# Track-to-Track Association Based on Structural Similarity in the Presence of Sensor Biases

Authors: Hongyan Zhu; Suying Han
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

The paper addresses the problem of track-to-track association in the presence of sensor biases. In some challenging scenarios, it may be infeasible to implement bias estimation and compensation in time due to the computational intractability or weak observability about sensor biases. In this paper, we introduce the structural feature for each local track, which describes the spatial relationship with its neighboring targets. Although the absolute coordinates of local tracks from the same target are severely different in the presence of sensor biases, their structural features may be similar. As a result, instead of using the absolute kinematic states only, we employee the structural similarity to define the association cost. When there are missed detections, the structural similarity between local tracks is evaluated by solving another 2D assignment subproblem. Simulation results demonstrated the power of the proposed approach.


---

## Body

## 1. Introduction

The potential advantages of fusing information from disparate sensor systems to achieve better surveillance have been recognized. Track-to-track association [1, 2] is a crucial step in the distributed estimation fusion system, which seeks to determine the correspondence between local tracks from different sensors. Unfortunately, sensor reports may be influenced by systematic errors (sensor biases) besides random errors. In this case, sensor registration [3, 4] is required to remove the sensor biases from the biased reports. When track association and sensor registration meet each other, there are more challenges. On the one hand, sensor registration is based on the common sensor reports from the same target. On the other hand, the removal of the sensor biases contributes greatly to making correct association assignments. In this sense, sensor registration and track association are tightly coupled together.An alternative way to tackle this problem is to implement bias estimates and track-to-track association jointly. Several heuristic algorithms, including centroid matching and singleton matching algorithms, for estimating the relative sensor biases in track-to-track association were explored in [5]. Considering the effects of biases, Stone et al. estimated the relative biases based on the fast Fourier transform (FFT) and multiple dimensional spatial cross-correlation function in [6]. In [7], Levedahl modeled this problem as a global nearest pattern (GNP) problem. It produced the bias estimates for each association hypothesis firstly, and the association probability was computed based on the given bias estimates. The final association hypothesis was determined with the maximum probability. In [8], Papageorgiou and Sergi optimized the computation complexity of the GNP algorithm to get the trading-off between the optimality and computation time. In [9, 10], Papageorgiou et al. computed the pure association probability by assuming a prior distribution of sensor biases. However, it is hard to get the prior distribution of sensor biases. A joint track association and relative bias estimation problem was formulated and a solution based on the Dijkstra search was developed in [11]. A solution based on the branch-and-bound framework for the same problem was described in [12]. In [13], the performance comparisons of several algorithms for estimating relative sensor bias were given. In [14], a feature called “reference topology” based on the relative coordinates is presented. However, its association performance depends on the granularity of cell partitions greatly. Moreover, no systematic method is given to handle the case of missed detections. In [15, 16], the joint approaches to data association, sensor registration, and track fusion were given by means of the expectation-maximization (EM) algorithm.In some challenging scenarios, it is impossible in practice to implement bias estimation and compensation in time due to the computational intractability or weak observability about sensor biases. In this paper, we turned to implement track-to-track association without bias estimation. As we know, it is extremely essential to establish a reasonable metric to measure the similarity of local tracks in track-to-track association problem. In general, the similarity of local tracks is based on the absolute kinematic parameters. However, the absolute kinematic parameters are corrupted greatly in the presence of sensor biases. For example, when given the azimuth bias of 3 degrees, the position deviation could reach up to 10 kilometers for a target located 200 km from the sensor. In this case, the similarity measure based on absolute coordinates is unbelievable anymore. In this paper, we introduce the structural feature for each local track describing the spatial relationship with its neighboring targets. Although the absolute coordinates of local tracks from the same target are severely different in the presence of sensor biases, their structural features may be similar. So, instead of using the absolute kinematic states only, we employ the structural similarity to measure the distance of two local tracks from different sensors.In this paper, we develop a structural similarity-based approach to deal with the problem of track association in the presence of sensor biases. Main contributions are given as follows. Firstly, under appropriate assumptions, the distance invariance between two targets detected by different sensors is verified in the presence of sensor biases. Secondly, the structural feature is introduced for each local track, which is represented by the distance set from the track to all the other tracks from the same sensor. Thirdly, a two-dimensional (2D) assignment model is established to implement track-to-track association in the presence of sensor biases. Instead of using the absolute kinematic states only, the structural similarity between local tracks is adopted to measure the association cost and is evaluated by solving another 2D assignment subproblem. Moreover, the dummy track is introduced for each sensor to cope with possibly missed detections. Simulation results demonstrated the power of the proposed approaches.The rest of the paper is organized as follows. In Section2 we describe the measurement model and formulate the problem of track-to-track association. Section 3 shows how to implement the track association based on structural similarity. Simulation results are given in Section 4 to demonstrate the effectiveness of the proposed approaches. Finally, concluding remarks and further research directions are in Section 5.


