Vardiational Bayesian Hybrid Multi-Bernoulli and CPHD Filters for Superpositional Sensors

Abstract

This paper addresses the problem of multi-target tracking with superpositional sensors, while the covariance matrices of measurement noise are not known. The proposed method is based on the hybrid multi-Bernoulli cardinalized probability hypothesis density (HMB-CPHD) filter, which has been developed for superpositional sensors-based multi-target tracking with known measurement noises. Specifically, we firstly propose the Gaussian mixture (GM) implementation of the HMB-CPHD filter, and then the covariance matrices of measurement noises are augmented into the target state vector, resulting in the Gaussian and inverse Wishart mixture (GIWM) representation of the augmented state. Then the variational Bayesian (VB) method is exploited to approximate the posterior distribution so that it maintains the same form as the prior distribution. A remarkable feature of the proposed method is that it can jointly perform multi-target tracking and measurement noise covariance estimation. The performance of the proposed algorithm is demonstrated via simulations.

1. Introduction

Multi-target tracking (MTT) is a class of algorithms that aim to estimate the number of targets and the state of each target. It is widely used in video surveillance [1], intelligent robots [2,3], and intelligent vehicles [4]. Traditional multi-target tracking algorithms mainly used data association techniques to extract multiple target states, such as joint probabilistic data association (JPDA) [5], joint integrated probabilistic data association (JIPDA) [6] and multiple hypothesis tracking (MHT) [7]. Recently, the random finite set (RFS) theory [8] has been introduced for multi-target filtering. The advantage of adopting RFS based multi-target trackers is that it is able to model target number and state variance into the Bayesian iterations, thus the principled approximation can be explored for alleviating the computational complexity.

In this paper, we are interested in performing multi-target tracking based on the superpositional sensors [9], e.g., direction of arrival (DOA) antenna arrays [10], acoustic amplitude sensors [11], and radio frequency tomographic tracking systems [12]. In [9], CPHD filters were used to track multiple targets under the superpositional sensors model, and particle and Gaussian mixture (GM) implementation were given in [13,14]. Building on this, ref. [15] implemented multi-target tracking under the superpositional sensors model using the multi-Bernoulli (MB) filter, and the particle implementation was given in [16]. The multi-Bernoulli filter models each target individually using a scalar existence probability and a state density function, allowing for a more precise description of the target state compared to the CPHD filter. However, the approximate multi-Bernoulli filter with superpositional sensors produces frequent cardinality errors, so a hybrid multi-Bernoulli CPHD filter (HMB-CPHD) was proposed in [17] to address this issue, where new targets are modeled with IIDC RFS and existing targets are modeled with multi-Bernoulli RFS.

The aforementioned methods were developed based on the assumption that sensor noise statistics have been known a priori which, however, is not always the case. Using incorrect noise statistics can lead to large estimation errors and even filter divergence. In [18], particle filtering is used for tracking under unknown noise, besides the assignment of a weight to every particle, two additional parameters are required to describe the unknown quantity. This method can generate a large amount of computation and low efficiency when the number of particles is large. Another approach is to use the variational Bayesian (VB) method to find an approximate posterior distribution that is the same as the prior form by minimizing the Kullback–Leibler divergence (KID), which has been widely used in [19,20,21,22].

In this paper, the HMB-CPHD filter [17] is extended to perform superpositional sensor-based multi-target tracking under unknown measurement noise covariance. In order to overcome the problem of the high computational cost of particle filtering in [17], a Gaussian Mixture (GM) implementation scheme is proposed for the HMB-CPHD filter. To address the problem of unknown noise, we add the noise coefficient in the target state and assume that the joint distribution of the hybrid state follows a Gaussian and inverse Wishart mixture (GIWM) distribution. In the update step, an approximate posterior distribution is obtained by the VB technique, and the performance of the algorithm is verified by simulation experiments.

The rest of this paper is organized as follows. We presented the problem formulation of the superpositional sensors model and outlined the basic framework of the HMB-CPHD filter in Section 2. The basic idea of the variational Bayesian is also introduced in Section 2. Section 3 presented the Gaussian mixture implementation of the HMB-CPHD filter, provided a solution for the case of unknown measurement noise using state extension and variational inference, and implemented the VB-HMB-CPHD filter. Section 4 compared the performance of the VB-HMB-CPHD filter and the HMB-CPHD filter for multi-target tracking in the radio frequency tomography sensor network model. We provide conclusions in Section 5.

2. Background

2.1. Problem Formulation

The multi-target state is a set of single-target state $x_{k,i}\in R^{n_x}$ and is represented by a random finite set $X_k=\{x_{k,1},\dots x_{k,n_k}\}$, where $n_k\ge 0$ is the number of targets existing at time k. In this paper, it is assumed that all targets move in the same plane, where the state vector x of a single-target is defined as

$$x\stackrel{\mathsf{\Delta }}{=}{[\zeta ,\theta ,\dot{\zeta },\dot{\theta }]}^{\top },$$

(1)

where $(\zeta ,\theta )$ represents the position vector of the target and $(\dot{\zeta },\dot{\theta })$ represents the velocity vector of the target.

In multi-target filtering, a common assumption is that the state follows a Markov process on the state space [23]. The movement of single-target $x\in X$ at time k follows the linear model

$$x_k=F_k{x}_{k-1}+w_k,$$

(2)

where $F_k$ is the transition matrix at time k, and $w_k$ is the additive white noise. Therefore, each target follows a linear Gaussian dynamical model, i.e.,

$$f_{k|k-1}\left(x\right|x_{k-1})=\mathcal{N}(x;F_k{x}_{k-1},Q_k)$$

(3)

where $\mathcal{N}(·;m,P)$ denotes a Gaussian density with mean m and covariance P, and $Q_k$ is the covariance of the process noise $w_k$.

The measurement model chosen in this paper is the superpositional sensors (SPs) model, where the measurement at each time step is determined by the sum of the measurements of all targets. The measurement at time k has the form

$$Z_k=\sum _{x\in X_k}{h}_k\left(x\right)+v_k,$$

(4)

where $h_k\left(x\right)$ is the real-valued measurement function, and $v_k\sim \mathcal{N}(v_k;0,R_k)$ is the measurement white noise with covariance $R_k$. Similar to [24], the likelihood $p_k$ of measurement $Z_k$ at time k can be written as

$$p_k\left(Z_k\right|X_k)=\mathcal{N}(Z_k;\sum _{x\in X_k}{h}_k\left(x\right),R_k).$$

(5)

In the superposition measurement model of this paper, it is also necessary to consider the case where the measurement noise $R_k$ may be unknown.

2.2. Hybrid Multi-Bernoulli/CPHD (HMB-CPHD) Filter for Superpositional Sensors

This section provides an overview of the hybrid multi-Bernoulli cardinalized probability hypothesis density filter, without considering the case where the noise covariance is unknown. There are different assumptions about multi-target states in the random finite set framework. The traditional PHD filter and CPHD filter model the multi-target states using Poisson RFS and independent and identically distributed cluster RFS, respectively [17]. Furthermore, the multi-Bernoulli filter uses a scalar existence probability and a state density function to model individual targets, which allows for a more accurate description of multi-target states.

