# An Extended Target Tracking Algorithm Based on Gaussian Surface Fitting

Authors: Yulan Han; Han Xu; Gangdun Hu; Chongzhao Han
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Although the existing algorithms have done a lot of progress in extended target tracking (ETT), most algorithms are limited by the assumption of the target shape, and there are problems with parameter calculations. To solve the problems above, an extended target tracking algorithm based on level set and Gaussian Surface fitting (GSF), is proposed. Different from the existing extended target tracking algorithms, the proposed algorithm uses a level set to implicitly describe the evolution of the estimated shape, eliminates the parameterization process, and uses the GSF to construct the measurement source spatial distribution function (MSSDF) of the extended target. The proposed algorithm can obtain the estimated shape of the extended target without knowing the prior shape and size of the target. The simulation results verify the effectiveness and reliability of the proposed algorithm.


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## Body

## 1. Introduction

The target tracking problem can be viewed as estimating the kinematic state of a target in the surveillance region based on a set of measurements collected by one or multiple sensors [1]. Limited by the resolution of early sensors, the traditional target tracking algorithms assume that the target is a point, i.e., that the target in the surveillance region produces at most one measurement per moment. As sensor accuracy continues to improve, this assumption is no longer established. In a real world scenario, the target may have multiple measurement sources at each moment, generating multiple measurements, such a target is called an extended target [2]. Multiple measurements of the extended target not only estimate the kinematic state of the target (position, velocity, acceleration, etc.) but also extract more characteristic information of the target, especially the shape information, which plays an important role in subsequent target identification and attitude estimation. Therefore, even for single extended target tracking in a clutter-free situation, the state evolution model and the measurement model of the extended target need to be remodeled according to the characteristics of the extended target and the sensors-target relationship, and the corresponding filtering algorithm needs to be established. In recent years, extended target tracking theory has been developed and widely applied to a range of military and civil applications such as aerospace, robot navigation, and vehicle tracking. Therefore, extended target tracking technology has received much attention from researchers and has become a very important research topic [3].Currently, extended target tracking research mainly focuses on multitarget tracking (MTT) and target shape estimation. MTT tracking algorithms are mainly divided into two categories: data association-based multitarget tracking algorithms and random finite set-based extended target tracking algorithms. Data association-based multitarget tracking algorithms need to address the relationship between measurements and targets and generally extend the point-target data association technique to extended target tracking [4–8]. As the number of targets increases, the required computational complexity of data association grows exponentially and becomes the bottleneck of multitarget tracking algorithms. The multitarget tracking algorithm builds a random finite set based on multitarget states and measurements separately [9–11], which avoids data correlation but requires consideration of the division of the measurement set at each moment [12–14]. In addition, this algorithm only provides the state set at each moment, and a reasonable track generation algorithm needs to be added [15].The shape estimation of extended targets is also a current research hotspot. Many scholars have proposed corresponding theories to address this issue. The Random Matrix (RM) proposed by Koch assumes extended targets as ellipses [16, 17] and uses Bayesian recursive methods to estimate the kinematic state and shape parameters of targets under a linear Gaussian model. However, only using an ellipsoidal representation of the contour characteristics of the target has limitations. To more accurately estimate the shape of the extended target, Lan and Li used multiple ellipses to represent the shape of the target and achieved RM tracking of nonelliptical extended targets [18, 19]. In addition, many scholars have conducted research on numerous problems in RM models [20–22]. Baum proposed the Random Hypersurface Model (RHM) [23], which represents the irregular shape of the extended target by a parameter vector composed of finite-order coefficients of the Fourier series expansion of the radial function, allowing the applicability of shape estimation to be broadened to star-convex shapes. On this basis, Wahlström and Özkan used the parameter-learning property of Gaussian Process Regression (GPR) to learn the radial function of the target shape [24], with the help of GPR, to complete the model of star-convex extended targets. Zea et al. proposed the level set-RHM to extend the RHM tracking technique to nonstar-convex target tracking [25]. In addition, many scholars have carried out extensive research on RHM-based target tracking techniques [26, 27]. Some scholars have also proposed the method of B-Spline Curve fitting [28] or the extension-deformation approach [29] to obtain the estimated shape of the extended target. These methods have achieved certain results in target shape estimation, but there are also drawbacks: different a priori shapes need to be predetermined for different targets, there are parameters to calculate during the shape evolution, and the details of the extended target are not sufficiently described.To address the shape estimation problem of extended target tracking, this paper uses a level set to implicitly describe the shape evolution of irregularly shaped extended targets and transforms the shape evolution of extended targets into the evolution of level set function in high-dimensional space, thus avoiding the parameterization process. On this basis, the measurement source spatial distribution function (MSSDF) of the extended target is established using Gaussian surfaces fitting (GSF) to more precisely describe the relationship between the measurements and the shape. The contributions of this paper are as follows.(1)
A new shape representation is proposed: the irregular shape of the extended target and its evolution are described by a level set, and the shape boundary of the extended target is directly extracted by the shape function, avoiding the complicated parameterization process.(2)
A new MSSDF is proposed: GSF is used to construct the MSSDF, which can describe the spatial distribution of measurement sources of an arbitrarily shaped extended target without a priori information. The MSSDF contains the shape information of the target, a connection is made between the MSSDF and the shape function in the level set, and the shape boundary is extracted directly from the level set.(3)
A shape recursion algorithm is proposed: the Gaussian mixture model is used to update the shape, which can retain the main characteristics of the shape, reduce the amount of calculation, and enhance the real-time performance of the algorithm.The remainder of the manuscript is organized as follows. Section2 introduces the general mathematical model of the proposed algorithm, including the extended target model, the measurement source model, the level set, and the spatial distribution function of the measurement sources. Section 3 will introduce the proposed CKF-GSF algorithm, which includes the cubature Kalman filter, the processing of the measurements, and the calculation of the level set threshold. In order to reduce the computational effort of the algorithm, the recursion of the extended target estimation shape is presented in Section 4. Section 5 shows the simulation experiments, where the simulation results of the proposed algorithm are analyzed, two different ways of calculating the covariance are discussed, and the effect of noise on the spatial distribution function of the measurement sources is also discussed. Section 6 shows the conclusion and provides an outlook on the next step.Notation: Unless otherwise noted, in this paper, we use lower-case (resp. upper-case) bold letters to denote vectors (resp. matrices), withxk denoting the state vector of the target at moment k; Nμ,∑ denotes a normal distribution with mean vector μ and covariance matrix ∑. Finally, the symbol ⟶ means “maps to.”


## 2. Mathematical Model

### 2.1. Extended Target Model

The state of the extended target includes kinematic state and shape information. The shape information of the extended target is transformed into the evolution of the level set. Therefore, the shape parameters and other related information will be introduced in detail in Section2.2.


#### 2.1.1. Dynamic Model

For kinematic states, the state evolution model for linear Gaussian systems can be described as follows:(1)xk+1=Fkxk+wk.The state evolution model for nonlinear systems can be described as follows:(2)xk+1=fxk,wk,where Fk is a state transfer matrix of the linear Gaussian system. fk is an n-dimensional vector function of the nonlinear Gaussian system, process noise wk is Gaussian white noise with a covariance of Qk, and xk represents the kinematic state of the target at moment k. That is,(3)xk=mkTxk∗TT,where mk is the position vector, xk∗ represents the speed vector of the extended target.


#### 2.1.2. Measurement Model

Assuming that each measurement is generated by a measurement source scattered on the surface of the extended target, for a given measurement source, the corresponding measurementzk,l is as follows:(4)zk,l=yk,l+vk,l,where yk,l represents the location of the measurement source, and the measurement noise vk,l is Gaussian white noise with a covariance of Rv. Assume that the set of all measurements generated by the target at moment k is as follows:(5)Zk=zk,ll=1Nz,k,where Nz,k represents the number of measurements produced by the target at moment k.


### 2.2. Level Set

The level set was jointly proposed by Osher and Sethian. The main idea is to transform the evolution of low-dimensional closed curves into the evolution of level set functions in a high-dimensional space, avoiding the parametric design process of deformed curves or surfaces. The shape estimation problem of an extended target is essentially a problem in the evolution of two-dimensional closed curves over time, and the level set is a powerful tool to solve such problems [30–32]. Therefore, the level set can describe the shape of an irregularly shaped extended target and its evolution.Suppose the shape of the extended target is represented byxkE, the set formed by all the points inside the extended target is denoted as SxkE, and the shape function ϕxkE,z defined on ℝd⟶ℝ satisfies the following formula:(6)SxkE=z|ϕxkE,z≥c,c≥0.The level set can be defined as follows:(7)ℒϕc=z|ϕxkE,z≥c,c≥0.The extended targetSxkE can be represented by the c≥0 level set. The difficulty of the level set method is in the construction of a reasonable shape function ϕxkE,z and determination of the threshold c. This paper proposes directly modeling the shape function using the measurements. Since the distribution of the measurements implicitly contains the shape information, the shape function can be modeled as an MSSDF, but the function should have several characteristics, as follows.(a)
SinceϕxkE,z is an unknown surface, this MSSDF should be an expression in three dimensions or a three-dimensional surface.(b)
This MSSDF can describe the main features of the extended target shape.Meeting these several characteristics, the MSSDF can be modeled as a level set of shape functions, that is,(8)ϕxkE,z=MSSDF.The modeling of MSSDF will be described in detail in Section2.3.


### 2.3. Measurement Source Spatial Distribution Function (MSSDF)

The measurement sourceyk is located on the extended target, and it is difficult to obtain a model for the prior distribution of its measurement source when tracking the extended target. In the real situation, the measurement source yk is more likely to be located at the corners, edges, etc., of the extended target, where the shape information of the target is implied. Since the measurement zk is generated by adding a Gaussian noise vk to the measurement source, the shape information of the target is implicit in the set of measurements obtained at each moment, and the MSSDF can be constructed according to the measurements obtained by the sensors.The MSSDF of an extended target can be viewed as a superposition of the spatial probability distribution of each measurement from the measurement source. Since each measurement occupies a certain spatial distribution, the MSSDF obtained by the superposition of multiple Gaussian distributions can represent an arbitrary target shape distribution. This is performed by generating multiple Gaussian surfaces with the measurements as the mean, which are superimposed and normalized by multiple Gaussian surfaces, as shown in the following equation:(9)hkz|Zk,Δk=1Zk∑l=1ZkNz|zk,l,Δk,where hkz|Zk,Δk is the MSSDF, Zk is the measurement set at moment k, Δk is the covariance of the measurement spatial distribution model at moment k, and Zk is the number of elements in the measurement set Zk. When there are enough measurements in Zk, the formula above can be rewritten as follows:(10)hkz|Zk,σ2=1Zk∑l=1Zk12πexp−12σ2z−zk,lz−zk,lT.We refer to the above process of modeling the MSSDF as Gaussian surface fitting (GSF). The actual situation of MSSDF constructed by equation (10) is given below.Figure1 is the actual shape of the extended target. Figure 2 is the MSSDF constructed of the GSF. The parameters are as follows: the MSSDF is constructed of 500 measurements; the sensor noise is 0.22000.22, and the Gaussian surface is generated by equation (10). Figures 1 and 2 show that it is feasible to use the GSF to construct the MSSDF.Figure 1
The actual shape of the target.Figure 2
The MSSDF constructed by GSF.By constructing the MSSDF of the extended target, a representation of the target can be obtained in 3-dimensional space using a surface, and the estimated shape of the target is implicit in this surface, in accordance with conditions (a) and (b) in Section2.2. Therefore, it is reasonable to use MSSDF as a shape function for the level set, and the estimated shape of the extended target can be expressed by the following equation:(11)ℒϕc=1Zk∑l=1zk12πexp−12σ2z−zk,lz−zk,lT,c≥0.The above equation shows that the thresholdc needs to be determined to obtain the estimated shape of the extended target. This process is described in detail in Section 3.2.3.


