A Novel Probabilistic Data Association for Target Tracking in a Cluttered Environment

The problem of data association for target tracking in a cluttered environment is discussed. In order to improve the real-time processing and accuracy of target tracking, based on a probabilistic data association algorithm, a novel data association algorithm using distance weighting was proposed, which can enhance the association probability of measurement originated from target, and then using a Kalman filter to estimate the target state more accurately. Thus, the tracking performance of the proposed algorithm when tracking non-maneuvering targets in a densely cluttered environment has improved, and also does better when two targets are parallel to each other, or at a small-angle crossing in a densely cluttered environment. As for maneuvering target issues, usually with an interactive multi-model framework, combined with the improved probabilistic data association method, we propose an improved algorithm using a combined interactive multiple model probabilistic data association algorithm to track a maneuvering target in a densely cluttered environment. Through Monte Carlo simulation, the results show that the proposed algorithm can be more effective and reliable for different scenarios of target tracking in a densely cluttered environment.


Introduction
As an important part of information fusion theory, target tracking has obtained more and more attention in wide applications. The main purpose of target tracking is to prevent false tracking or lost tracking, and ensure that the tracking is effective and accurate. Ship target tracking on the surface of the sea has been widely used in both military and civil fields. The performance of using radar to track surface ships are usually affected by sea clutter [1]. Due to the sea clutter, filter algorithms, such as a Kalman filter, and a series of improved filter algorithms like EKF (extended Kalman filtering) and UKF (unscented Kalman filter), are used in target tracking systems [2], and the data association is also adopted which can confirm the probability of measurement coming from the target. Previous studies have shown that the nearest-neighbor (NN) [3] algorithm works reasonably well with targets in sparse scenarios, and the probabilistic data association (PDA) [4] is suitable to track a single target in a cluttered environment, which, considering all of the measurements, falls into the validation gate. For the case of more than one target in the cluttered environment, an extension of probabilistic data association (PDA) was derived, called joint probabilistic data association (JPDA) [5], where the association probabilities are computed from the joint likelihood functions corresponding to the joint hypotheses associating all of the returns to different permutations of the targets and clutter points. Another advanced algorithm, multiple hypothesis tracking (MHT) [6], is categorized as a "deferred

Target Tracking Dynamic System in a Cluttered Environment
In a target tracking dynamic system, the major purpose is to estimate the parameters of the target that evolve sequentially with time. As measurement data become available, the unknown parameters forming a state vector are estimated sequentially using measurement data. Tracking in time with the problem of parameter estimation can be readily formalized in the following framework. In most surface ship tracking applications, the noise terms are additive, so the expression of the state equation and the measurement equation are defined as follows: X(k + 1) = F(k)X(k) + G(k)w 1 (k) (1) z(k) = H(k)X(k) + w 2 (k) (2) The state equation (Equation (1)) describes the evolution or transition of X(k) with k and assumes that the state follows a first-order Markov process. Function F(k) is a known function related with the state vector at time k − 1 to the time k. Function G(k) is a known noise matrix. w 1 (k) is the process noise and its covariance is Q(k).
The measurement Equation (2) relates function z(k) to state vector X(k) through a known function H(k), w 2 (k) is the measurement noise, and its covariance is R(k). The process noise and measurement noise are independent, zero mean noise, with known covariance, respectively. In the sea clutter environment, at time k, the measurements can come from the target or the clutter, The measurements at time k are z(k) = {z 1 (k), · · · z n (k)}. n is the measurement number at time k. The cumulative set of measurements until time k is Z k = {z(1), · · · z(k)}.