## 2. Problem Formulation

Consider a multitarget tracking scenario with two sensors andN
t targets in the surveillance region. In this paper, false alarms are not considered since they can also be viewed as the targets. In references with the radar sensor which measures the range and azimuth to the target, the measurement process is implemented in the local polar coordinate system (LPCS). Let z
m
,
i
≜
{
r
m
,
i
,
θ
m
,
i
} be theith measurement (i
=
1,2
,
…
,
n
m) from sensor m (m
=
1,2) at the time instant k, in which r
m
,
i and θ
m
,
i mean the range and angle measurement, respectively.Two kinds of sensor biases are considered in general: range bias and azimuth bias. Since there are systematic biases and random errors, the original measurement{
r
m
,
i
,
θ
m
,
i
} from sensorm can be modeled as
(1)
r
m
,
i
=
r
-
m
,
i
+
Δ
r
m
+
v
r
m
,
θ
m
,
i
=
θ
-
m
,
i
+
Δ
θ
m
+
v
θ
m
,
where r
-
m
,
i and θ
-
m
,
i denote the real range and azimuth of target i form sensor m, respectively. Δ
r
m and Δ
θ
m are systematic biases and v
r
m and v
θ
m are random noises. The random noises V
~
m
=
[
v
r
m
v
θ
m
]
T are white Gaussian with zero mean and variances of (
σ
r
m
)
2 and (
σ
θ
m
)
2, respectively.Figure1 depicts a scenario with two sensors and two targets. T
a
(
x
-
a
,
y
-
a
) and T
b
(
x
-
b
,
y
-
b
) are the real positions of targets. T
m
,
a
(
x
m
,
a
,
y
m
,
a
) and T
m
,
b
(
x
m
,
b
,
y
m
,
b
) are the position measurements given by sensor m (m
=
1,2).Figure 1
Sensor configuration geometry.Based on the biased measurements{
r
m
,
i
,
θ
m
,
i
}, each sensor m produces its own local tracks represented by two tuples {
x
^
m
,
i
,
P
m
,
i
} (m
=
1,2; i
=
1,2
,
…
,
n
m). x
^
m
,
i and P
m
,
i mean the state estimate and error covariance, respectively. For simplicity, we omit the time index here. Track-to-track association seeks to determine the correspondence between local tracks from different sensors. The key point is to measure the similarity of local tracks. Traditionally, it is defined by the weighted statistical distance under the independence assumption:
(2)
α
i
1
i
2
=
(
x
^
1
,
i
1
-
x
^
2
,
i
2
)
T
(
P
1
,
i
1
+
P
2
,
i
2
)
-
1
(
x
^
1
,
i
1
-
x
^
2
,
i
2
)
aaaaaaaaaaai
(
i
1
=
1,2
,
…
,
n
1
,
i
2
=
1,2
,
…
,
n
2
)
.However, local tracks are biased estimates in the presence of sensor biases. The direct association of biased estimates will not produce a satisfactory result.


## 3. Track Association Based on Structural Similarity

In traditional approaches to track-to-track association, only the absolute kinematic states of targets are considered. In the presence of sensor biases, it does not work well especially for the large azimuth bias, since it may result in severe deviations of absolute coordinates. In this section, the basic principle of the structural similarity is firstly verified, and then the structural similarity is used to measure the distance of local tracks instead of using the absolute kinematic states only. A complete 2D assignment model to deal with possible missed detections is also addressed.


### 3.1. Basic Principle of Structural Similarity

In what follows, by assuming small range biases, the distance invariance between two targets detected by different sensors is verified in the presence of sensor biases. We introduce a structural feature for each local track, which describes the spatial relationship with its neighboring targets. According to the distance invariance, the structural feature of local tracks from the same target is similar. As a result, the structural similarity is employed to define the association cost between local tracks.The random errorV
~
m (m
=
1,2) is ignored in the following derivation for the simplicity; thus we have
(3)
r
m
,
i
=
r
-
m
,
i
+
Δ
r
m
,
θ
m
,
i
=
θ
-
m
,
i
+
Δ
θ
m
.In the absence of sensor biases, the real position measurement of the target state is(4)
[
x
-
m
,
i
y
-
m
,
i
]
=
[
r
-
m
,
i
cos
⁡
(
θ
-
m
,
i
)
r
-
m
,
i
sin
(
θ
-
m
,
i
)
]
.In the presence of sensor biases, it becomes(5)
[
x
m
,
i
y
m
,
i
]
=
[
(
r
-
m
,
i
+
Δ
r
m
)
cos
⁡
(
θ
-
m
,
i
+
Δ
θ
m
)
(
r
-
m
,
i
+
Δ
r
m
)
sin
(
θ
-
m
,
i
+
Δ
θ
m
)
]
.Manipulating the items in the above equations, we have(6)
[
x
m
,
i
y
m
,
i
]
=
(
cos
⁡
Δ
θ
m
-
sin
Δ
θ
m
sin
Δ
θ
m
cos
⁡
Δ
θ
m
)
[
x
-
m
,
i
y
-
m
,
i
]
+
(
Δ
r
m
cos
⁡
⁡
(
θ
m
,
i
+
Δ
θ
m
)
Δ
r
m
sin
⁡
(
θ
m
,
i
+
Δ
θ
m
)
)
.From (6), it can be observed that the range bias Δ
r
m brings about the translation for targets, and the azimuth bias Δ
θ
m produces the rotation and also contributes to the translation of targets when the range bias is nonzero. Moreover, the azimuth bias is the critical factor compared with the range bias, which produces more severe influence on the target’s abstract coordinate and will be enlarged as the distance from the sensor to the target increases.The real squared distance between two targetsT
a and T
b is
(7)
d
2
(
T
a
,
T
b
)
=
(
x
-
a
-
x
-
b
)
2
+
(
y
-
a
-
y
-
b
)
2
.For sensorm (m
=
1,2), the squared distance d
m
2
(
T
a
,
T
b
) between two position measurements T
m
,
a and T
m
,
b is
(8)
d
m
2
(
T
a
,
T
b
)
≜
d
2
(
T
m
,
a
,
T
m
,
b
)
=
(
x
m
,
a
-
x
m
,
b
)
2
+
(
y
m
,
a
-
y
m
,
b
)
2
=
(
r
-
m
,
a
+
Δ
r
m
)
2
+
(
r
-
m
,
b
+
Δ
r
m
)
2
-
2
(
r
-
m
,
a
+
Δ
r
m
)
(
r
-
m
,
b
+
Δ
r
m
)
×
cos
⁡
(
θ
m
,
a
-
θ
m
,
b
)
.In this way, the difference betweend
m
2
(
T
a
,
T
b
) and d
2
(
T
a
,
T
b
) can be expressed by
(9)
D
≜
d
m
2
(
T
a
,
T
b
)
-
d
2
(
T
a
,
T
b
)
=
2
(
Δ
r
m
)
2
+
2
Δ
r
m
r
-
m
,
a
+
2
Δ
r
m
r
-
m
,
b
-
2
(
(
Δ
r
m
)
2
+
Δ
r
m
r
-
m
,
a
+
Δ
r
m
r
-
m
,
b
)
×
cos
⁡
(
θ
-
m
,
a
-
θ
-
m
,
b
)
=
2
(
(
Δ
r
m
)
2
+
Δ
r
m
r
-
m
,
a
+
Δ
r
m
r
-
m
,
b
)
×
[
1
-
cos
⁡
(
θ
-
m
,
a
-
θ
-
m
,
b
)
]
.From (9), it is seen that the difference D is only influenced by the range bias Δ
r
m and the real position of the target but is independent of azimuth bias Δ
θ
m. This reveals that although the azimuth bias may cause a severe impact on the absolute coordinate, it causes less impact on the relative coordinate. When the range bias is too large, the difference D may be severe. Fortunately, the range bias in practical applications is generally at a low level. That means that the distance between two targets detected by different sensors differs slightly, which is less sensitive to sensor biases. In what follows, we employ the structural similarity to measure the distance between two local tracks from different sensors.