However, the approximation of the multi-Bernoulli filter with superpositional sensors can lead to frequent cardinality errors and the computational cost of generating a new multi-Bernoulli term is high [17]. The hybrid multiple Bernoulli and CPHD filter, where new targets are modeled as IIDC RFS and existing targets are modeled as multi-Bernoulli RFS, can effectively address these issues.

(1)

Prediction Step

In the prediction step, the posterior distribution of the previous moment is modeled as a multi-Bernoulli RFS. Assume that there are $L_{k-1}$ targets at moment $k-1$ with parameter ${\{r_{k-1}^l,p_{k-1}^l\left(x\right)\}}_{l=1}^{L_{k-1}}$, where $r_{k-1}^l$ and $p_{k-1}^l\left(x\right)$ denote the scalar existence probability and the state density function of the l-th target. The multi-Bernoulli components are predicted using the motion model, and the predicted multi-Bernoulli components are recorded as the surviving target distribution. The parameter of the surviving target are recorded as ${\{r_{k|k-1}^l,p_{k|k-1}^l\left(x\right)\}}_{l=1}^{L_{k|k-1}}$, where $L_{k|k-1}=L_{k-1}$ and

$$\begin{array}{cc}\hfill & r_{k|k-1}^l=r_{k-1}^l\langle p_{k-1}^l,p_s\rangle ,\hfill \\ \hfill & p_{k|k-1}^l\left(x\right)=\frac{\langle q_{k|k-1}\left(x\right|·),p_{k-1}^l{p}_s\rangle }{\langle p_{k-1}^l,p_s\rangle },\hfill \end{array}$$

(6)

where $p_s\left(x\right)$ is the survival probability of the Bernoulli term and $q\left(x\right|·)$ is the Markov transition kernel. The mathematical symbol $\langle a,b\rangle$ represents the scalar product, defined as $\langle a,b\rangle =\int a\left(x\right)b\left(x\right)dx$ [17].

The predicted birth targets are directly modeled as IIDC RFS with a probability density function denoted as $p_{k|k-1}^c\left(x\right)$. Use ${\pi }_{k|k-1}^c\left(n\right)$ and ${\mu }_{k|k-1}^c$ to describe the cardinality distribution and expected cardinality of the birth target.

(2)

Update Step

The parameters of the multi-Bernoulli RFS and IIDC RFS need to be updated separately in the update step. The posterior parameters of the multi-Bernoulli and IIDC components are denoted as ${\{r_k^l,p_k^l\left(x\right)\}}_{l=1}^{L_k}$ and $\{p_k^c\left(x\right),{\pi }_k^c\left(n\right)\}$, where $L_k=L_{k|k-1}$. The parameter update of the l-th Bernoulli is given by the following approximate formula

$$r_k^l{p}_k^l\left(x\right)\approx r_{k|k-1}^l{p}_{k|k-1}^l\left(x\right)\frac{{\mathcal{N}}_{{\Sigma }_k^l+R_k}(Z_k-h_k\left(x\right)-{\sigma }_k^l)}{{\mathcal{N}}_{{\Sigma }_k+R_k}(Z_k-{\sigma }_k)}$$

(7)

where $Z_k$ is the observed data defined in (4), and

$$\begin{array}{cc}\hfill {\sigma }_k& =\sum _{l=1}^{L_{k|k-1}}{r}_{k|k-1}^l{s}_{k|k-1}^l+{\mu }_{k|k-1}^c{s}_{k|k-1}^c,\hfill \\ \hfill {\sigma }_k^l& ={\sigma }_k-r_{k|k-1}^l{s}_{k|k-1}^l,\hfill \\ \hfill {\Sigma }_k& =\sum _{l=1}^{L_{k|k-1}}(r_{k|k-1}^l{v}_{k|k-1}^l-{(r_{k|k-1}^l)}^2{s}_{k|k-1}^l{(s_{k|k-1}^l)}^{\top })\hfill \\ \hfill & +{\mu }_{k|k-1}^c{v}_{k|k-1}^c-({({\mu }_{k|k-1}^c)}^2-a)s_{k|k-1}^c{(s_{k|k-1}^c)}^T,\hfill \\ \hfill {\Sigma }_k^l& ={\Sigma }_k-(r_{k|k-1}^l{v}_{k|k-1}^l-{(r_{k|k-1}^l)}^2{s}_{k|k-1}^l{(s_{k|k-1}^l)}^{\top }),\hfill \\ \hfill s_k^l& =\langle p_{k|k-1}^l,h\rangle ,v_k^l=\langle p_{k|k-1}^l,h{\left(h\right)}^{\top }\rangle ,\hfill \\ \hfill s_k^c& =\langle p_{k|k-1}^c,h\rangle ,v_k^c=\langle p_{k|k-1}^c,h{\left(h\right)}^{\top }\rangle ,\hfill \\ \hfill a& =\sum _{n=0}^{\infty }n(n-1){\pi }_{k|k-1}^c\left(n\right).\hfill \end{array}$$

(8)

The parameter update of the IIDC RFS component is obtained by

$${\mu }_k^c{p}_k^c\left(x\right)\approx {\mu }_{k|k-1}^c{p}_{k|k-1}^c\left(x\right)\frac{{\mathcal{N}}_{{\Sigma }_k^c+R_k}(z_k-h\left(x\right)-{\sigma }_k^c)}{{\mathcal{N}}_{{\Sigma }_k+R_k}(z_k-{\sigma }_k)},$$

(9)

where

$$\begin{array}{cc}\hfill {\sigma }_k^c& =\sum _{l=1}^{L_{k|k-1}}{r}_{k|k-1}^l{s}_{k|k-1}^l+\frac{a}{{\mu }_{k|k-1}^c}{s}_{k|k-1}^c,\hfill \\ \hfill {\Sigma }_k^c& =\sum _{l=1}^{L_{k|k-1}}(r_{k|k-1}^l{v}_{k|k-1}^l-{(r_{k|k-1}^l)}^2{s}_{k|k-1}^l{(s_{k|k-1}^l)}^{\top })\hfill \\ \hfill & +\frac{a}{{\mu }_{k|k-1}^c}{v}_{k|k-1}^c-(\frac{a^2}{{({\mu }_{k|k-1}^c)}^2}-\frac{b}{{\mu }_{k|k-1}^c})s_{k|k-1}^c{(s_{k|k-1}^c)}^T,\hfill \\ \hfill b& =\sum _{n=0}^{\infty }n(n-1)(n-2){\pi }_{k|k-1}^c\left(n\right).\hfill \end{array}$$

(10)