## 2.1. Extended Target Model

The state of the extended target includes kinematic state and shape information. The shape information of the extended target is transformed into the evolution of the level set. Therefore, the shape parameters and other related information will be introduced in detail in Section2.2.


### 2.1.1. Dynamic Model

For kinematic states, the state evolution model for linear Gaussian systems can be described as follows:(1)xk+1=Fkxk+wk.The state evolution model for nonlinear systems can be described as follows:(2)xk+1=fxk,wk,where Fk is a state transfer matrix of the linear Gaussian system. fk is an n-dimensional vector function of the nonlinear Gaussian system, process noise wk is Gaussian white noise with a covariance of Qk, and xk represents the kinematic state of the target at moment k. That is,(3)xk=mkTxk∗TT,where mk is the position vector, xk∗ represents the speed vector of the extended target.


### 2.1.2. Measurement Model

Assuming that each measurement is generated by a measurement source scattered on the surface of the extended target, for a given measurement source, the corresponding measurementzk,l is as follows:(4)zk,l=yk,l+vk,l,where yk,l represents the location of the measurement source, and the measurement noise vk,l is Gaussian white noise with a covariance of Rv. Assume that the set of all measurements generated by the target at moment k is as follows:(5)Zk=zk,ll=1Nz,k,where Nz,k represents the number of measurements produced by the target at moment k.


## 2.1.1. Dynamic Model

For kinematic states, the state evolution model for linear Gaussian systems can be described as follows:(1)xk+1=Fkxk+wk.The state evolution model for nonlinear systems can be described as follows:(2)xk+1=fxk,wk,where Fk is a state transfer matrix of the linear Gaussian system. fk is an n-dimensional vector function of the nonlinear Gaussian system, process noise wk is Gaussian white noise with a covariance of Qk, and xk represents the kinematic state of the target at moment k. That is,(3)xk=mkTxk∗TT,where mk is the position vector, xk∗ represents the speed vector of the extended target.


## 2.1.2. Measurement Model

Assuming that each measurement is generated by a measurement source scattered on the surface of the extended target, for a given measurement source, the corresponding measurementzk,l is as follows:(4)zk,l=yk,l+vk,l,where yk,l represents the location of the measurement source, and the measurement noise vk,l is Gaussian white noise with a covariance of Rv. Assume that the set of all measurements generated by the target at moment k is as follows:(5)Zk=zk,ll=1Nz,k,where Nz,k represents the number of measurements produced by the target at moment k.


## 2.2. Level Set

The level set was jointly proposed by Osher and Sethian. The main idea is to transform the evolution of low-dimensional closed curves into the evolution of level set functions in a high-dimensional space, avoiding the parametric design process of deformed curves or surfaces. The shape estimation problem of an extended target is essentially a problem in the evolution of two-dimensional closed curves over time, and the level set is a powerful tool to solve such problems [30–32]. Therefore, the level set can describe the shape of an irregularly shaped extended target and its evolution.Suppose the shape of the extended target is represented byxkE, the set formed by all the points inside the extended target is denoted as SxkE, and the shape function ϕxkE,z defined on ℝd⟶ℝ satisfies the following formula:(6)SxkE=z|ϕxkE,z≥c,c≥0.The level set can be defined as follows:(7)ℒϕc=z|ϕxkE,z≥c,c≥0.The extended targetSxkE can be represented by the c≥0 level set. The difficulty of the level set method is in the construction of a reasonable shape function ϕxkE,z and determination of the threshold c. This paper proposes directly modeling the shape function using the measurements. Since the distribution of the measurements implicitly contains the shape information, the shape function can be modeled as an MSSDF, but the function should have several characteristics, as follows.(a)
SinceϕxkE,z is an unknown surface, this MSSDF should be an expression in three dimensions or a three-dimensional surface.(b)
This MSSDF can describe the main features of the extended target shape.Meeting these several characteristics, the MSSDF can be modeled as a level set of shape functions, that is,(8)ϕxkE,z=MSSDF.The modeling of MSSDF will be described in detail in Section2.3.


## 2.3. Measurement Source Spatial Distribution Function (MSSDF)

The measurement sourceyk is located on the extended target, and it is difficult to obtain a model for the prior distribution of its measurement source when tracking the extended target. In the real situation, the measurement source yk is more likely to be located at the corners, edges, etc., of the extended target, where the shape information of the target is implied. Since the measurement zk is generated by adding a Gaussian noise vk to the measurement source, the shape information of the target is implicit in the set of measurements obtained at each moment, and the MSSDF can be constructed according to the measurements obtained by the sensors.The MSSDF of an extended target can be viewed as a superposition of the spatial probability distribution of each measurement from the measurement source. Since each measurement occupies a certain spatial distribution, the MSSDF obtained by the superposition of multiple Gaussian distributions can represent an arbitrary target shape distribution. This is performed by generating multiple Gaussian surfaces with the measurements as the mean, which are superimposed and normalized by multiple Gaussian surfaces, as shown in the following equation:(9)hkz|Zk,Δk=1Zk∑l=1ZkNz|zk,l,Δk,where hkz|Zk,Δk is the MSSDF, Zk is the measurement set at moment k, Δk is the covariance of the measurement spatial distribution model at moment k, and Zk is the number of elements in the measurement set Zk. When there are enough measurements in Zk, the formula above can be rewritten as follows:(10)hkz|Zk,σ2=1Zk∑l=1Zk12πexp−12σ2z−zk,lz−zk,lT.We refer to the above process of modeling the MSSDF as Gaussian surface fitting (GSF). The actual situation of MSSDF constructed by equation (10) is given below.Figure1 is the actual shape of the extended target. Figure 2 is the MSSDF constructed of the GSF. The parameters are as follows: the MSSDF is constructed of 500 measurements; the sensor noise is 0.22000.22, and the Gaussian surface is generated by equation (10). Figures 1 and 2 show that it is feasible to use the GSF to construct the MSSDF.Figure 1
The actual shape of the target.Figure 2
The MSSDF constructed by GSF.By constructing the MSSDF of the extended target, a representation of the target can be obtained in 3-dimensional space using a surface, and the estimated shape of the target is implicit in this surface, in accordance with conditions (a) and (b) in Section2.2. Therefore, it is reasonable to use MSSDF as a shape function for the level set, and the estimated shape of the extended target can be expressed by the following equation:(11)ℒϕc=1Zk∑l=1zk12πexp−12σ2z−zk,lz−zk,lT,c≥0.The above equation shows that the thresholdc needs to be determined to obtain the estimated shape of the extended target. This process is described in detail in Section 3.2.3.


## 3. CKF-GSF Algorithm

In this section, the proposed extended target tracking algorithm is introduced. The algorithm will use the cubature Kalman filter (CKF) to calculate the kinematic state of the extended target and obtain the estimated shape of the target by extracting the boundary of the MSSDF from the level set. This section details the specific steps of the algorithm, including the processing of the measurements, the calculation of the covariance in the GSF, and the calculation of the thresholdc in the level set.


### 3.1. Kinematic Estimation of Extended Target

In the process of estimating the extended target shape, the accurate prediction of the target position is crucial. Kalman filtering and its improved algorithms are widely used to solve this problem. The literature [33, 34] is based on EKF and combines different scenarios to investigate this part of the work in depth with good results. In the EFK method, filtering has been performed by the analytical linearization and the first-order estimation of nonlinear equations. However, this first-order estimation resulted in the estimation error and even divergence. Compared with EKF, CKF has higher filtering accuracy and numerical stability, so the CKF used in this paper to estimate the motion state of the extended target [35]. CKF is derived from Bayesian estimation theory and the spherical-radial cubature criterion, which can effectively solve the state estimation problem of nonlinear systems using a set of cubature points to approximate the state mean and error covariance of a nonlinear system with additional Gaussian noise [36, 37]. The main purpose of using CKF in this paper is to estimate the kinematic state of the target, which is an important prerequisite to constructing the MSSDF.The CKF algorithm consists of a prediction step and an update step. The following section will introduce them separately.


#### 3.1.1. Prediction Step

Assume that the state vector of the target at the momentk−1 is xk−1|k−1, and the corresponding covariance matrix is Pk−1|k−1, then the predicted state vector and covariance of the target at moment k can be calculated by the following equations.Estimating the state prediction at momentk:(12)xk|k−1=Fkxk−1|k−1.Estimating state error covariance prediction at momentk:(13)Pk|k−1=FkPk−1|k−1FkT.


#### 3.1.2. Update Step

The core of the update step of the CKF tracking algorithm is to combine all the measurement information of the extended target at momentk to obtain the kinematics state of the target: First calculate the basic cubature point and the corresponding weight.(14)ξi=nc21i,wi=1nc,i=1,2,…,nc.According to the third-order cubature rule, the total number of cubature pointsnc is twice the dimension of the extended target state nx. The complete symmetric point set 1 can be obtained by fully permuting the elements of the unit vector e=1,0,0,…,0T and changing the elements’ conform, where the dimension of the unit vector e and the dimension of the extended target state nx are equal, 1i represents the i-th point of the complete symmetric point set.The extended target state vector isxk|k−1 and the covariance matrix is Pk|k−1, calculate the cubature point ξ˜k|k−1i according to the following, that is,(15)ξ˜k|k−1i=Hk|k−1ξi+xk|k−1,where Hk|k−1 is obtained by Cholesky decomposition of the covariance matrix Pk|k−1, Pk|k−1=Hk|k−1Hk|k−1T. ξi is the basic cubature point in the equation (14). Spread the cubature points according to the measurement equation:(16)ζk|k−1i=h∗ξ˜k|k−1i,i=1,2,…,nc.Calculate the measurement prediction:(17)znz=1nc∑i=1ncζk|k−1i.Compute the measured values of state variables and error covariance:(18)Pxz=1nc∑i=1ncξ˜k|k−1iζk|k−1iT−xx|k−1znzT,Pzz=1nc∑i=1ncζk|k−1iζk|k−1iT−znzznzT.Finally, calculate the gain matrix to update the state vector and covariance matrix of the extended target.Compute Kalman gain:(19)Kk=PxzPxzT.Update the state variable and the according to error covariance:(20)xk|k=xk|k−1+Kkzk,l−znz,(21)Pk|k=Pk|k−1+KkPzzKkT.From the above steps, the computational complexity of the CKF can be approximated asOn2.