### 3.2. 2D Assignment Model for Track Association

As we know, it is very important to establish a reasonable metric to measure the similarity of local tracks from different sensors. This problem becomes extremely complicated in the presence of sensor bias, random errors, and missed tracks. Figure2 illustrates a scenario with two sensors and six targets. Here, sensor 1 detects 4 targets, and sensor 2 detects 5 targets. That means that n
1
=
4, n
2
=
5, and there are 3 common targets.Figure 2
Distribution of measurements from different sensors.For the scenario in Figure2, a graphical representation of track association is shown in Figure 3. We introduce a node for each local track. To deal with possible missed detections, a dummy node is introduced for each local sensor which provides the access to link an isolated track with the dummy track. The edges mean possible links between different local tracks.Figure 3
Graphical representation of track association problem.In the following, a 2D assignment model with given constraints is established by minimizing the sum of all pairwise costs:(10)
min
⁡
∑
i
1
=
0
n
1
∑
i
2
=
0
n
2
c
i
1
i
2
ρ
i
1
i
2
i
s
.
t
a
i
∑
i
2
=
0
n
2
ρ
i
1
i
2
=
1
,
∀
i
1
=
1,2
,
…
,
n
1
a
a
a
a
a
∑
i
1
=
0
n
1
ρ
i
1
i
2
=
1
,
∀
i
2
=
1,2
,
…
,
n
2
a
a
ρ
i
1
i
2
∈
{
0,1
}
,
∀
i
1
=
0,1
,
2
,
…
,
n
1
,
aaaaaaaaaaaaaaaii
∀
i
2
=
0,1
,
2
,
…
,
n
2
.


