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A Gaussian Process Regression based Dynamical Models Learning Algorithm for Target Tracking

11/25/2022
by   Mengwei Sun, et al.
0

Maneuvering target tracking is a challenging problem for sensor systems because of the unpredictability of the targets' motions. This paper proposes a novel data-driven method for learning the dynamical motion model of a target. Non-parametric Gaussian process regression (GPR) is used to learn a target's naturally shift invariant motion (NSIM) behavior, which is translationally invariant and does not need to be constantly updated as the target moves. The learned Gaussian processes (GPs) can be applied to track targets within different surveillance regions from the surveillance region of the training data by being incorporated into the particle filter (PF) implementation. The performance of our proposed approach is evaluated over different maneuvering scenarios by being compared with commonly used interacting multiple model (IMM)-PF methods and provides around 90% performance improvement for a multi-target tracking (MTT) highly maneuvering scenario.

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I Introduction

Maneuvering target tracking is a fundamental task in sensor-based applications, such as radar, sonar, and navigation [LIU2016183, 6507656, 1413764]. Based on the Bayesian framework, target tracking is usually solved via a recursive update of the posterior probability density function (pdf) of the target states. The application of Bayesian filters is based on dynamical motion models and sensor measurement models [1413764, 5730505]. However, there may be significant motion model uncertainty when targets undergo unknown or mixed maneuvering behaviors [1561886, JLYang], such that the evolution of the target state is too complex to be approximated by a reasonable number of mathematical models. The uncertainty in motion behavior can also be due to various parameters in generative models not being known a priori or if they are time-varying. The tracking performance of Bayesian filters would degrade and even become unacceptable if incorrect models or parameters are applied. This paper focuses on the issue of learning uncertain motion models with higher flexibility and incorporating the learned models into the Bayesian framework for target tracking.

I-a State-of-the-Art in Bayesian Filters

MM methods are commonly used to deal with the model uncertainty problem [1561886]. The basic idea of multiple model (MM

) methods is to use a bank of elemental filters with different motion models to capture the mixed motion behaviors. Then, the overall estimate is generated based on the results achieved by each elemental filter

[1561886]. For the cases when the target’s behavior is a time-varying or doubly-stochastic process, [JLYang] and [LiuZX] incorporated adaptive parameter estimation methods into the Bayesian filter framework to deal with the uncertainty of unknown parameters, where the unknown parameter is estimated by approximating its distribution with particles and corresponding weights. MM methods may be limited in performance if the assumed trajectories are not modeled accurately enough to capture all aspects of the motion behaviors, and the estimation of unknown parameters results in increased computation complexity.

I-B State-of-the-Art in Machine Learning Approaches

As an alternative to traditional Bayesian methods, machine learning-based tracking algorithms have been proposed recently

[8099886, 7088657, 8096087, 9011258, Batch_nonlinear, 9185002, 9272174]. In [8099886], a quadruplet convolutional neural network (CNN) based algorithm was designed to learn object association for multi-object tracking via 2D image processing. An extended target tracking method using a Gaussian process (GP) was proposed in [7088657] and [8096087] to track the irregularly shaped object whose moving trajectory follows a fixed motion model. These papers use a GP for shape modeling but not for capturing the motion behavior. A long short-term memory (LSTM

) neural network was used in

[9011258] to perform the prediction step of target localization. The proposed filter with LSTM prediction can improve the tracking accuracy by using computationally efficient low-dimensional state spaces but only considers the single-target tracking (STT) scenario. In [9011258], the continuous-discrete estimation of the target’s trajectory is viewed as a one-dimensional sparse GP, with time as the independent variable and measurements acquired at discrete times. Recursive GP-based approaches for target tracking and smoothing were developed in [Batch_nonlinear, 9185002], with online training and parameter learning. Only STT with a linear observation model (i.e. -axis and -axis coordinates) is considered. Furthermore, it was assumed that the motion model could be decomposed into separate models for the X and Y axes. In [9272174], the motion behavior of a single target is modeled as a Markov process which switches between the constant velocity (CV) and coordinated turn (CT

) models with the specified transition probabilities. The complex mixed model is learned as joint

GPs for position and velocity prediction. Then the trained GPs are integrated into the PF for tracking the maneuvering targets in real-time.