### 3.2. Shape Estimation of Extended Target

In this section, this paper addresses the problem of MSSDF’s inability to be effectively constructed using single-moment measurements in the simulation scenario and proposes the use of multiple-moment measurements to construct the MSSDF. In addition, a detailed explanation of how to calculate the covarianceσ2 in equation (11) is given, and the problem of taking the threshold c in the level set.


#### 3.2.1. MSSDF Constructed by Multiple Moment Measurements

Given the situation at a certain moment, the MSSDF was constructed by the GSF.Figure3 shows the results when using the GSF to construct the MSSDF of the extended target at a certain moment. The parameter settings are roughly the same as those in Figure 2. In order to be close to the simulation scene, the number of measurements obtained at each moment is changed to obey a Poisson distribution with a parameter of 20. In this scenario, using the GSF to construct the MSSDF cannot describe the specific shape and main characteristics of the extended target. The main reason for this is that fewer measurements are obtained at each moment, and the shape information contained in these measurements is limited and cannot approximate the actual distribution of the target. Due to the limitation of the current sensor resolution, in actual situations, it is difficult to obtain enough measurements at one moment. To solve this problem, this paper intends to use multiple moment measurements to update the MSSDF of the extended target.Figure 3
MSSDF constructed by fewer measurements.Since the measurements at each moment are generated independently of each other, they may be generated by measurement sources at different locations of the target, so the MSSDF can be better constructed with measurements from multiple moments and thus better estimate the shape of the target.The optimal estimate of the extended target state obtained in equation (20) is xk|k, where the centroid estimate mk|k is included. Then, each measurement in the measurement set Zk at moment k is processed with the following:(22)dk,l=zk,l−mk.This can be interpreted as panning each measurement to the center point of the target estimate.Figure4 shows the simulation scenario of the same target at two consecutive moments. The gray part is the real shape of the extended target, the red “∗” on the target is the optimal estimate of the center coordinates calculated by the CKF, and the black “+” and the blue “x” are the measurements at two different moments. It can be seen that, after the calculation of equation (22), the measurements at two different moments are panned to an approximate position relative to the center of the extended target and constitute a new set of measurements. Compared with the measurements at a single moment, the new set of measurements contains more shape information, which will be more beneficial for the construction of the MSSDF.Figure 4
Panning measurements.Suppose that from the first moment to thek-th moment, through the calculation of equation (22), the dk calculated at each moment constitutes a new measurement set, which is represented by the set d1:k. This set can be regarded as the translation of the measurements generated at different positions on the target to the origin; then, equation (10) can be rewritten as follows:(23)hkz|d,σ2=1d1:k∑l=1d1:k12πexp−12σ2z−d1:k,lz−d1:k,lT.where d1:k is used as the mean value of the Gaussian surface. Section 3.2.2 will introduce how to solve the problem of calculating Gaussian surface covariance σ2.


#### 3.2.2. Calculation of the Covariance in GSF

When constructing the MSSDF, the covariance will affect the shape of the Gaussian surface. If the size of the covariance does not match the mean value, the shape of the Gaussian surface will be poor. Reference [38] established a Gaussian function when solving similar problem:(24)ε=exp−αRk2σ2,where Rk is the shortest distance to traverse all measurements in the measurement set, α is the given coefficient, and ε is the given minimum value. In this method, the Bellman-Ford algorithm can be used to calculate Rk to obtain the value of the covariance σ2. Experiments show that this method has good results when the number of measurements is not large, but this method needs to preset the values of ε and α. If the preset values are unreasonable, it will seriously affect the effect of MSSDF. When calculating Rk, this method leads to a significant increase in the computation if the number of measurements is large. Detailed experimental data are given in the experimental part of Section 5.2.The method in [38] cannot be applied to the calculation of the MSSDF proposed in this paper. Therefore, a method of calculating the covariance of Gaussian surfaces is proposed. From the geometric perspective, the covariance is closely related to the number of measurements and the distribution of the measurements. That is, the covariance should relate to the measurement set d1:k. All the boundary measurements in d1:k can be calculated (that is, the outermost measurement in d1:k). Since the boundary algorithm has been relatively mature, this paper omits the calculation process of this algorithm, and the results obtained are as in Figures 5 and 6), where the red “∗” is the calculated boundary measurements, connecting these measurements in turn and calculate the area Ak of the graph formed by the boundary measurements.Figure 5
Boundary measurements.Figure 6
Boundary area.DistributeAk evenly to all the measurements in d1:k and calculate the area occupied by each measurement. Assuming that the area occupied by each measurement is a circle, the average radius ra occupied by each measurement can be calculated, and this radius ra could be the value of σ2:(25)σ2=Ak|d1:k|×π.The covariances of MSSDF in Figures2 and 3 are both obtained by this method, which confirm the effectiveness of this method.


#### 3.2.3. Calculation of the Threshold in Level Set

In this section, equation (11) gives the calculation method for the estimated shape. The estimated shape of the extended target needs to be determined by the value of the threshold c. The shape ℒ∅ is a collection of the shape function ϕxkE,z, taking a certain value, c, which can be seen as the intersection of a surface ϕxkE,z and a plane c. In the absence of sensor noise, the measurement of the extended target comes from the interior and boundary, which is shown in equation (6): the value of c should be 0. However, in actual situations, due to the sensor noise factor, some measurements may come from outside the boundary of the target. Measurements outside these boundaries will also generate a Gaussian surface with a certain height and be normalized to the surface ϕxkE,z, so the threshold c cannot be 0.The size ofc directly determines the quality of the shape estimation of the extended target tracking. Since there is no a priori information about the shape of the target, it is difficult to calculate the c. To solve the problem so that the shape function ϕxkE,z can be obtained at each moment to get the optimal estimation of the shape, a method for the value of the adaptive threshold c is proposed.In Section3.2.2, the set of boundary points d1:k is obtained. If d1:k is the true boundary of the target, the threshold c is the average of the heights of the Gaussian surfaces where the boundary measurements d1:kBOU are located. However, in actual situations, due to noise interference, the outermost boundary points are generally not the true boundary of the target. They may also come from the outside or inside of the target. It is difficult to judge individually without prior knowledge. Average the height of the boundary measurements to obtain the value of the threshold c.(26)c=1d1:kBOU∑l=1dkBOUhdkBOU|d1:k,l,σ2.We have established an extended target shape representation model based on CKF and GSF. Bringing equations (22), (25) to (11), using a level set to describe the estimated shape of the target:(27)ℒϕc=z|1d1:k∑l=1d1:k12πexp−12σ2z−d1:k,lz−d1:k,lT=c,where c is calculated by equation (26); σ2 is calculated by equation (25); d1:k is calculated by equation (22).The specific steps of the algorithm atk moments are given below, which can better explain the algorithm proposed in this paper:(a)
Predict and update the target’s state information with CKF.(b)
Usingmk|k in the state information xk|k, the set d1:k is obtained by panning the k−th moment measurements by (22).(c)
Calculate the MSSDF by the GSF using equation (23); the covariance is calculated by (23).(d)
The threshold is calculated by (24), and the estimated shape of the target is obtained by (25).


## 3.1. Kinematic Estimation of Extended Target

In the process of estimating the extended target shape, the accurate prediction of the target position is crucial. Kalman filtering and its improved algorithms are widely used to solve this problem. The literature [33, 34] is based on EKF and combines different scenarios to investigate this part of the work in depth with good results. In the EFK method, filtering has been performed by the analytical linearization and the first-order estimation of nonlinear equations. However, this first-order estimation resulted in the estimation error and even divergence. Compared with EKF, CKF has higher filtering accuracy and numerical stability, so the CKF used in this paper to estimate the motion state of the extended target [35]. CKF is derived from Bayesian estimation theory and the spherical-radial cubature criterion, which can effectively solve the state estimation problem of nonlinear systems using a set of cubature points to approximate the state mean and error covariance of a nonlinear system with additional Gaussian noise [36, 37]. The main purpose of using CKF in this paper is to estimate the kinematic state of the target, which is an important prerequisite to constructing the MSSDF.The CKF algorithm consists of a prediction step and an update step. The following section will introduce them separately.


### 3.1.1. Prediction Step

Assume that the state vector of the target at the momentk−1 is xk−1|k−1, and the corresponding covariance matrix is Pk−1|k−1, then the predicted state vector and covariance of the target at moment k can be calculated by the following equations.Estimating the state prediction at momentk:(12)xk|k−1=Fkxk−1|k−1.Estimating state error covariance prediction at momentk:(13)Pk|k−1=FkPk−1|k−1FkT.


### 3.1.2. Update Step

The core of the update step of the CKF tracking algorithm is to combine all the measurement information of the extended target at momentk to obtain the kinematics state of the target: First calculate the basic cubature point and the corresponding weight.(14)ξi=nc21i,wi=1nc,i=1,2,…,nc.According to the third-order cubature rule, the total number of cubature pointsnc is twice the dimension of the extended target state nx. The complete symmetric point set 1 can be obtained by fully permuting the elements of the unit vector e=1,0,0,…,0T and changing the elements’ conform, where the dimension of the unit vector e and the dimension of the extended target state nx are equal, 1i represents the i-th point of the complete symmetric point set.The extended target state vector isxk|k−1 and the covariance matrix is Pk|k−1, calculate the cubature point ξ˜k|k−1i according to the following, that is,(15)ξ˜k|k−1i=Hk|k−1ξi+xk|k−1,where Hk|k−1 is obtained by Cholesky decomposition of the covariance matrix Pk|k−1, Pk|k−1=Hk|k−1Hk|k−1T. ξi is the basic cubature point in the equation (14). Spread the cubature points according to the measurement equation:(16)ζk|k−1i=h∗ξ˜k|k−1i,i=1,2,…,nc.Calculate the measurement prediction:(17)znz=1nc∑i=1ncζk|k−1i.Compute the measured values of state variables and error covariance:(18)Pxz=1nc∑i=1ncξ˜k|k−1iζk|k−1iT−xx|k−1znzT,Pzz=1nc∑i=1ncζk|k−1iζk|k−1iT−znzznzT.Finally, calculate the gain matrix to update the state vector and covariance matrix of the extended target.Compute Kalman gain:(19)Kk=PxzPxzT.Update the state variable and the according to error covariance:(20)xk|k=xk|k−1+Kkzk,l−znz,(21)Pk|k=Pk|k−1+KkPzzKkT.From the above steps, the computational complexity of the CKF can be approximated asOn2.


