Labeled Multi‐Bernoulli Maneuvering Target Tracking Algorithm via TSK Iterative Regression Model

Chinese Journal of ElectronicsVolume 31, Issue 2 p. 227-239 Signal ProcessingOpen Access Labeled Multi-Bernoulli Maneuvering Target Tracking Algorithm via TSK Iterative Regression Model Xiaoli WANG, Corresponding Author Xiaoli WANG xlwang@szu.edu.cn Department of Electronics and Information, Guangdong Polytechnic Normal University, Guangzhou, 510665 ChinaSearch for more papers by this authorWeixin XIE, Weixin XIE Automatic Target Recognition (ATR) Key Laboratory, Shenzhen University, Shenzhen, 518060 China Guangdong Key Laboratory of Intelligent Information Processing, Shenzhen University, Shenzhen, 518060 ChinaSearch for more papers by this authorLiangqun LI, Corresponding Author Liangqun LI lqli@szu.edu.cn Automatic Target Recognition (ATR) Key Laboratory, Shenzhen University, Shenzhen, 518060 China Guangdong Key Laboratory of Intelligent Information Processing, Shenzhen University, Shenzhen, 518060 ChinaSearch for more papers by this author Xiaoli WANG, Corresponding Author Xiaoli WANG xlwang@szu.edu.cn Department of Electronics and Information, Guangdong Polytechnic Normal University, Guangzhou, 510665 ChinaSearch for more papers by this authorWeixin XIE, Weixin XIE Automatic Target Recognition (ATR) Key Laboratory, Shenzhen University, Shenzhen, 518060 China Guangdong Key Laboratory of Intelligent Information Processing, Shenzhen University, Shenzhen, 518060 ChinaSearch for more papers by this authorLiangqun LI, Corresponding Author Liangqun LI lqli@szu.edu.cn Automatic Target Recognition (ATR) Key Laboratory, Shenzhen University, Shenzhen, 518060 China Guangdong Key Laboratory of Intelligent Information Processing, Shenzhen University, Shenzhen, 518060 ChinaSearch for more papers by this author First published: 01 March 2022 https://doi.org/10.1049/cje.2020.00.156 This work was supported in part by the National Natural Science Foundation of China (62171287, 61773267), the Major Scientific and Technological Project of Guangdong Province (2017B030308006), the Major Program for Tackling Key Problems of Guangzhou City, China (201704020144), and Science & Technology Program of Shenzhen (JCYJ20190808120417257). AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract Aiming at the problem that the existing labeled multi-Bernoulli (LMB) method has a single and fixed model set, an LMB maneuvering target tracking algorithm via Takagi-Sugeno-Kang (TSK) iterative regression multiple model is proposed. In the TSK iterative regression modeling, the feature information of the targets is analyzed and represented by multiple semantic fuzzy sets. Then the state is expanded to introduce model information, thereby the adaptive multi-model idea is incorporated into the framework of the LMB method to solve the uncertain maneuverability of moving targets. Finally, the simulation results show that the proposed algorithm can effectively achieve maneuvering target tracking in the nonlinear system. I. Introduction The traditional multi-target tracking algorithm is mainly to match the observation data with the target through the data association technology, and then to estimate the state and track the target in real time through the relevant filtering technology. Typical data association methods in the information fusion domain include nearest neighbor (NN)[1], joint probabilistic data association (JPDA)[2], etc., but the common research proposal of these related methods is to assume the number of targets is constant and known, which is usually difficult to meet in practical application systems. Another commonly used data association algorithm is multiple