### 3.3. Association Cost of Local Nodes Based on the Structural Similarity

The model (10) can be solved efficiently and optimally by the general 2D assignment algorithms, such as auction algorithm and Hungarian algorithm. Now, the remaining problem in solving (10) is how to define the connecting cost c
i
1
i
2. Instead of using the absolute kinematic states only, the structural similarity of local tracks is employed here. Three kinds of costs need to be specified: between local tracks, between dummy tracks, and between local track and the dummy track.Firstly, we define a structural feature for each nodeN
i
m (m
=
1,2; i
m
=
1
,
…
,
n
m), which is represented by the set I
i
m including all the distances from local track i
m to j (j
=
1,2
,
…
,
n
m, j
≠
i
m); that is,
(11)
I
i
m
=
{
d
(
j
,
i
m
)
}
j
=
1,2
,
…
,
n
m
,
j
≠
i
m
.We discuss the definition ofc
i
1
i
2 in terms of the following three cases.(
1) With Unity Detection Probability
P
D
m
(
P
D
m
=
1
,
m
=
1,2
). If P
D
m
=
1 (m
=
1,2), then n
1
=
n
2
=
n. We order the elements of set I
i
m from the smallest to largest: d
~
i
m
,
1
≤
d
~
i
m
,
2
≤
⋯
≤
d
~
i
m
,
n
-
1. In this way, the cost c
i
1
i
2 (for
all
i
1
=
1,2
,
…
,
n
1, for
all
i
2
=
1,2
,
…
,
n
2) can be defined by
(12)
c
i
1
i
2
=
D
(
I
i
1
,
I
i
2
)
=
∑
j
=
1
n
-
1
|
d
~
i
1
,
j
-
d
~
i
2
,
j
|
.(
2) Without Unity Detection Probability
P
D
m
(
P
D
1
·
P
D
2
≠
1
). If P
D
m
<
1 (m
=
1,2), the computation of c
i
1
i
2
(
for
all
i
1
=
1,2
,
…
,
n
1
,
for
all
i
2
=
1,2
,
…
,
n
2
) becomes complicated due to possible missed detections. To define the connection cost c
i
1
i
2, a 2D assignment subproblem (13) is constructed. As shown in Figure 4, a graphical representation is established to compute the cost c
23. As done in the optimization model (10), two dummy nodes are introduced to cope with the case of missed detections:
(13)
J
i
1
i
2
=
min
⁡
∑
j
1
=
0
,
j
1
≠
i
1
n
1
∑
j
2
=
0
,
j
2
≠
i
2
n
2
c
~
j
1
j
2
ρ
~
j
1
j
2
aai
s
.
t
aa
∑
j
2
=
0
,
j
2
≠
i
2
n
2
ρ
~
j
1
j
2
=
1
,
∀
j
1
=
1,2
,
…
,
n
1
,
j
1
≠
i
1
a
a
a
a
a
a
a
a
∑
j
1
=
0
,
j
1
≠
i
1
n
1
ρ
~
j
1
j
2
=
1
,
∀
j
2
=
1,2
,
…
,
n
2
,
j
2
≠
i
2
a
a
a
a
a
a
a
a
a
a
a
a
a
a
i
ρ
~
j
1
j
2
∈
{
0,1
}
,
∀
j
1
=
0,1
,
2
,
…
,
n
1
,
a
a
a
a
a
a
a
a
a
a
a
a
i
∀
j
2
=
0,1
,
2
,
…
,
n
2
,
j
1
≠
i
1
,
j
2
≠
i
2
.Figure 4
Graphical representation of 2D assignment subproblem to compute the costc
23.In this optimization model (13), the critical part is to define the cost c
~
j
1
j
2. For all j
1
=
1,2
,
…
,
n
1, j
1
≠
i
1, and for
all
j
2
=
1,2
,
…
,
n
2, j
2
≠
i
2, the cost c
~
j
1
j
2 is defined by the absolute value of distance difference:
(14)
c
~
j
1
j
2
=
|
d
(
j
1
,
i
1
)
-
d
(
j
2
,
i
2
)
|
.If track-pair{
i
1
,
j
1
} from sensor 1 and {
i
2
,
j
2
} from sensor 2 share the same two targets, then c
~
j
1
j
2 would be a small quantity due to the structural similarity. When given a threshold λ, we compare c
~
j
1
j
2 and the gate value λ. If c
~
j
1
j
2
>
λ, then it means that track-pair {
i
1
,
j
1
} from sensor 1 and {
i
2
,
j
2
} from sensor 2 do not share the same targets. To prevent them from associating, we set c
~
j
1
j
2
=
+
∞. In practical operations, we set c
~
j
1
j
2
=
Γ, where Γ is a large positive integer. In addition, we set c
~
0
j
2
=
c
~
j
1
0
=
1
0
-
2
Γ to encourage the isolated node to link with the dummy node. In addition, we set c
~
00
=
Γ.As far, a complete 2D assignment subproblem (13) is well defined. By solving it, the optimal fitness J
i
1
i
2
* can be obtained. It is noted that c
i
1
i
2 cannot be defined directly by J
i
1
i
2
*; further processing is required considering possible isolated tracks. We denote the number of valid associations by β, which is defined as the number of association assignments between local tracks (not with dummy nodes). If β is close enough to the expected number α of common targets (|
β
-
α
|
≤
ε, ε is a small positive integer), then c
i
1
i
2
=
J
i
1
i
2
*. Otherwise, set c
i
1
i
2
=
Γ, where the expected number α of common targets is computed by
(15)
α
=
P
D
1
P
D
2
(
n
1
+
n
2
)
(
1
+
P
D
1
P
D
2
)
.
It is proportional to the detection probability P
D
m (m
=
1,2) of sensors. If P
D
m
=
1 (m
=
1,2), then α
=
n
1
=
n
2.(
3) Connection Cost with Dummy Node. When a local node (i
1 or i
2) is declared isolated, we set c
0
i
2
=
c
i
1
0
=
1
0
-
2
Γ to encourage it to link with the dummy node. In addition, we set c
00
=
Γ to discourage the association between dummy tracks.


### 3.4. Discussions

(
1) Approximation Solution Based on the Nearest Neighborhood Method. A new approach to track-to-track association in the presence of sensor biases is given in Sections 3.2 and 3.3, which requires solving two 2D optimal assignment models. One serves to compute the connecting cost between two local tracks, and the other one is to determine the final association assignment between two sets of local tracks. To simplify, we call this method “structural similarity-based 2D assignment” (SS-based 2D assignment). Considering the complexity of the proposed model, an approximation solution can be obtained by using the nearest neighborhood (NN) method when determining the final association assignment, instead of using the optimal assignment. Such a scheme is denoted by structural similarity-based nearest neighborhood (SS-based NN). Simulation results provided before indicate that such an approximate method can also provide a better association result, because the structural property is considered when constructing the connection cost between local tracks.(
2) Possible Extensions. In practical fusion applications, heterogeneous sensors are commonly found. The proposed approach can also work well in this case, since what we handle here are local tracks after local processing. Although the measuring process is implemented in the local measurement coordinate system, local tracks from heterogeneous sensors may be in the common Cartesian coordinate system.In addition, in short-range applications or for maritime surveillance, different scattering centres of the objects under consideration may give rise to several distinct detections. At this time, what one is concerned with includes the state for both the target kinematics and the target extension. The proposed approach fails to deal with the case that the targets have more than one scattering centers. Readers may refer to literatures concerning extended target tracking [17, 18]. It is still a problem which few people get into to implement track-to-track association between extended targets.