## 3.1.1. Prediction Step

Assume that the state vector of the target at the momentk−1 is xk−1|k−1, and the corresponding covariance matrix is Pk−1|k−1, then the predicted state vector and covariance of the target at moment k can be calculated by the following equations.Estimating the state prediction at momentk:(12)xk|k−1=Fkxk−1|k−1.Estimating state error covariance prediction at momentk:(13)Pk|k−1=FkPk−1|k−1FkT.


## 3.1.2. Update Step

The core of the update step of the CKF tracking algorithm is to combine all the measurement information of the extended target at momentk to obtain the kinematics state of the target: First calculate the basic cubature point and the corresponding weight.(14)ξi=nc21i,wi=1nc,i=1,2,…,nc.According to the third-order cubature rule, the total number of cubature pointsnc is twice the dimension of the extended target state nx. The complete symmetric point set 1 can be obtained by fully permuting the elements of the unit vector e=1,0,0,…,0T and changing the elements’ conform, where the dimension of the unit vector e and the dimension of the extended target state nx are equal, 1i represents the i-th point of the complete symmetric point set.The extended target state vector isxk|k−1 and the covariance matrix is Pk|k−1, calculate the cubature point ξ˜k|k−1i according to the following, that is,(15)ξ˜k|k−1i=Hk|k−1ξi+xk|k−1,where Hk|k−1 is obtained by Cholesky decomposition of the covariance matrix Pk|k−1, Pk|k−1=Hk|k−1Hk|k−1T. ξi is the basic cubature point in the equation (14). Spread the cubature points according to the measurement equation:(16)ζk|k−1i=h∗ξ˜k|k−1i,i=1,2,…,nc.Calculate the measurement prediction:(17)znz=1nc∑i=1ncζk|k−1i.Compute the measured values of state variables and error covariance:(18)Pxz=1nc∑i=1ncξ˜k|k−1iζk|k−1iT−xx|k−1znzT,Pzz=1nc∑i=1ncζk|k−1iζk|k−1iT−znzznzT.Finally, calculate the gain matrix to update the state vector and covariance matrix of the extended target.Compute Kalman gain:(19)Kk=PxzPxzT.Update the state variable and the according to error covariance:(20)xk|k=xk|k−1+Kkzk,l−znz,(21)Pk|k=Pk|k−1+KkPzzKkT.From the above steps, the computational complexity of the CKF can be approximated asOn2.


## 3.2. Shape Estimation of Extended Target

In this section, this paper addresses the problem of MSSDF’s inability to be effectively constructed using single-moment measurements in the simulation scenario and proposes the use of multiple-moment measurements to construct the MSSDF. In addition, a detailed explanation of how to calculate the covarianceσ2 in equation (11) is given, and the problem of taking the threshold c in the level set.


### 3.2.1. MSSDF Constructed by Multiple Moment Measurements

Given the situation at a certain moment, the MSSDF was constructed by the GSF.Figure3 shows the results when using the GSF to construct the MSSDF of the extended target at a certain moment. The parameter settings are roughly the same as those in Figure 2. In order to be close to the simulation scene, the number of measurements obtained at each moment is changed to obey a Poisson distribution with a parameter of 20. In this scenario, using the GSF to construct the MSSDF cannot describe the specific shape and main characteristics of the extended target. The main reason for this is that fewer measurements are obtained at each moment, and the shape information contained in these measurements is limited and cannot approximate the actual distribution of the target. Due to the limitation of the current sensor resolution, in actual situations, it is difficult to obtain enough measurements at one moment. To solve this problem, this paper intends to use multiple moment measurements to update the MSSDF of the extended target.Figure 3
MSSDF constructed by fewer measurements.Since the measurements at each moment are generated independently of each other, they may be generated by measurement sources at different locations of the target, so the MSSDF can be better constructed with measurements from multiple moments and thus better estimate the shape of the target.The optimal estimate of the extended target state obtained in equation (20) is xk|k, where the centroid estimate mk|k is included. Then, each measurement in the measurement set Zk at moment k is processed with the following:(22)dk,l=zk,l−mk.This can be interpreted as panning each measurement to the center point of the target estimate.Figure4 shows the simulation scenario of the same target at two consecutive moments. The gray part is the real shape of the extended target, the red “∗” on the target is the optimal estimate of the center coordinates calculated by the CKF, and the black “+” and the blue “x” are the measurements at two different moments. It can be seen that, after the calculation of equation (22), the measurements at two different moments are panned to an approximate position relative to the center of the extended target and constitute a new set of measurements. Compared with the measurements at a single moment, the new set of measurements contains more shape information, which will be more beneficial for the construction of the MSSDF.Figure 4
Panning measurements.Suppose that from the first moment to thek-th moment, through the calculation of equation (22), the dk calculated at each moment constitutes a new measurement set, which is represented by the set d1:k. This set can be regarded as the translation of the measurements generated at different positions on the target to the origin; then, equation (10) can be rewritten as follows:(23)hkz|d,σ2=1d1:k∑l=1d1:k12πexp−12σ2z−d1:k,lz−d1:k,lT.where d1:k is used as the mean value of the Gaussian surface. Section 3.2.2 will introduce how to solve the problem of calculating Gaussian surface covariance σ2.


### 3.2.2. Calculation of the Covariance in GSF

When constructing the MSSDF, the covariance will affect the shape of the Gaussian surface. If the size of the covariance does not match the mean value, the shape of the Gaussian surface will be poor. Reference [38] established a Gaussian function when solving similar problem:(24)ε=exp−αRk2σ2,where Rk is the shortest distance to traverse all measurements in the measurement set, α is the given coefficient, and ε is the given minimum value. In this method, the Bellman-Ford algorithm can be used to calculate Rk to obtain the value of the covariance σ2. Experiments show that this method has good results when the number of measurements is not large, but this method needs to preset the values of ε and α. If the preset values are unreasonable, it will seriously affect the effect of MSSDF. When calculating Rk, this method leads to a significant increase in the computation if the number of measurements is large. Detailed experimental data are given in the experimental part of Section 5.2.The method in [38] cannot be applied to the calculation of the MSSDF proposed in this paper. Therefore, a method of calculating the covariance of Gaussian surfaces is proposed. From the geometric perspective, the covariance is closely related to the number of measurements and the distribution of the measurements. That is, the covariance should relate to the measurement set d1:k. All the boundary measurements in d1:k can be calculated (that is, the outermost measurement in d1:k). Since the boundary algorithm has been relatively mature, this paper omits the calculation process of this algorithm, and the results obtained are as in Figures 5 and 6), where the red “∗” is the calculated boundary measurements, connecting these measurements in turn and calculate the area Ak of the graph formed by the boundary measurements.Figure 5
Boundary measurements.Figure 6
Boundary area.DistributeAk evenly to all the measurements in d1:k and calculate the area occupied by each measurement. Assuming that the area occupied by each measurement is a circle, the average radius ra occupied by each measurement can be calculated, and this radius ra could be the value of σ2:(25)σ2=Ak|d1:k|×π.The covariances of MSSDF in Figures2 and 3 are both obtained by this method, which confirm the effectiveness of this method.


### 3.2.3. Calculation of the Threshold in Level Set

In this section, equation (11) gives the calculation method for the estimated shape. The estimated shape of the extended target needs to be determined by the value of the threshold c. The shape ℒ∅ is a collection of the shape function ϕxkE,z, taking a certain value, c, which can be seen as the intersection of a surface ϕxkE,z and a plane c. In the absence of sensor noise, the measurement of the extended target comes from the interior and boundary, which is shown in equation (6): the value of c should be 0. However, in actual situations, due to the sensor noise factor, some measurements may come from outside the boundary of the target. Measurements outside these boundaries will also generate a Gaussian surface with a certain height and be normalized to the surface ϕxkE,z, so the threshold c cannot be 0.The size ofc directly determines the quality of the shape estimation of the extended target tracking. Since there is no a priori information about the shape of the target, it is difficult to calculate the c. To solve the problem so that the shape function ϕxkE,z can be obtained at each moment to get the optimal estimation of the shape, a method for the value of the adaptive threshold c is proposed.In Section3.2.2, the set of boundary points d1:k is obtained. If d1:k is the true boundary of the target, the threshold c is the average of the heights of the Gaussian surfaces where the boundary measurements d1:kBOU are located. However, in actual situations, due to noise interference, the outermost boundary points are generally not the true boundary of the target. They may also come from the outside or inside of the target. It is difficult to judge individually without prior knowledge. Average the height of the boundary measurements to obtain the value of the threshold c.(26)c=1d1:kBOU∑l=1dkBOUhdkBOU|d1:k,l,σ2.We have established an extended target shape representation model based on CKF and GSF. Bringing equations (22), (25) to (11), using a level set to describe the estimated shape of the target:(27)ℒϕc=z|1d1:k∑l=1d1:k12πexp−12σ2z−d1:k,lz−d1:k,lT=c,where c is calculated by equation (26); σ2 is calculated by equation (25); d1:k is calculated by equation (22).The specific steps of the algorithm atk moments are given below, which can better explain the algorithm proposed in this paper:(a)
Predict and update the target’s state information with CKF.(b)
Usingmk|k in the state information xk|k, the set d1:k is obtained by panning the k−th moment measurements by (22).(c)
Calculate the MSSDF by the GSF using equation (23); the covariance is calculated by (23).(d)
The threshold is calculated by (24), and the estimated shape of the target is obtained by (25).


## 3.2.1. MSSDF Constructed by Multiple Moment Measurements

Given the situation at a certain moment, the MSSDF was constructed by the GSF.Figure3 shows the results when using the GSF to construct the MSSDF of the extended target at a certain moment. The parameter settings are roughly the same as those in Figure 2. In order to be close to the simulation scene, the number of measurements obtained at each moment is changed to obey a Poisson distribution with a parameter of 20. In this scenario, using the GSF to construct the MSSDF cannot describe the specific shape and main characteristics of the extended target. The main reason for this is that fewer measurements are obtained at each moment, and the shape information contained in these measurements is limited and cannot approximate the actual distribution of the target. Due to the limitation of the current sensor resolution, in actual situations, it is difficult to obtain enough measurements at one moment. To solve this problem, this paper intends to use multiple moment measurements to update the MSSDF of the extended target.Figure 3
MSSDF constructed by fewer measurements.Since the measurements at each moment are generated independently of each other, they may be generated by measurement sources at different locations of the target, so the MSSDF can be better constructed with measurements from multiple moments and thus better estimate the shape of the target.The optimal estimate of the extended target state obtained in equation (20) is xk|k, where the centroid estimate mk|k is included. Then, each measurement in the measurement set Zk at moment k is processed with the following:(22)dk,l=zk,l−mk.This can be interpreted as panning each measurement to the center point of the target estimate.Figure4 shows the simulation scenario of the same target at two consecutive moments. The gray part is the real shape of the extended target, the red “∗” on the target is the optimal estimate of the center coordinates calculated by the CKF, and the black “+” and the blue “x” are the measurements at two different moments. It can be seen that, after the calculation of equation (22), the measurements at two different moments are panned to an approximate position relative to the center of the extended target and constitute a new set of measurements. Compared with the measurements at a single moment, the new set of measurements contains more shape information, which will be more beneficial for the construction of the MSSDF.Figure 4
Panning measurements.Suppose that from the first moment to thek-th moment, through the calculation of equation (22), the dk calculated at each moment constitutes a new measurement set, which is represented by the set d1:k. This set can be regarded as the translation of the measurements generated at different positions on the target to the origin; then, equation (10) can be rewritten as follows:(23)hkz|d,σ2=1d1:k∑l=1d1:k12πexp−12σ2z−d1:k,lz−d1:k,lT.where d1:k is used as the mean value of the Gaussian surface. Section 3.2.2 will introduce how to solve the problem of calculating Gaussian surface covariance σ2.


