The multiple model multi-Bernoulli filter based track-before-detect using a likelihood based adaptive birth distribution

doi:10.1016/j.sigpro.2020.107501

Signal Processing

Volume 171, June 2020, 107501

https://doi.org/10.1016/j.sigpro.2020.107501 Get rights and content

Highlights

•
To address the tracking problem of multiple maneuvering targets in low SNR environment, a multi-Bernoulli filter based track-before-detect (MB-TBD) for multiple model (MM) is proposed.
•
To enhance the tracking performance of multi-Bernoulli filtering when tracking multiple maneuvering targets, a measurement driven adaptive birth distribution for MB filter is generalized to TBD measurement model.
•
A new likelihood based adaptive birth distribution for MB-TBD filter is proposed which is able to reduce the computational cost and eliminate the time delay during target birth.
•
The sequential Monte Carlo implementation of the proposed MB-TBD filtering method is presented to estimate multi-target states.

Abstract

In this paper, a novel multi-Bernoulli filter based track-before-detect (MB-TBD) algorithm is proposed, to solve the tracking problem of multiple maneuvering targets. We incorporate the multiple motion models into the basic MB-TBD filtering, and then derive the closed-form recursive equations including both prediction and update steps based on optimal Bayesian filtering. Moreover, to accommodate the case of unknown prior knowledge for target births, a likelihood based adaptive birth distribution for the MB-TBD is proposed. The implementation of the proposed algorithm with adaptive birth distribution is presented using sequential Monte Carlo (SMC) technique. The performance of the proposed algorithm is demonstrated in challenging scenarios including multiple highly maneuvering objects.

Introduction

Since the motion of maneuvering targets needs to be described by a combination of motion models but not a fixed one, the problem of jointly detecting and tracking maneuvering targets is more complex than that of nonmaneuvering targets, and has attracted great attention in recent years [1], [2], [3], [4]. In multi-target environment, the number of targets varies with time due to targets birth and death. Tracking multiple maneuvering targets involves jointly estimating the states of targets and their number at each time step under the condition that target maneuver, data association, existing noise and clutter. As such, multiple maneuvering targets tracking problem is extremely challenging in both theory and implementation.

Recently, many filtering algorithms with multiple models (MM) or jump Markov system (JMS) have been developed [5], [6], [7], of which the MM has been shown to be an effective approach for single maneuvering target tracking, in this approach the target can switch between a set of models in a Markovian fashion. Some algorithms generalize the MM to multi-target tracking field. Intuitively, these multi-target MM algorithms can be categorized into two groups. One group is based on traditional data association techniques such as joint probabilistic data association (JPDA) [8] and multiple hypothesis tracking (MHT) [9], whereas these data association-based approaches suffer from highly computation cost. The other group is random finite set (RFS) [10] based method which can avoid the data-association process.

The probability hypothesis density filter (PHD) [11] and the cardinalized PHD (CPHD) [12] are two famous multi-target tracking algorithms based on RFS theory, they are both moment approximations of the Bayes multi-target filter, which operate on the single target state space to avoid the computational problem that arises from data association. The PHD filering approaches for MM with Gaussian mixture implementations [1] have shown their capability in tracking multiple maneuvering targets.

In addition to PHD and CPHD filter, the multi-Bernoulli (MB) filter [10] is also a promising multi-target tracking algorithm in the RFS framework, unlike the PHD and CPHD filter, MB filter directly propagates the multi-target posterior density but not its moments and cardinality distributions, MB filter can be more efficient and accurate in problems that require particle implementations or target individual existence probabilities. Due to the excellent performance in multi-target tracking, MB filters have been successfully applied in a number of practical problems, such as radar target tracking [13], image target tracking [14], [15], sensor control [16], [17], visual tracking [18], [19] and information fusion [20], [21], [22]. Recently, the labeled multi-Bernoulli RFS [23] was frequently introduced to address the target trajectories and their uniqueness. Vo et al. proposed a particular multi-target density called generalized labeled multi-Bernoulli (GLMB) density [23]. Whereafter, the variant of GLMB called δ-GLMB [24] and two computational efficient δ-GLMB i.e. labeled multi-Bernoulli density (LMB) [25] and the marginalized δ-GLMB (Mδ-GLMB) [26] are also have been developed. It has been shown that the labels of multiple targets are of importance in [23], [24], [25], [26], [27], but there are still some cases that the labels of targets are not required such as in collision avoidance systems and in presence of closely spread targets. Hence the unlabeled multi-Bernoulli filters still have important significance and widespread application.

