Fuzzy track-to-track association and track fusion

ARTICLE IN PRESS

Signal Processing 87 (2007) 1474–1492 www.elsevier.com/locate/sigpro

Fuzzy track-to-track association and track fusion approach in distributed multisensor–multitarget multiple-attribute environment Ashraf M. Aziz Electrical Engineering Department, Military Technical College, Koubry Elkobba, Cairo, Egypt Received 9 April 2006; received in revised form 30 September 2006; accepted 2 January 2007 Available online 13 January 2007

Abstract A great deal of attention is currently focused on multisensor data fusion. Multisensor data fusion combines data from multiple sensor systems to achieve improved performance and provide more inferences than could be achieved using a single sensor system. One of the most important aspects of it is track-to-track-association. This paper develops a fuzzy data fusion approach to solve the problem of track-to-track association and track fusion in distributed multisensor–multitarget multiple-attribute environments in overlapping coverage scenarios. The proposed approach uses the fuzzy clustering means algorithm to reduce the number of target tracks and associate duplicate tracks by determining the degree of membership for each target track. It uses current sensor data and the known sensor resolutions for track-to-track association, track fusion, and the selection of the most accurate sensor for tracking fused targets. Numerical results based on Monte Carlo simulations are presented. The results show that the proposed approach significantly reduces the computational complexity and achieves considerable performance improvement compared to Euclidean clustering. We also show that the performance of the proposed approach is reasonable close to the performance of the Bayesian minimum mean square error criterion. r 2007 Elsevier B.V. All rights reserved. Keywords: Data fusion; Data association; Track fusion; Multisensor–multitarget tracking

1. Introduction In recent years, there has been an increasing interest in multisensor data fusion for both military and civilian applications. Multisensor data fusion is the process by which data from multiple/diverse sensors, sensing multiple objects, is combined to yield improved accuracy and more inferences than Tel./fax: +20 2 2318870.

E-mail address: [email protected].

could be achieved using a single sensor system [1–3]. Data refers to the measurements (attributes) obtained by the sensors and fusion is the process of combining the inferences into a single inference of the sensed object. Currently, multisensor data fusion is used extensively in military applications for target tracking [4–7]. Fusion for target tracking involves association and estimation [8,9]. There are two major types of data association in multisensor–multitarget (MSMT) tracking systems [10–14]; measurement-to-track association (MTTA)

0165-1684/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2007.01.001

ARTICLE IN PRESS A.M. Aziz / Signal Processing 87 (2007) 1474–1492

and track-to-track-association (TTTA). In MTTA [4,7,15], measurements are selected from many to update the tracks (implemented at the sensor level). In TTTA [16–22], all the measured tracks are processed in a data fusion center to decide whether two or more tracks represent the same target or not (implemented at the fusion center level). TTTA correlates redundant tracks, which are provided from multiple sensors on the same targets, into a unique set of tracks that represents the actual number of targets i.e. it employs track correlation for associating track information from one system with that from other system. The data fusion center fuses (combines) two or more tracks when it is decided that they represent the same target. This problem is called track fusion (TF). In [15], a fuzzy clustering approach, based on fuzzy clustering means (FCM) algorithm, is proposed to solve the problem of MTTA (at the sensor level). This paper proposes a computationally efficient and cost effective fuzzy TTTA (clustering) and TF (superclustering) method (at the data fusion center level). The proposed method depends on FCM algorithm. It uses the FCM equations to reduce the number of target tracks and associate duplicate tracks by determining the degree of membership for each target track. The proposed method uses current sensor data and the known sensor resolutions for TTTA and the selection of the most accurate sensor for tracking fused targets. The FCM algorithm is not used in its recursive procedures. Instead, the proposed method utilizes the FCM equations to generate a fuzzy likelihood measure (metric) instead of the classical measures. Thus, the proposed method uses the FCM equations to generate the degree of memberships of the sensor resolutions as well as the difference vector of the common state estimates. The obtained degrees of memberships are then compared to decide whether the state estimates (tracks) represent the same target or not. Using the proposed method, observations containing kinematics and non-kinematics data can be combined and updated estimates thereby formed. Results based on Monte Carlo simulations are presented. The proposed method is able to perform track correlation and fusion with a little prior knowledge. It can handle different types of information without excessive computation. It is also proved to be efficient. The remainder of this paper is organized as follows. A brief review of TTTA and TF approaches in MSMT tracking systems is addressed