## 3.2.2. Calculation of the Covariance in GSF

When constructing the MSSDF, the covariance will affect the shape of the Gaussian surface. If the size of the covariance does not match the mean value, the shape of the Gaussian surface will be poor. Reference [38] established a Gaussian function when solving similar problem:(24)ε=exp−αRk2σ2,where Rk is the shortest distance to traverse all measurements in the measurement set, α is the given coefficient, and ε is the given minimum value. In this method, the Bellman-Ford algorithm can be used to calculate Rk to obtain the value of the covariance σ2. Experiments show that this method has good results when the number of measurements is not large, but this method needs to preset the values of ε and α. If the preset values are unreasonable, it will seriously affect the effect of MSSDF. When calculating Rk, this method leads to a significant increase in the computation if the number of measurements is large. Detailed experimental data are given in the experimental part of Section 5.2.The method in [38] cannot be applied to the calculation of the MSSDF proposed in this paper. Therefore, a method of calculating the covariance of Gaussian surfaces is proposed. From the geometric perspective, the covariance is closely related to the number of measurements and the distribution of the measurements. That is, the covariance should relate to the measurement set d1:k. All the boundary measurements in d1:k can be calculated (that is, the outermost measurement in d1:k). Since the boundary algorithm has been relatively mature, this paper omits the calculation process of this algorithm, and the results obtained are as in Figures 5 and 6), where the red “∗” is the calculated boundary measurements, connecting these measurements in turn and calculate the area Ak of the graph formed by the boundary measurements.Figure 5
Boundary measurements.Figure 6
Boundary area.DistributeAk evenly to all the measurements in d1:k and calculate the area occupied by each measurement. Assuming that the area occupied by each measurement is a circle, the average radius ra occupied by each measurement can be calculated, and this radius ra could be the value of σ2:(25)σ2=Ak|d1:k|×π.The covariances of MSSDF in Figures2 and 3 are both obtained by this method, which confirm the effectiveness of this method.


## 3.2.3. Calculation of the Threshold in Level Set

In this section, equation (11) gives the calculation method for the estimated shape. The estimated shape of the extended target needs to be determined by the value of the threshold c. The shape ℒ∅ is a collection of the shape function ϕxkE,z, taking a certain value, c, which can be seen as the intersection of a surface ϕxkE,z and a plane c. In the absence of sensor noise, the measurement of the extended target comes from the interior and boundary, which is shown in equation (6): the value of c should be 0. However, in actual situations, due to the sensor noise factor, some measurements may come from outside the boundary of the target. Measurements outside these boundaries will also generate a Gaussian surface with a certain height and be normalized to the surface ϕxkE,z, so the threshold c cannot be 0.The size ofc directly determines the quality of the shape estimation of the extended target tracking. Since there is no a priori information about the shape of the target, it is difficult to calculate the c. To solve the problem so that the shape function ϕxkE,z can be obtained at each moment to get the optimal estimation of the shape, a method for the value of the adaptive threshold c is proposed.In Section3.2.2, the set of boundary points d1:k is obtained. If d1:k is the true boundary of the target, the threshold c is the average of the heights of the Gaussian surfaces where the boundary measurements d1:kBOU are located. However, in actual situations, due to noise interference, the outermost boundary points are generally not the true boundary of the target. They may also come from the outside or inside of the target. It is difficult to judge individually without prior knowledge. Average the height of the boundary measurements to obtain the value of the threshold c.(26)c=1d1:kBOU∑l=1dkBOUhdkBOU|d1:k,l,σ2.We have established an extended target shape representation model based on CKF and GSF. Bringing equations (22), (25) to (11), using a level set to describe the estimated shape of the target:(27)ℒϕc=z|1d1:k∑l=1d1:k12πexp−12σ2z−d1:k,lz−d1:k,lT=c,where c is calculated by equation (26); σ2 is calculated by equation (25); d1:k is calculated by equation (22).The specific steps of the algorithm atk moments are given below, which can better explain the algorithm proposed in this paper:(a)
Predict and update the target’s state information with CKF.(b)
Usingmk|k in the state information xk|k, the set d1:k is obtained by panning the k−th moment measurements by (22).(c)
Calculate the MSSDF by the GSF using equation (23); the covariance is calculated by (23).(d)
The threshold is calculated by (24), and the estimated shape of the target is obtained by (25).


## 4. Shape Recursion

When computing the MSSDF for the extended target, its complexity can be approximated asΟnk. When updating the MSSDF of the extended target, it is essentially a Gaussian surface fitting and normalization of each coordinate in the set of measures d1:k. Since the measures in d1:k can be approximated as growing linearly, the time complexity of updating the MSSDF is Οn. With the increase in time, the number of measurements also increases, and more measurements need to be used in GSF. When enough measurements (assume that the threshold value for the number of measurements is nd) are taken, the MSSDF is already close to the actual target.In order to increase the speed of the algorithm, the Gaussian mixture model (GMM) is used to obtain the recursive form of the estimated shape. The basic form of the Gaussian mixture model is as follows:(28)pz=∑u=1UψuNz|μku,Δku.


### 4.1. Prediction Step

Suppose that, at thek−1 moment, the spatial distribution function of the measurement source obtained by equation (9) is as follows:(29)hk|k−1z|dk−1,Δk−1=1d1;k−1∑l=1d1;k−1Nz|dk−1,l,Δk−1,l,where(30)hk|k−1z|dk−1,Δk−1=hk−1z|dk−1,l,Δk−1,l.


### 4.2. Update Step

The measurement set obtained at thek moment is Zk, and the MSSDF is constructed to the set Zk using equation (9) to obtain the following:(31)hk′z|Zk,Δk=1Zk∑l=1ZkNz|Zk,l,Δk,l.Bringing the above two Gaussian functions into the GMM, that is, whenu=2, (27) and (29) are brought into (26), the hk of MSSDF after update is available:(32)hkz=ψk1hk|k−1z|dk−1,Δk−1+ψk2hk′z|Zk,Δk,where hkz is the spatial distribution function of the measurement source obtained by updating the Gaussian mixture model at time k. Since hk|k−1z|dk−1,Δk−1 is known, hk′z|Zk,Δk only needs to calculate the MSSDF at the current moment, so the overall calculation is small. Only the proportional coefficients ψk1 and ψk2 need to be determined to obtain the updated hkz. The calculation methods of ψk1 and ψk2 are given below:(33)ψk1+ψk2=1,ψk2=ZkZ1:k.Similarly, at thek−1 and k moments, the thresholds c calculated by equation (30) are ck|k−1 and ck′, respectively. For the updated hkz, the threshold is determined by the following formula:(34)ck=ψk1×ck|k−1+ψk2×ck′.According to (30), the complexity of the MSSDF update can be reduced to Ο1 by using GMM. The effect of shape recursion will be discussed in Section 5. The flow charts of the algorithm are as follows, which include obtaining the estimated shape of the target directly using measurements, obtaining the estimated shape of the target using recursion, and the overall flow chart of the algorithm (Figures 7–9).Figure 7
Obtain the estimated shape of the target directly using measurements.Figure 8
Obtain the estimated shape of the target using recursion.Figure 9
Flow chart of the algorithm.


## 4.1. Prediction Step

Suppose that, at thek−1 moment, the spatial distribution function of the measurement source obtained by equation (9) is as follows:(29)hk|k−1z|dk−1,Δk−1=1d1;k−1∑l=1d1;k−1Nz|dk−1,l,Δk−1,l,where(30)hk|k−1z|dk−1,Δk−1=hk−1z|dk−1,l,Δk−1,l.


## 4.2. Update Step

The measurement set obtained at thek moment is Zk, and the MSSDF is constructed to the set Zk using equation (9) to obtain the following:(31)hk′z|Zk,Δk=1Zk∑l=1ZkNz|Zk,l,Δk,l.Bringing the above two Gaussian functions into the GMM, that is, whenu=2, (27) and (29) are brought into (26), the hk of MSSDF after update is available:(32)hkz=ψk1hk|k−1z|dk−1,Δk−1+ψk2hk′z|Zk,Δk,where hkz is the spatial distribution function of the measurement source obtained by updating the Gaussian mixture model at time k. Since hk|k−1z|dk−1,Δk−1 is known, hk′z|Zk,Δk only needs to calculate the MSSDF at the current moment, so the overall calculation is small. Only the proportional coefficients ψk1 and ψk2 need to be determined to obtain the updated hkz. The calculation methods of ψk1 and ψk2 are given below:(33)ψk1+ψk2=1,ψk2=ZkZ1:k.Similarly, at thek−1 and k moments, the thresholds c calculated by equation (30) are ck|k−1 and ck′, respectively. For the updated hkz, the threshold is determined by the following formula:(34)ck=ψk1×ck|k−1+ψk2×ck′.According to (30), the complexity of the MSSDF update can be reduced to Ο1 by using GMM. The effect of shape recursion will be discussed in Section 5. The flow charts of the algorithm are as follows, which include obtaining the estimated shape of the target directly using measurements, obtaining the estimated shape of the target using recursion, and the overall flow chart of the algorithm (Figures 7–9).Figure 7
Obtain the estimated shape of the target directly using measurements.Figure 8
Obtain the estimated shape of the target using recursion.Figure 9
Flow chart of the algorithm.


## 5. Simulation

This section verifies and compares the effectiveness of this algorithm, looking at three aspects.


### 5.1. Irregular Shape Extended Target Tracking

In this section, two extended targets with different shapes are used for experiments, both of which are more complex and irregular. The GPR-RHM algorithm is used to compare with the algorithm proposed in this paper.Setting the time intervalT=1, the extended target performs a uniform linear motion at 0–50 s in the X-Y coordinate system and a constant-speed turning motion at 51–100 s. The motion state of the target is expressed as mkTxk∗TT, and the initial state of the extended target is x0=1,1,20,0T. The parameters of uniform linear motion and uniform turning motion are introduced below.


#### 5.1.1. Constant Motion

The motion state equation of the target is as shown in equation (1) above, where the extended target has a system evolution equation when it moves in a uniform straight line as equation (12), where(35)F=1T01000T01001001.