After the update, the component still conforms to the IIDC distribution [17], and the base distribution is more accurate than the traditional multi-Bernoulli components. The cardinality distribution of the IIDC RFS component is updated as

$${\pi }_k^c\left(n\right)\approx {\pi }_{k|k-1}^c\left(n\right)\frac{{\mathcal{N}}_{{\Sigma }_k^{c,n}+R_k}(z_k-{\sigma }_k^{c,n})}{{\mathcal{N}}_{{\Sigma }_k+R_k}(z_k-{\sigma }_k)},$$

(11)

where

$${\sigma }_k^{c,n}=\sum _{l=1}^{L_{k|k-1}}{r}_{k|k-1}^l{s}_{k|k-1}^l+n{s}_{k|k-1}^c,$$

(12)

and

$$\begin{array}{cc}\hfill {\Sigma }_k^{c,n}& =\sum _{l=1}^{L_{k|k-1}}(r_{k|k-1}^l{v}_{k|k-1}^l-{(r_{k|k-1}^l)}^2{s}_{k|k-1}^l{(s_{k|k-1}^l)}^{\top })\hfill \\ \hfill & +n(v_{k|k-1}^c-s_{k|k-1}^c{(s_{k|k-1}^c)}^T).\hfill \end{array}$$

(13)

2.3. The Variational Bayesian Inference

In this paper, we use the variational Bayesian (VB) to approximate the joint distribution of the state and covariance matrices. Denote this hybrid state by $\tilde{x}=(x,R)$, where x represents the target state and R represents the noise covariance matrix. Similar to [25], the joint prior distribution is assumed to have the following form

$$p_k\left(\tilde{x}\right|Z_k)\approx Q_k\left(x\right)·Q_k\left(R\right)$$

(14)

where $Q_k\left(x\right)$ and $Q_k\left(R\right)$ are the state density and unknown covariance density, respectively.

The VB approximation is formed by minimizing the Kullback–Leibler (KL) scatter between the true and approximate distributions.

$$KL[Q_k\left(x\right)Q_k\left(R\right)\parallel p_k\left(\tilde{x}\right|Z_k)]=\int Q_k\left(x\right)Q_k\left(R\right)log\left(\frac{Q_k\left(x\right)Q_k\left(R\right)}{p_k\left(\tilde{x}\right|Z_k)}\right)dxdR.$$

(15)

Minimizing the KL divergence in the above equation yields the following equation:

$$\begin{array}{cc}\hfill & Q_k\left(x\right)\propto exp\left(\int logp\left(Z_k,x,R\mid Z_{k-1}\right)Q_k\left(R\right)dR\right),\hfill \\ \hfill & Q_k\left(R\right)\propto exp\left(\int logp\left(Z_k,x,R\mid Z_{k-1}\right)Q_k\left(x\right)dx\right).\hfill \end{array}$$

(16)

3. Robust HMB-CPHD Filter for Superpositional Sensors with Unknown Noise Covariance

In this paper, the covariance matrix of the noise density is used as an unknown quantity and estimated jointly with the multi-target state. The first part of this section presents the GM implementation of the HMB-CPHD filter, and the second part proposes a solution for the case where the measurement noise is unknown.

3.1. GM Implementations for the Original HMB-CPHD Method

In the superposition measurement model, the HMB-CPHD method cannot obtain an analytically tractable filter due to the involvement of integrals. Generally, it can be approximated by sequential Monte Carlo (SMC) or Gaussian mixture (GM) iterations, and in [17], sequential Monte Carlo is chosen for implementation. However, the particle approximation method is computationally complex; GM implementation is more suitable for the application purpose due to its effective reduction in computational cost. In this section, we present a general GM implementation of the HMB-CPHD filter (hereinafter referred to as GM-HMB-CPHD filter).

In the subsequent implementation steps, GM is used to approximate the distribution representing each Bernoulli term. The GM with $J^l$ Gaussian components (GCs) is used for the l-th Bernoulli term

$$p^l\left(x\right)=\sum _{i=1}^{J^l}{\alpha }_i^l·\mathcal{N}(x;m_i^l,P_i^l),$$

(17)

where $\alpha$, m, and P represent, respectively, the weight, mean and covariance of the i-th Gaussian component. The GM form of multi-Bernoulli density is abbreviated as $\pi ={\{r^l,{\{{\alpha }_i^l,m_i^l,P_i^l\}}_{i=1}^{J^l}\}}_{l=1}^L$, where L represents the number of multi-Bernoulli terms.

The predicted density function of the IIDC RFS component is also modeled with GM, and a mixture of $J^c$ Gaussian terms is adopted

$$p^c\left(x\right)=\sum _{i=1}^{J^c}{\alpha }_i^c·\mathcal{N}(x;m_i^c,P_i^c).$$

(18)

(1)

Prediction Step

The predictive distribution consists of mutually independent multi-Bernoulli RFS and IIDC RFS. Assume that the parameters of the prior multi-Bernoulli distribution at time k are

$${\pi }_{k-1}={\{r_{k-1}^l,{\{{\alpha }_{k-1,i}^l,m_{k-1,i}^l,P_{k-1,i}^l\}}_{i=1}^{J_{k-1}^l}\}}_{l=1}^{L_{k-1}}.$$

(19)

After prediction, the multi-Bernoulli predictive distribution still obeys the multi-Bernoulli distribution ${\pi }_{k|k-1}={\{r_{k|k-1}^l,{\{{\alpha }_{k|k-1,i}^l,m_{k|k-1,i}^l,P_{k|k-1,i}^l\}}_{i=1}^{J_{k|k-1}^l}\}}_{l=1}^{L_{k|k-1}}$. According to Formula (6), the prediction parameters can be calculated as

$$\begin{array}{cc}\hfill r_{k|k-1}^l& =P_S{r}_{k-1}^l,{\alpha }_{k|k-1,i}^l={\alpha }_{k-1,i}^l,\hfill \\ \hfill m_{k|k-1,i}^l& =F_k{m}_{k-1,i}^l,\hfill \\ \hfill P_{k|k-1,i}^l& =F_k{P}_{k-1,i}^l{F}_k^{\top }+Q_k,\hfill \end{array}$$

(20)

where $J_{k|k-1}^l=J_{k-1}^l$ and the number of multi-Bernoulli remains unchanged.

In this paper, newborn targets are initialized using IIDC RFS, and it is assumed that the predictive IIDC distribution has the parameters ${\{{\alpha }_{k|k-1,i}^c,m_{k|k-1,i}^c,P_{k|k-1,i}^c\}}_{i=1}^{J_{k|k-1}^c}$. Let ${\pi }_{k|k-1}^c\left(n\right)$ and ${\mu }_{k|k-1}^c$ denote the cardinality distribution and expected cardinality of the predicted IIDC RFS component.