The learning-based tracking algorithms proposed in [Batch_nonlinear, 9185002, 9272174] were for motion models based on Cartesian positions which are problematic because positions are not translationally invariant, i.e. the motion model needs to be constantly updated as the target moves. This paper proposes learning a naturally shift invariant motion (NSIM) model. Specifically, the change in the target’s Cartesian positions on the -axis and -axis in one instant (i.e. Cartesian velocities), instead of the Cartesian positions, is learned as multi-independent GPs.

I-C Proposed Method and Contributions

This work follows a learning-to-track framework for tracking problems with uncertain motion models and proposes a new learning method with which the motion behaviors of targets are learned as GPs based on a training data set. Then we show how the learned motion models are integrated into the PF and belief propagation (BP) to realize target tracking. The three main contributions are:

  • We propose a new GP-based dynamical motion model learning method. The NSIM behavior of a target is learned using Gaussian process regression (GPR) based on the Cartesian velocities. The learned GPs are translationally invariant and can be applied to track objects within different surveillance regions from the surveillance region of the training data.

  • We explore the potential of using the learned motion model within the PF [978374, article] for STT and PF based BP [910572, koller2009probabilistic, 6301723] for multi-target tracking (MTT) to demonstrate that the proposed algorithm is a plug-and-play learning method in target tracking systems.

  • The proposed learning method is firstly tested in STT with different motion scenarios to demonstrate its capability to learn uncertain motion models with higher flexibility. Then, the learned motion model is plugged into MTT, which shows significant improvement in tracking performance compared with interacting multiple model (IMM)-PF methods. Additionally, offline learning is performed only once and then applied for MTT. Hence, the online tracking time is not affected by the learning process.

Compared with the available learning-based tracking methods using target positions, the motivations and advantages of the proposed GP-based NSIM model include:

  • The information available for use in the tracking problems may be limited depending on how we envisage getting the training data. For example, if the data was taken from other previously tracked targets, it could be challenging to access speed or acceleration.

  • The existing methods learn the motion models based on target positions, which are not translationally invariant and need to update the model as the target moves constantly. Moreover, the tested targets’ surveillance regions may differ from the training data’s surveillance region. On the other hand, the motion of an object is usually controlled by the application of a force, e.g. aircraft control surfaces. As such, the motion is translationally invariant. Therefore, the proposed NSIM model based on Cartesian velocities will be translationally invariant and can reduce the requirement of training data size and algorithm complexity.

  • The learned GP-based NSIM model can provide confidence intervals for particle sampling in the prediction step of the

    PF to provide a reliable estimate of model uncertainty.

I-D Paper Organization and Notations

The remainder of this paper is organized as follows. The problem solved in this investigation is defined in Section II. Section III reviews the GPR and proposes the GP-based learning method of NSIM behavior. The proposed algorithm, which incorporates the learned GPs

into the target tracking, is developed in Section IV. Simulation results and performance comparisons are presented in Section V. Conclusions are provided in Section VI. This paper presents matrices with uppercase letters, vectors with boldface lowercase letters and scalars with lowercase letters. The notations are defined and summarized in Table I.

Notation Meaning
Target states of training data sequence
Observations of training data sequence
Target states of test data sequence
Observations of test data sequence
Training time index
Test tracking time index
Target index
Cartesian positions at 2-D plane
Cartesian velocities at 2-D plane
Ground speed
Velocity heading angle
Along-track acceleration
Cross-track acceleration
Covariance matrix of process noise
Covariance matrix of measurement noise
Training data set
Motion function
Measurement function
-axis velocity motion function for training
-axis velocity motion function for training
-axis velocity motion function for testing
-axis velocity motion function for testing
Table I: The meaning of notations.

