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Sensor systems are not always equipped with the ability to track targets. Sudden maneuvers of a target can have a great impact on the sensor system, which will increase the miss rate and rate of false target detection. The use of the generic particle filter (PF) algorithm is well known for target tracking, but it can not overcome the degeneracy of particles and cumulation of estimation errors. In this paper, we propose an improved PF algorithm called PF-RBF. This algorithm uses the radial-basis function network (RBFN) in the sampling step for dynamically constructing the process model from observations and updating the value of each particle. With the RBFN sampling step, PF-RBF can give an accurate proposal distribution and maintain the convergence of a sensor system. Simulation results verify that PF-RBF performs better than the Unscented Kalman Filter (UKF), PF and Unscented Particle Filter (UPF) in both robustness and accuracy whether the observation model used for the sensor system is linear or nonlinear. Moreover, the intrinsic property of PF-RBF determines that, when the particle number exceeds a certain amount, the execution time of PF-RBF is less than UPF. This makes PF-RBF a better candidate for the sensor systems which need many particles for target tracking.

For surveillance, a sensor system is installed to search for targets and provide reliable detection within the given region. The sensor system can measure the range and bearing of the targets, but it can not track them. In spite of the recent advances in sensor technology, there are no devices that can detect the manned maneuvers of a tracked target in surveillance and guidance systems [

For implementing target tracking in a sensor system, the extended Kalman filter (EKF) was introduced [

This paper proposes a PF-RBF algorithm for target tracking in sensor systems. It is an improved PF algorithm which uses the radial-basis function network (RBFN) in sampling. According to the observations, PF-RBF uses RBFN to approximate the moving trajectory, construct the process model, perform sampling and decrease the cumulated effect of errors. We compare PF-RBF with UKF, PF and UPF. The results show that PF-RBF can track targets effectively, especially when the observation model of the sensor system is nonlinear, and it also displays good real time performance.

The remainder of this paper is organized as follows: target motion and observation model of sensor systems are introduced in section 2. The proposed PF-RBF algorithm is presented in section 3. The experimental results are described in section 4. Finally, the conclusions are given in section 5.

In a sensor system, the measurement sequences are in polar or Cartesian coordinates, but the target dynamics are best described in Cartesian coordinates, so we assume that measurement sequences described in Cartesian coordinates are available. The target tracking is always formulated as a dynamic state space model. The general state space model can be broken down into a state transition and state measurement model:
_{k}_{k}_{k}_{k}_{k}_{k}_{k}

It is always assumed that the states follow a first order Markov process and the observations are assumed to be independent given the states

In target tracking, the posterior density _{k}_{1:}_{k}_{1:}_{k}_{1}, _{2}, ⋯, _{k}

The generic PF algorithm is a sequential importance sampling method which is based on Monte Carlo simulation and Bayesian sampling estimation theories. Various PFs all contain three important steps: sampling current value of each particle, evaluation of the recursive important weights and resampling. Recent research has focused almost exclusively on the weighting and resampling for improving the tracking accuracy [

In Bayesian sampling estimation theory, the posterior density _{k}_{1:}_{k}_{k}_{1:}_{k}_{−1})

Then PF uses the Monte Carlo simulation method to approximate the posterior density by

For solving the difficulty of sampling from the posterior density function, the sequential importance sampling method is used, which samples from a known, easy-to-sample, proposal distribution _{0:}_{k}_{1:}_{k}_{0}_{:}_{k}_{1:}_{k}

The recursive estimate for the importance weights of particle

Then the estimated state can be approximated by

Radial basis functions are a special class of function. Their responses decrease (or increase) monotonically with the distance from a central point [_{j}_{2} denotes the Euclidean norm, _{j}_{j}_{j}

RBFNs present good approximation properties. The RBFN family is broad enough to uniformly approximate any continuous function on a compact set. For any continuous input-output function _{p}_{j}^{m}^{0} ∈

Because the trajectory of target is a typical continuous function, RBFN can approximate it and construct the dynamic process model for state estimation in PF algorithm well. In PF-RBF, we use previous observations and current prediction to train RBFN; the details will be discussed below.

