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Article

A Robust Trajectory Multi-Bernoulli Filter for Superpositional Sensors

by
Huaguo Zhang
1,*,
Wenting Luo
1,
Xu Zhou
1,2,
Hao Mu
3,
Lin Gao
1 and
Xiaodong Wang
2
1
School of Information and Communication Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China
2
The 29th Research Institute of China Electronics Technology Group Corporation, Chengdu 610036, China
3
National Key Laboratory of Electromagnetic Space Security, Chengdu 610036, China
*
Author to whom correspondence should be addressed.
Electronics 2024, 13(20), 4001; https://doi.org/10.3390/electronics13204001
Submission received: 20 August 2024 / Revised: 4 October 2024 / Accepted: 10 October 2024 / Published: 11 October 2024
Browse Figures
Figure 1
<p>The values of the measurement noise covariance parameters.</p> "> Figure 2
<p>The actual trajectories of multiple radiation sources and the estimated trajectories using the robust VB-TMB algorithm.</p> "> Figure 3
<p>GOSPA error of VB-TMB algorithm and GM-TMB algorithm under different parameters.</p> "> Figure 4
<p>The GOSPA distances for the VB-TMB algorithm under different L-scan lengths.</p> "> Figure 5
<p>Covariance estimation of the VB-TMB algorithm under low noise time-varying rates.</p> "> Figure 6
<p>Covariance estimation of the VB-TMB algorithm under high noise time-varying rates.</p> "> Figure 7
<p>GOSPA error of VB-HMB-CPHD algorithm, VB-TPHD algorithm, and VB-TMB algorithm.</p> "> Figure 8
<p>Covariance estimation errors of VB-HMB-CPHD algorithm, VB-TPHD algorithm, and VB-TMB algorithm.</p> ">
Versions Notes

Abstract

:
This paper proposes a trajectory multi-Bernoulli filter applied to the superpositional sensor model for multi-target tracking in the presence of unknown measurement noise. This filter can provide a Multi-Bernoulli approximation of the posterior density on a set of alive trajectories at the current time step. We also provide a Gaussian mixture (GM) implementation of this filter, employing a mixture of Gaussian and inverse Wishart distributions to represent the combined state of measurement noise and target information. Subsequently, the variational Bayesian (VB) method is employed to approximate the posterior distribution, ensuring its form remains consistent with the prior distribution. This method is capable of directly generating trajectory estimates and can jointly estimate both multi-object tracking and measurement noise covariance. The performance of this algorithm is verified through simulation. Finally, a computationally more efficient L-scan approximation is provided. The simulation results indicate that the filter can achieve robust tracking performance, adapting to unknown measurement noise.

1. Introduction

The problem of multi-target tracking (MTT) [1] was initially generated by military applications. In recent years, with the rapid development of sensor technology, signal detection and processing technology, and computer science, multi-target tracking algorithms have also been widely applied in different fields, such as ground missile defense systems, battlefield monitoring, search and recognition, traffic control, machine vision, and biomedicine.
The MTT problem is generally described as a modeling and estimation problem for stochastic systems [2], which involves estimating and determining the number, motion state, or trajectory of targets in the monitoring area at the current time based on the measurement information and prior information received by sensors. However, the appearance and disappearance of targets are highly random, and their mobility makes it difficult for a single deterministic model to describe their state. In addition, sensors are affected by their own performance and environmental factors, resulting in false alarms, missed detections, clutter interference information, etc., which can lead to uncertainty in multi-target measurement. These issues will make it difficult to track multiple targets in practical applications.
The state and measurement of multiple targets are modeled as a random finite set (RFS) [3]. Probability Hypothesis Density (PHD) filtering is a method for multi-target tracking based on RFS theory. The biggest advantage of PHD algorithm is that it simplifies the data association problem [4] and can simultaneously estimate the state and number of multiple targets, which is suitable for solving the various complex and uncertain problems mentioned above. Among them, the MB filtering algorithm approximates the probability density of multiple targets directly by combining multiple Bernoulli processes [5], making the solution and application of multi-objective state estimation problems more intuitive. Compared with PHD filtering [6], this method does not require additional clustering algorithms [7] to extract multi-objective states and can achieve more stable estimation performance, bringing great convenience to multi-target tracking.
The multi-target RFS filter based on superpositional sensors has the same prediction step as the classical multi-target RFS filter based on superpositional sensors, both modeling the multi-target state as RFS. However, the update steps of the two filters are different. The former models the measurement as a random vector, while the latter models the measurement as RFS. Mahler derived the precise filtering equation for the general superposition CPHD [8], but this equation is difficult to implement due to its computational complexity. Hauschildt proposed an approximate closed-form solution for this problem, but it is only applicable to a small number of dense target situations [9]. Thouin, Nannullil, and Coates proposed the additive likelihood moment filter for recursive superposition PHD filters in 2011 [10]. This approximation method is effective, but the observation update equation is only applicable to discrete state spaces. Mahler extended the approximate superposition CPHD filter based on this, and then summarized the filter and its special case (i.e., PHD filter) with Nannum and Coates, and provided computable auxiliary particle implementation forms for approximate superposition PHD and superposition CPHD [11].
This paper proposes a trajectory multi-Bernoulli filter applied to superpositional sensor models for multi-target tracking in the presence of unknown measurement noise. The rationale of exploiting the trajectory multi-Bernoulli filter is that this method can reduce computational complexity compared to other methods. The variational Bayes method is exploited for estimating the measurement noise as well as target states. The robust tracking performance of the proposed algorithm versus existing algorithms in the aspects of both measurement noise parameters and multi-target tracking is evaluated, showing the superiority of the proposed algorithm.
The rest of this paper is arranged as follows. In Section 2, the trajectory set, superpositional sensor model and random finite set of multi-Bernoulli trajectories are introduced. Then, the prediction and update process of the trajectory multi-Bernoulli filter of the superpositional sensor model is introduced in Section 3. In Section 4, the Gaussian mixture (GM) implementation of the filter is also provided. Subsequently, the variational Bayesian (VB) method is used to approximate the posterior distribution and jointly estimate the noise covariance of multi-target tracking and measurement in Section 5. In Section 6, we evaluate the performance of our method via several simulations. Finally, we draw a conclusion in Section 7.