(2)

Update Step

Considering that the measurement function $h_k\left(x\right)$ adopted in this paper is non-linear. To calculate the updated distribution parameters, the measurement function is linearly expanded by Taylor expansion

$$h_k\left(x\right)\cong h_k\left(x_0\right)+H_k\left(x_0\right)(x-x_0),$$

(21)

where $H_k$ denotes the Jacobian matrix defined as

$$H_k\left(x_0\right)=\left[\frac{\partial h_k\left(x\right)}{\partial \zeta },\frac{\partial h_k\left(x\right)}{\partial \theta },\frac{\partial h_k\left(x\right)}{\partial \dot{\zeta }},\frac{\partial h_k\left(x\right)}{\partial \dot{\theta }}\right]{|}_{x=x_0}.$$

(22)

The Gaussian mixture parameters of the updated multi-Bernoulli density are calculated as

$${\pi }_k={\{(r_k^l,{\{{\alpha }_{k|k,i}^l,m_{k|k,i}^l,P_{k|k,i}^l\}}_{i=1}^{J_k^l})\}}_{l=1}^{L_k},$$

(23)

where the number of multi-Bernoulli terms $L_k=L_{k|k-1}$, the number of Gaussian components $J_k^l=J_{k|k-1}^l$, and

$$\begin{array}{cc}\hfill & r_k^l={\beta }_k·\frac{r_{k|k-1}^l{\sum }_{i=1}^{J_{k|k-1}^l}{\tilde{\alpha }}_{k,i}^l}{{max}_{l^{\prime }\le L_{k|k-1}}\int r_{k|k-1}^{l^{\prime }}{\sum }_{i=1}^{J_{k|k-1}^{\prime }}{\tilde{\alpha }}_{k,i}^{l^{\prime }}dx},\hfill \\ \hfill & {\tilde{\alpha }}_{k,i}^l={\alpha }_{k|k-1,i}^l·\mathcal{N}(Z_k-h_k(m_{k|k-1,i}^l)-{\sigma }_k^l;0,S_k^l),\hfill \\ \hfill & {\alpha }_{k|k,i}^l=\frac{{\tilde{\alpha }}_{k,i}^l}{{\sum }_{j=1}^{J_{k|k-1}^l}{\tilde{\alpha }}_{k,j}^l},\hfill \\ \hfill & m_{k|k,i}^l=m_{k|k-1,i}^l+K_{k,i}^l(Z_k-h_k(m_{k|k-1,i}^l)-{\sigma }_k^l),\hfill \\ \hfill & P_{k|k,i}^l=(I-K_{k,i}^l{H}_k(m_{k|k-1,i}^l))P_{k|k-1,i}^l,\hfill \\ \hfill & K_{k,i}^l=P_{k|k-1,i}^l{H}_k^{\top }(m_{k|k-1,i}^l){(S_k^l)}^{-1},\hfill \\ \hfill & S_k^l={\Sigma }_k^l+R_k+H_k(m_{k|k-1,i}^l)P_{k|k-1,i}^l{H}_k^{\top }(m_{k|k-1,i}^l).\hfill \end{array}$$

(24)

The PHD update for the IIDC RFS component can be calculated according to Formula (9). The GM form of the update IIDC density is computed as

$${\pi }_k^c={\{{\alpha }_{k|k,i}^c,m_{k|k,i}^c,P_{k|k,i}^c\}}_{i=1}^{J_k^c},$$

(25)

where the number of GCs $J_k^c=J_{k|k-1}^c,$ and

$$\begin{array}{cc}\hfill & {\alpha }_{k|k,i}^c={\alpha }_{k|k-1,i}^c·\mathcal{N}(Z_k-h_k(m_{k|k-1,i}^c)-{\sigma }_k^c;0,S_k^c),\hfill \\ \hfill & m_{k|k,i}^c=m_{k|k-1,i}^c+K_{k,i}^c(Z_k-h_k(m_{k|k-1,i}^c)-{\sigma }_k^c),\hfill \\ \hfill & P_{k|k,i}^c=(I-K_{k,i}^c{H}_k(m_{k|k-1,i}^c))P_{k|k-1,i}^c,\hfill \\ \hfill & K_{k,i}^c=P_{k|k-1,i}^c{H}_k^{\top }(m_{k|k-1,i}^c){(S_k^c)}^{-1},\hfill \\ \hfill & S_k^c={\Sigma }_k^c+R_k+H_k(m_{k|k-1,i}^c)P_{k|k-1,i}^c{H}_k^{\top }(m_{k|k-1,i}^c).\hfill \end{array}$$

(26)

The cardinality distribution ${\pi }_k^c(n)$ of the IIDC RFS component is updated according to Formula (11), and the the expected cardinality ${\mu }_k^c$ is calculated as

$${\mu }_k^c=\sum _{n=0}^{\infty }n{\pi }_k^c(n).$$

(27)

The posterior distribution obtained at this time is the union of the multi-Bernoulli distribution and the IIDC distribution, whose parameters include the PHD of the multi-Bernoulli RFS, the PHD of the IIDC RFS, and the cardinality distribution of the IIDC RFS [17]. In order to maintain the form consistency in the loop of the algorithm, the IIDC component is approximated by the multi-Bernoulli component after the update step. The parameters of the approximated multi-Bernoulli component are recorded as

$${\pi }_k^b={\{(r_k^l,\{{\alpha }_{k|k}^l,m_{k|k}^l,P_{k|k}^l\})\}}_{l=1}^{L_k^b},$$

(28)

where the number of the Bernoulli component $L_k^b=J_k^c$, and

$$\begin{array}{cc}\hfill & r_k^l={\alpha }_{k|k,l}^c{\mu }_k^c,{\alpha }_{k|k}^l=1,\hfill \\ \hfill & m_{k|k}^l=m_{k|k,l}^c,P_{k|k}^l=P_{k|k,l}^c.\hfill \end{array}$$

(29)

Remark 1.

In the update step, to address the non-linearity of the measurement function in (4), the extended Kalman filter (EKF) approach was used for approximation. However, it should be noted that this method may introduce higher bias in highly non-linear scenarios. Alternatively, unscented transform or cubature rule, among other methods, can also be used for parameter updates. However, these sampling-based methods generally require a higher computational load compared to EKF.

3.2. The Proposed Method

Since there may be no prior information on the measured noise in practical applications, a robust filtering algorithm is needed. This section considers the VB-HMB-CPHD filter, which can jointly estimate the target state and the measurement noise covariance matrix in the case of unknown measurement noise. This paper considers the following mixed states

$$\tilde{x}=(x,\tilde{R}),$$

(30)

where $\tilde{R}$ is the sum of the noise covariance matrix R and the predicted covariance matrix $\tilde{\Sigma }$. According to the update Formulas (7) and (9) of the GM-HMB-CPHD filter, the predictive covariance matrix $\tilde{\Sigma }$ is represented by ${\Sigma }^l$ and ${\Sigma }^c$, respectively, in the multi-Bernoulli components and IIDC components.