#### 5.1.2. Constant Turning Motion

The extended target has the same system evolution equation when it moves in a uniform circular motion in a two-dimensional plane, and the state transition matrix is as follows:(36)F=1sinωTω0−1−cosωTω0cosωT0−sinωT10−1−cosωTω11sinωTω0sinωT.The turning angle isω=0.0196π.Figure10 shows the real trajectory of the target in 100 seconds.Figure 10
True tracking.The number of measurement points generated by the targetZk obeys the Poisson distribution with 20 as the mean [39]. In the simulation scenario, the covariance of sensor noise is Q=diag0.22,0.22. In this experiment, equation (21) is used to update MSSDF in 0–50 s, and equation (30) is used to recurse MSSDF in 51–100 s. The boundary function in MATLAB was used to calculate the boundary measurements, and the shrinkage factor is 0.1. In order to objectively verify the performance of the algorithm, 100 Monte Carlo simulation experiments will be constructed to verify the filter performance. The results of the shape estimation from MC are randomly selected at three different moments and the simulation results are shown below.


#### 5.1.3. Aircraft Shape Single Extended Target

Given an aircraft-shaped extended target, the single tracking effect is shown in Figure11:Figure 11
Extended target tracking in aircraft shape.In this figure, the blue line is the estimated shape obtained by the proposed algorithm, the yellow dashed line is the estimated shape obtained by the GPR-RHM algorithm, the black “+” are the measurements of the extended target obtained at the current moment, and the blue “x” is the estimated position.Starting from the fiftieth second, equation (30) is used to extend the recurrence of the target shape, distinguishing it in the figure by the red solid line.Figure11 is the trajectory of the extended target. Figures 12–14, respectively, show the single tracking result of the extended target at moments 5–7, 49–51, and 71–72. The initial shape of the GPR-RHM algorithm is a circle with a radius of 5. In the starting phase of the target, due to the small number of measurements, the MSSDF constructed from these measurements has a large difference compared to the actual spatial distribution of the target, and the estimated shape obtained is not good. However, more effective measurements can be obtained over time, and the MSSDF is constantly updated and gradually approaches the spatial distribution of the actual shape. In Figure 11, a better shape estimation is obtained, and the number of measurements at this moment is about 450. Through the calculation of (21), the shape size obtained by equation (11) is similar to the actual size, and a better shape effect can be achieved. The figures above can clearly show that the estimated shape calculated by the proposed algorithm can better represent the actual shape of the extended target, and the details of the nose, wings, and tail can be seen more intuitionally. Compared with the GPR-RHM algorithm, this algorithm can better describe the shape details of the extended target, especially the shape feature part of the extended target. The proposed algorithm does not need to preset the a priori shape of the extended target and can represent any extended target shape. But the GPR-RHM algorithm needs to preset a circle with the same size as the extended target, which requires certain prior information.Figure 12
Steps 5–7 partial effect of the extended target tracking.Figure 13
Steps 49–51 partial effect of the extended target tracking.Figure 14
Steps 71–72 partial effect of the extended target tracking.Figure14 shows results the 50-th moment, using GMM to recurse the estimating shape of the extended target. Experiments show that the proposed GMM is effective in recursively extending the shape of the target.To verify the stability of the algorithm, two more randomly selected shape estimates from 100 Monte Carlo experiments at moments 49–51 are compared with the true shape of the target, and the results are as in Figures15 and 16).Figure 15
Results of a random Monte Carlo.Figure 16
Results of a random Monte Carlo.From Figures15 and 16, it can be seen that the proposed algorithm can estimate the shape of the target better and has good stability.The RMSE of the estimated positions obtained using the CKF for 100MC simulation trials are given in Figure17. It should be noted that in the calculation of the kinematic state of the extended target, the CKF is used for both algorithms to obtain the estimates, so there is only one RMSE in the figure. Figure 17 shows that the accuracy of the CKF tracking is high. The RMSE fluctuates more in several moments after the 50th moment due to the kinematic state transition of the extended target. However, with the accumulation of measurements, the center-of-mass estimation of the extended target will be more accurate and the RMSE will decrease and smoothen.Figure 17
RMSE of position.The MSSDF fitted by the GSF is given in Figure18.Figure 18
MSSDF fitted by GSF.Figure18 shows that the spatial measurement distribution fitted by the GSF is very similar to the actual spatial distribution of the target, which further proves the effectiveness of the proposed algorithm.


#### 5.1.4. Extended Target Tracking with Irregular Triangles

The simulation parameters are the same as the scenario above. Figure19 shows the real trajectory of the target in 100 seconds:Figure 19
Extended target tracking in irregular triangles shape.Section5.1.2 adopts an extended target, and the shape is shown in Figures 20–22. The experimental parameters are the same as those in Section 5.1.1, and the initial shape of GPR-RHM is transformed into a circle with a radius of 6. Compared with the GPR-RHM extended target tracking algorithms, the proposed algorithm can better describe the shape and characteristics of the target. For example, the triangle and the two rectangles at the tail can be clearly seen in the estimated shape, and the size is closer than the actual situation of the extended target.Figure 20
Steps 5–7 partial effect of the extended target tracking.Figure 21
Steps 49–51 partial effect of the extended target tracking.Figure 22
Steps 71–72 partial effect of the extended target tracking.Figures23 and 24 are randomly selected shape estimates from 100 Monte Carlo experiments at moments 49–51, and the results were as follows (Figure 25).Figure 23
Results of a random Monte Carlo.Figure 24
Results of a random Monte Carlo.Figure 25
RMSE of position.Similar to 5.1.1, the RMSE fluctuates in a few moments because the motion state of the target changes from the 50th moment onwards. It should be noted that because the shape of the extended target is larger in Section5.1.2 (about five times larger than in Section 5.1.1, the absolute value of the RMSE is larger than that in Section 5.1.1)To illustrate the effectiveness of the shape recursion algorithm in Section4, a comparison of the time consumed by the proposed algorithm and the GPR-RHM algorithms is given in this paper, as shown in Figure 26.Figure 26
Comparison of the time consumption of the two algorithms.Figure26 shows a time consumption comparison of the two algorithms. At moments 1–50, the proposed CKF-GSF algorithm needs to compute the MSSDF using the measurements of each moment, and the computation time keeps increasing because of the increasing number of measurements. Excluding the effect of fluctuations in the adjacent moments, it can be seen that the time spent by the proposed algorithm grows linearly overall in these moments from 1–50, while in the remaining moments, the time spent varies little overall and can be approximated as varying in a constant order, which is consistent with the hypothesis presented in Section 4. At these moments 51–100, the shape recursion algorithm is used, so the computation is significantly reduced and even lower than the GPR-RHM algorithm, which also shows the effectiveness of the proposed shape recursion algorithm.


### 5.2. Discussion on the Calculation Method of Covariance

To illustrate the effectiveness of the covariance calculation proposed in Section4.2, this paper compares the shortest distance method in [38] with the edge area method proposed in this paper. The main comparison is the value of covariance at different moments, the effect of MSSDF constructed with different covariances, and the time required to calculate the effectiveness of the proposed method.From (23), the covariance can be calculated as follows:(37)σ2=−αRk2lnε,where according to the [38], the preset parameters α=1, ε=0.01, Rk is the shortest distance to traverse for all the measurements in the measurement set, which can be obtained using the Bellman-Ford algorithm. The constructed object is the measurement set d1:k at the fiftieth moment in Section 5.1.2. Both methods take the measurement in d1:k as the mean and give the MSSDF with different covariance fits calculated by the two methods.Figure27 is obtained by the edge area method, and Figure 28 is obtained by the shortest distance method. As can be seen from the figures above, the MSSDF surface of Figure 27 is relatively smooth overall, with good integrity, fewer peaks, and a more uniform average height, while the surface of Figure 28 is not coherent, with more peaks on the upper part and uneven height, which is not conducive to the description of the estimated shape. Therefore, the covariance calculated by this method cannot well describe the measurement source distribution of the extended target. Figure 29 gives the value of the covariance at each moment:Figure 27
Proposed method.Figure 28
Literature proposed method.Figure 29
Sigma calculated by two different methods.Figure29 shows the corresponding covariance value at each moment. The covariance calculated by the shortest distance method decreases faster than the edge area method, which may cause the MSSDF to change more quickly, resulting in a decrease in the robustness of the entire algorithm. Moreover, the covariance calculated by the shortest distance method is small, which may cause the MSSDF to produce many peaks (as shown in Figure 28), and there is a big gap between the estimated shape and the real shape.A time comparison of the two methods under different measurement quantities is given below. Each group of measurement quantities is calculated 10 times, and the average value is taken as the result.As can be seen in Figure30, the time taken by the shortest distance method increases exponentially as the number of measurements increases, while the proposed edge area method does not require complex calculations and has a better tracking performance.Figure 30
Time required for the two methods.


### 5.3. The Influence of Different Noise on MSSDF and Estimated Shape

#### 5.3.1. The Influence of Sensor Noise on MSSDF

Take the shape of the airplane in Section5.1.1 as an example to provide the shape estimation effect of different sensor noise at a certain moment.Figures31 and 32 show the MSSDF under different sensor noise. Figure 31 is the result under low noise (Q=diag0.12,0.12), and Figure 32 is the result under higher noise (Q=diag0.52,0.52). Obviously, it can be seen that the MSSDF effect of Figure 31 is significantly better than that of Figure 32. Due to the low noise, the measurements are relatively concentrated, and most are inside the extended target or not far from the actual boundary of the extended target. The effect of using GSF is better, especially in the boundary part of MSSDF, which is very similar to the actual shape of the target. However, in Figure 32, due to the higher noise, the obtained measurements are more scattered, especially the measurements near the boundary, which are far from the actual boundary of the target, so the MSSDF obtained is not satisfying. It is not conducive to subsequent shape estimation or to show of the details of the target shape.Figure 31
MSSDF fitted in the low noise.Figure 32
MSSDF fitted in the high noise.


#### 5.3.2. The Influence of Different Noise on the Estimated Shape

Figures33 and 34 are the estimated shape obtained by using the MSSDF of Figures 31 and 32 with different thresholds. The blue line is the result of the proposed algorithm, and the others are the results under different thresholds. The result of shape estimation is closely related to MSSDF, and the sensitivity to the threshold is not strong in the case of low noise. In Figure 33, the percentage of plane height difference between blue and yellow is 86.7%, but the yellow result can still retain most of the shape features of the expanded target except for the difference in the actual size of the expanded target. In Figure 34, the actual distribution of MSSDF and the target is quite different. The estimated shape obtained is not significant and is very sensitive to the threshold. The percentage of the plane height difference between blue and yellow is 9.83%, but the difference in shape size is huge, and the estimated shape of yellow has only a small amount of useful shape information. In summary, noise has a great impact on the proposed algorithm, and it not only has an impact on the fitting of MSSDF but also directly affects the final shape estimation result.Figure 33
Shapes corresponding to different thresholds in the low noise.Figure 34
Shapes corresponding to different thresholds in the high noise.