Ii Problem Definition and Background

In this section, we will introduce the system model and conventional Bayesian approaches of target tracking.

Ii-a System Model

The dynamic state-space model (DSSM) for target tracking is given by the following motion and measurement models:

(1)
(2)

Here, represents the time index, , is the target index, . For STT scenarios, . The state motion model and the measurement model are represented as and , respectively, where , , and are dimensions of the state, process noise, observation, and measurement noise vectors, respectively. The process noise and the measurement noise are assumed to be additive white Gaussian noise (AWGN). At each time instant, the sensor node detects each target with detection probability . Owing to missed detections and clutter, the number of detected measurements at time may not be equal to . is defined as the number of measurements at time and is time-varying. The set of measurements detected at time is denoted as . The measurement index is represented as and . The number of clutter points is modeled as a Poisson point process with intensity . The corresponding clutter measurements are independent and identically distributed (iid) with pdf .

Ii-B Sequential Bayesian Filters for Target Tracking

Based on the Bayesian theorem, most target tracking methods are sequential Bayesian filters which consist of the prediction and the update steps. In the prediction step, the prior distribution of targets’ states at time is calculated via the Chapman–Kolmogorov equation as [8290605]

(3)

In the update step, the joint posterior pdf of the targets’ states is calculated as [8290605]

(4)

For STT in non-linear systems, the PF uses random particles with associated weights to approximate the posterior pdf as

(5)

where is the Dirac delta function, the number of particles used at a given time is represented as . The particles are drawn from an importance density which can be set as [978374].

For MTT, the association between targets and measurements is described by two data association (DA) vectors shown in Fig. 1 [6978890, 7889057], i.e. the target-oriented association vector and the measurement-oriented association vector where is the number of measurements at time . The DA vectors are defined as [6302287]

  • if at time , the -th target generates the -th measurement.

  • if at time , the -th target does not generate a measurement, i.e. missed detection occurs.

  • if at time , the -th measurement is hypothesized to be associated with -th target.

  • if at time , the -th measurement is hypothesized not to be generated by a target, i.e. clutter.

Figure 1: Factor graphs for single-sensor MTT, is the target-oriented association vector and is the measurement-oriented association vector.

The objective of MTT is to calculate the marginal posterior pdf by

(6)

where the marginal association probability, i.e. , approximates based on the loopy BP algorithm, as given by Eqn. (22) in [6978890].

It should be noted that most of the existing STT/MTT filters are fully model-driven and, therefore, of use when the DSSM is explicitly available. However, assuming that the state motion function is given as a simple deterministic model is restrictive. These filters are intractable for some applications where the motion model is very complicated. Instead, information about the state transitions can be inferred from examples of preceding states and current states pairs. Hence, we assume that the motion model is learned using the non-parametric Gaussian process regression (GPR) [books/lib/RasmussenW06]. The measurement model is deterministic by different sensor types with no correlation over time and is in a known parametric form, i.e. we know the acquisition model. In the next section, we will discuss the application of GPR to learn motion behavior.

Iii Gaussian Progress Regression for Motion Behavior Learning

GP models are widely used to perform Bayesian non-linear regression. A

GP

model takes a training data set and a kernel as the input and outputs a joint Gaussian distribution that characterizes the value of the function at some given points

[phdthesisgp]. One practical advantage of GP is having confidence intervals for predictions that provide a reliable estimate of uncertainty. In this section, we briefly review the standard theory of GPR and then introduce the proposed GP-based method for motion behavior learning.