Similar to the handling of the measurement noise of the unknown variables in [25], an inverse Wishart (IW) distribution is used to model $\tilde{R}$. Here, the assumption of [19] is still followed, assuming that the target state x and the covariance matrix $\tilde{R}$ are independent of each other. Since the covariance matrix is unknown, the Gaussian mixture form in the GM-HMB-CPHD filter cannot meet the requirements and the following operations are all extended by Gaussian Inverse Wishart (GIW) to a single Bernoulli item or IIDC item

$$\begin{array}{cc}\hfill p(\tilde{x})& =\mathcal{N}(x;m,P)·\mathcal{I}\mathcal{W}(\tilde{R};u,U)\stackrel{\mathsf{\Delta }}{=}\mathcal{N}\mathcal{I}\mathcal{W}(\tilde{x};\vartheta ),\hfill \end{array}$$

(31)

where $\mathcal{IW}(\tilde{R};u,U)$ represents the IW distribution with degree of freedom u and scale matrix U, and $\vartheta =(m,P,u,U)$ denote all parameters of the GIW distribution. The transfer processes of the target state and covariance parameters are assumed to be independent of each other [19].

Now assume that the prior distribution is GIW. In the prediction process of Bayesian filtering, the density form of GIW distribution does not change, and the GIW distribution after prediction is as follows

$$p_{k|k-1}(\tilde{x})=\mathcal{N}\mathcal{I}\mathcal{W}(\tilde{x};{\vartheta }_{k|k-1}),$$

(32)

where ${\vartheta }_{k|k-1}=(m_{k|k-1},P_{k|k-1},u_{k|k-1},U_{k|k-1})$ is the parameter of the GIW distribution, using the heuristic algorithm [26] and Gaussian mixture PHD prediction formula, the parameter calculation is as follows

$$\begin{array}{cc}\hfill & m_{k|k-1}=F_{k-1}{m}_{k-1},\hfill \\ \hfill & P_{k|k-1}=Q_{k-1}+F_{k-1}{P}_{k-1}{(F_{k-1})}^{\top },\hfill \\ \hfill & u_{k|k-1}=\lambda u_{k-1},\hfill \\ \hfill & U_{k|k-1}=\lambda (U_{k-1}-M_z-1)+M_z+1,\hfill \end{array}$$

(33)

where $\lambda \in (0,1]$ denotes a forgetting factor. For the l-th single Bernoulli item, its measurement information is $Z_k^l=Z_k-{\sigma }_k^l$; while for the IIDC item, its measurement information is $Z_k^c=Z_k-{\sigma }_k^c$, so the likelihood function is uniformly written as

$${\phi }_k({\tilde{Z}}_k|x)=\mathcal{N}({\tilde{Z}}_k-h_k(x);0,{\tilde{R}}_k),$$

(34)

where the measurement ${\tilde{Z}}_k$ is represented by $Z_k^l$ and $Z_k^c$, respectively, in the multiple Bernoulli components and IIDC components. Linearizing $h_k(x)$ according to Formula (21) can yield an approximate formula

$${\phi }_k({\tilde{Z}}_k|x)\cong \mathcal{N}({\tilde{Z}}_k-h_k(x_0)+H_k(x_0)x_0;H_k(x_0)x,{\tilde{R}}_k).$$

(35)

After Bayesian updating, the posterior distribution is different from the prior distribution, so it is necessary to approximate the posterior distribution through the variational Bayesian method. The principle of VB is to find the distribution that is closest to the expected form by minimizing the Kullback–Leibler (KL) divergence [25]. Assuming that the approximated distribution is

$$p_k(\tilde{x}|{\tilde{Z}}_k)=Q_k(x)·Q_k(\tilde{R})=\mathcal{N}(x;m_{k|k},P_{k|k})·\mathcal{IW}(\tilde{R};u_{k|k},U_{k|k}).$$

(36)

The calculation of parameters $m_{k|k}$, $p_{k|k}$, $u_{k|k}$, $U_{k|k}$ is obtained by J fixed-point iteration [27], and the j-th iteration formula is as follows

$$\begin{array}{cc}\hfill u_{k|k}^{(j)}& =u_{k|k-1}+1,\hfill \\ \hfill U_{k|k}^{(j)}& =U_{k|k-1}+{{\rm Y}}_k^{(j)}\hfill \\ \hfill {\tilde{R}}_k^{(j)}& =U_{k|k}^{(j)}/(u_{k|k}-M_z-1),\hfill \\ \hfill S_k^{(j)}& ={\tilde{R}}_k^{(j)}+H_k(m_{k|k-1})P_{k|k-1}{(H_k(m_{k|k-1}))}^{\top },\hfill \\ \hfill K_k^{(j)}& =P_{k|k-1}{(H_k(m_{k|k-1}))}^{\top }{(S_k^{(j)})}^{-1},\hfill \\ \hfill m_{k|k}^{(j)}& =m_{k|k-1}+K_k^{(j)}({\tilde{Z}}_k-h_k(m_{k|k-1})),\hfill \\ \hfill P_{k|k}^{(j)}& =(I-K_k^{(j)}{H}_k(m_{k|k-1}))P_{k|k-1},\hfill \end{array}$$

(37)

where

$$\begin{array}{cc}\hfill {{\rm Y}}_k^{(j)}& =({\tilde{Z}}_k-H_k(m_{k|k-1})m_{k|k}^{(j)}){({\tilde{Z}}_k-H_k(m_{k|k-1})m_{k|k}^{(j)})}^{\top }\hfill \\ \hfill & +H_k(m_{k|k-1})P_{k|k}^{(j)}{(H_k(m_{k|k-1}))}^{\top }.\hfill \end{array}$$

(38)

After J iterations, the posterior distribution parameters are given by $m_{k|k}=m_{k|k}^{(J)}$, $P_{k|k}=P_{k|k}^{(J)}$, $u_{k|k}=u_{k|k}^{(J)}$, and $U_{k|k}=U_{k|k}^{(J)}$. The variational lower bound of the likelihood distribution can be obtained by the following equation

$$\begin{array}{cc}\hfill Q({\tilde{Z}}_k)=& exp\{-\frac{M_z}{2}ln\pi +\frac{1}{2}ln|P_{k|k}|-\frac{1}{2}ln|P_{k|k-1}|-\frac{1}{2}tr[{(P_{k|k-1})}^{-1}{P}_{k|k}]\hfill \\ \hfill & -\frac{1}{2}tr(((m_{k|k}-m_{k|k-1}){(m_{k|k}-m_{k|k-1})}^{\top })·{(P_{k|k-1})}^{-1})\hfill \\ \hfill & +\frac{(u_{k|k-1}-M_z-1)}{2}ln|U_{k|k-1}|-\frac{(u_{k|k}-M_z-1)}{2}ln\left|U_{k|k}\right|\hfill \\ \hfill & -ln{\Gamma }_{M_z}(\frac{u_{k|k-1}-M_z-1}{2})+ln{\Gamma }_{M_z}(\frac{u_{k|k}-M_z-1}{2})\}.\hfill \end{array}$$

(39)

where the proof of (37)–(39) has been given in [28] earlier.