2. Problem Formulation

2.1. Trajectory Set

This section presents a summary of the theory presented in [12]. For a single target, its state (such as position and velocity) can be represented by x ∈ R n x , where n x denotes the dimensionality of the state space. The corresponding multi-target state can be represented by x ∈ F ( R n x ) , where F ( R n x ) denotes the set of all finite subsets of R n x . Here, our primary interest lies in the trajectory of the target state, which can be represented as a finite sequence of the target’s states from the initial to the final time. In this context, we use the variable X = ( t , x 1 : i ) to represent a trajectory that originates at time t, while x 1 : i represents a continuous sequence of target states with a length of i over time. If we consider all trajectories X = ( t , x 1 : i ) (including both alive and terminated trajectories) until time k, their variables should belong to the set I ( k ) = { ( t , i ) : 0 ≤ t ≤ k & t + i − 1 ≤ k } . We define the single-trajectory space until time k as T ( k ) . When considering only alive trajectories, their variables should belong to the set S ( k ) = { ( t , i ) : 0 ≤ t ≤ k & t + i − 1 = k } . The multi-trajectory state can be represented by a random finite set X k ∈ F ( T ( k ) ) .
For a function π ( · ) defined on the single-trajectory space T ( k ) , its integral can be computed as
∫ π ( X ) d X = ∑ ( t , i ) ∈ I ( k ) ∫ π t , x 1 : i d x 1 : i .
If a function π ( · ) is defined on the space of trajectory sets F ( T ( k ) ) , its set integral can be calculated as
∫ π ( X ) δ X = ∑ n = 0 ∞ 1 n ! ∫ π X 1 , … , X n d X 1 : n = ∑ n = 0 ∞ 1 n ! ∑ t 1 , i 1 ∈ I k ′ … ∑ t n , i n ∈ I k ′ ∫ … ∫ π ( { ( t 1 , x 1 1 : i 1 ) , … , ( t n , x n 1 : i n ) } ) d x 1 1 : i 1 … d x n 1 : i n .

2.2. Superpositional Sensor Model

The measurement model used in this paper is the superpositional sensor (SP) model [13], where the measurement value at each time step is determined as the sum of measurement values from all alive trajectories. The measurement form at time k is given by
Z k = r ( X ) + v k = ∑ X ∈ X k h k ( X ) + v k ,
where
h k ( X ) = h k ( t , x ( 1 : i ) ) = h k ( x i ) δ k t + i − 1 ,
and v k is the measured white noise with covariance R k . The likelihood function in this case can be expressed as:
L ( Z k | X k ) = N ( Z k ; r ( X k ) , R k ) ,
where N ( · ; m , P ) denotes a Gaussian density [14] with mean m and covariance P.

2.3. Random Finite Set of Multi-Bernoulli Trajectories

In this paper, multi-trajectory states are modeled as a multi-Bernoulli random finite set. The multiple-trajectory Bernoulli RFS can be seen as a union of a finite number of trajectory Bernoulli RFSs, and the distribution of the trajectory multi-Bernoulli [15] RFS can be written as
f X = ∑ X 1 ⊎ … ⊎ X n = X ∏ l = 1 n f l X l ,
where ⊎ stands for the union of mutually disjoint sets and
f l X l = 1 − r l X l = ∅ r l p l ( X ) X l = { X } 0 otherwise ,
where r l is the probability of existence and p l ( · ) is the single-trajectory density function.

3. Trajectory Multi-Bernoulli Filters for Superpositional Sensors

3.1. Prediction Step

As the influence of alive trajectories is much greater than that of dead trajectories during the filtering process, this paper mainly considers a multi-Bernoulli filter that is designed for tracking only alive trajectories [16].
Let us assume that the alive multi-trajectory state at time k − 1 is a trajectory multi-Bernoulli RFS of the form
f k − 1 X k − 1 = ∑ X 1 ⊎ … ⊎ X n k − 1 = X k − 1 ∏ l = 1 n k − 1 f k − 1 l X l ,
where the density of the l-th Bernoulli component is
f k − 1 l X l = 1 − r k − 1 l X l = ∅ r k − 1 l p k − 1 l ( X ) X l = { X } 0 otherwise ,
where X = ( t , x ( 1 : i ) ) and t + i − 1 = k − 1 .
Given the set of alive trajectories sets X k − 1 at time step k − 1 , each trajectory ( t , x 1 : i ) ∈ X k − 1 , where t + i − 1 = k − 1 , either survives with probability p s ( t , x 1 : i ) = p s ( x i ) or dies with probability 1 − p s ( t , x 1 : i ) at time step k, where the transition density is
g ( t ´ , x ´ 1 : i ´ ∣ t , x 1 : i ) = δ t [ t ´ ] δ i + 1 [ i ´ ] δ x 1 : i ( x ´ 1 : i ´ − 1 ) × g ( x ´ i ´ ∣ x i ) .
The set X k is the union of the set of surviving trajectories and the set of newly born trajectories, where the newly born trajectories are independent of each other and have density given by
f k b X k = ∑ X 1 ⊎ … ⊎ X n k b = X k ∏ l = 1 n k b f k b , l X l ,
where
f k b , l X l = 1 − r k b , l X l = ∅ r k b , l p k b , l ( x ) X l = { ( k , x 1 ) } 0 otherwise .
Based on the above assumption, the predicted density at time k is trajectory MB
f k | k − 1 X k | k − 1 = ∑ X 1 ⊎ … ⊎ X n k | k − 1 = X k | k − 1 ∏ l = 1 n k | k − 1 f k | k − 1 l X l ,
where n k | k − 1 = n k − 1 + n k b , and for previously existing Bernoulli components ( 1 ≤ l ≤ n k − 1 ), the Bernoulli parameters are
r k | k − 1 l = r k − 1 l · ∫ p k − 1 l ( X ) · p s ( X ) d X = r k − 1 l · ∫ p k − 1 l ( t , x 1 : i ) · p s ( x i ) d x 1 : i
and
p k | k − 1 l ( X ) = ∫ g k X | X ′ p s X ′ p k − 1 l X ′ d X ′ ∫ p k − 1 l ( X ) · p s ( X ) d X = ∫ g k t , x 1 : i | t ˙ , x ˙ 1 : i ˙ p s x ˙ i ˙ p k − 1 l t ˙ , x ˙ 1 : i ˙ d x ˙ 1 : i ˙ ∫ p k − 1 l ( t , x 1 : i ) · p s ( x i ) d x 1 : i
and for newly born Bernoulli components ( n k − 1 + 1 ≤ l ≤ n k | k − 1 ), the densities are
f k − 1 | k l X l = f k b , l − n k − 1 X l − n k − 1