Iii-a Gaussian Process Regression

The GP model for regression can be considered as a distribution over a function and is defined as [7088657]

(7)

where and represent the input variables. The mean function and the covariance function of two variables and are defined as [8882261]

(8)
(9)

Let us consider a general GPR problem with noisy observations from an unknown function described as

(10)

Here,

is the variance of the noise. The training data set is denoted as

, where are the inputs with the corresponding outputs . The purpose of GPR is to derive the latent distributions of the vector at the test inputs , conditioned on the test measurements and the training data set

. Specifically, the joint distribution of the training measurements

and the function at one test point, i.e. , is assumed to be Gaussian and is given by

(11)

Here, , is the covariance of , and denotes the covariance matrix for the training input data [9272174]. The conditional distribution of the function is then derived from the joint density in (11) and written as the following standard forms:

(12a)
(12b)
(12c)

where is the abbreviation for . The most widely used kernel function is the squared exponential (SE) covariance function [books/lib/RasmussenW06], which is also used in this paper,

(13)

Here, the hyperparameters

and represent the prior variance of the signal amplitude and the kernel function length scale, respectively.

Iii-B Gp-based Motion Behaviors Learning — Limited Information of Training State

This paper proposes a new GP-based learning method. I.e., the naturally shift invariant motion (NSIM) behavior of a target is learned as GPs based on the Cartesian velocities. The Cartesian positions and are converted into the Cartesian velocities as and , respectively. The NSIM models, i.e. and , are learned as GPs, where . As most target maneuvers are coupled across coordinates, the learned models capture the crucial characteristics of the target behavior during maneuvers by using both Cartesian velocities as the input of the GPs. Specifically, the motion behavior of the targets in (1) is rewritten as

(14a)
(14b)
(14c)
(14d)

The proposed GP-based method for prediction shown in Fig. 2 involves offline training and online prediction steps. The training data is used offline to capture the statistical characteristics of the motion behavior of the targets, which are then transferred to online tracking for state prediction.

  • [leftmargin=1.5mm]

  • Offline training: To train the Cartesian velocity motion functions, we take training data sets and . Specifically, the training inputs for both GPs are the same as

    while the output data are different and set as

    Here, denotes the number of training samples. The Cartesian / velocity motion functions in the offline training process, i.e. and , are modeled as two GPs shown in (15) and (16).

    (15)
    (16)

    Here, and are the input variables, the squared exponential (SE) defined in (13) are used to calculate and with the same hyperparameters and . Examples of the learned GPs are shown in Fig. 3, where the motion behavior of the target follows the gradual coordinated turns (GCT) model, defined in Section V. The turn rate of the target follows the left and coordinated right turns (/s for 10s). In Fig. 3, the and axes represent the input of the GP, i.e., Cartesian velocities, and the Z axis represents the prediction value. The training samples are shown in blue points. The orange-edge and green surfaces represent the mean value of the predictive Gaussian distribution for the -axis and -axis velocity, respectively, which are joint functions of current Cartesian velocities.

    Figure 2: The factor graph of the proposed GP-based learning algorithm for state prediction mainly includes offline training and online prediction processes. Legend: Circles – variable nodes; Squares – factor nodes. The process for the prediction on -axis is marked in blue. The process for the prediction on -axis is marked in orange.
  • Online Prediction: In the following, we describe the estimation of the -axis position . The estimation of the -axis position is done similarly and so is not discussed. The estimated parameters at time include the Cartesian position and velocity estimates represented as . The predictive distribution of the tracking state is partitioned as

    (17)

    Here, . The pdfs in (17) are further calculated in (18) – (19),

    (18)
    (19)

    where the predictive mean and covariance are calculated in (20) and (21), respectively.

    (20)
    (21)

    where and are the abbreviations for and .

Figure 3: Example of two-dimensional GPR. Legend: Points – training samples; Surface – the mean function of the trained GPs. (a) GP for -axis velocity. (b) GP for -axis velocity.

Iv Incorporation of Learned Gp-based Nsim Model into Target Tracking

For nonlinear and non-Gaussian motion and observation models, the pdf of the hidden states in (6) and (17) cannot be evaluated in closed form. Therefore, particle-based implementation algorithms are commonly used for target tracking. The proposed GP-based NSIM model learning method is plugged into the PF and PF-based belief propagation (BP) algorithms in this section. When the parametric motion model is unavailable, particles are drawn based on the learned GPs. The learned GP-based PF-MP implementation for MTT is summarized in Algorithm I. The GP-based PF for STT is implemented the same way without executing Step 5, i.e. the loopy BP. Note that the data association for MTT is obtained by running an iterative BP algorithm following [6978890]. Therefore, the proposed algorithm is still valid when the target number is unknown and time varying.