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in Section 2. The problem formulation and the proposed fuzzy TTTA and TF method is presented in Section 3. Performance evaluation and numerical results based on Monte Carlo simulations are reported in Section 4. Performance and required computation comparisons with other TTTA approaches described in [17,18,20,28,30–35,62,63] are also presented in Section 4. The advantages of the proposed approach are listed in Section 5. Finally Section 6 contains conclusions. 2. TTTA and TF in MSMT tracking systems In 1970, the first computer TTTA technique was developed by Singer and Kanyuck [23]. Their correlation technique simply represents a gating technique. Two track estimates, from two different systems, are said to be correlated if and only if the difference between all their attributes fall within certain gates. The gate sizes depend on the system accuracy in terms of the attribute noise standard deviations. Singer et al. [24,25] considered the same problem and developed a TTTA technique based on a test statistic assuming that the estimation errors of different systems are independent. The common test statistic is a weighted estimate difference that depends on the covariance associated with each estimate. Willner et al. [26] addressed the problem of TF of two track estimates assuming independent estimation errors. Bar-Shalom [10,16] retreated the same problems of TTTA and TF assuming that the estimation errors of different systems are correlated (dependent). The results show that taking into consideration the cross correlation between the two estimates reduces the estimation error [16,19,20]. The problem of TF of sensors with dissimilar accuracies is discussed in several papers (see [17,19,27–36] for examples). Several others have also made significant contributions. The results show that under certain conditions the performance of the fused track may perform worse than the performance of the better quality sensor estimate. Saha et al. [18,37,38] showed that the performance of the fused estimate is marginally better than that of the better quality sensor estimate when the sensors are dissimilar (with different sensor accuracies). The best performance of the fused estimate occurs when the two sensors are similar. The performance of the fused track is worse than the performance of the better quality estimate when the two sensor noise variances vary widely [17,20].

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A.M. Aziz / Signal Processing 87 (2007) 1474–1492

In this case, it is recommended to adopt the estimate of the better quality sensor and the fusion of both sensor estimates is not recommended. Representative TF algorithms and track association metrics are quantitatively compared in [85,86]. In general, the computational cost in generating the optimal solutions to the problems of TTTA and TF is usually excessive and infeasible for real-time surveillance systems. Furthermore, they assume idealized modeling assumptions and a prior knowledge of the signal environment, which is limited in practice. We can dispute that the use of an optimum, complicated, TTTA and TF techniques under idealized assumptions may be not better than the use of suboptimal, simple, techniques, that require a little prior information [1,2]. Unlike the optimal solutions, the suboptimal solutions provide approximate solutions to the problems of TTTA and TF. The approximate solutions are based on neural network and fuzzy logic techniques. Sengupta and Iltis [39] developed an analog neural network to solve the problem of TTTA. Their approach is capable of handling six targets and 20 measurements at most. The implementation of their approach is difficult due to the heuristic nature of the approach. Brown et al. [40] described neural network implementations for the data association algorithms in a MSMT environment. The major drawbacks to the neural network implementations are that they require unreasonable numbers of neurons and require training with a very large set of tracks [41–45]. Fuzzy systems have been proven very successfully in many important application areas such as medical imaging, robot vision, remote sensing, sonar systems and pattern recognition [46–50] and are rapidly growing to become a powerful technique for multisensor data fusion [51–54]. Fuzzy systems are well suited to manage uncertainty and to model decision-making processes [55–58] and offer the advantage of a clear understanding of their operation, because they construct knowledge by rules that resemble human thinking. For the problems of TTTA and TF in MSMT environment, the distributed sensors use their input data to form an opinion, in the form of a fuzzy membership, on the environment [59]. The features are fuzzified using membership functions. The outputs from the fuzzification are values between 0 and 1 and represent the correlation between all the tracks. These outputs are called fuzzy outputs. The fuzzy outputs from the fuzzification process are processed

using fuzzy rules represented as IF THEN rules. The defuzzification process converts the fuzzy outputs to non-fuzzy outputs, which are called crisp data. The defuzzification outputs are analyzed and compared with each other or with thresholds to determine whether two/more tracks, obtained from two or more different sensors, represent the same target. Several studies have been done in the application of fuzzy techniques to TTTA and TF. Wide et al. [60] developed a fuzzy technique for classification of measurements in different known quality profiles. Hossam et al. [61] developed a fuzzy approach for solving the data association problem in target tracking. In their approach, the measurement that has maximum degree of membership is chosen as the true measurement. Smith [62–64] developed a fuzzy logic association approach for TTTA in MSMT environments. He utilizes the fuzzy clustering algorithm to determine the grades of membership of all observations to a known number of targets. His approach requires initialization of either the prototype values or the grade of memberships. Tummala et al. [65,66] developed a fuzzy TTTA algorithm. It is applied to a real scenario comprising multiple sensors and vessels within the United States coast guard (USCG) Vessel Traffic Services (VTS) system. In their algorithm, the differences between attributes of two tracks are fuzzified using fuzzy membership functions and compared to that of the fuzzification outputs of the sensor accuracy limitations. A membership function for each attribute is chosen according to the corresponding system error. Their algorithm is tested using simulated and real data and is proved to be efficient [67,68]. Unfortunately, the extension of fuzzy TTTA to the case of a large number of sensors/targets is fairly complex due to the required large number of IF THEN rules [15]. Furthermore, as the system complexity increases, it becomes difficult to determine the right set of rules and membership functions to describe the system behavior. In addition, the solution of the conventional fuzzy logic approach to the TTTA problem is an approximate solution, and the accuracy depends on several factors including the number of input variables, the number of linguistic variables, the choice of membership function, and the accuracy of the fuzzy rules and statements. Although Singh and Bailey [69] addressed the critical problem of constructing the optimal membership function for