## 3.1. Basic Principle of Structural Similarity

In what follows, by assuming small range biases, the distance invariance between two targets detected by different sensors is verified in the presence of sensor biases. We introduce a structural feature for each local track, which describes the spatial relationship with its neighboring targets. According to the distance invariance, the structural feature of local tracks from the same target is similar. As a result, the structural similarity is employed to define the association cost between local tracks.The random errorV
~
m (m
=
1,2) is ignored in the following derivation for the simplicity; thus we have
(3)
r
m
,
i
=
r
-
m
,
i
+
Δ
r
m
,
θ
m
,
i
=
θ
-
m
,
i
+
Δ
θ
m
.In the absence of sensor biases, the real position measurement of the target state is(4)
[
x
-
m
,
i
y
-
m
,
i
]
=
[
r
-
m
,
i
cos
⁡
(
θ
-
m
,
i
)
r
-
m
,
i
sin
(
θ
-
m
,
i
)
]
.In the presence of sensor biases, it becomes(5)
[
x
m
,
i
y
m
,
i
]
=
[
(
r
-
m
,
i
+
Δ
r
m
)
cos
⁡
(
θ
-
m
,
i
+
Δ
θ
m
)
(
r
-
m
,
i
+
Δ
r
m
)
sin
(
θ
-
m
,
i
+
Δ
θ
m
)
]
.Manipulating the items in the above equations, we have(6)
[
x
m
,
i
y
m
,
i
]
=
(
cos
⁡
Δ
θ
m
-
sin
Δ
θ
m
sin
Δ
θ
m
cos
⁡
Δ
θ
m
)
[
x
-
m
,
i
y
-
m
,
i
]
+
(
Δ
r
m
cos
⁡
⁡
(
θ
m
,
i
+
Δ
θ
m
)
Δ
r
m
sin
⁡
(
θ
m
,
i
+
Δ
θ
m
)
)
.From (6), it can be observed that the range bias Δ
r
m brings about the translation for targets, and the azimuth bias Δ
θ
m produces the rotation and also contributes to the translation of targets when the range bias is nonzero. Moreover, the azimuth bias is the critical factor compared with the range bias, which produces more severe influence on the target’s abstract coordinate and will be enlarged as the distance from the sensor to the target increases.The real squared distance between two targetsT
a and T
b is
(7)
d
2
(
T
a
,
T
b
)
=
(
x
-
a
-
x
-
b
)
2
+
(
y
-
a
-
y
-
b
)
2
.For sensorm (m
=
1,2), the squared distance d
m
2
(
T
a
,
T
b
) between two position measurements T
m
,
a and T
m
,
b is
(8)
d
m
2
(
T
a
,
T
b
)
≜
d
2
(
T
m
,
a
,
T
m
,
b
)
=
(
x
m
,
a
-
x
m
,
b
)
2
+
(
y
m
,
a
-
y
m
,
b
)
2
=
(
r
-
m
,
a
+
Δ
r
m
)
2
+
(
r
-
m
,
b
+
Δ
r
m
)
2
-
2
(
r
-
m
,
a
+
Δ
r
m
)
(
r
-
m
,
b
+
Δ
r
m
)
×
cos
⁡
(
θ
m
,
a
-
θ
m
,
b
)
.In this way, the difference betweend
m
2
(
T
a
,
T
b
) and d
2
(
T
a
,
T
b
) can be expressed by
(9)
D
≜
d
m
2
(
T
a
,
T
b
)
-
d
2
(
T
a
,
T
b
)
=
2
(
Δ
r
m
)
2
+
2
Δ
r
m
r
-
m
,
a
+
2
Δ
r
m
r
-
m
,
b
-
2
(
(
Δ
r
m
)
2
+
Δ
r
m
r
-
m
,
a
+
Δ
r
m
r
-
m
,
b
)
×
cos
⁡
(
θ
-
m
,
a
-
θ
-
m
,
b
)
=
2
(
(
Δ
r
m
)
2
+
Δ
r
m
r
-
m
,
a
+
Δ
r
m
r
-
m
,
b
)
×
[
1
-
cos
⁡
(
θ
-
m
,
a
-
θ
-
m
,
b
)
]
.From (9), it is seen that the difference D is only influenced by the range bias Δ
r
m and the real position of the target but is independent of azimuth bias Δ
θ
m. This reveals that although the azimuth bias may cause a severe impact on the absolute coordinate, it causes less impact on the relative coordinate. When the range bias is too large, the difference D may be severe. Fortunately, the range bias in practical applications is generally at a low level. That means that the distance between two targets detected by different sensors differs slightly, which is less sensitive to sensor biases. In what follows, we employ the structural similarity to measure the distance between two local tracks from different sensors.


## 3.2. 2D Assignment Model for Track Association

As we know, it is very important to establish a reasonable metric to measure the similarity of local tracks from different sensors. This problem becomes extremely complicated in the presence of sensor bias, random errors, and missed tracks. Figure2 illustrates a scenario with two sensors and six targets. Here, sensor 1 detects 4 targets, and sensor 2 detects 5 targets. That means that n
1
=
4, n
2
=
5, and there are 3 common targets.Figure 2
Distribution of measurements from different sensors.For the scenario in Figure2, a graphical representation of track association is shown in Figure 3. We introduce a node for each local track. To deal with possible missed detections, a dummy node is introduced for each local sensor which provides the access to link an isolated track with the dummy track. The edges mean possible links between different local tracks.Figure 3
Graphical representation of track association problem.In the following, a 2D assignment model with given constraints is established by minimizing the sum of all pairwise costs:(10)
min
⁡
∑
i
1
=
0
n
1
∑
i
2
=
0
n
2
c
i
1
i
2
ρ
i
1
i
2
i
s
.
t
a
i
∑
i
2
=
0
n
2
ρ
i
1
i
2
=
1
,
∀
i
1
=
1,2
,
…
,
n
1
a
a
a
a
a
∑
i
1
=
0
n
1
ρ
i
1
i
2
=
1
,
∀
i
2
=
1,2
,
…
,
n
2
a
a
ρ
i
1
i
2
∈
{
0,1
}
,
∀
i
1
=
0,1
,
2
,
…
,
n
1
,
aaaaaaaaaaaaaaaii
∀
i
2
=
0,1
,
2
,
…
,
n
2
.