Probabilistic Data Association (PDA) Algorithm
As for PDA, there are two assumptions: (1) a measurement can only have one source; (2) no more than one measurement can originate from a target. Suppose the set of validated measurements at time k , m k is the validated measurements at time k. The cumulative set of measurements until time k is Z k = {z(j)} k j=1 . The validation region is centered on the predicted measurement of the target, which is set up to accept the measurement at a certain probability. The validation region is defined as follows: Here, we use a Kalman filter to estimate the state of the target. The prediciton of the state and the measurement at time k are defined, respectively, asX(k|k T is the covaiance of the prediction state. γ is the threshold which can be get from the chi-square distribution. Denote association events as θ i (k) = {z i (k) is originated from the target}, i = 1, 2 · · · m k θ i (k) = {none of the measurement is originated from the target}, i = 0. The probability of either the measurements originating from the target or clutter (false alarm) is expressed as the likelihood (Equation (4)) condition that all of the measurements lie in the validation region: As we know, the definition of events are mutually exclusive and limited: If there is no measurement that falls into the validation region, namely i = 0, we use the state prediction to approximate the updated values of the target state.
The error covariance associated with the updated state estimation can be expressed as follows:

A New Association Probability of PDA
The association probability is the likelihood of θ i (k) on the condition that all of the measurement lie in the validation region as explained by Equation (3). Using the parametric PDA [4], the association probability is as follows: is the probability of detection, and P g is the probability that the target measurement falls into the validation region. β i (k) is an important parameter which directly influences the estimation of the target state. The PDA algorithm considers all of the measurements falling into the validation region, and there is no clear and effective relationship between the validation measurement and the prediction measurement. Since the clutter is distributed randomly in position, number, and density in the tracking space, the measurement originating from the target is more likely near the predicted measurement. Thus, this important prior information is introduced as a distance weight to optimize the association probability. The weight calculation method in the literature [21] has certain limitations, namely, when the number of validation measurements is just one, the value will be infinite, which can lead to filter failure. Here we propose a new expression of distance weight: where d i (k) is the Mahalanobis distance between measurement i and the prediction measurement at time k, the expression is is the prediction measurement, and the expression isẑ i (k|k − 1) = H(k)X(k|k − 1). S(k) is the covariance matrix of innovation in the Kalman filter, and its expression is The association probability expression with the distance weighting is show below: The normalized association probability expression is show as Equation (10): Here, with the new association probability, we use a Kalman filter to estimate the target state. The calculation is shown in Equations (5) and (6).

Improved C-IMM-PDA Algorithm
The traditional algorithm is just a combination of the interactive multiple model and the probabilistic data association together. On the one hand, each model uses an independent validation region to obtain the validated measurement. However, for the entire system, the validation region may not be optimal, which can increase the target estimation error, and finally lead to a lost target or tracking the wrong target. On the other hand, with the increase in the number of targets and the number of target motion modes, the computational cost will also increase. Thus, the literature [20] proposes a new IMM-PDA structure (combined interactive multiple-model probabilistic data association algorithm, C-IMM-PDA), which uses the same validation region to obtain the validated measurement, and also uses comprehensive measurement to update and estimate the target state. This method can better realize target tracking and save on the computational cost. Here, based on C-IMM-PDA, we use the improved PDA filter to obtain the association probability which can enhance the accuracy of the estimation target state. The improved C-IMM-PDA algorithm implementation stated is as follows.

Interaction Mixed with the Estimate from the Previous Time
Starting withX j (k − 1|k − 1), its covariance, P j (k − 1|k − 1) and the model probability u j (k − 1), the mixed initial condition for the filter matched to model j can be expressed as follows: normalizing factor, and c j = M ∑ j=1 p ij u j (k − 1). p ij is the switch probability from model i to model j.

Submodel Filter and Prediction
With the mixed initial stateX 0j (k − 1|k − 1) and its covariance P 0j (k − 1|k − 1), here we use a Kalman filter to estimate the target state, the predicted stateX j (k|k − 1), and its corresponding covariance P j (k|k − 1), the predicted measurementẑ j (k), and the covariance matrix of innovation S j (k) for each motion model j can be expressed as follows: Here, we use the comprehensive prediction of the measurement and the comprehensive prediction covariance matrix tectonic tracking region. Its definitions are as follows: Here, using definitions from Equations (13) and (14), the parameters of the comprehensive validation region can be given as: obeys the normal distribution so that d 2 (k) obeys the chi-square distribution with the degree of freedom n z , we use the method of hypothesis testing to determine whether z(k) falls into the validation region.
Suppose m k is the number of validated measurements at time k. Each filter using comprehensive measurement as the prediction measurement to update the state, the update equations are the same as the normal Kalman filter. Comprehensive measurement can be expressed as follows: where β j l (k) can be obtained from Equation (10). Using Equation (16) to update each filter will lead to the estimation covariance being large; here we use Equation (17) to correct its covariance:

Model Probability Update
The model probability is updated by u l (k) as in Equation (18): Here, considering model probabilities update as an independent part of IMM,

Interaction Ouput Estimation Results
Finally, the model-conditioned estimates are calculate using Equation (19).

The Dynamic Model
Assume the target state is X(k) = [x, .
x, y, . y]. The two-dimensional plane dynamic model of the target and the measurement equation can be described by the following equations: In the constant velocity model (CV model), the movement characteristic is the velocity of the target, which remains the same, and the model parameters are as follows: where The turning motion of the target is usually referred to as a coordinated turn (CT model). With a known target turning rate ω, the model parameters are as follows: where

Simulation and Analysis of Non-Maneuvering Target Tracking
In this section, the performance of the improved PDA algorithm are evaluated and compared with the existing method of PDA. Assume detection probability P d = 0.9, the clutter density λ = 1/km 2 . The gate threshold is set to γ = 9, which corresponds to a two-dimensional gating probability of P g = 0.989, sampling interval T = 1 s, the process noise variance q cv = 0.05 m , and the measurement noise variance r = 100 m . In order to compare the performance, 100 Monte Carlo simulations have been performed.  Figure 1. Figure 1 shows the RMS position errors, which is computed with clutter densities of 1, 10, and 50, respectively. From Figure 1a, we can see the two algorithms exhibit similar RMS position error when clutter density is λ = 1, while when the clutter density increase, the RMS position error of the improved PDA is smaller than the PDA, especially when clutter density λ = 50, which can be seen from Figure 1c. In the improved PDA algorithm, we use the new association probability to realize the data association which can optimize the probability of the measurement originating from the target, so the improved algorithm can perform better to estimate the target state accurately. From the above, we know that improved PDA can improve the tracking performance in densely cluttered environments when tracking a single non-maneuvering target.  Figure 1 shows the RMS position errors, which is computed with clutter densities of 1, 10, and 50, respectively. From Figure 1a, we can see the two algorithms exhibit similar RMS position error when clutter density is 1   , while when the clutter density increase, the RMS position error of the improved PDA is smaller than the PDA, especially when clutter density 50   , which can be seen from Figure 1c. In the improved PDA algorithm, we use the new association probability to realize the data association which can optimize the probability of the measurement originating from the target, so the improved algorithm can perform better to estimate the target state accurately. From the above, we know that improved PDA can improve the tracking performance in densely cluttered environments when tracking a single non-maneuvering target.   Figures 2b and 3b show that the RMS position of the improved PDA algorithm is always smaller than the PDA algorithm. Figures 2a and 3a show that the improved PDA can be more accurate to realize target tracking. As for two crossing targets tracking, we can obtain similar results from Figures 4 and 5. From the above simulation results, we can see that the improved PDA algorithm performs better than the PDA algorithm whenever the two targets move in parallel or cross in the densely cluttered environment. This can be attributed to the reduction of the association probability of false measurement and, at the same time, enhance the association probability originating from the target. The improved algorithm can estimate the target state more accurately to avoid track   Figures 2b and 3b show that the RMS position of the improved PDA algorithm is always smaller than the PDA algorithm. Figures 2a and 3a show that the improved PDA can be more accurate to realize target tracking. As for two crossing targets tracking, we can obtain similar results from Figures 4 and 5. From the above simulation results, we can see that the improved PDA algorithm performs better than the PDA algorithm whenever the two targets move in parallel or cross in the densely cluttered environment. This can be attributed to the reduction of the association probability of false measurement and, at the same time, enhance the association probability originating from the target. The improved algorithm can estimate the target state more accurately to avoid track coalescence whatever two targets are in parallel or at a small-angle crossing in a densely cluttered environment.