## 5.1. Irregular Shape Extended Target Tracking

In this section, two extended targets with different shapes are used for experiments, both of which are more complex and irregular. The GPR-RHM algorithm is used to compare with the algorithm proposed in this paper.Setting the time intervalT=1, the extended target performs a uniform linear motion at 0–50 s in the X-Y coordinate system and a constant-speed turning motion at 51–100 s. The motion state of the target is expressed as mkTxk∗TT, and the initial state of the extended target is x0=1,1,20,0T. The parameters of uniform linear motion and uniform turning motion are introduced below.


### 5.1.1. Constant Motion

The motion state equation of the target is as shown in equation (1) above, where the extended target has a system evolution equation when it moves in a uniform straight line as equation (12), where(35)F=1T01000T01001001.


### 5.1.2. Constant Turning Motion

The extended target has the same system evolution equation when it moves in a uniform circular motion in a two-dimensional plane, and the state transition matrix is as follows:(36)F=1sinωTω0−1−cosωTω0cosωT0−sinωT10−1−cosωTω11sinωTω0sinωT.The turning angle isω=0.0196π.Figure10 shows the real trajectory of the target in 100 seconds.Figure 10
True tracking.The number of measurement points generated by the targetZk obeys the Poisson distribution with 20 as the mean [39]. In the simulation scenario, the covariance of sensor noise is Q=diag0.22,0.22. In this experiment, equation (21) is used to update MSSDF in 0–50 s, and equation (30) is used to recurse MSSDF in 51–100 s. The boundary function in MATLAB was used to calculate the boundary measurements, and the shrinkage factor is 0.1. In order to objectively verify the performance of the algorithm, 100 Monte Carlo simulation experiments will be constructed to verify the filter performance. The results of the shape estimation from MC are randomly selected at three different moments and the simulation results are shown below.


### 5.1.3. Aircraft Shape Single Extended Target

Given an aircraft-shaped extended target, the single tracking effect is shown in Figure11:Figure 11
Extended target tracking in aircraft shape.In this figure, the blue line is the estimated shape obtained by the proposed algorithm, the yellow dashed line is the estimated shape obtained by the GPR-RHM algorithm, the black “+” are the measurements of the extended target obtained at the current moment, and the blue “x” is the estimated position.Starting from the fiftieth second, equation (30) is used to extend the recurrence of the target shape, distinguishing it in the figure by the red solid line.Figure11 is the trajectory of the extended target. Figures 12–14, respectively, show the single tracking result of the extended target at moments 5–7, 49–51, and 71–72. The initial shape of the GPR-RHM algorithm is a circle with a radius of 5. In the starting phase of the target, due to the small number of measurements, the MSSDF constructed from these measurements has a large difference compared to the actual spatial distribution of the target, and the estimated shape obtained is not good. However, more effective measurements can be obtained over time, and the MSSDF is constantly updated and gradually approaches the spatial distribution of the actual shape. In Figure 11, a better shape estimation is obtained, and the number of measurements at this moment is about 450. Through the calculation of (21), the shape size obtained by equation (11) is similar to the actual size, and a better shape effect can be achieved. The figures above can clearly show that the estimated shape calculated by the proposed algorithm can better represent the actual shape of the extended target, and the details of the nose, wings, and tail can be seen more intuitionally. Compared with the GPR-RHM algorithm, this algorithm can better describe the shape details of the extended target, especially the shape feature part of the extended target. The proposed algorithm does not need to preset the a priori shape of the extended target and can represent any extended target shape. But the GPR-RHM algorithm needs to preset a circle with the same size as the extended target, which requires certain prior information.Figure 12
Steps 5–7 partial effect of the extended target tracking.Figure 13
Steps 49–51 partial effect of the extended target tracking.Figure 14
Steps 71–72 partial effect of the extended target tracking.Figure14 shows results the 50-th moment, using GMM to recurse the estimating shape of the extended target. Experiments show that the proposed GMM is effective in recursively extending the shape of the target.To verify the stability of the algorithm, two more randomly selected shape estimates from 100 Monte Carlo experiments at moments 49–51 are compared with the true shape of the target, and the results are as in Figures15 and 16).Figure 15
Results of a random Monte Carlo.Figure 16
Results of a random Monte Carlo.From Figures15 and 16, it can be seen that the proposed algorithm can estimate the shape of the target better and has good stability.The RMSE of the estimated positions obtained using the CKF for 100MC simulation trials are given in Figure17. It should be noted that in the calculation of the kinematic state of the extended target, the CKF is used for both algorithms to obtain the estimates, so there is only one RMSE in the figure. Figure 17 shows that the accuracy of the CKF tracking is high. The RMSE fluctuates more in several moments after the 50th moment due to the kinematic state transition of the extended target. However, with the accumulation of measurements, the center-of-mass estimation of the extended target will be more accurate and the RMSE will decrease and smoothen.Figure 17
RMSE of position.The MSSDF fitted by the GSF is given in Figure18.Figure 18
MSSDF fitted by GSF.Figure18 shows that the spatial measurement distribution fitted by the GSF is very similar to the actual spatial distribution of the target, which further proves the effectiveness of the proposed algorithm.


### 5.1.4. Extended Target Tracking with Irregular Triangles

The simulation parameters are the same as the scenario above. Figure19 shows the real trajectory of the target in 100 seconds:Figure 19
Extended target tracking in irregular triangles shape.Section5.1.2 adopts an extended target, and the shape is shown in Figures 20–22. The experimental parameters are the same as those in Section 5.1.1, and the initial shape of GPR-RHM is transformed into a circle with a radius of 6. Compared with the GPR-RHM extended target tracking algorithms, the proposed algorithm can better describe the shape and characteristics of the target. For example, the triangle and the two rectangles at the tail can be clearly seen in the estimated shape, and the size is closer than the actual situation of the extended target.Figure 20
Steps 5–7 partial effect of the extended target tracking.Figure 21
Steps 49–51 partial effect of the extended target tracking.Figure 22
Steps 71–72 partial effect of the extended target tracking.Figures23 and 24 are randomly selected shape estimates from 100 Monte Carlo experiments at moments 49–51, and the results were as follows (Figure 25).Figure 23
Results of a random Monte Carlo.Figure 24
Results of a random Monte Carlo.Figure 25
RMSE of position.Similar to 5.1.1, the RMSE fluctuates in a few moments because the motion state of the target changes from the 50th moment onwards. It should be noted that because the shape of the extended target is larger in Section5.1.2 (about five times larger than in Section 5.1.1, the absolute value of the RMSE is larger than that in Section 5.1.1)To illustrate the effectiveness of the shape recursion algorithm in Section4, a comparison of the time consumed by the proposed algorithm and the GPR-RHM algorithms is given in this paper, as shown in Figure 26.Figure 26
Comparison of the time consumption of the two algorithms.Figure26 shows a time consumption comparison of the two algorithms. At moments 1–50, the proposed CKF-GSF algorithm needs to compute the MSSDF using the measurements of each moment, and the computation time keeps increasing because of the increasing number of measurements. Excluding the effect of fluctuations in the adjacent moments, it can be seen that the time spent by the proposed algorithm grows linearly overall in these moments from 1–50, while in the remaining moments, the time spent varies little overall and can be approximated as varying in a constant order, which is consistent with the hypothesis presented in Section 4. At these moments 51–100, the shape recursion algorithm is used, so the computation is significantly reduced and even lower than the GPR-RHM algorithm, which also shows the effectiveness of the proposed shape recursion algorithm.


## 5.1.1. Constant Motion

The motion state equation of the target is as shown in equation (1) above, where the extended target has a system evolution equation when it moves in a uniform straight line as equation (12), where(35)F=1T01000T01001001.


## 5.1.2. Constant Turning Motion

The extended target has the same system evolution equation when it moves in a uniform circular motion in a two-dimensional plane, and the state transition matrix is as follows:(36)F=1sinωTω0−1−cosωTω0cosωT0−sinωT10−1−cosωTω11sinωTω0sinωT.The turning angle isω=0.0196π.Figure10 shows the real trajectory of the target in 100 seconds.Figure 10
True tracking.The number of measurement points generated by the targetZk obeys the Poisson distribution with 20 as the mean [39]. In the simulation scenario, the covariance of sensor noise is Q=diag0.22,0.22. In this experiment, equation (21) is used to update MSSDF in 0–50 s, and equation (30) is used to recurse MSSDF in 51–100 s. The boundary function in MATLAB was used to calculate the boundary measurements, and the shrinkage factor is 0.1. In order to objectively verify the performance of the algorithm, 100 Monte Carlo simulation experiments will be constructed to verify the filter performance. The results of the shape estimation from MC are randomly selected at three different moments and the simulation results are shown below.


## 5.1.3. Aircraft Shape Single Extended Target

Given an aircraft-shaped extended target, the single tracking effect is shown in Figure11:Figure 11
Extended target tracking in aircraft shape.In this figure, the blue line is the estimated shape obtained by the proposed algorithm, the yellow dashed line is the estimated shape obtained by the GPR-RHM algorithm, the black “+” are the measurements of the extended target obtained at the current moment, and the blue “x” is the estimated position.Starting from the fiftieth second, equation (30) is used to extend the recurrence of the target shape, distinguishing it in the figure by the red solid line.Figure11 is the trajectory of the extended target. Figures 12–14, respectively, show the single tracking result of the extended target at moments 5–7, 49–51, and 71–72. The initial shape of the GPR-RHM algorithm is a circle with a radius of 5. In the starting phase of the target, due to the small number of measurements, the MSSDF constructed from these measurements has a large difference compared to the actual spatial distribution of the target, and the estimated shape obtained is not good. However, more effective measurements can be obtained over time, and the MSSDF is constantly updated and gradually approaches the spatial distribution of the actual shape. In Figure 11, a better shape estimation is obtained, and the number of measurements at this moment is about 450. Through the calculation of (21), the shape size obtained by equation (11) is similar to the actual size, and a better shape effect can be achieved. The figures above can clearly show that the estimated shape calculated by the proposed algorithm can better represent the actual shape of the extended target, and the details of the nose, wings, and tail can be seen more intuitionally. Compared with the GPR-RHM algorithm, this algorithm can better describe the shape details of the extended target, especially the shape feature part of the extended target. The proposed algorithm does not need to preset the a priori shape of the extended target and can represent any extended target shape. But the GPR-RHM algorithm needs to preset a circle with the same size as the extended target, which requires certain prior information.Figure 12
Steps 5–7 partial effect of the extended target tracking.Figure 13
Steps 49–51 partial effect of the extended target tracking.Figure 14
Steps 71–72 partial effect of the extended target tracking.Figure14 shows results the 50-th moment, using GMM to recurse the estimating shape of the extended target. Experiments show that the proposed GMM is effective in recursively extending the shape of the target.To verify the stability of the algorithm, two more randomly selected shape estimates from 100 Monte Carlo experiments at moments 49–51 are compared with the true shape of the target, and the results are as in Figures15 and 16).Figure 15
Results of a random Monte Carlo.Figure 16
Results of a random Monte Carlo.From Figures15 and 16, it can be seen that the proposed algorithm can estimate the shape of the target better and has good stability.The RMSE of the estimated positions obtained using the CKF for 100MC simulation trials are given in Figure17. It should be noted that in the calculation of the kinematic state of the extended target, the CKF is used for both algorithms to obtain the estimates, so there is only one RMSE in the figure. Figure 17 shows that the accuracy of the CKF tracking is high. The RMSE fluctuates more in several moments after the 50th moment due to the kinematic state transition of the extended target. However, with the accumulation of measurements, the center-of-mass estimation of the extended target will be more accurate and the RMSE will decrease and smoothen.Figure 17
RMSE of position.The MSSDF fitted by the GSF is given in Figure18.Figure 18
MSSDF fitted by GSF.Figure18 shows that the spatial measurement distribution fitted by the GSF is very similar to the actual spatial distribution of the target, which further proves the effectiveness of the proposed algorithm.