3.2. Update Step

The PHD of (13) can be shown as [17]
D k | k − 1 ( X ) = ∫ f k | k − 1 ( { X } ∪ W ) δ W = ∑ l = 1 n k | k − 1 r k | k − 1 l · p k | k − 1 l ( t , x 1 : i )
where t + i − 1 = k .
We introduce the conditional PHD as defined in [18] and denote the conditional PHD of the l-th Bernoulli trajectory as D k | k − 1 l ( X ) . The symbol ( X ← l ) is used to represent this event, and
D k | k − 1 l ( X ) = ∫ f k | k − 1 ( { X } ∪ W | X ← l ) δ W = ∑ n = 0 ∞ 1 n ! ∫ f k | k − 1 X , Y 1 , ⋯ Y n ∣ X ← i d Y 1 : n = r k | k − 1 l · p k | k − 1 l ( t , x 1 : i )
Suppose that the conditional PHD of the l-th Bernoulli trial is updated to
D k | k l ( X ) = r k | k l · p k | k l ( t , x 1 : i ) .
By applying the definition of PHD and using the Bayesian update formula, we can obtain
r k | k l · p k | k l ( X ) = ∫ f k ∣ k ( { X } ∪ W ∣ X ← l ) δ W = ∫ f k Z k ∣ { X } ∪ W , X ← i f k ∣ k − 1 ( { X } ∪ W ∣ X ← l ) δ W f k Z k ∣ Z [ k − 1 ] , X ← l = r k | k − 1 l · p k | k − 1 l ( X ) ∫ f k Z k ∣ { X } ∪ W f k ∣ k − 1 l ( W ) δ W f k Z k ∣ Z [ k − 1 ] , X ← l
where f k ∣ k − 1 l ( W ) is defined as
f k ∣ k − 1 l ( W ) = f k ∣ k − 1 ( { X } ∪ W ∣ X ← l ) r k | k − 1 l · p k | k − 1 l ( X )
and
f k Z k ∣ Z [ k − 1 ] , X ← l = ∫ f k Z k ∣ W f k ∣ k − 1 ( W ) δ W .
After rearranging, we can obtain
r k | k l · p k | k l ( X ) = r k | k − 1 l · p k | k − 1 l ( X ) × ∫ L k Z k | { X } ∪ W l f k | k − 1 l ( W l ) δ W l ∫ L k Z k | W f k | k − 1 ( W ) δ W
The likelihood function here is obtained from Equation (5), and under the SP model, Equation (23) can be written as
r k | k l ·   p k | k l ( X ) = r k | k − 1 l · p k | k − 1 l ( X ) × ∫ N ( Z k ; h ( X ) + r ( W l ) , R k ) f k | k − 1 l ( W l ) δ W l ∫ N ( Z k ; r ( W ) , R k ) f k | k − 1 ( W ) δ W .
By introducing the transformations y l = r ( W l ) and y = r ( W ) , using the formula for the transformation of variables, we obtain
r k | k l ·   p k | k l ( X ) = r k | k − 1 l · p k | k − 1 l ( X ) × ∫ N ( Z k ; h ( X ) + y l , R k ) Q k | k − 1 l ( y l ) d y l ∫ N ( Z k ; y , R k ) Q k | k − 1 ( y ) d y .
where Q k | k − 1 l ( y l ) and Q k | k − 1 ( y ) are probability distributions of random vectors y l and y, respectively. Assuming Gaussian distributions for Q k | k − 1 l ( y l ) and Q k | k − 1 ( y ) , the updated conditional PHD can be approximated as follows:
r k | k l ·   p k | k l ( X ) = r k | k − 1 l · p k | k − 1 l ( X ) × ∫ N ( Z k ; h ( X ) + y l , R k ) N ( y l ; σ k l , Σ k l ) d y l ∫ N ( Z k ; y , R k ) N ( y ; σ k , Σ k ) d y = r k | k − 1 l p k | k − 1 l ( X ) · N ( Z k ; h ( X ) + σ k l , R k + Σ k l ) N ( Z k ; σ k , R k + Σ k )
where σ k l and Σ k l are the mean and covariance matrix of the distribution Q k | k − 1 l ( y l ) , and σ k and Σ k are the mean and covariance matrix of the distribution Q k | k − 1 ( y ) . The values can be obtained using Campbell’s theorem [19] as follows
σ k = ∑ l = 1 L k | k − 1 r k | k − 1 l s k | k − 1 l , σ k l = σ k − r k | k − 1 l s k | k − 1 l , Σ k = ∑ l = 1 L k | k − 1 ( r k | k − 1 l v k | k − 1 l − ( r k | k − 1 l ) 2 s k | k − 1 l ( s k | k − 1 l ) ⊤ ) , Σ k l = Σ k − ( r k | k − 1 l v k | k − 1 l − ( r k | k − 1 l ) 2 s k | k − 1 l ( s k | k − 1 l ) ⊤ ) , s k l = ∫ p k | k − 1 l ( X ) · h k ( X ) d X = ∫ p k | k − 1 l ( t , x 1 : i ) · h k ( x i ) δ k t + i − 1 d x 1 : i , v k l = ∫ p k | k − 1 l ( X ) · h k ( X ) h k ( X ) ⊤ d X = ∫ p k | k − 1 l ( t , x 1 : i ) · h k ( x i ) h k ( x i ) ⊤ δ k t + i − 1 d x 1 : i

4. GM Implementations for the Trajectory Multi-Bernoulli Filters

This section completes the Gaussian implementation of the trajectory multi-Bernoulli filter for the superposition measurement model.