Considering a single Bernoulli term and IIDC term jointly, the Gaussian and inverse Wishart mixture (GIWM) distribution is used to represent the Probability density of each Bernoulli term and IIDC component. The l-th Bernoulli term is represented by $J^l$ GIW distributions, with PHD as

$${\stackrel{`}{r}}^l{\stackrel{`}{p}}^l(\tilde{x})={\stackrel{`}{r}}^l\sum _{i=1}^{J^l}{\alpha }_i^l·\mathcal{N}\mathcal{IW}(\tilde{x};{\vartheta }_i^l),$$

(40)

where ${\vartheta }_i^l=(m_i^l,P_i^l,u_i^l,U_i^l)$ and the symbol $(\stackrel{`}{·})$ is used to differentiate from the previous GM-HMB-CPHD filter. The GIWM form of the multi-Bernoulli density becomes $\stackrel{`}{\pi }={\{(r^l,{\{{\alpha }_i^l,{\vartheta }_i^l,\}}_{i=1}^J)\}}_{l\le L}$. The PHD of IIDC RFS modeled by GIWM is

$${\stackrel{`}{r}}^c{\stackrel{`}{p}}^c(\tilde{x})={\stackrel{`}{r}}^c\sum _{i=1}^{J^c}{\alpha }_i^c·\mathcal{NIW}(\tilde{x};{\vartheta }_i^c).$$

(41)

In the prediction step, assuming that the parameter of the prior Bernoulli distribution is ${\stackrel{`}{\pi }}_{k-1}={\{(r_{k-1}^l,{\{{\alpha }_{k-1,i}^l,{\vartheta }_{k-1,i}^l\}}_{i=1}^{J_{k-1}^l})\}}_{l=1}^{L_{k-1}}$, the predictive multi-Bernoulli density can be calculated as

$${\stackrel{`}{\pi }}_{k|k-1}={\{r_{k|k-1}^l,{\{{\alpha }_{k|k-1,i}^l,{\vartheta }_{k|k-1,i}^l\}}_{i=1}^{J_{k-1}^l}\}}_{l=1}^{L_{k|k-1}},$$

(42)

where

$$\begin{array}{cc}\hfill & r_{k|k-1}^l=P_S{r}_{k-1}^l,{\alpha }_{k|k-1,i}^l={\alpha }_{k-1,i}^l,\hfill \\ \hfill & m_{k|k-1,i}^l=F_k{m}_{k-1,i}^l,\hfill \\ \hfill & P_{k|k-1,i}^l=F_k{P}_{k-1,i}^l{F}_k^{\top }+Q_k,\hfill \\ \hfill & u_{k|k-1,i}^l=\lambda u_{k-1,i}^l,\hfill \\ \hfill & U_{k|k-1,i}^l=\lambda (U_{k-1,i}^l-M_z-1)+M_z+1.\hfill \end{array}$$

(43)

Model the newborn targets using the IIDC component and record its predictive density as ${\stackrel{`}{\pi }}_{c,k|k-1}={\{{\alpha }_{k|k-1,i}^c,{\vartheta }_{k|k-1,i}^c\}}_{i=1}^{J_{k|k-1}^c}$. In addition, the cardinality distribution and expected cardinality of the predicted IIDC RFS component are given as ${\stackrel{`}{\pi }}_{k|k-1}^c(n)$ and ${\stackrel{`}{\mu }}_{k|k-1}^c$.

Using the variational Bayesian approach for updating, the posterior multi-Bernoulli density after updating is

$${\stackrel{`}{\pi }}_k={\{(r_k^l,{\{{\alpha }_{k|k,i}^l,{\vartheta }_{k|k,i}^l\}}_{i=1}^{J_k^l})\}}_{l=1}^{L_k},$$

(44)

where

$$\begin{array}{cc}\hfill & {\vartheta }_{k|k,i}^l=(m_{k|k,i}^l,P_{k|k,i}^l,u_{k|k,i}^l,U_{k|k,i}^l),\hfill \\ \hfill & u_{k|k,i}^l=u_{k|k-1,i}^l+1,\hfill \\ \hfill & U_{k|k,i}^l=U_{k|k-1,i}^l+{{\rm Y}}_{k,i}^l,\hfill \\ \hfill & {\tilde{R}}_{k,i}^l=U_{k|k,i}^l/(u_{k|k,i}^l-M_z-1),\hfill \\ \hfill & {\widehat{R}}_{k,i}^l={\tilde{R}}_{k,i}^l-{\Sigma }_k^l,\hfill \\ \hfill & r_k^l={\beta }_k\frac{r_{k|k-1}^l{\sum }_{i=1}^{J_{k|k-1}^l}{\tilde{\alpha }}_{k,i}^l}{{max}_{l^{\prime }\le L_{k|k-1}}\int r_{k|k-1}^l{\sum }_{i=1}^{J_{k|k-1}^{\prime }}{\tilde{\alpha }}_{k,i}^{l^{\prime }}},\hfill \\ \hfill & {\tilde{\alpha }}_{k,i}^l={\alpha }_{k|k-1,i}^l·Q({\tilde{Z}}_k^l),\hfill \\ \hfill & {\alpha }_{k|k,i}^l=\frac{{\tilde{\alpha }}_{k,i}^l}{{\sum }_{j=1}^{J_{k|k-1}^l}{\tilde{\alpha }}_{k,j}^l},\hfill \\ \hfill & S_{k,i}^l={\tilde{R}}_{k,i}^l+H_k(m_{k|k-1,i}^l)P_{k|k-1,i}^l{(H_k(m_{k|k-1,i}^l))}^{\top },\hfill \\ \hfill & K_{k,i}^l=P_{k|k-1,i}^l{(H_k(m_{k|k-1,i}^l))}^{\top }{(S_{k,i}^l)}^{-1},\hfill \\ \hfill & m_{k|k,i}^l=m_{k|k-1,i}^l+K_{k,i}^l({\tilde{Z}}_k^l-h_k(m_{k|k-1,i}^l)),\hfill \\ \hfill & P_{k|k,i}^l=(I-K_{k,i}^l{H}_k(m_{k|k-1,i}^l))P_{k|k-1,i}^l,\hfill \end{array}$$

(45)

where ${{\rm Y}}_{k,i}^l$ can be obtained from (38), ${\widehat{R}}_{k,i}^l$ represents the estimated measurement noise covariance matrix, and the predictive likelihood $Q({\tilde{Z}}_k^l)$ can be obtained from (39). The parameters of the IIDC component after the update can be represented as

$${\stackrel{`}{\pi }}_k^c={\{{\alpha }_{k|k,i}^c,{\vartheta }_{k|k,i}^c\}}_{i=1}^{J_k^c},$$