0:  Measurement model and training data .
1:  Off-line training: The X/Y-axis velocity motion functions are modeled as two GPs, ,
2:  On-line tracking:
3:  for  do
4:     Draw the particle vector as: , , , . Calculate the approximation of the predictive state as .
5:     Execute the loopy BP by passing messages repeatedly, , . Then, approximate the marginal association probability as .
6:     Update associated weights, .
7:     Weights normalize and resampling: The particles with negligible weights are replaced by new particles in the proximity of the particles with higher weights.
8:  end for
Algorithm 1 Learned GP based PF-BP filter

Iv-a Draw Particles based on Learned GPs

The particle vectors for the -th target at the preceding time slot are represented as

with the corresponding posterior weights , . The particle vector consists of the Cartesian position particles on the -axis and -axis, i.e. , and the Cartesian velocity particles on the -axis and -axis, i.e.

. The posterior probability of

-th target’s positions and velocities at time is approximated as

(22)

The new particles at time are drawn based on the importance densities represented as . Specifically, the particles of the Cartesian velocities are sampled from the Gaussian distributions based on the learned GPs as

(23)
(24)

Here, is the training data. The mean and covariance are calculated based on (20) and (21), respectively. Next, the position particles and are sampled according to (25) and (26) as

(25)
(26)

Then, the prediction of the position state at time is approximated as,

(27)

where .

Iv-B Data Association

In the PF-based BP, the marginal association probabilities, and , are calculated based on the loopy BP, i.e. updating the messages between factor nodes and , as shown in Fig. 1. Specifically, the two sets of messages are represented as and to denote the message passed from node to node and the message passed from the node to the node , respectively. The messages are updated as [6978890]

(28)
(29)

where the factor indicates the association probability calculated based on the likelihood ratio (LR) as

(30)

where when no measurement is correctly associated to -th target, and the predictive distribution is approximated as (27). The messages are passed between the latent association variables until convergence, i.e. (28) and (29) are repeatedly processed until the conditions of convergence is satisfied [DBLP:journals/jmlr/IhlerFW05]. Finally, the approximations of the marginal association probabilities are obtained as

(31)
(32)

Iv-C PF-based State Estimation

After these marginal association probabilities in (31) and (32) are obtained, the marginal posterior pdf in (6) is then approximated as

(33)

where the updated weights are calculated by

(34)
(35)

Based on the key idea of the PF [978374, 1232326], the maximum a posteriori (MAP) estimation of hidden states for the -th target at time , i.e. , are approximated as

(36)

where represents the estimates of the -th target’s position, and represents the estimates of -th target’s Cartesian velocities between and .

Based on the above descriptions, the implementation of the proposed learned GP based PF-BP filter for target tracking is summarized in Algorithm I.

V Simulation Results

This section introduces the dynamic state-space model (DSSM) used to generate simulation data and different tracking maneuvering scenarios in the first two subsections. Then, the performance comparisons for STT and MTT are presented. Specifically, the proposed learned GP method is applied for STT over different maneuvering scenarios. The tracking performance demonstrates the validity and robustness of the proposed training method to track the targets within different surveillance regions and motion behaviors. Then, the data-driven MTT algorithm is compared with the oracle PF with fully known system information and the interacting multiple model-particle filter (IMM-PF) methods that know the correct DA [1561886]. The numbers of the used particle are set to be the same for all compared filters.

V-a Dynamic State-space Model (DSSM)

This paper generates the simulated target trajectories and measurements following the standard curvilinear-motion kinematics model [1261132]. The state of -th target at time in (1) and (2) is represented as , where are the target Cartesian position, and represent the ground speed and the velocity heading angle, respectively. The along-track and cross-track accelerations in the horizontal plane are denoted as and , respectively. By setting different values of