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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

In order to detect and track multiple maneuvering dim targets in sensor systems, an improved dynamic programming track-before-detect algorithm (DP-TBD) called penalty DP-TBD (PDP-TBD) is proposed. The performances of tracking techniques are used as a feedback to the detection part. The feedback is constructed by a penalty term in the merit function, and the penalty term is a function of the possible target state estimation, which can be obtained by the tracking methods. With this feedback, the algorithm combines traditional tracking techniques with DP-TBD and it can be applied to simultaneously detect and track maneuvering dim targets. Meanwhile, a reasonable constraint that a sensor measurement can originate from one target or clutter is proposed to minimize track separation. Thus, the algorithm can be used in the multi-target situation with unknown target numbers. The efficiency and advantages of PDP-TBD compared with two existing methods are demonstrated by several simulations.

For surveillance applications, a sensor system can be installed to detect and track targets within a given region [

Previously developed techniques for TBD include Hough transform [

However, traditional DP-TBD methods perform poorly for maneuvering target detection and tracking. In [

The above TBD algorithms consider only a single target, whereas many practical situations require the detection of multiple targets [

In this paper, a modified algorithm (PDP-TBD) is proposed to detect and track multiple maneuvering dim targets of unknown target number in sensor systems. For PDP-TBD, the performances of tracking techniques are used as a feedback to the detection part. Thus, the algorithm combines traditional tracking techniques with DP-TBD and its performance of detection and tracking can be improved. Two technical modifications are proposed in this paper, which are described as follows.

The proposed algorithm uses the performances of tracking techniques as a feedback to the detection part, and the feedback is constructed by a penalty term in the merit function. The penalty term is a function of the possible target state estimation, which is obtained by traditional tracking techniques. If the tracking techniques perform well (the estimation errors are small), the target will have high influence on the merit function. For a larger merit function is more likely to be originated from the target [

The track separation phenomenon has a bad influence on the tracking accuracy and target number estimation [

The outline of the paper is as follows: in Section 2, the target model and the sensor measurement model are formulated. Section 3 gives the description of the merit function with a penalty term. The procedure of PDP-TBD algorithm is given in Section 4. The simulation results and conclusions are presented in Section 5 and 6, respectively.

We consider here the problem of detecting and tracking multiple dim point targets in sensor systems. Suppose that there are _{k}_{k}

In practical applications, the target can be various aircrafts, such as airplanes, helicopters or missiles, and the sensor can be radar, sonar, _{P,k}_{x,k}_{y,k}^{th} position measurement and _{I,k}_{P,k}_{I,k}_{P,k}

If the ^{th} sensor measurement _{k}_{P,k}_{k}_{k}_{k}

If the _{k}_{P,k}_{I}

Target-originated measurements:

Clutter-originated measurements:
_{SNR}

The problem of multi-target detection and tracking is as follows. For a given sensor measurement sequence of

Traditional DP-TBD methods integrate the measurements along possible target trajectories, returning as possible targets those trajectories for which the merit function exceeds a threshold. However, it is proposed based on a single target with slowly maneuvering motion [

In this Section, a new algorithm named PDP-TBD is proposed to detect and track multiple maneuvering dim targets. By integrating a penalty term into the merit function, the algorithm combines traditional tracking techniques with DP-TBD. With this modification, the new algorithm has the advantages of the tracking techniques for different target motions.

For DP-TBD, _{k}_{k}_{k}_{k}_{k}_{k}_{k}

For each _{k}_{K}_{T}_{T}

The construction of the merit function is a key problem for DP-TBD. One approach is to use a likelihood-ratio function in the merit function [

For another approach, the merit function is calculated only depending on the reflected power, therefore, no knowledge of the signal and noise statistics is required [_{k}_{k}_{k}_{−1} ∈ _{k}_{−1} the merit function is given by:
_{k}_{k}_{k}_{−1}) and _{k}_{k}_{−1} and _{k}_{k}_{k}_{k}_{−1} for which a transition to _{k}