hypothesis tracking (MHT)[3], but in practical applications, it is very difficult to complete the pairing between the observation data and the target, it can be seen that the actual effect of data association technology is not ideal when applied to multi-target tracking. With the rapid development of science and technology and the increasingly complex environment of modern warfare, the maneuverability of the targets has become more complex and changeable, and the demand for tracking performance of maneuvering multiple targets have been increasing. The traditional maneuvering target tracking algorithm based on data association has been hindered in practical applications. American scholar Mahler put forward the theory of stochastic finite sets in 1994, which made the data fusion of multi-target and multi-sensor had a unified and scientific mathematical basis. The emergence of random finite set (RFS) theory has brought the research of multi-target tracking technology into a new field. The multi-target tracking algorithm of finite set statistics (FISST)[4] did not need to associate the observed data with the target during the tracking, which broke through the difficulty of data association. It is to describe the state and observation of multiple targets with RFS respectively, and recursively process the posterior probability density of multiple targets according to the Bayesian filtering principle. The main difference between this method and the traditional method is that it completely avoids data association, so that the bottleneck of data association in multi-target tracking can be solved. Since the stochastic finite set theory was proposed, domestic and foreign scholars have conducted in-depth research in the field of multi-target tracking based on this theory, mainly based on the further expansion and application of Mahler's "first-order moment filter" (probability hypothesis density filter, PHD)[5], and widely used in the following aspects: sonar tracking[6,7], sound source tracking[8], video target tracking[9,10], aerial image processing[11] and synchronous positioning[12], etc. However, these filters do not record and maintain the target's movement trajectory, resulting in the inability to distinguish targets during tracking. Vo et al.[13] introduced the concept of label random sets, and the published paper successfully solved the above problems. A new type of RFS distribution[14] was proposed in 2013, which was conjugated to the multi-target observation likelihood and was closed under the Chapman-Kolmogorov equation, and the effect of the random label finite set and multi-target conjugate prior algorithm was tested on Bayesian multi-target tracking algorithm[15], but the algorithm still had the problem of biased target number estimation. In the same year, a generalized labeled multi-Bernoulli algorithm (GLMB)[16], also known as labeled multi-Bernoulli (LMB) algorithm, effectively solved the problem of biased estimation of the number of targets. In the prediction and update process, the k-shortest path and m-best method were used to cut off the hypothetical targets below the existence threshold, which not only inherited the advantages of the δ-GLMB filter, but also achieved better performance. In order to more efficiently implement the LMB algorithm, Vo et al.