(46)

where

$$\begin{array}{cc}\hfill & {\vartheta }_{k|k,i}^c=(m_{k|k,i}^c,P_{k|k,i}^c,u_{k|k,i}^c,U_{k|k,i}^c),\hfill \\ \hfill & u_{k|k,i}^c=u_{k|k-1,i}^c+1,\hfill \\ \hfill & U_{k|k,i}^c=U_{k|k-1,i}^c+{{\rm Y}}_{k,i}^c,\hfill \\ \hfill & {\tilde{R}}_{k,i}^c=U_{k|k,i}^c/(u_{k|k,i}^c-M_z-1),\hfill \\ \hfill & {\widehat{R}}_{k,i}^c={\tilde{R}}_{k,i}^c-{\Sigma }_k^c,\hfill \\ \hfill & {\alpha }_{k|k,i}^c={\alpha }_{k|k-1,i}^c·\mathcal{N}(Z_k-h_k(m_{k|k-1,i}^c)-{\sigma }_k^c;0,S_k^c),\hfill \\ \hfill & S_{k,i}^c={\tilde{R}}_{k,i}^c+H_k(m_{k|k-1,i}^c)P_{k|k-1,i}^c{(H_k(m_{k|k-1,i}^c))}^{\top },\hfill \\ \hfill & K_{k,i}^l=P_{k|k-1,i}^c{(H_k(m_{k|k-1,i}^c))}^{\top }{(S_{k,i}^c)}^{-1},\hfill \\ \hfill & m_{k|k,i}^c=m_{k|k-1,i}^c+K_{k,i}^c({\tilde{Z}}_k^c-h_k(m_{k|k-1,i}^c)),\hfill \\ \hfill & P_{k|k,i}^c=(I-K_{k,i}^c{H}_k(m_{k|k-1,i}^c))P_{k|k-1,i}^c.\hfill \end{array}$$

(47)

The cardinality distribution of IIDC components after VB update is ${\stackrel{`}{\pi }}_k^c(n)$, and the expected cardinality mu calculation is ${\stackrel{`}{\mu }}_k^c={\sum }_{n=0}^{\infty }n{\stackrel{`}{\pi }}_k^c(n)$.

The posterior density obtained after the VB update consists of multiple Bernoulli components following GIWM distribution and one IIDC component following GIWM distribution. To be consistent with the a priori form at the next moment, the two components need to be approximated as a multi-Bernoulli RFS.

Assume that the multi-Bernoulli distribution after the IIDC component approximation is as follows

$${\stackrel{`}{\pi }}_k^b={\{(r_k^l,\{{\alpha }_{k|k}^l,{\vartheta }_{k|k}^l\})\}}_{l=L_k+1}^{L_k+L_k^b},$$

(48)

where the number of the Bernoulli component $L_k^b=J_k^c$, and

$$\begin{array}{cc}\hfill & r_k^l={\alpha }_{k|k,l-L_k}^c{\mu }_k^c,{\alpha }_{k|k}^l=1,\hfill \\ \hfill & {\vartheta }_{k|k}^l={\vartheta }_{k|k,l-L_k}^c.\hfill \\ \hfill \end{array}$$

(49)

The final posterior distribution obtained is the multi-Bernoulli components as

$$\begin{array}{cc}\hfill {\dot{\pi }}_k& ={\pi }_k\cup {\stackrel{`}{\pi }}_k^b\hfill \\ \hfill & ={\{(r_k^l,{\{{\alpha }_{k|k,i}^l,{\vartheta }_{k|k,i}^l\}}_{i=1}^{J_k^l})\}}_{l=1}^{L_k}\hfill \\ \hfill & \cup {\{(r_k^l,\{{\alpha }_{k|k}^l,m_{k|k}^l,P_{k|k}^l\})\}}_{l=L_k+1}^{L_k+L_k^b}\hfill \\ \hfill & ={\{(r_k^l,{\{{\alpha }_{k|k,i}^l,{\vartheta }_{k|k,i}^l\}}_{i=1}^{J_k^l})\}}_{l=1}^{L_k+L_k^b}.\hfill \end{array}$$

(50)

Algorithm 1 gives the pseudo-code for the Gaussian mixture implementation of the VB-HMB-CPHD filter. In the final step of the algorithm, filter out the Bernoulli components with probabilities smaller than the threshold, and denote the number of remaining Bernoulli components as ${\widehat{L}}_k$. From the selected Bernoulli items, target state estimation and covariance matrix estimation are performed, and ${\widehat{R}}_k$ is computed as

$${\widehat{R}}_k=\frac{{\sum }_{l\in {\widehat{L}}_k}{r}_k^l{\sum }_{i=1}^{l_k^l}{\alpha }_{k\mid k,i}^l·{\widehat{R}}_{k,i}^l}{{\sum }_{l\in {\widehat{L}}_k}{r}_k^l}$$

(51)

where ${\widehat{R}}_{k,i}^l$ is obtained by (45) and (47).   

Algorithm 1: Pseudocode for Gaussian mixture implementation of VB-HMB-CPHD filter

For the HMB-CPHD filter implemented with auxiliary particles [17], the main computational costs involve updating particle weights and cardinality distribution, as well as clustering at the k-th iteration. The weight update requires estimating the covariance matrix of size $M_z\times M_z$ and inverting it, with a complexity of $O(N_k{N}_p{M}_z^2+N_k{M}_z^3)$(where $M_z$ is the measurement dimension, $N_k$ is the number of targets at moment k and $N_p$ is the number of approximate particles used for each Bernoulli term or IIDC term). The cardinality update also involves inverting the covariance matrix, with a complexity of $O(N_0{M}_z^3)$(where $N_0$ is the maximum number of targets). Clustering is performed only once, using the k-means algorithm, and has the computational complexity $O(N_p{N}_0)$. Therefore, the overall complexity of the HMB-CPHD filter implemented with auxiliary particles at the k-th iteration is $O(N_k{N}_p{M}_z^2+N_k{M}_z^3+N_0{M}_z^3+N_p{N}_0)$. In comparison, the HMB-CPHD filter implemented with Gaussian mixtures (GM) has a lower computational complexity of $O(N_k{M}_z^2+N_k{M}_z^3+N_0{M}_z^3)$. From this, it can be seen that the computation of the covariance matrix using particle filtering requires $N_p$ times the implementation of the GM approach, and does not require the use of the k-means algorithm for clustering. This indicates that GM implementation is superior to auxiliary particle implementation in terms of processing time and efficiency.

4. Simulations

This section demonstrates the application of the VB-HMB-CPHD filter in multi-target tracking through simulation, using a radio frequency (RF) tomography approach as the sensor model.

4.1. Simulation Scene Setup

Radio frequency tomography is a popular superpositional measurement sensor and has been widely used in single-target tracking [29] and multi-target tracking [30,31]. The sensors can communicate with each other and the receiving value of each sensor is the signal strength. At each time, M sensor networks can be combined into $M_z=M(M-1)/2$ different sensor pairs and generate $M_z$ measurement values.

If there is a set of targets X in the space, the $M_z$ measurements can be obtained using the model mentioned in [29], where the i-th measurement generated by sensors a and b has the following form

$$z^i=\sum _{x\in X}\varphi exp(-\frac{{\chi }_i(x)}{{\sigma }_{\chi }})+v^i=\sum _{x\in X}h(x)+v^i.$$

(52)

where $\varphi$ and ${\sigma }_{\chi }$ are fixed parameters based on the physical properties of the sensors and targets, $v^i$ is zero-mean Gaussian sensor noise with unknown covariance matrix R, and ${\chi }_i(x)$ is a parameter related to the relative position between the sensor and the target [17], which can be represented as

$${\chi }_i(x)=d_a^i(x)+d_b^i(x)-d_{ab}^i,$$

(53)

where $d_a^i(x)$ and $d_b^i(x)$ represent the distance between the target and the two sensors, and $d_{ab}^i$ represents the relative distance between the sensor a and b.

