .In order to represent the interior of the star-convex target, RHM introduces a scaling mechanism that reduces the profile by a certain scale, so that extended object can be described as a radial function of the distance to the object boundary point to k m about the angleφ , expressed in following form where ( )  ), when k s equal to 1, it means that the measurement sources are located on the contours of target.
Next, expand radial function into the following form with an N-order Fourier series: Where, ( ) In turn, equation ( 1) becomes (5) Shape estimate is related to the number of Fourier levels.Generally, setting the Fourier coefficient to about ten can achieve the tracking effect.) 8) Square both sides of equation ( 8) to derive the following pseudo-measurement equation.
Now, derives the relationship between , , , , and , k l z , gives the final measurement model.

Movement models
Assuming that two parameters are independent of each other, the kinematic model can be expressed as k F is the state transfer matrices, k Q is process noise.

Problem Statement
The RHM can describe the object shape when the prior shape is unknown.In the tracking process, circle as a priori needs to consume more time, in some environments with high real-time requirements, tracking requirements of this method cannot be met.Therefore, this paper needs to develop a new method to solve this problem.

UKF based on star convex shape mutation
From the pseudo measurement equation ( 9) can understand the nonlinear relationship between the measured values and the states satisfies, so nonlinear Filter Unscented Kalman Filter (UKF) is adopted in this paper.Since the covariance matrix at the current moment cannot be obtained directly, so UKF uses calculate the covariance matrix by Unscented Transform.Firstly, sampling is carried out according to equations (11~13), (the number of sampling points is 2n+1), and the x and P in the formula refers to the mean value and variance of states.0 , 0 Then determine the parameters of sampling point weight according to equation (14~16), where n is the dimension of the state vector.Updating All sigma points states [12] , then weighting its prediction estimates and covariances.
(18) Next, calculate covariance matrix of state measurement the Kalman gain.Then combine the idea of a linear Kalman filter [13] to update posterior state.
In this paper, center contour method [14] is adopted to obtain the measured value at initial moment as the prior shape information of star-convex object.Firstly, measurement information is divided into different regions according to appropriate Angle, and then the Euclidean distance between all point and measurement distribution center is calculated in each area.The measurement with the largest distance from the center point of each area is found out to form an ordered set, then Fourier transform is carried out on this set, so as to generate prior shape information.
In some fields that pay much attention to the shape of each time step, when the object has shape mutation due to some special circumstances, it must be able to quickly converge to the changed shape of the object with shape mutation.In this paper, using IOU method to judge the moment when the object has shape mutation and update the state of the object.First, should calculate estimated shape at the current moment ^k x S ( ) ，then calculated the intersecting and union area ratios with the true shape )) , when the result is 1, it means that the estimated shape overlaps with the real shape completely, when result is equal to 0, it means that estimated shape is completely invalid.After the result is obtained, After the result is obtained, set a suitable value to judgement, and when result is lower than this value, it means that star convex shape is likely to have changed abruptly.At this time, the state of the current moment is updated twice, that is, the updated value is taken as an estimate and filtered again, this method is represented as RHM-CC-IOU-UKF.

Evaluation indicators for extended objects
In this paper, using RMSE and quasi-Jaccard distance to judge the estimation result of centroid position and irregular shape of extended object respectively.Quasi-Jaccard distance at time k is calculated according to equation (20), so as to determine the similarity between real contour of extended object and estimated contour.

Simulation experiments
The star convex state transition follows the linear Gaussian model, sampling period T is 1 second, sampling time lasts 50 seconds.Fourier expansion order is 11.Process noise standard deviation is 0.06.Measurement noise covariance matrix ( ) , scaling factor s ~(0.7, 0.06)

N
.The number of measurements produced in each sampling time follows Poisson distribution of λ = 30, and select the appropriate state transition matrix k F and process noise k Q to make the object motion approximately uniform motion.
Experiment 1: sampling time duration of 30 seconds.Initial shape of the star convex object is set to a cross, the variant shape is a pentagram.prior circle of a radius 3. The mean and variance of initial time are 0 x and 0 P respectively.By comparison, it is clear that the algorithm in this paper can track the true shape of the star convex shape at a faster rate at the initial moment, and at the 15th second, the object changes from the cross shape to the pentagram shape, the RHM-CC-IOU-UKF algorithm is possible to converge to the approximate contour of the object in a much shorter time step.RMSE method and the statistical value of quasi-Jaccard distance change over time given in figure 3 show that the similarity between the estimated shape and real shape at initial moment is the lower, the matching degree is greatly enhanced over time.Eventually, it steadily becomes stable.For location estimates, the error in (a) is also kept within a small range.(24) Here is a polygon of 30 objects that mutates at some point during its movement, forming a cruciform shape.Similarly, the algorithm in this paper is compared with the traditional algorithm RHM-UKF.The red line is the contour formed by the group object, while the blue one is the shape tracked by two algorithms after filtering.As given in Figure 4, following part of the picture shows the tracking local magnification of first few seconds of the two methods.It can be clearly seen that the method of center contour algorithm is added in this paper, which is more suitable for the real shape.Above figure 4 gives the local enlarged image before and after the shape change in the process of group movement.RHM-CC-IOU-UKF algorithm can accurately judge the moment of mutation and converge to the real shape in time according to the measurement information.Position and shape estimation effect are analyzed in Figure 5.It is demonstrated that this algorithm can precisely estimate state of extended object.

Conclusion
Aiming at the problem of tracking extended targets with irregular convex mutation, this paper proposes the RHM-CC-IOU-UKF algorithm and gives its detailed implementation process.Finally, tracking simulation of single target and group target with shape mutation is carried out to verify the superiority of this algorithm.This algorithm is also of great significance in practical application, especially in the field of aerial reconnaissance and combat, which requires real-time attention to the extended state of enemy fighters.It can be found in the results that the estimated shapes of objects are described by smooth is a radial function describing the shape.φ is angle formed between the line link k m to contour point and the X-axis.( ) [cos( ), sin( )]

Figure 1 .
Figure 1.Star-convex random hypersurface model the influence of higher order on the model.
Figure2shows the overall tracking graph of the object as well as the local zoomed-in graphs at the initial moment and the moment of abrupt change within 30 seconds for the two algorithms.Figure (a)   shows the target tracking effect of the traditional algorithm (RHM-UKF) combined with UKF and random hypersurface model.Figure (b)  shows the tracking result of the proposed algorithm.By comparison, it is clear that the algorithm in this paper can track the true shape of the star convex shape at a faster rate at the initial moment, and at the 15th second, the object changes from the cross shape to the pentagram shape, the RHM-CC-IOU-UKF algorithm is possible to converge to the approximate contour of the object in a much shorter time step.RMSE method and the statistical value of quasi-Jaccard distance change over time given in figure3show that the similarity between the estimated shape and real shape at initial moment is the lower, the matching degree is greatly enhanced over time.Eventually, it steadily becomes stable.For location estimates, the error in (a) is also kept within a small range.
(a) Statistical results of RMSE (b) Statistical results of quasi-Jaccard Figure3.Evaluation indexes of RHM-CC-IOU-UKF In this scenario, the sampling period is 50 seconds, 30 objects in the group, and the initial shape of the objects set as polygons and the mutated shape as crosses.The mean and variance of the initial state obedience were (a) RHM-UKF tracking result (b) RHM-CC-IOU-UKF tracking result Figure 4. Tracking results for a Group objects (a) Statistical results of RMSE (b) Statistical results of quasi-Jaccard Figure 5. Evaluation indexes of RHM-CC-IOU-UKF