[17] combined the prediction and update into an iterative process in 2017, and used only one truncation method (Gibbs sampling) during the iterative process, which effectively reduced the complexity of the algorithm. It is well known that for maneuvering target tracking, the multi-model algorithm is one of the effective ways to deal with maneuvering problem. In recent years, there have been proposed many multi-maneuvering target tracking algorithms based on multi-model algorithms and RFS filters[18-20]. Due to the inherent limitations of PHD/cardinality probability hypothesis density (CPHD)[19] and multi-target multi-Bernoulli filter (MeMber)/cardinality Balanced multi-target multi-Bernoulli filter (CBMeMber)[20], the performance of such algorithms is severely degraded in low signal-to-noise ratio detection environment. To solve this problem, the multi-model idea was introduced into the LMB algorithm, and a multi-model LMB algorithm[21,22] was proposed, which was suitable for multi-maneuvering target tracking. Cament et al.[22] proposed LMB distribution to model the multi-target state, together with a Poisson RFS distribution to model target birth. However, the above algorithms are executed under the assumption that the target maneuver is known and the weak maneuver is performed, and the number of models in these methods is fixed. When the target maneuver is complex, diverse and unknown, the performance of the algorithm will be greatly affected. In summary, it is a problem to be solved urgently to research an adaptive modeling and tracking algorithm suitable for the case of strong target maneuvering. The TSK fuzzy model[23-25] is the key technology to solve such problem. In the case of single target strong maneuver, we have proposed TSK iterative regression multiple model[26-29], which effectively solved the problem of uncertainty of target motion in nonlinear systems. Therefore, we extend this algorithm to the multi-target tracking, and effectively integrate with the LMB algorithm proposed by Vo et al.[17], and propose a LMB maneuvering target method based on TSK iterative regression fuzzy model technology to solve the problem of maneuvering complexity. The main contributions of this article are as follows: 1) In view of the uncertainty modeling problem of the target dynamic model, it is proposed to use spatial constraint information to construct a TSK fuzzy model, in which spatial feature information is represented by multiple semantic fuzzy sets; 2) Expand the dimension of the target state and introduce model information, so as to integrate the multi-model idea into the framework of the LMB method; 3) The TSK iterative regression filtering method is used to predict and update the state of the targets, which can effectively improve filtering accuracy when the turning rate is unknown. And a TSK iterative regression multi-model LMB algorithm is constructed to solve the problem of biased target number estimation and data association difficulties in the strong maneuvering target tracking. The rest of this paper is organized as follows. Some background knowledge is introduced in the Section II. The proposed algorithm is presented in the Section III. The simulation results that compare the performance of all algorithms are described in the Section IV. Finally, some conclusions of the proposed algorithm are given in the Section V. II. Background 1. Nonlinear system model Consider the nonlinear discrete-time dynamic system: x k = f k x k − 1 + e k − 1 (1) z k = h k x k + v k (2) In this system, k ∈ N denotes discrete time, f k : R n x × R n v → R n x and h k : R n x × R n e → R n z denote nonlinear functions, x k ∈ R n x is n x dimensional state vector. z k ∈ R n z is n z dimensional observation vector. e k − 1 ∈ R n e is the process noise, with zero mean and covariance Q e k and v k ∈ R n v is the non-Gaussian observation noise. 