4.1. Prediction Step

We can use a Gaussian mixture model to represent the probability density function of a single trajectory
p ( X ) = ∑ j = 1 J α j · N ( t , x 1 : i ; t ′ , m j , P j ) ,
where
N ( t , x 1 : i ; t ′ , m j , P j ) = N ( x 1 : i ; m j , P j ) δ t ′ [ t ]
and ∑ j = 1 J α j = 1 with m j ∈ R i n x and P j ∈ R i n x × i n x . For the distribution of a single trajectory, its initial time t ′ remains unchanged and is a constant parameter that is tied to the trajectory. If the dimensions of x 1 : i are different from those of m j , the output is zero.
The assumptions used in this section are as follows. The survival probability of the Bernoulli trial is a constant parameter independent of the state p s ( X ) = p s . The single-target transition function g ( x ´ i ´ | x i ) in the trajectory transition density (10) is equal to N ( x ´ i ´ , F x i , Q ) .
We assume that the alive l-th Bernoulli trajectory SPDF in (9) are
p k − 1 l ( X ) = ∑ j = 1 J l α j , k − 1 l · N ( t , x 1 : i ; t l , m j , k − 1 l , P j , k − 1 l )
where t l + d i m ( m j , k − 1 l ) / n x − 1 = k − 1 . We also assume that the newly born l-th Bernoulli trajectory SPDF in (13) is given by
p k b , l ( X ) = ∑ j = 1 J b , l α j , k b , l · N ( t , x 1 : i ; t b , l , m j , k b , l , P j , k b , l )
where t b , l = k and d i m ( m j , k b , l ) / n x = 1 . From the previous section, we can infer the calculation of the Bernoulli parameter in Equation (13). We also assume that the PHD in Formula (13) can be written in the following form:
D k | k − 1 ( X ) = ∑ l = 1 L k | k − 1 r k | k − 1 l · p k | k − 1 l ( t , x 1 : i )   = ∑ l = 1 L k | k − 1 r k | k − 1 l · ∑ j = 1 J l α j , k | k − 1 l · N ( t , x 1 : i ; t l , m j , k | k − 1 l , P j , k | k − 1 l )
where for previously existing Bernoulli components ( 1 ≤ l ≤ n k − 1 ), and by substituting the distribution (30) into Equations (14) and (15), we obtain
r k | k − 1 l = r k − 1 l · ∫ p k − 1 l ( t , x 1 : i ) · p s d x 1 : i = p s · r k − 1 l
and
∑ j = 1 J l α j , k | k − 1 l · N ( t , x 1 : i ; t l , m j , k | k − 1 l , P j , k | k − 1 l ) = ∫ g k t , x 1 : i | t ˙ , x ˙ 1 : i ˙ p s p k − 1 l t ˙ , x ˙ 1 : i ˙ d x ˙ 1 : i ˙ ∫ p k − 1 l ( t , x 1 : i ) · p s d x 1 : i
where
α j , k | k − 1 l = α j , k − 1 l , m j , k | k − 1 l = ( m j , k − 1 l ) ⊤ ( F ˙ k − 1 ( j , l ) m j , k − 1 l ) ⊤ ] ⊤ P j , k | k − 1 l = P j , k − 1 l P j , k − 1 l ( F ˙ k − 1 ( j , l ) ) ⊤ F ˙ k − 1 ( j , l ) P j , k − 1 l F ˙ k − 1 ( j , l ) P j , k − 1 l ( F ˙ k − 1 ( j , l ) ) ⊤ + Q k F ˙ k − 1 ( j , l ) = 0 1 , i j , k − 1 l − 1 , 1 ⊗ F k − 1 .
For the newly born Bernoulli component ( n k − 1 + 1 ≤ l ≤ n k | k − 1 ), its parameter is given by Equation (31).

4.2. Update Step

The function h k ( x i ) in Equation (4) is nonlinear, so when updating the Gaussian mixture parameters, the first step is to linearize it using a Taylor expansion [20].
h k ( x i ) ≅ h k ( x 0 ) + H k ( x 0 ) ( x i − x 0 ) ,
where H k denotes the Jacobian matrix [21] defined as
H k ( x 0 ) = ∂ h k ( x i ) ∂ ζ , ∂ h k ( x i ) ∂ θ , ∂ h k ( x i ) ∂ ζ ˙ , ∂ h k ( x i ) ∂ θ ˙ | x i = x 0 .
The conditional PHD of the l-th Bernoulli trajectory in Formula (18) can be written in the following form:
D k | k − 1 l ( X ) = r k | k − 1 l · p k | k − 1 l ( t , x 1 : i ) = r k | k − 1 l · ∑ j = 1 J l α j , k | k − 1 l · N ( t , x 1 : i ; t l , m j , k | k − 1 l , P j , k | k − 1 l )
Let us assume that the updated conditional PHD of the l-th Bernoulli trajectory is given by:
D k l ( X ) = r k l · ∑ j = 1 J l α j , k l · N ( t , x 1 : i ; t l , m j , k l , P j , k l )
The likelihood generated by the l-th Bernoulli trajectory is given by
l ( Z k | X ˜ ) = N ( Z k ; h k ( x i ) + σ k l , Σ k l + R k ) ≈ N ( Z k − h k ( x 0 ) + H k ( x 0 ) x 0 − σ k l ; H k ( x 0 ) x , Σ k l + R k ) .
Substituting the updated conditional PHD into Equation (26), we obtain
r k l = r k | k − 1 l ∑ j = 1 J k | k − 1 l α ˜ j , k l N ( Z k − σ k ; 0 , Σ k + R k ) , α ˜ j , k l = α j , k | k − 1 l · N ( Z k − h k ( m j , k | k − 1 l ) − σ k l ; 0 , S k l ) , α j , k l = α ˜ j , k l ∑ j = 1 J k | k − 1 l α ˜ j , k l , m j , k l = m j , k | k − 1 l + K k , j l ( Z k − h k ( m j , k | k − 1 l ) − σ k l ) , P j , k l = ( I − K k , j l H ˙ k j , l ( m j , k | k − 1 l ) ) P j , k | k − 1 l , K j , k l = P j , k | k − 1 l ( H ˙ k j , l ) ⊤ ( m j , k | k − 1 l ) ( S k l ) − 1 , S k l = Σ k l + R k + H ˙ k j , l ( m j , k | k − 1 l ) P j , k | k − 1 l ( H ˙ k j , l ) ⊤ ( m j , k | k − 1 l ) , H ˙ k j , l = 0 1 , i j , k | k − 1 l − 1 , 1 ⊗ H k ( m j , k | k − 1 l ) .
where the definitions of parameters σ k , σ k l , Σ k , and Σ k l can be obtained from Equation (27).

5. Robust Trajectory Multi-Bernoulli Filters with Unknown Noise Covariance

In practical scenarios, the covariance matrix of measurement noise is often unknown. Therefore, in this section, we propose extending the state of the trajectory to include the joint representation of the covariance matrix and spatial information. The following hybrid target trajectory state is considered
X ˜ = ( t , x 1 : i , R ) ,
where R represents the noise covariance matrix. In this section, all the assumptions made in the previous section continue to be followed.

