Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 8742897, 9 pages

https://doi.org/10.1155/2017/8742897

## Efficient Approximation of the Labeled Multi-Bernoulli Filter for Online Multitarget Tracking

College of Mechanical and Electrical Engineering, Harbin Engineering University, Harbin, Heilongjiang, China

Correspondence should be addressed to Liang Ma

Received 4 March 2017; Revised 26 May 2017; Accepted 1 June 2017; Published 20 July 2017

Academic Editor: Huanqing Wang

Copyright © 2017 Ping Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Online tracking time-varying number of targets is a challenging issue due to measurement noise, target birth or death, and association uncertainty, especially when target number is large. In this paper, we propose an efficient approximation of the Labeled Multi-Bernoulli (LMB) filter to perform online multitarget state estimation and track maintenance efficiently. On the basis of the original LMB filer, we propose a target posterior approximation technique to use a weighted single Gaussian component representing each individual target. Moreover, we present the Gaussian mixture implementation of the proposed efficient approximation of the LMB filter under linear, Gaussian assumptions on the target dynamic model and measurement model. Numerical results verify that our proposed efficient approximation of the LMB filer achieves accurate tracking performance and runs several times faster than the original LMB filer.

#### 1. Introduction

Online tracking of time-varying number of targets is a challenging issue in the presence of measurement noise, target birth or death, and association uncertainty [1]. Recently, the random finite set (RFS) based Bayesian framework has been proved to be unified elegant approach for multisensor multiobject estimation [2], other than the traditional Joint Probabilistic Data Association (JPDA) [3] and Multiple Hypothesis Tracking (MHT) [4] methods in the tracking area [5]. The Probability Hypothesis Density (PHD) filter [6], Cardinalized PHD (CPHD) filter [7], and multi-Bernoulli filter [8] were established for multitarget state estimation by avoiding data association, which are incapable of track maintenance.

With the help of the labeled RFS, the -Generalized Labeled Multi-Bernoulli (-GLMB) filter has been proposed lately in order to handle multitarget state estimation and track maintenance simultaneously [9, 10]. Then, the Marginalized -Generalized Labeled Multi-Bernoulli (M-GLMB) [11] and the Labeled Multi-Bernoulli (LMB) filter [12] were proposed to perform multitarget tracking more efficiently, respectively, under different approximations and update paradigms. However, the efficiency issue of multitarget tracking method still remains challenging for online applications, especially for multisensor scenarios [13, 14]. Recently, [15] has proposed an efficient implementation of the GLMB filter via Gibbs sampling to solve the ranked assignment problem stochastically, which has solution complexity quadratic in the number of hypothesized labels and linear in the number of measurements. Nevertheless, this efficient implementation of the GLMB filter cannot enjoy the benefit of the parallelizability of the LMB filter, which makes the number of hypothesized labels and the number of measurements very large when tracking large number of targets.

In this paper, we further study the efficiency issue of multitarget tracking problem. We propose an efficient approximation of the LMB filter for online multitarget tracking, in which a single weighted Gaussian component is used to approximate the posterior state of each target. We present the Gaussian mixture implementation of proposed filter for the linear and Gaussian target dynamic model and observation model. The paper is organised as follows. Section 2 presents a short review of the basic knowledge about the RFS based Bayesian filtering, the labeled RFS, and the -GLMB filter. Section 3 illustrates the target posterior approximation approach and explains its efficiency. Section 4 provides the Gaussian mixture implementation of the approximation of the LMB filter in detail. Section 5 shows numerical results that verify the effectiveness and efficiency of proposed approximation.

#### 2. Background

This section presents the basic knowledge of the following content in this paper. We first give the RFS based Bayesian filtering in Section 2.1. Then, we present typical types of the labeled RFS in Section 2.2. Section 2.3 provides the LMB filter which lays out the fundamentals for our proposed approximation.