The process model of the generic PF algorithm always uses a single motion model which is static and cannot offer a dynamic and consistent approximation of state variables for target tracking [

The objective of our algorithm is to perform a robust and accurate approximation of state variables and decrease the execution time. Our algorithm contains five steps: (1) constructing RBFN based on previous observations _{0:}_{k}_{−1} and current prediction _{k}_{0:}_{k}_{−1} do not equal to the states of target, which just have relationship to the state variable _{0:}_{k}_{−1} according to the observation model _{0:}_{k}_{−1} should be inferred from observations _{0:}_{k}_{−1}. Because of the observation noise, we can only get the approximate state variable _{0:}_{k}_{−1} instead of _{0:}_{k}_{−1}; (2) sampling new value of each particle based on RBFN; (3) evaluating the recursive importance weights; (4) resampling; (5) output. The pseudo-code for our algorithm is outlined in Algorithm 1:

Algorithm 1

Initialization:

_{0})) / draw the states from the prior

Constructing RBFN

Use the previous observations _{0:}_{k}_{−1} to infer the previous approximate state variable _{0:}_{k}_{−1};

Predict the current state by kinematic theory with the given time interval

Construct RBFN with the previous real state of target _{0:}_{k}_{−1} and current state prediction _{k}

Sampling

Sample

Evaluating

Evaluate the recursive importance weights by (

Normalize the importance weights

Resampling

Multiply / suppress samples

Output

Output the estimated state

Set

As illustrated in Algorithm 1, in the PF-RBF algorithm, the RBFN is dynamically trained as the process model for the sampling step. For approximating the state of the target, we use the previous observations and current prediction based on observations as the samples to train the RBFN. Then, each particle uses the estimated state sequence

Because RBFN is trained with the observations and prediction based on observations, it will not be impacted by the posterior distribution error of each particle, and can approximate the real trajectory of target effectively. With the guidance of RBFN, each particle can estimate the probability distribution more effectively and keep the multimodality at the same time.

Although the computation complexity of RBFN is O(n^{3}) [_{k}_{−}_{m:k}_{−1} to train the RBFN, which can significantly reduce the computation complexity.

The estimation improvement obtained by PF-RBF algorithm is illustrated in this section. The UKF is more suitable than the EKF for proposal distribution generation within the particle filter framework [

For simulation, a time-series is generated by the following process model
_{k}_{−1} + _{k}_{−1} ·

For comparing tracking performance, a non-stationary observation model is used as in [_{2} = 0.5. The observation noise, _{k}_{k}_{k}_{k}_{−}_{m:k}_{−1} used for training the RBFN is 20. The time interval is 1s. The root-mean-square-errors (RMSEs) of the estimation values are acquired for comparison.

In

As illustrated in

The RMSEs of PF-RBF are all small whether the observation model is linear or nonlinear. This means that the PF-RBF is robust and effective for both linear and nonlinear observation models. When a maneuver occurs ( 20 <

Then we compared the mean and variance of RMSEs generated by each algorithm with 200 particles over 100 independent runs. As shown in

In each PF algorithm, the execution time and accuracy are partially determined by the particle number. In this section, we compare the average RMSE and execution time of PF, UPF and PF-RBF. The particle number increases from 10 to 200.

The average RMSE calculated over 100 independent runs, which decreases when the particle number increases, are shown in

In this paper, we focused on target tracking in sensor systems. The PF-RBF algorithm was proposed, which trains RBFN to approximate the trajectory of target and constructs dynamic process model according to the previous observations and current predictions. The trained RBFN is used to perform sampling steps for each particle, instead of the classical process model. With the guidance of RBFN, each particle can give an accurate proposal distribution, the cumulated effect of errors can be decreased, and the sensor system remains convergent even if the target maneuvers. The target tracking experiment results verify that when the observation model is linear, PF-RBF perform better than UKF, PF and UPF, and for a nonlinear observation model, PF-RBF is most robust and accurate, except when the particle number is less than 20. Moreover, the rate of change of execution time of PF-RBF is about seven times less than that of UPF, it this makes the execution time of PF-RBF less than that of UPF when the particle number exceeds a certain number, which in this paper is 50. It should be noted that the critical particle number thresholds depend on the given problem, and may be different in other problems. However, it still implies that PF-RBF is suitable for dealing with sensor systems which need many particles for multi-target tracking.

This paper is sponsored National Grand Research 973 Program of China (No. 2006CB303000) and National Natural Science Foundation of China (No. 60673176, No.60373014, No. 50175056).

The shape of function

Plot of estimated results generated from a single run of the different filters with non-stationary observation model.

Errors of different filters versus the process noises in a single run.

The average RMSEs of three algorithms calculated over 100 independent runs with the linear and non-linear observation model.

The average execution time of three particle filters for 60 steps tracking over 100 independent runs.

Estimated results generated by each algorithm where the mean and variance of RMSEs are calculated over 100 independent runs. The particle number is 200.

Algorithm | Mean of RMSEs | |
---|---|---|

| ||

Nonlinear | Linear | |

UKF | 0.58461 | 0.0067026 |

Generic PF | 0.47292 | 0.1485 |

UPF | 0.085445 | 0.0080179 |

PF-RBF | 0.0187598 | 0.0066431 |