Following the same arrangement as in [17], the sensors are uniformly distributed around the measurement area, as shown in Figure 1, where 20 sensors are distributed around the measurement area.

In the simulation, the measurement parameters are the same as [28], and are set as follows. The measurement area is a square area with a side length of 20 m × 20 m; the number of RF sensors is 20; the dimension of the measurement vector at each moment is 190; the fixed parameters $\varphi =5$ and ${\sigma }_{\chi }=0.2$; the covariance matrix of the measurement noise is $R_k=\eta I_{M_z}$, where $I_{M_z}$ is an $M_z$-dimensional unit matrix. Assuming that the targets move in the monitoring area and that each target’s motion is independent, and the motion model following approximately constant velocity model [32]. The form of the motion Equation (2) can be rewritten as

$$x_k=F_k{x}_{k-1}+B{\omega }_k,$$

(54)

where ${\omega }_k={[\zeta ,\theta ]}^{\top }$ is the process noise, $\zeta$ and $\theta$ are zero-mean Gaussian white noise with respective variance ${\sigma }_{\zeta }^2$ and ${\sigma }_{\theta }^2$, and

$$F=\left[\begin{array}{cccc}1\hfill & 0\hfill & T\hfill & 0\hfill \\ 0\hfill & 1\hfill & 0\hfill & T\hfill \\ 0\hfill & 0\hfill & 1\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & 1\hfill \end{array}\right],B=\left[\begin{array}{cc}\frac{T^2}{2}& 0\\ 0& \frac{T^2}{2}\\ T& 0\\ 0& T\end{array}\right],$$

(55)

where T represents the sampling interval.

The target motion parameters are as follows. The sampling time T is set to 0.25 s; the noise covariance ${\sigma }_{\zeta }^2$ and ${\sigma }_{\theta }^2$ is set to 0.2; the number of iterations is set to 50; and there are 6 targets appearing and disappearing in the simulation time [28].

The newborn targets are modeled as IIDC RFS $\left\{({\stackrel{`}{\pi }}_c={\{{\alpha }_i^c,m_i^c,P_i^c,u_i^c,U_i^c\}}_{i=1}^{L^c})\right\}$, where $L^c=6$, ${\alpha }_i^c=1/6$, and

$$\begin{array}{cc}\hfill & \left[\begin{array}{c}{(m_1^c)}^{\top }\hfill \\ {(m_2^c)}^{\top }\hfill \\ {(m_3^c)}^{\top }\hfill \\ {(m_4^c)}^{\top }\hfill \\ {(m_5^c)}^{\top }\hfill \\ {(m_6^c)}^{\top }\hfill \end{array}\right]=\left[\begin{array}{cccc}0.6\hfill & 18\hfill & 0\hfill & 0\hfill \\ 8\hfill & 15\hfill & 0\hfill & 0\hfill \\ 10\hfill & 18\hfill & 0\hfill & 0\hfill \\ 10.4\hfill & 8\hfill & 0\hfill & 0\hfill \\ 11.2\hfill & 11.2\hfill & 0\hfill & 0\hfill \\ 19.2\hfill & 6.6\hfill & 0\hfill & 0\hfill \end{array}\right],\hfill \\ \hfill & P_i^c=\left[\begin{array}{cccc}1\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 1\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 1\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & 1\hfill \end{array}\right],\hfill \\ \hfill & u_i^c=2+M_z+1,U_i^c=0.95I_{M_z}.\hfill \end{array}$$

(56)

In the following simulations, the VB-HMB-CPHD filtering algorithm proposed in this paper will be compared with the standard HMB-CPHD filter with manually set noise covariance. The number of variational fixed-point iterations J is set to 10, the heuristic factor is set to $\beta =0.95$, and the forgetting factor is set to $\lambda =0.74$. The optimal subpattern assignment (OSPA) distance [33] (with cut-off c = 1.5 and order p = 2 ) and target number estimation are used as metrics to evaluate the performance of the two algorithms. Finally, the effect of the forgetting factor on the performance of the algorithm is discussed.

4.2. Simulation Results

In this section, the true measurement noise covariance matrix is set to $0.4I_{M_z}$, and the multi-target tracking results of the VB-HMB-CPHD filter are shown in Figure 1. Performance evaluation of the proposed algorithm was conducted for the case of unknown noise covariance, and the GM-HMB-CPHD algorithm was run with different assumed prior noise covariance. The two algorithms were simulated 100 times using Monte Carlo simulation, and the estimation results and estimation error of the number of targets are shown in Figure 2, and the average OSPA distance is shown in Figure 3.

It can be seen from Figure 2 and Figure 3 that the performance of the proposed algorithm is comparable to that of the GM-HMB-CPHD filter with a known noise covariance matrix. However, when the noise covariance used by the GM-HMB-CPHD algorithm does not match the true value, its performance drops rapidly.

To evaluate the estimation performance of the noise covariance, the following estimation error is defined as

$$\left.R_{\mathrm{error}}=tr\left({\widehat{R}}_k-R_k\right)\right)/M_z$$

(57)

where ${\widehat{R}}_k$ is computed with (51), and the simulation results are shown in Figure 4.

It can be seen from the simulation that the estimation error of noise covariance will suddenly increase when there is a real newborn target, which is due to the instability of the newborn target in the estimation. This sudden increase in error can be solved by adjusting the IW distribution of the newborn target and generating the parameters of the newborn target with the estimated noise covariance of the previous moment.

The effect of different $\lambda$ values on the performance of the algorithm is unknown. Figure 5 shows the average OSPA error under different $\lambda$ values in 20 Monte Carlo simulations. The figure shows that when $\lambda$ is close to 0.74, the average OSPA error is the smallest, and when $\lambda$ is in the range of [0.66, 0.98], a lower error and better performance can be obtained. Although $\lambda$ has a great influence on the accuracy of the algorithm, this influence can only be obtained by experiments on different $\lambda$ values, and cannot be obtained in advance by mathematical analysis. Therefore, in the application of multi-target tracking, the optimal $\lambda$ value should be selected in advance by data experiments, and then tracking should be carried out.

5. Conclusions

In this paper, the multi-target tracking problem under the superpositional sensors is studied. First, we propose a Gaussian mixture implementation of the HMB-CPHD filter, and then extend the method to the problem of unknown measurement noise covariance. To address this, the target state and noise covariance matrix are jointly modeled as a mixed state, and the Gaussian and inverse Wishart mixture (GIWM) distribution is used for modeling. In the state update step, the variational Bayesian (VB) method is used to approximate the posterior distribution, thus ensuring the consistency of its prior and posterior forms, and the robustness of the algorithm under unknown noise covariance is proved by simulation.