^{1}

^{2}

^{*}

^{1}

^{1}

^{1}

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/).

An analytic method for predicting the performance of track-to-track association (TTTA) with biased data in multi-sensor multi-target tracking scenarios is proposed in this paper. The proposed method extends the existing results of the bias-free situation by accounting for the impact of sensor biases. Since little insight of the intrinsic relationship between scenario parameters and the performance of TTTA can be obtained by numerical simulations, the proposed analytic approach is a potential substitute for the costly Monte Carlo simulation method. Analytic expressions are developed for the global nearest neighbor (GNN) association algorithm in terms of correct association probability. The translational biases of sensors are incorporated in the expressions, which provide good insight into how the TTTA performance is affected by sensor biases, as well as other scenario parameters, including the target spatial density, the extraneous track density and the average association uncertainty error. To show the validity of the analytic predictions, we compare them with the simulation results, and the analytic predictions agree reasonably well with the simulations in a large range of normally anticipated scenario parameters.

In distributed multi-sensor surveillance systems, the process of associating sets of local estimates from multiple sensors is a fundamental problem, named track-to-track association (TTTA) [

With respect to the analytic performance prediction of data association, Sea and Singer did pioneering work on the performance analysis of the nearest neighbor (NN) algorithm [

In performing multi-target tracking with a sensor network, the systematic errors,

A number of TTTA algorithms are built upon the GNN criterion [

The rest of the paper is structured as follows. Section 2 describes the single-scan TTTA problem. Section 3 presents the existing analytic prediction results of the bias-free situation. Section 4 provides the analytic prediction results for the GNN association algorithm with biased data. These results are verified via simulations in Section 5. Section 6 concludes the paper.

In this paper, we consider a two-sensor single-scan TTTA problem. Although TTTA using multi-scan data has been theoretically proven to be better than that using single-scan data in [^{m}

At a given time (without the time argument, for simplicity), the track set available at sensor A is denoted by
_{i}^{m}_{a}_{j}^{m}_{b}_{0} and z_{0} represent the dummy tracks of sensor A and B, respectively. When a track in a list is not associated with any track from the other list, it will be assigned to the dummy track. Following [_{A}_{B}_{0}, while tracks in
_{0}.

Let _{i}_{A}_{B}_{i}_{i}

The association can also be defined as an (_{A}_{B} + 1) matrix _{ij}_{A}_{B}

The relationship between _{ij}_{i}_{,0} = 1; if _{0,}_{j}_{i,j}

The GNN algorithm determines the best association decision from all possibilities by minimizing the overall cost [

Constraints _{ij}_{i}_{j}_{M} denotes a semi-norm of a vector, _{i,j}_{A}_{B}

In the GNN algorithm, the probability of correct association, _{C}

Mori _{C}_{A}_{B}_{C}

The relevant parameters and assumptions used in

_{m}_{m}

_{T}^{m}

Targets are assumed to be independently identically distributed with a common distribution, which is uniform on a _{T}_{T}

_{m}

_{F}

The states of the extraneous objects share the same distribution with targets. The number of false objects, _{F}_{F}

Actually, the results of

In this section, we consider the impact of sensor biases and propose the analytic prediction results for the GNN algorithm.

First, we start with the two-target case, target

Considering track model

Then,

Thus, based on the detailed derivations in

It is very difficult to extend _{C}_{ij}_{ij}

Generally,

To maintain consistency with [

In Section 4.1, we assumed that the cardinalities of the two given random point sets are identical. In reality, false tracks may disturb the situation. In this section, we discuss the effects of false tracks on the TTTA performance with biased data, in which there are false tracks only in the track set of sensor B. The event, _{C}_{CT}_{Cf}_{i}

We can use the result of _{CT}_{CF}

Given an extraneous track, _{e}_{a*(i)}_{ie}_{e}_{i}

Without loss of generality, let _{i}

According to the track model

Consequently, together with _{a}*_{(}_{i}_{)} ˜

Assuming that the extraneous track, _{e}

Let

Let
_{X}_{e}_{a*(i)}

Then, we have:

If

When there are _{F}_{i}

Under the assumption of the false track number in

By substituting

When there are no extraneous tracks, _{F} = 0, _{m}

In addition, if sensors do not have biases (or have the same translational biases), i.e., Δ

When false tracks exist in both track sets, with a simple generalization of _{CFA}_{i}_{FB}_{j}_{FA}_{FB}

Comparing _{C}_{FA}_{FB}

To show the validity of our theoretical analysis, we test the above analytic results by comparing them to the simulation results. We consider the single-scan TTTA problem and make use of the Munkres algorithm to solve the GNN assignment problem [_{T}_{A}_{B}_{x}_{y}

When we study the validity of _{T}^{2}, with the target extent radius ^{2}, while Δ_{x}_{y}_{C}_{C}_{C}_{x}_{y}^{2}, varies from 0 ^{2} to 2 ^{2} in increments of 0.4 ^{2}. _{C}^{2}, and each point is compared with the prediction (solid line) by

When we test _{T}^{2}, and the target extent radius, _{x}_{y}_{F}_{T}_{F}_{F}

When we test ^{2}, and Δ_{x}_{y}_{F}_{F}_{FA}_{FB}_{FA}_{F}_{T}_{F}_{T}_{F}_{T}_{T}_{F}_{T}_{F}_{T}

Since a series of assumptions are made in the derivation of _{F}_{T}_{F}_{T}_{T}_{T}_{F}_{T}_{x}_{FA}_{FB}_{C}

To sum up, in a large range of normally anticipated operating conditions,

In this paper, we proposed an analytic performance prediction method for the GNN association algorithm with biased data in multi-sensor multi-target tracking applications. The probability of correct association is adopted as the performance criteria. The novelty of the method lies in that it accounts for the sensor biases in the TTTA problem, and analytic approaches are developed to reveal the intrinsic relationship between a set of key scenario parameters and the performance of the optimal TTTA algorithm. To verify the validity of the predictions, we compared them with the results of Monte-Carlo simulations. These shows that the analytic predictions agree reasonably well with the simulation results. In a word, the main results of the paper are as follows:

When both track sets are free from extraneous tracks, the correct association probability of the GNN association algorithm in

When one track set suffers from extraneous tracks, an exponential law was given in

When both track sets contain extraneous tracks, an analytic upper bound of the correct association probability for the GNN association algorithm was proposed in

A series of assumptions were made to get concise and explicit mathematical results, including no missed detection, the translational bias model, identical association uncertainty covariance on all tracks,

To make the proposed analytic method more practical, a lot of work is needed in the following directions:

Further investigations are needed to improve the prediction accuracy of our method in the scenario where false tracks exist in both sensors. The statistics concerning the more complicated association error,s such as multi-object transpositions, can be addressed in the future.

For simplicity, we assumed that the track detection probabilities of both sensors are one. However, missed detections and false tracks do happen in real environments. To complicate matters, when missed detections occur at one sensor, the corresponding tracks that are supposed to associate with these missed tracks in the track set of the other sensor are equivalent to extraneous tracks for the remaining tracks of the former sensor. When sensors suffer from both missed detections and extraneous tracks, accounting for the missed detections in the formula of the correct association probability needs to be investigated in the future.

We only examined the impact of translational biases on the performance of TTTA. Since the impact of translational biases on the tracks is uncoupled with the states of targets, the translational biases can be named state-independent biases. However, in some applications, sensors can have biases on their range and azimuth measurements. The impact of these biases on the position estimates of targets is dependent on the real states of targets. Therefore, we name the range and azimuth biases the state-dependent biases. Under the scenario with state-dependent biases, the bias terms cannot be canceled out in

The authors would like to thank the anonymous reviewers for their careful review and detailed comments, which have greatly improved the readability and contents of this paper.

In this appendix, we give the derivation of

Consider the case where target ^{m}

Then, based on _{m}_{i}

In this appendix, a kind of spherical integral, used in the derivation of ^{m}_{m}^{m}

The authors declare no conflicts of interest.