Simulation and Analysis of Maneuvering Target Tracking
In this section, the performance of the proposed algorithm is evaluated and compared with the

Simulation and Analysis of Maneuvering Target Tracking
In this section, the performance of the proposed algorithm is evaluated and compared with the existing method of C-IMM-PDA. The initial states x = [1000 m 10 m/s 400 m 5 m/s], the target initially stays at a constant velocity between 0 s and 50 s, with a coordinated turn ω = −1 rad/s between 50 s and 100 s, then with a coordinated turn ω = 1 rad/s between 100 s and 150 s and, finally, a straight line with constant velocity between 150 s and 200 s. Assuming the detection probability P d = 0.9, the clutter measurement density λ = 1/km 2 . The gate threshold is set to γ = 16 which corresponds to a two-dimensional gating probability of P g = 0.9997. Sampling interval T = 1 s, the process noise variance for CV model q cv = 0.05 m , the CT model q ct = 0.015 m , and the measurement noise variance r = 100 m . The transition probability matrix of IMM models is   Figures 6 and 7 show the trajectories of a single maneuvering target and the RMS position error statistics using C-IMM-PDA and improved C-IMM-PDA with clutter densities of 1 and 10, respectively. As it would be expected, along with the increase of clutter density, the RMS position error shows the two algorithms exhibiting a decrease in performance. However, the degradation in performance is less for the improved C-IMM-PDA algorithm than for the C-IMM-PDA algorithm. In the improved C-IMM-PDA algorithm, using the same validated region can obtain the effective validated region for the whole system, and using the improved PDA algorithm can make the estimation of target more accurate. Additionally, when using the same validated region, this algorithm can reduce the computational cost. Simulation results show that the improved C-IMM-PDAF algorithm can be more effective to achieve high reliability in target tracking in densely cluttered environments.  Figures 6 and 7 show the trajectories of a single maneuvering target and the RMS position error statistics using C-IMM-PDA and improved C-IMM-PDA with clutter densities of 1 and 10, respectively. As it would be expected, along with the increase of clutter density, the RMS position error shows the two algorithms exhibiting a decrease in performance. However, the degradation in performance is less for the improved C-IMM-PDA algorithm than for the C-IMM-PDA algorithm. In the improved C-IMM-PDA algorithm, using the same validated region can obtain the effective validated region for the whole system, and using the improved PDA algorithm can make the estimation of target more accurate. Additionally, when using the same validated region, this algorithm can reduce the computational cost. Simulation results show that the improved C-IMM-PDAF algorithm can be more effective to achieve high reliability in target tracking in densely cluttered environments.

Conclusions
In this paper, we proposed a novel probabilistic data association algorithm based on distance weighting for target tracking. Due to clutter distribution being random, and the measurement originating from the target is more likely closer to the predicted measurement, this important prior information is introduced as a distance weight to optimize the association probability, which can enhance the association probability of the measurement originating from the target. When the improved algorithm is applied to non-maneuvering target tracking, the algorithm can be more accurate to realize the single non-maneuvering target tracking in a densely cluttered environment. For two-target tracking, it can avoid tracking coalescence whatever two targets are in parallel or at a small-angle crossing in a densely cluttered environment. For maneuvering target tracking in densely cluttered environments, based on the frame of C-IMM-PDA, we use the improved PDA to realize the data association and a Kalman filter to estimate the target state. Simulation results show the performance of the improved C-IMM-PDA algorithm is better than the C-IMM-PDA when tracking maneuvering targets in densely cluttered environments. Above all, with the increase of clutter density, the performance of the improved algorithm did not degrade significantly, which can ensure the real-time processing and accuracy of target tracking in densely cluttered environments.