2. TSK iterative regression fuzzy model The TSK fuzzy model is a nonlinear mathematical model for complex nonlinear and dynamic system proposed by Takagi, Sugeno and Kang in 1985. It has good nonlinear approximation capabilities and is widely used in system identification, pattern identification, image processing, data mining and so on. The model regards a multiple-input multiple-output system as a combination of several multiple-input single-output systems, and implements the rule of the model through fuzzy logic inference. The TSK fuzzy model not only has strong approximation capabilities, but also simplifies the rule of the fuzzy model. These advantages bring convenience to system modeling and analysis. Thus, this model has become a common frame for solving various practical problems. The TSK fuzzy model is modeled according to the feature information of targets. The detailed TSK fuzzy model can be found in Refs.[26-29]. This section only briefly describes the TSK fuzzy model. In general, the T-S fuzzy model can be described by fuzzy linear rules: Rule i: IF θ k 1 is A k i , 1 and ⋯ and θ k G is A k i , G , then x k i = Φ k − 1 i x k − 1 i + e k − 1 i (3) z k i = H k i x k i + v k i , i = 1 , 2 , … , N f (4) where θ k = θ k 1 , θ k 2 , … , θ k G is the premise parameter, A k i , G denotes the fuzzy set of G t , h premise parameter in the rule i, Φ k − 1 i and H k i are the state transition matrix and the observation matrix, respectively. N f is the total number of rules. The consequent part is updated iteratively by the extended forgetting factor least squares method (EFFLSM)[26], the global fuzzy model can be expressed as follows: x k = ∑ i = 1 N f μ k i θ k Φ k − 1 i x k − 1 i + e k − 1 i (5) z k = ∑ i = 1 N f μ k i θ k H k i x k i + v k (6) μ k i θ k is the weight of the fuzzy linear model, it is calculated as follows: μ k i θ k = μ ¯ k i θ k ∑ j = 1 N f μ ¯ k j θ k , μ ¯ k i θ k = ∏ m = 1 G p A k im θ k m (7) where p A k im θ k m ∈ [ 0 , 1 ] is membership degree of the premise parameter θ k m belonging to model set A k i , m is designed as the Gaussian type function: p A k im θ k m = exp − 1 2 θ k m − τ k i , m σ k i , m 2 (8) where τ k i , m and σ k i , m are the mean and standard deviation of the membership function, respectively. 3. Labeled multi-Bernoulli filter The LMB filter can be interpreted as an effective approximation of the δ-generalized label multi-Bernoulli ( δ-GLMB) filter. It inherits the advantages of multi-Bernoulli filter and δ-GLMB filter. The GLMB random set is a random set of labels with state space X and label space L. Its density function is: π ( X ) = Δ ( X ) ∑ c ∈ C ω ( c ) ( L ( X ) ) p ( c ) X (9) where C is the discrete index set. ∑ L ∈ L ∑ c ∈ C ω ( c ) ( L ) = 1 ∫ p ( c ) ( x , l ) d x = 1 LMB random set is a special GLMB random set, it can be written as: p ( c ) ( x , l ) = p ( l ) ( x ) (10) ω ( c ) ( L ) = ∏ i ∈ L 1 − r ( i ) ∏ l ∈ L 1 L ( l ) r ( l ) 1 − r ( l ) (11) In the Ref.[17], the prediction and update are combined, and only one truncation method is used during the iteration process, which greatly reduces the complexity of the algorithm. Based on this inspiration, we introduce the TSK iterative regression multiple model filtering technology, and propose a labeled multi-Bernoulli target tracking algorithm based on the TSK iterative regression multiple model, which effectively solves the maneuver tracking problem under unknown turning rate. III. The Proposed Algorithm Aiming at the problem of the standard LMB algorithm that only considers a single motion model, a TSK iterative regression multi-model LMB target tracking algorithm is proposed. It is mainly to expand the target state to introduce model and label information, so as to integrate the TSK iterative regression multi-model idea into the LMB algorithm to obtain new prediction and update equations. The block diagram