## 3.3. Association Cost of Local Nodes Based on the Structural Similarity

The model (10) can be solved efficiently and optimally by the general 2D assignment algorithms, such as auction algorithm and Hungarian algorithm. Now, the remaining problem in solving (10) is how to define the connecting cost c
i
1
i
2. Instead of using the absolute kinematic states only, the structural similarity of local tracks is employed here. Three kinds of costs need to be specified: between local tracks, between dummy tracks, and between local track and the dummy track.Firstly, we define a structural feature for each nodeN
i
m (m
=
1,2; i
m
=
1
,
…
,
n
m), which is represented by the set I
i
m including all the distances from local track i
m to j (j
=
1,2
,
…
,
n
m, j
≠
i
m); that is,
(11)
I
i
m
=
{
d
(
j
,
i
m
)
}
j
=
1,2
,
…
,
n
m
,
j
≠
i
m
.We discuss the definition ofc
i
1
i
2 in terms of the following three cases.(
1) With Unity Detection Probability
P
D
m
(
P
D
m
=
1
,
m
=
1,2
). If P
D
m
=
1 (m
=
1,2), then n
1
=
n
2
=
n. We order the elements of set I
i
m from the smallest to largest: d
~
i
m
,
1
≤
d
~
i
m
,
2
≤
⋯
≤
d
~
i
m
,
n
-
1. In this way, the cost c
i
1
i
2 (for
all
i
1
=
1,2
,
…
,
n
1, for
all
i
2
=
1,2
,
…
,
n
2) can be defined by
(12)
c
i
1
i
2
=
D
(
I
i
1
,
I
i
2
)
=
∑
j
=
1
n
-
1
|
d
~
i
1
,
j
-
d
~
i
2
,
j
|
.(
2) Without Unity Detection Probability
P
D
m
(
P
D
1
·
P
D
2
≠
1
). If P
D
m
<
1 (m
=
1,2), the computation of c
i
1
i
2
(
for
all
i
1
=
1,2
,
…
,
n
1
,
for
all
i
2
=
1,2
,
…
,
n
2
) becomes complicated due to possible missed detections. To define the connection cost c
i
1
i
2, a 2D assignment subproblem (13) is constructed. As shown in Figure 4, a graphical representation is established to compute the cost c
23. As done in the optimization model (10), two dummy nodes are introduced to cope with the case of missed detections:
(13)
J
i
1
i
2
=
min
⁡
∑
j
1
=
0
,
j
1
≠
i
1
n
1
∑
j
2
=
0
,
j
2
≠
i
2
n
2
c
~
j
1
j
2
ρ
~
j
1
j
2
aai
s
.
t
aa
∑
j
2
=
0
,
j
2
≠
i
2
n
2
ρ
~
j
1
j
2
=
1
,
∀
j
1
=
1,2
,
…
,
n
1
,
j
1
≠
i
1
a
a
a
a
a
a
a
a
∑
j
1
=
0
,
j
1
≠
i
1
n
1
ρ
~
j
1
j
2
=
1
,
∀
j
2
=
1,2
,
…
,
n
2
,
j
2
≠
i
2
a
a
a
a
a
a
a
a
a
a
a
a
a
a
i
ρ
~
j
1
j
2
∈
{
0,1
}
,
∀
j
1
=
0,1
,
2
,
…
,
n
1
,
a
a
a
a
a
a
a
a
a
a
a
a
i
∀
j
2
=
0,1
,
2
,
…
,
n
2
,
j
1
≠
i
1
,
j
2
≠
i
2
.Figure 4
Graphical representation of 2D assignment subproblem to compute the costc
23.In this optimization model (13), the critical part is to define the cost c
~
j
1
j
2. For all j
1
=
1,2
,
…
,
n
1, j
1
≠
i
1, and for
all
j
2
=
1,2
,
…
,
n
2, j
2
≠
i
2, the cost c
~
j
1
j
2 is defined by the absolute value of distance difference:
(14)
c
~
j
1
j
2
=
|
d
(
j
1
,
i
1
)
-
d
(
j
2
,
i
2
)
|
.If track-pair{
i
1
,
j
1
} from sensor 1 and {
i
2
,
j
2
} from sensor 2 share the same two targets, then c
~
j
1
j
2 would be a small quantity due to the structural similarity. When given a threshold λ, we compare c
~
j
1
j
2 and the gate value λ. If c
~
j
1
j
2
>
λ, then it means that track-pair {
i
1
,
j
1
} from sensor 1 and {
i
2
,
j
2
} from sensor 2 do not share the same targets. To prevent them from associating, we set c
~
j
1
j
2
=
+
∞. In practical operations, we set c
~
j
1
j
2
=
Γ, where Γ is a large positive integer. In addition, we set c
~
0
j
2
=
c
~
j
1
0
=
1
0
-
2
Γ to encourage the isolated node to link with the dummy node. In addition, we set c
~
00
=
Γ.As far, a complete 2D assignment subproblem (13) is well defined. By solving it, the optimal fitness J
i
1
i
2
* can be obtained. It is noted that c
i
1
i
2 cannot be defined directly by J
i
1
i
2
*; further processing is required considering possible isolated tracks. We denote the number of valid associations by β, which is defined as the number of association assignments between local tracks (not with dummy nodes). If β is close enough to the expected number α of common targets (|
β
-
α
|
≤
ε, ε is a small positive integer), then c
i
1
i
2
=
J
i
1
i
2
*. Otherwise, set c
i
1
i
2
=
Γ, where the expected number α of common targets is computed by
(15)
α
=
P
D
1
P
D
2
(
n
1
+
n
2
)
(
1
+
P
D
1
P
D
2
)
.
It is proportional to the detection probability P
D
m (m
=
1,2) of sensors. If P
D
m
=
1 (m
=
1,2), then α
=
n
1
=
n
2.(
3) Connection Cost with Dummy Node. When a local node (i
1 or i
2) is declared isolated, we set c
0
i
2
=
c
i
1
0
=
1
0
-
2
Γ to encourage it to link with the dummy node. In addition, we set c
00
=
Γ to discourage the association between dummy tracks.