PDP-TBD uses the tracking performances as a feedback to the detection part, and the feedback is constructed by a penalty term in the merit function. Then for all _{k}_{k}_{k}_{−1} ∈ _{k}_{−1}, the new merit function is designed as:
_{k∣k} is the possible target estimation of _{k}_{k∣k}, _{k}_{k∣k}, _{k}_{k∣k} and _{k}_{k∣k} and _{k}_{k}_{k}_{k∣k}, _{k}_{k}

By the penalty term, the feedback of the tracking performances is constructed and traditional tracking techniques are combined with DP-TBD. If the tracking techniques perform well (the estimation errors are small), the penalty term will be small for clutter and big for targets. Therefore, the target will have much higher influence on the penalty term than the clutter. For a larger measurement is more likely to have originated form the target than a smaller measurement[_{k∣k} is far from _{k}

For the efficiency of the feedback relies on the performances of tracking techniques, the proper tracking methods should be chosen according to the target motions. In this paper, PDP-TBD is applied to detect and track maneuvering targets in sensor systems, hence, the tracking techniques used here are based on IMM [

Suppose that the transition from _{k}_{−1} to _{k}_{k}_{k∣k}, _{k}_{k}_{k∣k}. _{k∣k} is obtained by IMM tracking techniques. Then the merit function _{k}_{k}_{k}_{−1}), penalty term _{k∣k}, _{k}_{k}

DP-TBD is a modified version of the Viterbi algorithm. It is equivalent to an exhaustive search of all possible target trajectories, returning all state sequences for which the final stage merit function exceeds a specific threshold [

For DP-TBD, when calculating the merit function using _{k}_{−1} for which a transition to _{k}

Let _{k}_{k}_{k}, i.e._{k}_{k}_{k}_{min}, _{max}] is the range of target velocity.

_{k}_{k}_{k}_{k}_{xk}

Step 1: initialization. _{1} ∈ _{1},

Step 2: recursion. 2 ≤ _{k}_{k}_{k}_{k}_{k}_{−1}), _{k∣k}, _{k}_{k}_{k}_{−1}) is the merit function at scan _{k∣k}, _{k}_{k∣k}, _{k}

where _{k}_{k}_{k∣k} is the estimation of _{k}_{k∣k} = 0 represents no estimation being obtained for _{k}_{k∣k} and _{k}

Step 3: repetition. The track separation is a key problem for DP-TBD algorithm [

Regardless of the resolution influence and some other factors, we suppose that a sensor measurement cannot be originated by two targets. Then a reasonable constraint is proposed in Step 3 to eliminate the track separation.

Constraint: a sensor measurement can originate from one target or clutter,

Therefore, a possible state of the target can transit to only one state in the next scan,

The times of repetition is influenced by the clutter density. If the clutter number is below 200 in the surveillance region of 10 km × 10 km, the repetition will be terminated within six cycles. An example is given in

Although the above constraint is used in Step 3, the clutter and target measurements may coincide with practical application. Thus, after Step 3, the situation that no state is associated with a state at next scan may exist. And if this state is originated from the target, the target will be probably lost. Considering this situation, if no state is associated with a state at next scan after Step 3, the possible state estimation obtained will be used as the associated state.

Furthermore, although PDP-TBD can be applied to deal with multi-target situations by Step 3, it may be worse when the preferred association is not the one that best matches the past target dynamics. To solve this problem, a penalty term is constructed to combine DP-TBD with traditional tracking methods, which is described in Section 3.2. According to Section 3.2, if the tracking methods work properly, the target will be much more likely to be detected than the clutter. Hence, the disadvantage by using Step 3 can be greatly alleviated.

Step 4: Termination and backtracking. For the final stage merit function of PDP-TBD consists of a penalty term, the threshold is difficult to be determined. In this step, a decision function is applied to replace the final stage merit function when determining the target trajectories. The decision function is a sum of reflected power. Therefore, the threshold can be determined without influence of the penalty term. For all _{k}_{K}

All possible trajectories (state sequences) of the targets are obtained by _{1}, _{2}, …, _{K}

Termination: for all _{K}_{T}

Backtracking: for all _{K}, k

The trajectories are recovered using

When calculating the penalty term using _{k∣k} is essential and it is obtained by the tracking techniques. For the tracking techniques, state initiation is referred to [