Correct association probability as a function of the average association uncertainty variance—analytic prediction results against simulation results. Target spatial density _{T}^{2}. Sensor biases are set as Δ_{x}_{y}

Correct association probability as a function of _{F}_{T}_{T}_{x}_{y}

Correct association probability as a function of sensor biases when both track sets contain extraneous tracks—analytic prediction results _{T}_{F}_{T}

Correct association probability as a function of sensor biases when both track sets contain extraneous tracks—theoretical prediction _{T}_{F}_{T}

Correct association probability as a function of sensor biases when both track sets contain extraneous tracks—analytic prediction results _{T}_{F}_{T}

Correct association probability as a function of sensor biases when both track sets contain extraneous tracks—analytic prediction results _{T}_{F}_{T}

The impact of sensor biases on correct association probability—the prediction by _{T}^{2}.

P_{C} (%) |
Δb_{x} | |||||||
---|---|---|---|---|---|---|---|---|

−3km | −2km | −1km | 0km | 1km | 2km | 3km | ||

Δ_{y} |
-3 km | 96.87 | 97.16 | 96.63 | 97.23 | 97.48 | 97.55 | 97.09 |

-2 km | 96.85 | 96.86 | 97.03 | 97.23 | 97.26 | 97.10 | 97.06 | |

-1 km | 97.21 | 96.80 | 96.37 | 96.91 | 96.62 | 96.95 | 97.00 | |

0 km | 97.43 | 97.37 | 96.79 | 97.24 | 97.26 | 97.43 | 96.95 | |

1 km | 97.19 | 97.04 | 97.67 | 97.52 | 97.11 | 97.16 | 97.17 | |

2 km | 97.07 | 97.20 | 97.17 | 96.90 | 97.62 | 97.67 | 97.32 | |

3 km | 97.26 | 97.08 | 97.28 | 96.88 | 97.19 | 97.17 | 97.11 |

The impact of sensor biases on correct association probability—the prediction by _{T}^{2}.

P_{C} (%) |
Δb_{x} | |||||||
---|---|---|---|---|---|---|---|---|

−3km | −2km | −1km | 0km | 1km | 2km | 3km | ||

Δ_{y} |
-3 km | 92.98 | 93.14 | 93.07 | 93.36 | 93.47 | 92.96 | 92.5 |

-2 km | 93.54 | 92.98 | 93.55 | 93.03 | 93.25 | 93.18 | 93.27 | |

-1 km | 92.87 | 94.02 | 92.83 | 93.15 | 93.35 | 92.98 | 92.21 | |

0 km | 93.03 | 92.41 | 93.30 | 93.21 | 93.00 | 93.02 | 92.89 | |

1 km | 92.14 | 93.67 | 92.87 | 92.95 | 93.44 | 93.44 | 93.92 | |

2 km | 93.17 | 92.80 | 92.40 | 93.11 | 93.24 | 92.76 | 93.15 | |

3 km | 92.87 | 92.91 | 93.58 | 92.83 | 93.14 | 93.17 | 92.77 |

The impact of sensor biases on the correct association probability—the prediction by _{T}^{2}.

P_{C} (%) |
Δb_{x} | |||||||
---|---|---|---|---|---|---|---|---|

−3km | −2km | −1km | 0km | 1km | 2km | 3km | ||

Δ_{y} |
-3 km | 87.81 | 86.85 | 87.20 | 87.79 | 86.73 | 87.03 | 86.62 |

-2 km | 87.51 | 86.83 | 86.48 | 87.34 | 86.31 | 86.29 | 87.63 | |

-1 km | 86.88 | 87.78 | 86.66 | 86.63 | 86.61 | 87.28 | 86.25 | |

0 km | 87.24 | 87.07 | 86.39 | 88.13 | 87.64 | 86.46 | 86.98 | |

1 km | 87.17 | 87.01 | 87.10 | 86.63 | 87.02 | 86.95 | 88.18 | |

2 km | 86.73 | 86.68 | 86.74 | 87.27 | 86.42 | 87.10 | 86.86 | |

3 km | 86.48 | 86.49 | 87.28 | 86.85 | 86.57 | 87.34 | 86.71 |