of the TSK iterative regression multi-model LMB target tracking algorithm is shown in Fig.1, which is mainly divided into the following parts: 1) TSK joint prediction and update. Label and model information are introduced to expand the target state, the TSK iterative regression method is used to modify the transition density of the maneuver model, and a new state transition density function is derived to decrease the influence of the wrong maneuver model on the target state prediction. At the same time, the joint observation likelihood function of target position and motion feature information is established to increase the discrimination between target and clutter, thereby enhancing the clutter suppression ability. Then we use the TSK modeling method to achieve state prediction and update, the update process includes TSK fuzzy model parameter identification, model probability update and fusion. For the identification of model parameters, please refer to the Refs.[26,27], the Ref.[26] used the extended forgetting factor least square method (EFFLSM) to estimate the consequent parameter, and Wang et al.[27] used the fuzzy expectation maximization (EM) method to identify the premise parameter; 2) Gibbs sampling; 3) Track management. Fig. 1Open in figure viewerPowerPoint Block diagram of TSK iterative regression multi-model LMB target tracking algorithm 1. TSK iterative regression fuzzy model LMB algorithm 1. Prediction Suppose the target state is m = ( x , l , n ) , x is the initial state of the target, l is the label of the target, and n is the motion model. The expanded LMB distribution is: π = r ( l ) , p ( l ) ( x , n ) l ∈ L (12) where p ( l ) ( x , n ) = p ( l ) ( x ∣ n ) p ( l ) ( n ) , p ( l ) ( n ) is the probability of the trajectory l in the model n, and p ( l ) ( x ∣ n ) is the spatial distribution function. The predicted LMB random set of multi-model LMB filter is expressed as follows: π k + 1 ∣ k = r j , k + 1 ∣ k ( l ) , p j , k + 1 k ( l ) l ∈ L k + 1 ∣ k = r j , k + 1 , S ( l ) , p j , k + 1 , S ( l ) x k + 1 , n k + 1 l ∈ L k ∪ r B ( l ) , p B ( l ) x k + 1 , n k + 1 l ∈ B k + 1 (13) where r k + 1 , S ( l ) = η S ( l ) r ( l ) p k + 1 , S ( l ) x k + 1 , n k + 1 = ∫ p S ( l ) ( x ) f m k + 1 ∣ m k p ( l ) ( x , n ) d ( x , n ) η S ( l ) η S ( l ) = ∫ p S ( l ) ( x ) p ( l ) ( x , n ) d ( x , n ) The state transition equation containing model information is expressed as follows: f m k + 1 ∣ m k = p n k + 1 ∣ n k f k + 1 ∣ k l x k + 1 ∣ x k , n k + 1 (14) where p n k + 1 ∣ n k is the model transition probability function, f k + 1 ∣ k l x k + 1 ∣ x k , n k + 1 is the state transition function, p S ( l ) ( · ) is the survival probability of the target. 2. Update The predicted LMB random set is expressed as follows: π k + 1 ∣ k = r j , k + 1 k ( l ) , p j , k + 1 ∣ k ( l ) x k + 1 , n k + 1 l ∈ L k + 1 ∣ k (15) Thus the posterior probability LMB random set is expressed as follows: π ( · ∣ Z ) = r ( l ) , p ( l ) ( · , n ) l ∈ L k + 1 (16) r ( l ) = ∑ I k + 1 , θ k + 1 ∈ F L k + 1 × Ξ I k + 1 w I k + 1 , θ k + 1 Z k + 1 1 I k + 1 ( l ) (17) p ( l ) ( · , n ) = 1 r ( l ) ∑ I k + 1 , θ k + 1 ∈ F L k + 1 × Ξ I k + 1 w I k + 1 , θ k + 1 × Z k + 1 1 I k + 1 ( l ) p l , θ k + 1 ( · , n ) (18) where w I k + 1 , θ k + 1 ( · ) is the weight function of the updated part. Similar to the traditional multi-model filtering algorithm, the TSK iterative regression model LMB target tracking algorithm also adaptively predicts and updates the state of the targets. 