In _{k−1∣k−1} which are obtained at scan _{distance} of the _{distance} >

IMMMHT is chosen as IMM multi-target tracking algorithm in

PDA is efficient and its computation cost for tracking a single target is small [

In practical application, the CV (Constant velocity) and CT (Coordinate turn) models are two of the most common forms of target motion in the Cartesian plane [_{k}^{T}

For the CV model:

For the CT model:
_{k} in each coordinate are equal, the measurement noise covariance matrix is denoted as ^{2}, ^{2}). The measurement matrix in

The surveillance region covers an area of 10,000 m on x axes and 10,000 m on y axes. At each scan, the number of clutters is Poisson distributed with parameter _{max} = 400m/s, _{min} = 0m/s and the threshold is _{T}

The performance of PDP-TBD is compared against IMMPDAF-AI and DP-TBD [

In this section, the probability of correct target number estimation is used to illustrate the detection performance. For example, two targets appear in the surveillance region, after 200 runs, if two targets are declared in 150 runs, the probability of correct target number estimation will be calculated as 150/200 = 75%. After the target number is declared, we use correct track probability to illustrate the tracking accuracy. Considering the difference of the three algorithms, for PDP-TBD and DP-TBD, when the detected position state is equal to the state originated from the actual target, the detected state is correctly tracked. For IMMPDAF-AI, when the estimated error between the detected position state (

We consider the estimated state is correctly tracked. Therefore, correct track probability is calculated as the rate of correct tracks over all tracks. For example, there are 400 target states obtained, if 360 states are correctly tracked using PDP-TBD, the correct track probability will be calculated as 360/400 = 90%.

Two targets appear in the surveillance region for the first scenario. The target initial states are [4500, −100,6000, −200,0.15] and [2000,150,5000,100,0.12] with assumed positions in meters, velocities in m/s and turn rates in rad/s. Target 1 makes an approximate CT motion when

_{1} and _{2} denote the SNR of Target 1 and Target 2 respectively. When _{1} = _{2} = 3

The performance of PDP-TBD is compared against DP-TBD and IMMPDAF-AI for target SNR values of 13 dB, 7 dB, 3 dB and 2 dB, and for average clutter numbers of 50, 100 and 150. After 200 Monte Carlo trials are performed,

In this scenario, the two targets are well separated, hence, this scenario can be considered as a single target situation, and DP-TBD and IMMPDAF-AI can be applied to detect and track these two targets. For PDP-TBD, the tracking techniques are combined with DP-TBD through constructing a penalty term. By the penalty term, PDP-TBD has the advantages of IMM tracking methods for different target motions and the target can be more likely to be detected than the clutter. Thus, the detection performance and tracking accuracy can be improved. Inspection of

Two targets have crossing trajectories in this scenario. The target initial states are [3467,150,3684,50,0] and [4500,−100,6000,−200,0.15] with assumed positions in meter, velocities in m/s and turn rates in rad/s. Target 1 makes approximate CV motion. Target 2 makes approximate CT motion when _{1} = _{2} = 3

After 200 Monte Carlo trials are performed, the comparison results of correct target number estimation by using these three algorithms in different SNR values and clutter densities are shown in

Inspection of

DP-TBD is proposed based on a single target scenario, and it performs the maximization over the state transition range in the step of recursion [

In this scenario, the performance of PDP-TBD is compared with DP-TBD in different turn rates (different speed of the target maneuvering). A target appears in the surveillance region with initial position [4500,6000] and initial velocity [−100,−200]. The target makes an approximate CT motion, and the SNR is 3 dB. The average clutter number is

Three targets appear in this scenario. The target initial states are [3467,150,3684,50,0], [4500,−100,6000,−200,0.15] and [2000,150,5000,100,0.12] with assumed positions in meters, velocities in m/s and turn rates in rad/s respectively. Target 1 makes an approximate CV motion. Target 2 makes an approximate CT motion when _{1} = _{2} = 3

A new algorithm called PDP-TBD has been presented for simultaneously detecting and tracking multiple maneuvering dim targets with unknown target number. The algorithm uses the performances of tracking techniques as a feedback to the detection part, and the feedback is constructed by a penalty term. The penalty term is a function of the possible target state estimation, which can be obtained by the tracking methods. With this feedback, traditional tracking techniques can be combined with DP-TBD. Meanwhile, a constraint that a measurement can originate from one target or clutter is proposed to minimize track separation. With the above two technical modifications, the application scope of PDP-TBD is significantly extended compared with DP-TBD. Simulation results justified the performance of PDP-TBD under a variety of conditions.