5.1. Prediction Step

Assuming that the noise covariance matrix is independent of the position information, the distribution of a single trajectory can be expressed as p ( X ˜ ) = p ( X ) · p ( R ) .
The PHD of the prior multi-Bernoulli hybrid trajectory distribution at time k − 1 is
D k − 1 ( X ˜ ) = ∑ l = 1 L k − 1 r k − 1 l · p k − 1 l ( X ) · p k − 1 l ( R k − 1 )
In this paper, the inverse Wishart (IW) [22] distribution is adopted to model the noise covariance at each time step. Therefore, for the surviving l-th trajectory in Equation (9), its SPDF takes the form of a mixture of Gaussian-inverse Wishart distribution
p k − 1 l ( X ˜ ) = ∑ j = 1 J l α j , k − 1 l · N ( t , x 1 : i ; t l , m j , k − 1 l , P j , k − 1 l ) × IW ( R k − 1 ; u j , k − 1 l , U j , k − 1 l )
where IW ( R ; u , U ) represents the IW distribution with degree of freedom u and scale matrix U. Furthermore, the Gaussian parameters, t l + d i m ( m j , k − 1 l ) / n x − 1 = k − 1 , remain consistent with the Section 4.1. Therefore, further elaboration on this matter is unnecessary. The SPDF of the l-th newly born Bernoulli trajectory is given by the following expression
p k b , l ( X ˜ ) = ∑ j = 1 J b , l α j , k b , l · N ( t , x 1 : i ; t b , l , m j , k b , l , P j , k b , l ) × IW ( R k | k − 1 ; u j , k b , l , U j , k b , l ) .
This assumption, as stated in [23], considers that the noise covariance and the motion of the target spatial position information are mutually independent and do not interfere with each other.
Then, the predicted PHD of the trajectories at time k is
D k | k − 1 ( X ˜ ) = ∑ l = 1 n k | k − 1 r k | k − 1 l · ∑ j = 1 J l α j , k | k − 1 l × N ( t , x 1 : i ; t l , m j , k | k − 1 l , P j , k | k − 1 l ) × IW ( R k | k − 1 ; u j , k | k − 1 l , U j , k | k − 1 l )
The parameters for the targets that survived in the previous time step ( 1 ≤ l ≤ n k − 1 ) can be obtained using the heuristic algorithm [24] and the Gaussian mixture PHD prediction formula
r k | k − 1 l = p s r k − 1 l , α j , k | k − 1 l = α j , k − 1 l , m j , k | k − 1 l = ( m j , k − 1 l ) ⊤ ( F ˙ k − 1 ( j , l ) m j , k − 1 l ) ⊤ ] ⊤ P j , k | k − 1 l = P j , k − 1 l P j , k − 1 l ( F ˙ k − 1 ( j , l ) ) ⊤ F ˙ k − 1 ( j , l ) P j , k − 1 l F ˙ k − 1 ( j , l ) P j , k − 1 l ( F ˙ k − 1 ( j , l ) ) ⊤ + Q k u j , k | k − 1 l = λ u j , k − 1 l , U j , k | k − 1 l = λ ( U j , k − 1 l − M z − 1 ) + M z + 1 . F ˙ k − 1 ( j , l ) = 0 1 , i j , k − 1 l − 1 , 1 ⊗ F k − 1
where λ ∈ ( 0 , 1 ] denotes a forgetting factor. For the newly born Bernoulli component ( n k − 1 + 1 ≤ l ≤ n k | k − 1 ), its parameter is given by Equation (44).