## 5.1.4. Extended Target Tracking with Irregular Triangles

The simulation parameters are the same as the scenario above. Figure19 shows the real trajectory of the target in 100 seconds:Figure 19
Extended target tracking in irregular triangles shape.Section5.1.2 adopts an extended target, and the shape is shown in Figures 20–22. The experimental parameters are the same as those in Section 5.1.1, and the initial shape of GPR-RHM is transformed into a circle with a radius of 6. Compared with the GPR-RHM extended target tracking algorithms, the proposed algorithm can better describe the shape and characteristics of the target. For example, the triangle and the two rectangles at the tail can be clearly seen in the estimated shape, and the size is closer than the actual situation of the extended target.Figure 20
Steps 5–7 partial effect of the extended target tracking.Figure 21
Steps 49–51 partial effect of the extended target tracking.Figure 22
Steps 71–72 partial effect of the extended target tracking.Figures23 and 24 are randomly selected shape estimates from 100 Monte Carlo experiments at moments 49–51, and the results were as follows (Figure 25).Figure 23
Results of a random Monte Carlo.Figure 24
Results of a random Monte Carlo.Figure 25
RMSE of position.Similar to 5.1.1, the RMSE fluctuates in a few moments because the motion state of the target changes from the 50th moment onwards. It should be noted that because the shape of the extended target is larger in Section5.1.2 (about five times larger than in Section 5.1.1, the absolute value of the RMSE is larger than that in Section 5.1.1)To illustrate the effectiveness of the shape recursion algorithm in Section4, a comparison of the time consumed by the proposed algorithm and the GPR-RHM algorithms is given in this paper, as shown in Figure 26.Figure 26
Comparison of the time consumption of the two algorithms.Figure26 shows a time consumption comparison of the two algorithms. At moments 1–50, the proposed CKF-GSF algorithm needs to compute the MSSDF using the measurements of each moment, and the computation time keeps increasing because of the increasing number of measurements. Excluding the effect of fluctuations in the adjacent moments, it can be seen that the time spent by the proposed algorithm grows linearly overall in these moments from 1–50, while in the remaining moments, the time spent varies little overall and can be approximated as varying in a constant order, which is consistent with the hypothesis presented in Section 4. At these moments 51–100, the shape recursion algorithm is used, so the computation is significantly reduced and even lower than the GPR-RHM algorithm, which also shows the effectiveness of the proposed shape recursion algorithm.


## 5.2. Discussion on the Calculation Method of Covariance

To illustrate the effectiveness of the covariance calculation proposed in Section4.2, this paper compares the shortest distance method in [38] with the edge area method proposed in this paper. The main comparison is the value of covariance at different moments, the effect of MSSDF constructed with different covariances, and the time required to calculate the effectiveness of the proposed method.From (23), the covariance can be calculated as follows:(37)σ2=−αRk2lnε,where according to the [38], the preset parameters α=1, ε=0.01, Rk is the shortest distance to traverse for all the measurements in the measurement set, which can be obtained using the Bellman-Ford algorithm. The constructed object is the measurement set d1:k at the fiftieth moment in Section 5.1.2. Both methods take the measurement in d1:k as the mean and give the MSSDF with different covariance fits calculated by the two methods.Figure27 is obtained by the edge area method, and Figure 28 is obtained by the shortest distance method. As can be seen from the figures above, the MSSDF surface of Figure 27 is relatively smooth overall, with good integrity, fewer peaks, and a more uniform average height, while the surface of Figure 28 is not coherent, with more peaks on the upper part and uneven height, which is not conducive to the description of the estimated shape. Therefore, the covariance calculated by this method cannot well describe the measurement source distribution of the extended target. Figure 29 gives the value of the covariance at each moment:Figure 27
Proposed method.Figure 28
Literature proposed method.Figure 29
Sigma calculated by two different methods.Figure29 shows the corresponding covariance value at each moment. The covariance calculated by the shortest distance method decreases faster than the edge area method, which may cause the MSSDF to change more quickly, resulting in a decrease in the robustness of the entire algorithm. Moreover, the covariance calculated by the shortest distance method is small, which may cause the MSSDF to produce many peaks (as shown in Figure 28), and there is a big gap between the estimated shape and the real shape.A time comparison of the two methods under different measurement quantities is given below. Each group of measurement quantities is calculated 10 times, and the average value is taken as the result.As can be seen in Figure30, the time taken by the shortest distance method increases exponentially as the number of measurements increases, while the proposed edge area method does not require complex calculations and has a better tracking performance.Figure 30
Time required for the two methods.


## 5.3. The Influence of Different Noise on MSSDF and Estimated Shape

### 5.3.1. The Influence of Sensor Noise on MSSDF

Take the shape of the airplane in Section5.1.1 as an example to provide the shape estimation effect of different sensor noise at a certain moment.Figures31 and 32 show the MSSDF under different sensor noise. Figure 31 is the result under low noise (Q=diag0.12,0.12), and Figure 32 is the result under higher noise (Q=diag0.52,0.52). Obviously, it can be seen that the MSSDF effect of Figure 31 is significantly better than that of Figure 32. Due to the low noise, the measurements are relatively concentrated, and most are inside the extended target or not far from the actual boundary of the extended target. The effect of using GSF is better, especially in the boundary part of MSSDF, which is very similar to the actual shape of the target. However, in Figure 32, due to the higher noise, the obtained measurements are more scattered, especially the measurements near the boundary, which are far from the actual boundary of the target, so the MSSDF obtained is not satisfying. It is not conducive to subsequent shape estimation or to show of the details of the target shape.Figure 31
MSSDF fitted in the low noise.Figure 32
MSSDF fitted in the high noise.


### 5.3.2. The Influence of Different Noise on the Estimated Shape

Figures33 and 34 are the estimated shape obtained by using the MSSDF of Figures 31 and 32 with different thresholds. The blue line is the result of the proposed algorithm, and the others are the results under different thresholds. The result of shape estimation is closely related to MSSDF, and the sensitivity to the threshold is not strong in the case of low noise. In Figure 33, the percentage of plane height difference between blue and yellow is 86.7%, but the yellow result can still retain most of the shape features of the expanded target except for the difference in the actual size of the expanded target. In Figure 34, the actual distribution of MSSDF and the target is quite different. The estimated shape obtained is not significant and is very sensitive to the threshold. The percentage of the plane height difference between blue and yellow is 9.83%, but the difference in shape size is huge, and the estimated shape of yellow has only a small amount of useful shape information. In summary, noise has a great impact on the proposed algorithm, and it not only has an impact on the fitting of MSSDF but also directly affects the final shape estimation result.Figure 33
Shapes corresponding to different thresholds in the low noise.Figure 34
Shapes corresponding to different thresholds in the high noise.


## 5.3.1. The Influence of Sensor Noise on MSSDF

Take the shape of the airplane in Section5.1.1 as an example to provide the shape estimation effect of different sensor noise at a certain moment.Figures31 and 32 show the MSSDF under different sensor noise. Figure 31 is the result under low noise (Q=diag0.12,0.12), and Figure 32 is the result under higher noise (Q=diag0.52,0.52). Obviously, it can be seen that the MSSDF effect of Figure 31 is significantly better than that of Figure 32. Due to the low noise, the measurements are relatively concentrated, and most are inside the extended target or not far from the actual boundary of the extended target. The effect of using GSF is better, especially in the boundary part of MSSDF, which is very similar to the actual shape of the target. However, in Figure 32, due to the higher noise, the obtained measurements are more scattered, especially the measurements near the boundary, which are far from the actual boundary of the target, so the MSSDF obtained is not satisfying. It is not conducive to subsequent shape estimation or to show of the details of the target shape.Figure 31
MSSDF fitted in the low noise.Figure 32
MSSDF fitted in the high noise.


## 5.3.2. The Influence of Different Noise on the Estimated Shape

Figures33 and 34 are the estimated shape obtained by using the MSSDF of Figures 31 and 32 with different thresholds. The blue line is the result of the proposed algorithm, and the others are the results under different thresholds. The result of shape estimation is closely related to MSSDF, and the sensitivity to the threshold is not strong in the case of low noise. In Figure 33, the percentage of plane height difference between blue and yellow is 86.7%, but the yellow result can still retain most of the shape features of the expanded target except for the difference in the actual size of the expanded target. In Figure 34, the actual distribution of MSSDF and the target is quite different. The estimated shape obtained is not significant and is very sensitive to the threshold. The percentage of the plane height difference between blue and yellow is 9.83%, but the difference in shape size is huge, and the estimated shape of yellow has only a small amount of useful shape information. In summary, noise has a great impact on the proposed algorithm, and it not only has an impact on the fitting of MSSDF but also directly affects the final shape estimation result.Figure 33
Shapes corresponding to different thresholds in the low noise.Figure 34
Shapes corresponding to different thresholds in the high noise.


## 6. Conclusion

In this paper, the level set is applied to the shape estimation of extended target tracking, and the measurements at multiple moments are used to construct the MSSDF by the GSF, which reduces the difficulty of building any shape of the spatial distribution of measurement sources. The shape description equation of the extended target is obtained from the level set, and the shape recursion formula under the Bayesian framework is obtained by combining the GMM.The proposed algorithm can show more details of the target and is more robust. Compared with the existing algorithms, the algorithm in this paper does not need to preset the contour equation of the target, nor does it need to preset the size of the extended target and can estimate the target of any shape. This algorithm has a wider range of application scenarios.In conclusion, the proposed algorithm is more robust and has a better shape estimation performance, but its computational cost is higher than the GPR-RHM algorithm, and it is suitable for tracking and recognition systems with higher accuracy. The next step is to extend the proposed algorithm to the field of multiextended target tracking to solve the problem of multitarget state and shape estimation. It is also necessary to solve the problem of optimizing the spatial distribution function of the measurement source in the case of high noise. The proposed algorithm can accurately estimate the shape of the target without high noise, so the next work can be based on this to identify and classify the target and get more information about the target.



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*Source: 2246522-2022-04-22.xml*