Therefore, the proposed algorithm can be applied to simultaneously detect and track dim targets using sensor data. In practical applications, the sensor can be radar, sonar,

For PDP-TBD, the scale factor in the penalty term is affected by the measurement noise and the estimation errors. It is a constant and is empirically determined in this paper. However, the performances of traditional tracking techniques may change in different practical applications. Therefore, the scale factor should be adaptive according to the tracking performances. The Cramér Rao Low Bound (CRLB) theories may be a feasible method to estimate the tracking accuracy, it sets a lower bound on the various of any unbiased estimator [

Furthermore, the constraint of a measurement having only one source is used in this paper. However, this constraint can be relaxed considering practical applications. For example, when a target moves into the surveillance region, a sensor with high resolution may provide several measurements, which are all originated from the target. And a sensor with low resolution may provide a measurement, which is actually originated from the target and clutter,

This research was supported by the National Basic Research Program of China Project (No2009CB320600) and the NFSC project of (60805013).

Merit function calculation block diagram at scan

State transition range.

An example (

Tracking techniques block diagram.

Recovered trajectories and estimated target number.

Comparison of the probabilities of correct target number estimation (200 runs, Scenario-1) (

Recovered trajectories and estimated target number.

Comparison of the probabilities of correct target number estimation (200 runs, Scenario-2) (

Comparison of the performances of the algorithms (200 runs) (

Recovered trajectories and estimated target number (Scenario-3, “-o-” represents the recovered trajectories of the targets, “·” represents the measurements).

Comparison of correct track probabilities (200 runs, Scenario-1).

| ||||||||
---|---|---|---|---|---|---|---|---|

Target 1(%) | PDP-TBD | 98.20 | 96.25 | 94.79 | 94.00 | 98.46 | 96.33 | 94.79 |

DP-TBD | 96.95 | 92.95 | 83.76 | 82.24 | 95.14 | 89.55 | 83.76 | |

IMMPDAF-AI | 89.17 | 84.65 | 79.15 | 73.90 | 94.70 | 91.72 | 79.15 | |

| ||||||||

Target 2(%) | PDP-TBD | 98.96 | 96.86 | 94.36 | 94.62 | 98.41 | 97.11 | 94.36 |

DP-TBD | 96.95 | 90.94 | 81.21 | 81.18 | 94.61 | 89.82 | 81.21 | |

IMMPDAF-AI | 94.33 | 90.70 | 87.13 | 79.19 | 95.23 | 88.80 | 87.13 |

Comparison of correct track probabilities (200 runs, Scenario-2).

| ||||||||
---|---|---|---|---|---|---|---|---|

Target 1 (%) | PDP-TBD | 90.95 | 92.63 | 93.87 | 93.11 | 96.49 | 95.29 | 93.87 |

DP-TBD | 69.47 | 66.23 | 55.65 | 50.93 | 73.04 | 59.86 | 55.65 | |

IMMPDAF-AI | 87.35 | 81.15 | 79.86 | 71.20 | 97.61 | 89.49 | 79.86 | |

| ||||||||

Target 2 (%) | PDP-TBD | 90.59 | 91.61 | 91.24 | 91.25 | 95.82 | 93.69 | 91.24 |

DP-TBD | 61.06 | 63.40 | 51.29 | 55.25 | 69.02 | 63.65 | 51.29 | |

IMMPDAF-AI | 78.47 | 63.46 | 75.56 | 35.20 | 93.12 | 86.15 | 75.56 |

Probabilities of correct target number estimation (200 runs, Scenario-3).

Probability statistics (%) | 0.0 | 1.0 | 8.5 | 90.0 | 0.5 |

Correct track probabilities (200 runs, Scenario-3).

Probability statistics (%) | 92.50 | 91.53 | 95.14 |