5.2. Update Step

In the preceding prediction step, due to the independence of motion between the trajectory’s spatial state and noise covariance, the resulting distribution after prediction maintains its form unchanged. However, in the update step, since both are coupled together, the posterior form is inevitably altered. To maintain consistency in the form of the trajectory state, it is necessary to approximate the posterior distribution in order to proceed to the next iteration.
The approximation method used here is variational Bayesian approximation, which involves forming an expected distribution that is closest to the true distribution by minimizing the Kullback–Leibler (KL) divergence [25] between the true distribution and the approximate distribution.
The likelihood generated by the l-th Bernoulli trajectory is given by
l ( Z k | t , x 1 : i , R ) = N ( Z k ; h k ( x i ) + σ k l , Σ k l + R ) ≈ N ( Z k − h k ( x 0 ) + H k ( x 0 ) x 0 − σ k l ; H k ( x 0 ) x , Σ k l + R k ) .
According to Formula (26), it can be inferred that the PHD of the updated l-th Bernoulli trajectory is given by
D ˙ k l ( X ˜ ) = r ˙ k | k − 1 l p ˙ k | k − 1 l ( X ˜ ) × N ( Z k − h k ( x 0 ) + H k ( x 0 ) x 0 − σ k l ; H k ( x 0 ) x , Σ k l + R k ) N ( Z k ; σ k , R k + Σ k )
where
p ˙ k | k − 1 l ( X ˜ ) = ∑ j = 1 J l α j , k | k − 1 l · N ( t , x 1 : i ; t l , m j , k | k − 1 l , P j , k | k − 1 l ) × IW ( R k | k − 1 ; u j , k | k − 1 l , U j , k | k − 1 l )
and the parameters for the above equation have already been obtained in the previous steps.
Here, we assume the desired PHD of the l-th Bernoulli trajectory to be obtained as
D ˙ k l ( X ˜ ) = r ˙ k l p ˙ k l ( X ) p ˙ k l ( R k ) = r ˙ k l ∑ j = 1 J l α j , k l · N ( t , x 1 : i ; t j , k l , m j , k l , P j , k l ) · IW ( R k ; u j , k l , U j , k l )
In this case, the VB approach is used to find the parameters of the approximate distribution, specifically to find the best distribution of (51) such that r ˙ k l p ˙ k l ( X ) p ˙ k l ( R k ) ≈ D ˙ k l ( X ˜ ) , while minimizing the Kullback–Leibler divergence described by the following equation
K L [ r ˙ k l p ˙ k l ( X ) p ˙ k l ( R k ) ∥ D ˙ k l ( X ˜ ) | Z k ) ] = ∫ r ˙ k l p ˙ k l ( X ) p ˙ k l ( R k ) log ( r ˙ k l p ˙ k l ( X ) p ˙ k l ( R k ) D ˙ k l ( X ˜ ) | Z k ) ) d X d R
Minimizing the above equation leads to the following equation
r ˙ k l p ˙ k l ( X ) ∝ exp ∫ log p ˙ Z k , X , R ∣ Z k − 1 p ˙ k l ( R k ) d R , p ˙ k l ( R k ) ∝ exp ∫ log p ˙ Z k , X , R ∣ Z k − 1 r ˙ k l p ˙ k l ( X ) d X .
In this case, the parameters in Formula (51) are calculated by performing VB approximation using fixed-point iteration through N iterations. The formula for the n-th iteration is given by
u j , k l , n = u j , k | k − 1 l + 1 , U j , k l , n = U j , k | k − 1 l + Υ j , k l , n R ^ j , k l , n = U j , k l , n / ( u j , k | k − 1 l − M z − 1 ) , S j , k l , n = R ^ j , k l , n + H ˙ j , k l P j , k | k − 1 l ( H ˙ j , k l ) ⊤ , K j , k l , n = P j , k | k − 1 l ( H ˙ j , k l ) ⊤ ( S j , k l , n ) − 1 , m j , k | k l , n = m j , k | k − 1 l + K j , k l , n ( Z k − h k ( m j , k | k − 1 l ) − σ k l ) , P j , k | k l , n = ( I − K j , k l , n H ˙ j , k l ) P j , k | k − 1 l , H ˙ j , k l = 0 1 , i j , k | k − 1 l , 1 ⊗ H k ( m j , k | k − 1 l )
where
Υ j , k l , n = ( Z k − H ˙ j , k l m j , k | k l , n − σ k l ) ( Z k − H ˙ j , k l m j , k | k l , n − σ k l ) ⊤ + H ˙ j , k l P j , k | k l , n ( H ˙ j , k l ) ⊤ .
After iteration, the parameters in Equation (51) can be obtained as follows
u j , k l = u j , k | k − 1 l + 1 , U j , k l = U j , k | k − 1 l + Υ j , k l , N , R ^ j , k l = U j , k l / ( u j , k l − M z − 1 ) − Σ k l , r k l = r k | k − 1 l ∑ j = 1 J k | k − 1 l α ˜ j , k l N ( Z k − σ k ; 0 , Σ k + R ^ j , k l ) , α ˜ j , k l = α j , k | k − 1 l · Q ( Z k l − σ k l ) , α j , k l = α ˜ j , k l ∑ j = 1 J k | k − 1 l α ˜ j , k l ,
and
m j , k | k l = m j , k | k − 1 l + K k , j l ( Z k − h k ( m j , k | k − 1 l ) − σ k l ) , P j , k | k l = ( I − K k , j l H ˙ k j , l ( m j , k | k − 1 l ) ) P j , k | k − 1 l , K j , k l = P j , k | k − 1 l ( H ˙ k j , l ) ⊤ ( m j , k | k − 1 l ) ( S k l ) − 1 , S k l = Σ k l + R ^ j , k l + H ˙ k j , l ( m j , k | k − 1 l ) P j , k | k − 1 l ( H ˙ k j , l ) ⊤ ( m j , k | k − 1 l ) , H ˙ k j , l = 0 1 , i j , k | k − 1 l − 1 , 1 ⊗ H k ( m j , k | k − 1 l ) ,
where the form of the likelihood in this case can be expressed by the following equation
Q ( Z k − σ k l ) = exp { − M z 2 ln π + 1 2 ln | P k | k | − 1 2 ln | P k | k − 1 | − 1 2 tr [ ( P k | k − 1 ) − 1 P k | k ] − 1 2 tr ( ( ( m k | k − m k | k − 1 ) ( m k | k − m k | k − 1 ) ⊤ ) · ( P k | k − 1 ) − 1 ) + ( u k | k − 1 − M z − 1 ) 2 ln | U k | k − 1 | − ( u k | k − M z − 1 ) 2 ln | U k | k | − ln Γ M z ( u k | k − 1 − M z − 1 2 ) + ln Γ M z ( u k | k − M z − 1 2 ) } .
As the number of Bernoulli components in the filter increases over time, it is necessary to perform pruning and merging before executing multi-target state extraction and noise covariance estimation. The pruning method involves removing components with weights below the pruning threshold and then merging the pruned posterior distributions. It is possible to select the Bernoulli components with probabilities greater than a given threshold ( r t ) for further processing. The merging principle prioritizes absorbing components with larger weights. Assuming the weight of the e-th component in Equation (51) is the largest, ( m j , k l − m e , k l ) ⊤ ( P e , k l ) − 1 ( m j , k l − m e , k l ) is compared with the merging threshold to determine whether to absorb the j-th component. After completing the first round of absorption, the next largest weight component is selected for absorption. This iterative process completes the pruning and merging steps of the VB-TMB filter.
For target state estimation and covariance matrix estimation in this case, the selected posterior Bernoulli components can be used. The calculation method for R ^ k is as follows:
R ^ k = ∑ l ≤ L ^ k r k l ∑ j = 1 J k l α j , k l · R ^ j , k l ∑ l ≤ L ^ k r k l
where R ^ j , k l is obtained by (56), and L ^ k is the number of Bernoulli components satisfying the probability condition.

5.3. L-Scan Implementations

In this section, the L-scan method is adopted to efficiently implement Gaussian mixture filtering. As the filtering progresses, not only does the number of Bernoulli components increase, but also the length of each trajectory increases, which introduces computational challenges.
In fact, for the current time step, the measurements only have a significant impact on the estimation of the most recent trajectory states. Therefore, the L-scan implementation approach is used here, where only the joint density of the last L time steps of each component is propagated, while earlier time steps are propagated using independent state densities. This approach has been proven in literature [26] to minimize the Kullback–Leibler divergence (KLD).
Algorithm 1 is the pseudocode for the Gaussian Mixture implementation of the VB-TMB filter.
Algorithm 1: Pseudocode for Gaussian mixture implementation of VB-TMB filter
Electronics 13 04001 i001

6. Simulation

This section conducts simulation experiments on the VB-TMB algorithm, using a centralized intensity superimposition measurement scenario. The simulation employs a log-normal shadowing attenuation model to represent the path loss of radiation source signals.