2. Gaussian mixture implementation of TSK iterative regression LMB method 1. TSK iterative regression multi-model joint prediction and update a. Prediction The state and covariance of the target in the rule are estimated as follows, where F j , k i is the state transition matrix, Q j , k i is the process noise covariance: m j , k + 1 ∣ k i = F j , k i m j , k i , i = 1 , 2 , … , N f (19) P j , k + 1 ∣ k i = F j , k i P j , k i F j , k i + Q j , k i (20) Suppose the state space of targets at the time k is L k , and the predicted state set X k + 1 ∣ k is composed of the survival targets and the newborn targets, then the predicted label space is L k + 1 ∣ k = L k ∪ B k + 1 , L k ∩ B k + 1 ≠ Φ . If the probability parameter and spatial distribution of the LMB random set at time k are r j , k ( l ) and p j , k ( l ) , respectively, then its posterior density is: π k = r j , k ( l ) , p j , k ( l ) x k , n k l ∈ L k (21) Thus the predicted posterior density is π k + 1 ∣ k = r j , k + 1 ∣ k ( l ) , p j , k + 1 ∣ k ( l ) l ∈ L k + 1 ∣ k = r j , k + 1 , S ( l ) , p j , k + 1 , S ( l ) l ∈ L k ∪ r B ( l ) , p B ( l ) l ∈ B k + 1 (22) where r j , k + 1 , S ( l ) = η j , S ( l ) r j , k ( l ) η j , S ( l ) = p S ( l ) , p j , k ( l ) f x j , k + 1 ∣ x j , k , l = N x j , k + 1 ; F i , k j , Q i , k j p j , k + 1 ∣ k ( l ) = p S ( l ) f x j , k + 1 ∣ · , p j , k ( l ) η j , S ( l ) For the label I k + 1 ∈ L ( X ) at the time k + 1 , the label L ( X ) is the label set of the random set X. Then the weight of the predicted LMB is: w j , k + 1 ∣ k I k + 1 = w j , k + 1 , B I k + 1 ∩ B k + 1 w j , k + 1 , S I k + 1 ∩ L k (23) where w j , k + 1 , B ( L ) = ∏ v ∈ B k + 1 1 − r B ( ν ) ∏ l ∈ L 1 B k + 1 r B ( l ) 1 − r B ( l ) w j , k + 1 , S ( L ) = ∏ v ∈ L k 1 − r j , k + 1 , S ( v ) ∏ l ∈ L 1 L k r j , k + 1 , S ( l ) 1 − r j , k + 1 , S ( l ) b. Update The updated state and covariance are: m j , k + 1 = ∑ i = 1 N f μ j , k + 1 i m ^ j , k + 1 i (24) P j , k + 1 = ∑ i = 1 N f μ j , k + 1 i P j , k + 1 i + ( m j , k + 1 − m ^ j , k + 1 i ) ( m j , k + 1 − m ^ j , k + 1 i ) T (25) m ^ j , k + 1 i , P j , k + 1 i , μ j , k + 1 i are the state estimation, covariance estimation and rule weight of TSK rule i. μ j , k + 1 i is calculated by the entropy regression clustering method via the fuzzy EM, and the specific process is referred to Ref.[27]. m ^ j , k + 1 i , P j , k + 1 i are calculated by the EFFLSM, the implementation process is as follows: m ^ i , k + 1 j = F j , k i m j , k + 1 ∣ k i + K j , k + 1 i ( z j , k + 1 i − H j , k + 1 i F j , k i m j , k + 1 ∣ k i ) (26) K j , k + 1 i = P j , k + 1 ∣ k i ( F j , k i ) T ( H j , k + 1 i ) T ( λ I + H j , k + 1 i F j , k i P j , k + 1 ∣ k i ( F j , k i ) T ( H j , k + 1 i ) T ) − 1 (27) P j , k + 1 i = λ − 1 F j , k i T I − K j , k + 1 i H j , k + 1 i F j , k i P j , k + 1 ∣ k i F j , k i T (28) λ = κ · T · v 2 · σ v 2 ( k ) σ 2 ( k ) (29) In Eq.(29), T , v are time interval and residual. Now, we use the linear matrix inequalities (LMI) to analyze the stability of the proposed algorithm. Problem: Consider the system constituted by Eqs.(1), (2) and (26). Establish an LMI condition for overall system be asymptotically stable. Theorem 1[30,31]: Consider the system constituted by Eqs.(3), (4) and (33), if there exist symmetric positive definite matrix P such that the LMI − P ( F − K H F ) T P P ( F − K H F ) − P is negative definite, then the system is asymptotically stable. Proof: let e k be e k = m k − m ^ k = F k − 1 m k − 1 − F k − 1 m ^ k − 1 + K k z k − H k F k − 1 m ^ k − 1 = F k − 1 − K k H k F k − 1 m k − 1 − F k − 1 − K k H k F k − 1 x ^ k − 1 = F k − 1 − K k H k F k − 1 e k − 1 Let the Lyapunov function candidate representing the energy function V e k − 1 V e k − 1 = e k − 1 T P e k − 1 , P = P T > 0 Next we calculate the variation of energy, which is defined as Δ V = V e k − V e k − 1 < 0 Thus we have e k − 1 T F k − 1 − K k H k F k − 1 T P F k − 1 − K k H k F k − 1 − P e k − 1 < 0 Hence F k − 1 − K k H k F k − 1 T P F k − 1 − K k H k F k − 1 − P < 0 Thus by applying the Schur-Lemma complement, the LMI − P ( F − K H F ) T P P ( F − K H F ) − P is negative definite, then the system is asymptotically stable. Schur-Lemma: The Hermitian block matrix Q S S T R is negative definite, If and only if Q < 0 , and, R − S T Q − 1 S < 0 . If and only if R < 0 , and, Q − S T R − 1 S < 0 . Suppose that Θ is a one-to-one mapping set, in which the mapping relationship is θ k + 1 : L k + 1 → 0 , 1 , … , Z k + 1 . For example, if θ k + 1 ( e ) = θ k + 1 e T > 0 , then e = e T . The mapping subset of I k + 1 is Θ I k + 1 , the finite subset of X is F ( X ) , the historical space of association matching is expressed as a discrete space Ξ, and the posterior distribution of the updated LMB is: π k + 1 ( X ) = r j , k + 1 ( l ) , p j , k + 1 ( l ) l ∈ L k + 1 (30) where r j , k + 1 ( l ) = ∑ I k + 1 , θ k + 1 ∈ F L k + 1 × Ξ I k + 1 I k + 1 , θ k + 1 Z k + 1 1 I k + 1 ( l ) p j , k + 1 ( l ) = 1 r j , k + 1 ( l ) ∑ I k + 1 , θ k + 1 ∈ F L k + 1 × Ξ I k + 1 I k + 1 , θ k + 1 Z k + 1 1 I k + 1 ( l ) p j , k + 1 l , θ k + 1 w j , k + 1 I k + 1 , θ k + 1 ∝ w j , k + 1 ∣ k I k + 1 η j , Z θ i + 1 τ k + 1 p j , k + 1 l , θ k + 1 = p j , k + 1 ∣ k ( l ) ψ j , Z l , θ k + 1 η j , Z ( l , θ k + 1 ) η j , Z l , θ k + 1 = p j , k + 1 ∣ k ( l ) , ψ j , Z l , θ k + 1 ψ j , Z ( l , θ k + 1 ) = p D ( l ) m j , k + 1 g j , k + 1 ( l ) Z θ k + 1 ( l ) ∣ m j , k + 1 κ z θ k + 1 ( l ) , if θ k + 1 ( l ) > 0 ; q D ( l ) m j , k + 1 , if θ k + 1 ( l ) = 0 g j , k + 1 ( l ) z θ k + 1 ( l ) ∣ m j , k + 1 = N z j , k + 1 ; H j , k + 1 i m j , k + 1 k i , λ I + H j , k + 1 i F j , k i P j , k + 1 k i F j , k i T H j , k + 1 i T ) where p D ( l ) is the detection probability of the trajectory L, p D ( l ) = 1 − p D ( l ) is the probability of missed detection, κ ( · ) is the Poison noise density function, g j , k + 1 ( l ) ( z ∣ m ) is the likelihood probability density function; the inclusion function 1 I k + 1 ( · ) refers to the definition in Ref.[16]. 2. Gibbs sampling Verification shows that the L 1 error between the original δ-GLMB density and the truncated δ-GLMB density can be minimized. This shows that the target associated with the observation is mainly determined by the hypothetical event, which has a higher probability. To avoid calculating all hypothetical events, it is only necessary to find hypothetical events whose probability is greater than the threshold, and form a new hypothetical event space instead of the original hypothetical event space, ignoring hypothetical events with a lower probability, which reduces computational complexity. The Gibbs sampling method adopted in Ref.[17] has low complexity and can map high-dimensional to simple low-dimensional problem. A detailed introduction is described in Ref.[17]. Here is a brief introduction to the implementation of this method in the paper. The hypothetical target at time k is ( ξ , I ) , the observation at the time k + 1 is Z k + 1 = z 1 : M , the label of the surviving target is I = l 1 : R , and the label of the new target is B k + 1 = l R + 1 : P . For all n ∈ { 1 : P } , the existence probability of the trajectory is p S ξ l n ∈ [ 0 , 1 ] , define the following expression: η n ( m ) = 1 − p S ξ l n , 1 ≤ n ≤ R , m < 0 p S ξ l n η j , Z l n , θ k + 1 , 1 ≤ n ≤ R , m ≥ 0 1 − r B l n , R + 1 ≤ n ≤ P , m < 0 r B l n η j , Z l n , θ k + 1 , R + 1 ≤ n ≤ P , m ≥ 0 (31) where m ∈ { − 1 : M } is the index associated with the observation and the track label l n . When m = 0 means l n misse