## 3.4. Discussions

(
1) Approximation Solution Based on the Nearest Neighborhood Method. A new approach to track-to-track association in the presence of sensor biases is given in Sections 3.2 and 3.3, which requires solving two 2D optimal assignment models. One serves to compute the connecting cost between two local tracks, and the other one is to determine the final association assignment between two sets of local tracks. To simplify, we call this method “structural similarity-based 2D assignment” (SS-based 2D assignment). Considering the complexity of the proposed model, an approximation solution can be obtained by using the nearest neighborhood (NN) method when determining the final association assignment, instead of using the optimal assignment. Such a scheme is denoted by structural similarity-based nearest neighborhood (SS-based NN). Simulation results provided before indicate that such an approximate method can also provide a better association result, because the structural property is considered when constructing the connection cost between local tracks.(
2) Possible Extensions. In practical fusion applications, heterogeneous sensors are commonly found. The proposed approach can also work well in this case, since what we handle here are local tracks after local processing. Although the measuring process is implemented in the local measurement coordinate system, local tracks from heterogeneous sensors may be in the common Cartesian coordinate system.In addition, in short-range applications or for maritime surveillance, different scattering centres of the objects under consideration may give rise to several distinct detections. At this time, what one is concerned with includes the state for both the target kinematics and the target extension. The proposed approach fails to deal with the case that the targets have more than one scattering centers. Readers may refer to literatures concerning extended target tracking [17, 18]. It is still a problem which few people get into to implement track-to-track association between extended targets.


## 4. Simulation Results

### 4.1. Scenario Setup and Results

In this section, we provide some simulation results to illustrate the efficiencies of the proposed approaches (“SS-based 2D assignment” and “SS-based NN”). Here, we consider a multiple target tracking scenario with two sensors. At the beginning of the tracking process, 30 targets are distributed uniformly in the region[
20
km
,
40
km
]
×
[
20
km
,
40
km
]. The motion of all the targets follows the constant acceleration (CA) model with acceleration 1 m/s
2. The process noise is assumed white Gaussian with zero mean and covariance matrix Q
=
(
0.1
m
/
s
3
)
I
2, where I
2 is an 2
×
2 identity matrix. The initial direction of velocity occurs randomly in [
0
,
2
π
] with the amplitude 100 m/s. The sample interval T is set to 1 s, and the tracking time is 30 s. Two sensors are located at (40000 m, 5000 m) and (15000 m, 10000 m) in the global Cartesian coordinate system (GCCS), respectively. Each sensor measures the range and angle to the target. The random range error for both sensors is modelled as white Gaussian noises with the standard deviations σ
r
1
= 30 m and σ
r
2
= 40 m, respectively. The random angle error is also modelled as white Gaussian noises with standard deviations σ
θ
1
= 0.2° and σ
θ
2
= 0.1°. Both of the sensors employ EKF to obtain the local tracks based on their own measurement information. The measurements are supported by 50 Monte-Carlo runs performed on the same target trajectories but with independently generated measurements for each trial. Except for the special declaration in the following simulations, the detection probabilities of the two sensors are P
d
1
=
0.8 and P
d
2
=
0.7; the systematic biases are set to Δ
r
1
=
400 m, Δ
r
2
=
500 m, Δ
θ
1
= 6°, Δ
θ
2
=
-5°.Performance comparisons are made between the proposed approaches and the competing algorithms. In competing algorithms, the square Mahalanobis distance is chosen as the similarity measure of two local tracks. The final association assignment is determined based on the 2D assignment algorithm and the nearest neighborhood (NN) algorithm, respectively. To simplify, they are called “bias-ignorant 2D assignment” and “bias-ignorant NN.”We adopt the probability of correct associationP
c
(
t
) at the time t to evaluate the association performance:
(16)
P
c
(
t
)
=
∑
i
=
1
L
C
i
(
t
)
L
×
N
,
where C
i
(
t
) denotes the number of tracks with correct association at time t at the ith Monte-Carlo run, L is the total number of Monte-Carlo runs, and N is the number of targets.The average probability of correct associationA
P
c is computed by
(17)
A
P
c
=
∑
i
=
1
K
P
c
(
t
)
K
,
where K is the overall tracking time.Figure5 shows sensor locations and target trajectories during the whole tracking time in one run. The probability comparison of the correct association via the tracking time is shown in Figure 6.Figure 5
Sensors and target trajectories.Figure 6
Probability of correct association via the whole tracking time.Figures7–10 illustrate the average probabilities of correct association via different detection probability, angle bias, range bias, and the total number of targets, respectively. In Figure 7, the detection probability P
D
1 of sensor 1 is set to 0.8, and the detection probability P
D
2 of sensor 2 varies from 0.5 to 1. In Figure 8, the angle bias Δ
θ
1 of sensor 1 is 1°, and the angle bias Δ
θ
2 of sensor 2 varies from 0° to 7°. In Figure 9, the range bias of sensor 1 Δ
r
1 varies from 0 m to 1000 m, and the one of sensor 2 Δ
r
2 is 500 m. In Figure 10, the total number of targets varies from 10 to 70.Figure 7
Average probability of correct association via different detection probabilities.Figure 8
Average probability of correct association via different angle biases.Figure 9
Average probability of correct association via different range biases.Figure 10
Average probability of correct association via the total number of targets.


### 4.2. Result Analysis

From Figures6–10, it is shown that the proposed “SS-based 2D assignment” and “SS-based NN” outperform the competing algorithms. Moreover, “SS-based NN” is inferior to “SS-based 2D assignment” just a little bit due to the fact that the structural property is considered when constructing the association cost. From Figure 8, it is seen that, as the angle bias increases, the association performances of the “bias-ignorant 2D assignment” and “bias-ignorant NN” experience severe degradations. However, the proposed approaches still perform well, which are insensitive to the angle bias. In addition, it is observed from Figure 10 that the probabilities of correct association given by all methods decrease to some extent as the total number of targets becomes large.Although the proposed approach performs well in the simulation experiments, it has some flaws. Firstly, the range bias is assumed to be a small one when using the proposed approach; otherwise, the distance invariance between two targets detected by different sensors cannot hold. Secondly, it is required to solve a 2D assignment model when computing the association cost. Doing so may result in a high computational burden, especially for a large number of targets.