In [14], a MB filter based track-before-detect (MB-TBD) is proposed which utilizes the raw, unthreshold measurements (also called TBD measurements [28]) but not the traditional threshold measurements to cope with the joint detection and tracking problem. TBD measurements contain more information, i.e. amplitude, than the traditional measurements, it can obviously enhance the tracking performance [29], [30]. On account of this advantage, TBD measurement model is widely used especially in low signal-to-noise ratio (SNR) environment by incorporating dynamic programming (DP) [31], particle filtering (PF) [32] and RFS based methods [33].

In this paper, we focus on solving the multiple maneuvering targets tracking problem by combining MB-TBD with MM. The major contributions are as follows:

a) We present a multi-Bernoulli filter based track-before-detect for MM. We combine the basic MB-TBD with MM to accommodate target birth, death and switching dynamics so that the proposed filtering algorithm is capable to deal with the tracking problem of multiple maneuvering targets. Since the basic MM is not general enough to cope with multi-target tracking, the MM multi-target model for MB is introduced. By using the MM multi-target model, the target state of each multi-Bernoulli component is augmented with an additional motion model label, and the augmented state of each component evolves with time according to a finite state Markov chain. The specific closed-form solution for multiple model extension based MM approach called MM-MB-TBD filter is derived based on optimal Bayes filtering.

b) We present an efficient adaptive birth distribution for the proposed MB-TBD filter. In the standard MB filter, the region where targets appear in the surveillance area is usually known as prior knowledge. However, in practice, there exist some cases that the prior knowledge of the birth information is completely unknown, making the prior knowledge based birth procedure difficultly implemented. The filtering algorithm using prior knowledge based birth procedure is hardly to re-initialize targets after the corresponding Bernoulli components have been truncated possibly due to the successive miss-detection¹. To this end, we generalize the measurement driven adaptive birth distribution for basic MB filter [25], [34] to TBD measurement model. The adaptive method in this paper means that the multi-Bernoulli birth distributions are built only depend on measurements, there are some other adaptive methods applied in communication engineering and control [35], [36], [37], [38]. While our analysis shows that this birth distribution initialization method is not rigorous enough so that most clutters are allowed to create birth components. We proceed proposing a likelihood based adaptive birth distribution which is able to pick out the measurements which are likely to arise from real targets.

In simulations, the performance of the proposed MB-TBD filtering algorithm with SMC implementation is tested in challenging multiple maneuvering tracking scenarios.

Previous work have been demonstrated in the conference paper [39]. This paper focuses on a more complete numerical study and the adaptive birth distributions. In Section 2, the necessary background on RFS, MM and TBD measurement model is given. Our MM-MB-TBD filter and the adaptive birth distributions are proposed in Section 3. The implementation for the proposed MB-TBD filtering algorithm is presented in Section 4. In Section 5, the performance of the proposed algorithm is analyzed. Concluding remarks are given in Section 6.

Section snippets

Target dynamic model

In this paper, two typical target kinematic models are considered, the standard constant velocity (CV) model and the co-ordinated turn (CT) model [40] with a known turn rate.

Given the kinematic state of individual target $x_{k} =$ [ p_x, ${\dot{p}}_{x},$ p_y, ${\dot{p}}_{y}$ ]′ $\in R^{υ}$ consists of the position (p_x, p_y) and velocity $({\dot{p}}_{x}, {\dot{p}}_{y}),$ where “′” denotes the matrix transpose, for CV model, the kinematic state transition equation is described as $x_{k} = F_{k - 1}^{1} x_{k - 1} + w_{k - 1},$ with $w_{k - 1} \sim N (\cdot; 0, Q_{k - 1}^{1}),$ $N (\cdot; R, Ω)$ denotes the Gaussian

MM-MB-TBD Algorithm with adaptive birth distributions

In this section, we introduce the MM method at first, since the MM method is not general enough to deal with the multiple maneuvering targets tracking problem, an multi-target MM model based on MB is presented, then the MM-MB-TBD algorithm and the adaptive birth distributions are proposed.