6.1. Simulation Scene Setup

The method of using intensity overlay measurement information for localization and tracking does not require complex clock synchronization or high transmission bandwidth [27]. It has now gained widespread development and application [28,29]. The model in this paper uses fictitious points near the radiation source as references. There is a mapping relationship between the RSS information and the distance between the sensors and the radiation source.
Assuming that the number of intensity reconnaissance sensors in the space is M, the intensity measurement contribution of the l-th radiation source to the m-th sensor at time k is denoted as n k ( y m , y l 1 : k ) . Here, y m [ ζ , θ ] ⊤ represents the position coordinates of the m-th sensor, and y l 1 : k = [ y l 1 , … , y l k ] denotes the historical state vectors of the l-th radiation source. The specific expression for n ( · ) is as follows:
n k ( y m , y 1 : k l ) = A l − 10 ι l g ( d ( y m , y 1 : k l ) )
where A l represents the received signal strength from the l-th radiation source at a distance of 1 m, ι is the path loss exponent, and d ( y m , y 1 : k l ) denotes the distance from the m-th sensor to all historical positions of the l-th radiation source. In this paper, the difference in received signal strength between two sensors is used as the measurement information. The intensity difference contribution of the l-th radiation source to the m 1 -th and m 2 -th sensors at time k is given by:
n k ( y m 1 , y 1 : k l ) − n k ( y m 2 , y 1 : k l ) = 10 ι l g ( d ( y m 2 , y 1 : k l ) ) − 10 ι l g ( d ( y m 1 , y 1 : k l ) ) = 10 ι l g ( d ( y m 2 , y 1 : k l ) d ( y m 1 , y 1 : k l ) )
Therefore, in this paper, M sensors generate a total of M z = M ( M − 1 ) 2 intensity difference measurements at each time step. The i-th measurement generated by sensor m 1 and sensor m 2 is formulated as follows:
z k i = ∑ l = 1 L k h ( y 1 : k l ) + v k = ∑ l = 1 L k 10 ι l g ( d ( y m 2 , y 1 : k l ) d ( y m 1 , y 1 : k l ) ) + v k
where v k is the additive noise in the measurement model at time k.
In the simulation, the fixed path loss exponent parameter is set to ι = 3 . The reference signal strengths of the radiation sources are specified as [ A 1 , … , A 8 ] = [ 10 , 12 , 14 , 16 , 16 , 14 , 12 , 10 ] in dBm units. The covariance matrix of measurement noise is R k = η ˙ k I M z , where I M z denotes the M z -dimensional identity matrix. It is assumed that the motion of radiation sources within the monitoring area follows a constant velocity model [30].
The form of the motion equation can be written as:
x k = F k x k − 1 + B ω k
where ω k = [ ζ ¨ , θ ¨ ] ⊤ represents the process noise, where ζ ¨ and θ ¨ are Gaussian white noises with zero mean, and variances σ ζ ¨ 2 and σ θ ¨ 2 , respectively [31]. We have
F = 1 0 T 0 0 1 0 T 0 0 1 0 0 0 0 1 , B = T 2 2 0 0 T 2 2 T 0 0 T
where T = 0.25 s denotes the sampling interval.
Noise covariances σ ζ ¨ 2 and σ θ ¨ 2 are set to 10. The number of iterations is 50. The newly born radiation source targets are modeled as an IIDC RFS { ( π ` c = { α i c , m i c , P i c , u i c , U i c } i = 1 L c ) } , where L c = 3 , α i c = 1 / 3 , u i c = 2 + M z + 1 , and U i c = 0.95 I M z , and initialized with the following parameters:
( m 1 c ) ⊤ ( m 2 c ) ⊤ ( m 3 c ) ⊤ = 220 700 0 0 700 230 0 0 730 720 0 0 , P i c = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 .
The non-one values of P i c exist because assuming that each dimension is independent, position and velocity are not related.
The VB-TMB algorithm will be compared with the GM-TMB algorithm, with the number of variational fixed-point iterations J set to 10 and the forgetting factor λ set to 0.7. In the simulation experiments, the GOSPA distance is used as the metric to evaluate algorithm performance. The sensitive indicators of GOSPA include two adjustable parameters, distance sensitivity parameter p = 2 (which can adjust the degree to which the distance error between the fused trajectory and the true trajectory affects the accuracy index; the value is usually set to 1 or 2) and correlation sensitivity parameter c = 40 (which can adjust the impact of the different number of trajectories generated by trajectory correlation operations between real trajectories and fused trajectories on accuracy indicators; the value is usually greater than 0), which can comprehensively reflect the errors of different components in the trajectory fusion algorithm and ultimately obtain a normalized evaluation result [32]. After calculating the GOSPA distance, we take the average of GOSPA at all simulation moments to obtain the trajectory GOSPA distance. The gamma parameter for the GOSPA measurement of trajectory is set to 1, which means that different attributes in the target have the same weight. Finally, the discussion addressed the impact of trajectory correlation length, noise time-varying rate, and noise power on the performance of the L-scan algorithm.

6.2. Simulation Results

In this section, simulations were conducted using time-varying noise parameters, as illustrated in Figure 1. The trajectory length for the L-scan was set to L = 4 , and the tracking performance of VB-TMB is depicted in Figure 2.
Figure 1 shows the measurement noise covariance parameters during sampling times 0 to 10, 10 to 30, and 30 to 50.
Figure 2 is a two-dimensional scene with a size of a square with a side length of 1000 m. There are a total of six radiation sources in the scene, and the actual and estimated trajectories are shown in the figure.
Under unknown time-varying noise parameters, the performance of the proposed TMB algorithm was evaluated and compared with the GM-TMB algorithm using different noise parameter settings. To implement the GM-TMB filtering algorithm with different initial noise parameters η 1 , the noise parameters were set to 10, initial value, and 70, respectively. Monte Carlo simulations were conducted for both algorithms over 100 runs, and the average GOSPA distances are shown in Figure 3. The reason why we use GOSPA for evaluation is that it is a metric for trajectories.
From Figure 3, it can be observed that the GOSPA error of the VB-TMB algorithm is comparable to that of the GM-TMB algorithm with known true measurement noise distribution and significantly better than the GM-TMB algorithm with incorrectly initialized noise parameters. This indicates that the VB-TMB algorithm exhibits strong adaptability to unknown time-varying measurement noise.
From Table 1, we calculate the average GOSPA error within 50 s as the trajectory GOSPA error. Similarly, the conclusion is consistent with Figure 1.
To investigate the impact of trajectory correlation length on the performance of the L-scan algorithm, the lengths of the L-scan were varied as 1, 2, 3, and 4. The results of 100 Monte Carlo simulations using the VB-TMB algorithm are depicted in Figure 4.
From Figure 4, it can be observed that varying the trajectory correlation length in the L-scan algorithm significantly affects the GOSPA distance for tracking. When L = 1 , the algorithm reduces to the standard FS-LMB algorithm. As L increases, the GOSPA error gradually decreases. However, the advantage of increasing L becomes less pronounced when L > 3 .
From Table 2, we calculate the average GOSPA distances within 50 s as the trajectory GOSPA distances. Similarly, the conclusion is consistent with Figure 4.
Further discussion on the noise estimation performance of the proposed algorithm involves varying the noise time-varying rate and noise power range in the VB-TMB algorithm. The results of noise covariance estimation are depicted in Figure 5 and Figure 6.
From Figure 5 and Figure 6, it is evident that the covariance estimation performance of the proposed algorithm deteriorates when the noise power is too high. Moreover, significant noise time-varying rates can lead to covariance estimation lag.
From Figure 7, it can be seen that the GOSPA error of VB-HMB-CPHD algorithm is the largest, while the GOSPA error of VB-TMB algorithm is the smallest. All three robust algorithms can adapt to unknown time-varying measurement noise, and the VB-TMB algorithm has the best performance. Therefore, it can be demonstrated that adopting the approach proposed in this paper can improve tracking performance under the same signal-to-noise ratio.
From Table 3, we calculate the average GOSPA distances within 50 s as the trajectory GOSPA distances. It is evident that all three robust algorithms can adapt to unknown time-varying measurement noise, and the VB-TMB algorithm has the best performance.
From Figure 8, it can be seen that the VB-TMB algorithm has the smallest estimation error and obtains the most accurate noise covariance estimate. Through simulation comparison, we found that the computation time of the method proposed in this paper is 1 s, while the computation time of other methods is 9 s. The proposed method’s computational complexity is significantly lower than that of other methods.