##### 2.1. Random Finite Sets Based Bayesian Filtering

The RFS is a random variable in which the number of its elements and the value of each element are both random process. With regard to multitarget tracking problem, the RFS has been shown to be a more natural and powerful description for multitarget state compared to the conventional vector representation [16]. Let and denote the state set and observation set with time-varying cardinalities and , respectively:

In a typical target tracking scenario, at every time step, the multitarget state RFS is composed of two parts: an RFS for existing targets and an RFS for spontaneous birth targets . Hence, at time step , the multitarget state RFS . The observation RFS is composed of two parts as well: target-oriented measurements and clutter , which gives . By modeling the multitarget state RFS and the observation RFS using different types of probability distributions such as Binomial, Poisson, and multi-Bernoulli, the RFS based Bayesian framework for optimal estimation is given as follows:which represent the prediction and update process of Bayesian recursion, respectively. Notice that the key to solve the RFS based Bayesian filtering is the set integrals in (3) and (4), which can be found in the finite set statistics [2]. Under different assumptions of the RFS type, the PHD filter [6], CPHD filter [7], and multi-Bernoulli filter [8] have been derived from (3) and (4) to perform multitarget state estimation without data association.

##### 2.2. Labeled Random Finite Set

With respect to the labeled RFS, a unique label is assigned to each element in the multitarget state RFS in order to perform track maintenance in a multitarget tracking scenario. Assume that is a countable label space; then target state vector is augmented with label . Hence, the labeled target state for each .

There are several types of the labeled RFS whose density is conjugate with standard multiobject likelihood function and is closed under the multiobject Chapman-Kolmogorov equation using the standard multiobject dynamic model, such as the GLMB family and the LMB family. In other words, the Bayesian recursion can be derived using these labeled RFSs. Here, we first introduce some useful symbols for illustration and provide the GLMB family and the LMB family as two types of the labeled RFSs in the following. Let represent the projection and denote the distinct label indicator function.

A GLMB is a labeled RFS on with the following distribution: in which is a discrete index set, and and satisfy

A GLMB RFS comprising a single component for each unique label can be simplified into a LMB RFS. So, the index set is singleton and the index is omitted, and a LMB is a labeled RFS on with the following distribution: and and satisfy in which is a given finite parameter set, with representing the existence probability of target and denoting the probability density of the kinematic state of target given its existence [12]. Remark that the LMB family is a special case of the GLMB family with only one term for each target.

##### 2.3. The LMB Filter

The LMB filter is derived from the Bayesian filtering by assuming multitarget state to be the LMB family RFS. In the following, we briefly recall the prediction and update process for the LMB filter which was proposed in [12].

*Prediction.* Suppose that the multitarget posterior density is an LMB RFS on space , and the parameter set . Besides, the multitarget birth model is also an LMB RFS on space , and the parameter set . Then, the predicted multitarget density is also an LMB RFS on space , and the parameter set is given as follows: where

is target state dependent survival probability function. is the integral of two real-valued function, that is, . Then, is the survival probability of target . is the state transition model for single target.

*Update.* Suppose that the multitarget predicted density is an LMB RFS on space , and the parameter set . Then, the posterior multitarget density is also an LMB RFS on space , and the parameter set is given as follows:where

Here, is the subsets union drawn from space . is the space of mappings such that only exists if . is target state dependent detection probability function. is the likelihood function for single target, and is the intensity of Poisson clutter. is the Dirac delta function.

#### 3. Target Posterior Approximation

##### 3.1. Gaussian Approximation

The LMB filter recursion described in Section 2.3 is an approximation of the Bayes multitarget tracking filter by only preserving the unlabeled intensity of the multitarget posterior density [15]. Due to its simplification, the LMB filter can be parallelized by grouping and gating [12] and runs much faster than the -GLMB filter, especially for tracking large number of targets. However, in every update step, the component number to represent target posterior state, that is, Gaussian component number or particle number, would increase linearly with the number of measurements caused by the association mapping . As a result, the computation of the LMB filter would grow exponentially over time if no component pruning techniques were applied for each target. Therefore, we propose to use a single weighted Gaussian component to approximate the posterior state of each target in order to perform the recursion more efficiently with small sacrifice on the tracking performance.

Let denote the posterior distribution of single target state; we assume that at every time step can be approximated by a single Gaussian distribution , that is, . Figure 1 shows the procedure of using a single Gaussian distribution to approximate an arbitrary distribution . The Expectation Maximization (EM) algorithm is a classical approach to fit a discrete distribution into a Gaussian Mixture Model (GMM), and there are other more advanced alternatives [17, 18].