SMC Implementation of MM-MB-TBD

In the following, we present the SMC implementation for the proposed multi-Bernoulli prediction step and update step.

SMC Prediction: Suppose that at time k, the multi-Bernoulli posteriors with the mode-independent parameters are given $π_{k - 1} = {(r_{k - 1}^{(ℓ)}, p_{k - 1}^{(ℓ)} (x, e))}_{ℓ = 1}^{N_{k - 1}}$ . According to the proposed MM-MB-TBD algorithm, the set of weighted samples is generated not only by the kinematic state but also the target motion model, a new parameter $e_{k - 1}^{(i, j)}$ is used to denote the label of model in effect

Simulations

The performance of the proposed MB-TBD filtering algorithms are evaluated in two scenarios in term of estimating the number of targets and optimal subpattern assignment (OSPA) distance [49]. Comparisons with KP-MB-TBD and ABD-MB-TBD will also be given as benchmark.

We consider a two-dimensional scenario over 120 × 120 resolution cells with cell length $Δ x = Δ y = 15$ m. Measurement in each cell is Rayleigh distribution as (5). Up to 4 targets are included in the scenario whose trajectories during the

Conclution

In this paper, we addressed the problem of multiple maneuvering targets tracking with multi-Bernoulli filter based track-before-detect (MB-TBD) for multiple model (MM). Firstly, we proposed a MB-TBD filtering algorithm for MM by extending the basic MB-TBD with MM to accommodate target birth, death and switching dynamics, the sequential Monte Carlo (SMC) implementation was also provided. Secondly, we analyzed the drawbacks of the birth distribution with known target locations, then an adaptive

CRediT authorship contribution statement

Lei Chai: Conceptualization, Methodology, Software, Investigation, Writing - original draft, Writing - review & editing. Lingjiang Kong: Validation, Formal analysis, Supervision. Suqi Li: Conceptualization, Methodology, Validation, Formal analysis, Resources, Supervision. Wei Yi: Conceptualization, Methodology, Validation, Formal analysis, Resources, Supervision, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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^☆: This work was supported in part by the National Natural Science Foundation of China under grant 61771110, in part by the Chang Jiang Scholars Program, in part by the 111 Project No. B17008.

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The multiple model multi-Bernoulli filter based track-before-detect using a likelihood based adaptive birth distribution☆

Highlights

Abstract

Introduction

Section snippets

Target dynamic model

MM-MB-TBD Algorithm with adaptive birth distributions

SMC Implementation of MM-MB-TBD

Simulations

Conclution

CRediT authorship contribution statement

Declaration of Competing Interest

Signal Process.

Signal Process.

Neurocomputing

Pattern Recognit.

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IEEE Trans. Aerosp. Electron. Syst.

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IEEE Trans. Signal Process.

Hybrid grid multiple-model estimation with application to maneuvering target tracking

IEEE Trans. Aerosp. Electron. Syst.

The multiple model Vo–Vo filter

IEEE Trans. Aerosp. Electron. Syst.

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Multiple hypothesis tracking for multiple target tracking

IEEE Aerosp. Electron. Syst.

Statistical Multisource-Multitarget Information Fusion

The gaussian mixture probability hypothesis density filter

IEEE Trans. Signal Process.

Analytic implementations of the cardinalized probability hypothesis density filter

IEEE Trans. Signal Process.

Robust multi-Bernoulli filtering

IEEE J Sel Top Signal Process

Joint detection and estimation of multiple objects from image observations

IEEE Trans. Signal Process.

Multi-Bernoulli based track-before-detect with road constraints

2012 15th International Conference on Information Fusion

Robust multi-Bernoulli sensor selection for multi-target tracking in sensor networks

IEEE Signal Process. Lett.

Multi-Bernoulli sensor control via minimization of expected estimation errors

IEEE Trans. Aerosp. Electron. Syst.

Visual tracking in background subtracted image sequences via multi-Bernoulli filtering

IEEE Trans. Signal Process.

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IEEE Trans. Signal Process.

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IEEE Trans. Signal Process.