7. Conclusions

This paper investigates the adaptive tracking problem of multiple radiation sources under a trajectory set using superpositioned measurements. Initially, we propose a Gaussian mixture implementation of the TMB filter and subsequently extend this method to address scenarios involving unknown measurement noise. We apply trajectory random finite sets (RFSs) to the superpositioned measurement model, using the Gaussian and inverse Wishart mixture (GIWM) to model the states of mixed trajectories. The algorithm is closed-looped using the variational Bayesian (VB) method. The simulation verifies the adaptive capability of the proposed algorithm and discusses the impact of trajectory correlation length on tracking performance, as well as the effects arising from variations in noise power. However, the method proposed in this paper also has some drawbacks. It might not perform efficiently when the signal-to-noise ratio is particularly low and the radiation source is moving vigorously. In the future, we will optimize our methods to address this drawback.

Author Contributions

Conceptualization, H.Z.; methodology, H.Z. and W.L.; software, H.Z.; validation, H.Z.; formal analysis, H.Z. and W.L.; investigation, W.L.; resources, W.L., X.Z., H.M., L.G. and X.W.; data curation, L.G.; writing—original draft preparation, H.Z.; writing—review and editing, H.Z. and W.L.; supervision, X.Z. and H.M.; project administration, W.L. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

The data presented in this study are available on request from the corresponding author. The data are not publicly available due to restrictions, e.g., privacy or ethical.

Acknowledgments

The authors thank the anonymous reviewers and editor whose valuable comments and suggestions have improved the quality of this paper.

Conflicts of Interest

Author Xu Zhou and Xiaodong Wang were employed by the company The 29th Research Institute of China Electronics Technology Group Corporation. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Figure 1. The values of the measurement noise covariance parameters.
Figure 1. The values of the measurement noise covariance parameters.
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Figure 2. The actual trajectories of multiple radiation sources and the estimated trajectories using the robust VB-TMB algorithm.
Figure 2. The actual trajectories of multiple radiation sources and the estimated trajectories using the robust VB-TMB algorithm.
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Figure 3. GOSPA error of VB-TMB algorithm and GM-TMB algorithm under different parameters.
Figure 3. GOSPA error of VB-TMB algorithm and GM-TMB algorithm under different parameters.
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Figure 4. The GOSPA distances for the VB-TMB algorithm under different L-scan lengths.
Figure 4. The GOSPA distances for the VB-TMB algorithm under different L-scan lengths.
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Figure 5. Covariance estimation of the VB-TMB algorithm under low noise time-varying rates.
Figure 5. Covariance estimation of the VB-TMB algorithm under low noise time-varying rates.
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Figure 6. Covariance estimation of the VB-TMB algorithm under high noise time-varying rates.
Figure 6. Covariance estimation of the VB-TMB algorithm under high noise time-varying rates.
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Figure 7. GOSPA error of VB-HMB-CPHD algorithm, VB-TPHD algorithm, and VB-TMB algorithm.
Figure 7. GOSPA error of VB-HMB-CPHD algorithm, VB-TPHD algorithm, and VB-TMB algorithm.
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Figure 8. Covariance estimation errors of VB-HMB-CPHD algorithm, VB-TPHD algorithm, and VB-TMB algorithm.
Figure 8. Covariance estimation errors of VB-HMB-CPHD algorithm, VB-TPHD algorithm, and VB-TMB algorithm.
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Table 1. The trajectory GOSPA distances for the VB-TMB algorithm and the GM-TMB algorithm under different parameters.
Table 1. The trajectory GOSPA distances for the VB-TMB algorithm and the GM-TMB algorithm under different parameters.
AlgorithmTrajectory GOSPA
GM-TMB ( η 1 = 70)13.75
GM-TMB ( η 1 = η )6.25
GM-TMB ( η 1 = 10)20.04
VB-TMB6.35
Table 2. The trajectory GOSPA distances for the VB-TMB algorithm under different L-scan lengths.
Table 2. The trajectory GOSPA distances for the VB-TMB algorithm under different L-scan lengths.
Algorithm Trajectory GOSPA
VB-TMB (L = 1)9.96
VB-TMB (L = 2)7.94
VB-TMB (L = 3)7.32
VB-TMB(L = 4)7.13
Table 3. The trajectory GOSPA distances for the VB-HMB-CPHD algorithm, the VB-TPHD algorithm, and the VB-TMB algorithm.
Table 3. The trajectory GOSPA distances for the VB-HMB-CPHD algorithm, the VB-TPHD algorithm, and the VB-TMB algorithm.
AlgorithmTrajectory GOSPA
VB-HMB-CPHD13.19
VB-TPHD9.35
VB-TMB6.36
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Zhang, H.; Luo, W.; Zhou, X.; Mu, H.; Gao, L.; Wang, X. A Robust Trajectory Multi-Bernoulli Filter for Superpositional Sensors. Electronics 2024, 13, 4001. https://doi.org/10.3390/electronics13204001

AMA Style

Zhang H, Luo W, Zhou X, Mu H, Gao L, Wang X. A Robust Trajectory Multi-Bernoulli Filter for Superpositional Sensors. Electronics. 2024; 13(20):4001. https://doi.org/10.3390/electronics13204001

Chicago/Turabian Style

Zhang, Huaguo, Wenting Luo, Xu Zhou, Hao Mu, Lin Gao, and Xiaodong Wang. 2024. "A Robust Trajectory Multi-Bernoulli Filter for Superpositional Sensors" Electronics 13, no. 20: 4001. https://doi.org/10.3390/electronics13204001

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