## 4.1. Scenario Setup and Results

In this section, we provide some simulation results to illustrate the efficiencies of the proposed approaches (“SS-based 2D assignment” and “SS-based NN”). Here, we consider a multiple target tracking scenario with two sensors. At the beginning of the tracking process, 30 targets are distributed uniformly in the region[
20
km
,
40
km
]
×
[
20
km
,
40
km
]. The motion of all the targets follows the constant acceleration (CA) model with acceleration 1 m/s
2. The process noise is assumed white Gaussian with zero mean and covariance matrix Q
=
(
0.1
m
/
s
3
)
I
2, where I
2 is an 2
×
2 identity matrix. The initial direction of velocity occurs randomly in [
0
,
2
π
] with the amplitude 100 m/s. The sample interval T is set to 1 s, and the tracking time is 30 s. Two sensors are located at (40000 m, 5000 m) and (15000 m, 10000 m) in the global Cartesian coordinate system (GCCS), respectively. Each sensor measures the range and angle to the target. The random range error for both sensors is modelled as white Gaussian noises with the standard deviations σ
r
1
= 30 m and σ
r
2
= 40 m, respectively. The random angle error is also modelled as white Gaussian noises with standard deviations σ
θ
1
= 0.2° and σ
θ
2
= 0.1°. Both of the sensors employ EKF to obtain the local tracks based on their own measurement information. The measurements are supported by 50 Monte-Carlo runs performed on the same target trajectories but with independently generated measurements for each trial. Except for the special declaration in the following simulations, the detection probabilities of the two sensors are P
d
1
=
0.8 and P
d
2
=
0.7; the systematic biases are set to Δ
r
1
=
400 m, Δ
r
2
=
500 m, Δ
θ
1
= 6°, Δ
θ
2
=
-5°.Performance comparisons are made between the proposed approaches and the competing algorithms. In competing algorithms, the square Mahalanobis distance is chosen as the similarity measure of two local tracks. The final association assignment is determined based on the 2D assignment algorithm and the nearest neighborhood (NN) algorithm, respectively. To simplify, they are called “bias-ignorant 2D assignment” and “bias-ignorant NN.”We adopt the probability of correct associationP
c
(
t
) at the time t to evaluate the association performance:
(16)
P
c
(
t
)
=
∑
i
=
1
L
C
i
(
t
)
L
×
N
,
where C
i
(
t
) denotes the number of tracks with correct association at time t at the ith Monte-Carlo run, L is the total number of Monte-Carlo runs, and N is the number of targets.The average probability of correct associationA
P
c is computed by
(17)
A
P
c
=
∑
i
=
1
K
P
c
(
t
)
K
,
where K is the overall tracking time.Figure5 shows sensor locations and target trajectories during the whole tracking time in one run. The probability comparison of the correct association via the tracking time is shown in Figure 6.Figure 5
Sensors and target trajectories.Figure 6
Probability of correct association via the whole tracking time.Figures7–10 illustrate the average probabilities of correct association via different detection probability, angle bias, range bias, and the total number of targets, respectively. In Figure 7, the detection probability P
D
1 of sensor 1 is set to 0.8, and the detection probability P
D
2 of sensor 2 varies from 0.5 to 1. In Figure 8, the angle bias Δ
θ
1 of sensor 1 is 1°, and the angle bias Δ
θ
2 of sensor 2 varies from 0° to 7°. In Figure 9, the range bias of sensor 1 Δ
r
1 varies from 0 m to 1000 m, and the one of sensor 2 Δ
r
2 is 500 m. In Figure 10, the total number of targets varies from 10 to 70.Figure 7
Average probability of correct association via different detection probabilities.Figure 8
Average probability of correct association via different angle biases.Figure 9
Average probability of correct association via different range biases.Figure 10
Average probability of correct association via the total number of targets.


## 4.2. Result Analysis

From Figures6–10, it is shown that the proposed “SS-based 2D assignment” and “SS-based NN” outperform the competing algorithms. Moreover, “SS-based NN” is inferior to “SS-based 2D assignment” just a little bit due to the fact that the structural property is considered when constructing the association cost. From Figure 8, it is seen that, as the angle bias increases, the association performances of the “bias-ignorant 2D assignment” and “bias-ignorant NN” experience severe degradations. However, the proposed approaches still perform well, which are insensitive to the angle bias. In addition, it is observed from Figure 10 that the probabilities of correct association given by all methods decrease to some extent as the total number of targets becomes large.Although the proposed approach performs well in the simulation experiments, it has some flaws. Firstly, the range bias is assumed to be a small one when using the proposed approach; otherwise, the distance invariance between two targets detected by different sensors cannot hold. Secondly, it is required to solve a 2D assignment model when computing the association cost. Doing so may result in a high computational burden, especially for a large number of targets.


## 5. Conclusions

The sensor reports are disturbed by not only random errors, but also sensor biases. When track-to-track association and sensor registration meet each other, some challenging problems arise since they are tightly coupled together. In this paper, instead of using the absolute kinematic states only, the structural features of local tracks are employed for track-to-track association in the presence of sensor biases. A complete 2D assignment model for track-to-track association is given to deal with possibly missed detections. Lots of simulation results demonstrated the power of the proposed approaches. Further work includes the utilization of multiscan information and the extension of the proposed approach to the multisensor case.



---

*Source: 294657-2014-03-25.xml*