a given distribution of the data, constructing membership functions from statistical data requires an analytic expression for the statistical distribution of the data and assumes stationarity of the statistical environment. Furthermore, the optimal membership function in fuzzy systems design is constructed by approximating a number of closely submembership functions [70–72]. 3. Proposed TTTA and TF approach 3.1. Problem formulation We formulate the problem with a simple example. We assume a MSMT environment in an overlapping coverage scenario as shown in Fig. 1. In this scenario, we assume two sensors observe three targets. Due to overlapping coverage, the number of tracks, reported to the fusion center, is four tracks (although we only have three targets). We assume that each track/report, Rij ; i ¼ 1; 2; 3, j ¼ 1; 2, has two attributes, which are the x and y positions of the observed targets. Each report, Rij , represents the track from sensor j due to observing target i. Let us construct a data matrix, as shown in Table 1, which contains all the information (tracks) reported to the fusion center at certain scan. The columns of the data matrix stand for tracks, the things whose similarities to each other we want to estimate. Its rows stand for attributes, the features

(properties) of the tracks. Our goal is to find out which tracks are similar (i.e. represent the same target) and dissimilar (i.e. represent different targets) to each other (TTTA). The second goal is to fuse two tracks together, when it is decided that they are similar (TF). With a small data matrix like the one shown in Table 1, there is ordinarily no need for cluster analysis. We can simply look at the data matrix and find the similar and dissimilar tracks. Two tracks that have about the same values, attribute for attribute, are more similar than two objects that do not. Based on this observation, it is clear that tracks R21 and R22 are similar. By finding out that tracks R21 and R22 are similar, we automatically find out that R11 and R32 are dissimilar. But for a large data matrix (say, a hundred tracks and ten attributes) visual inspection fails us and cluster analysis becomes useful. Furthermore, the number of clusters (targets) is not known a priori, which makes the problem more complicated for a large number of data (tracks). The number of ways of clustering n tracks into c clusters is a Stirling number of the second kind [80] and is given by S ðcÞ n ¼

k¼c 1X c n k . c! k¼0 k

target 2

Sensor 2

R11

R22 R32

R21

(2)

which is very large. The problem is compounded by the fact that the number of clusters is usually unknown in practice. When the number of clusters is unknown, the number of possibilities increases markedly to

target 3

Sensor 1

(1)

For even the relatively simple problem of sorting 25 tracks into known five clusters, the number of possibilities is the quantity S ð5Þ 25 ¼ 2; 436; 684; 974; 110; 751,

target 1

1477

j¼25 X

18 S ðjÞ 25 44 10 .

(3)

j¼1

Fusion Center

This example shows the essential need of clustering.

Fig. 1. MSMT environment with two sensors and three targets.

3.2. Fuzzy track-to-track association (TTTA) Table 1 Data matrix Attribute/track

R11

R21

R22

R32

x-Position (m) y-Position (m)

100 300

200 400

202 403

300 500

The FCM algorithm [46,47,73,78–80,82] determines a partition matrix U of elements mik , which represents the degree of membership of a data point xk in a fuzzy cluster i (with a cluster center vi ) [74–77,81]. The degrees of membership are established by minimizing the sum of the squared errors


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weighted by the corresponding mth power of the degree of membership. The results are 1 mik ¼ Pc ½ j¼1 ðd ik =d jk Þ2=ðm1Þ Pn ðmik Þm xk vi ¼ Pk¼1 n m k¼1 ðmik Þ

8i; k,

(4)

(5)

The degrees of membership describe the similarities between the elements of the matrix DFCM . It is required to utilize the FCM to match our problem. Let Ri be the column vector of na attributes with corresponding resolution Di ; i ¼ 1; 2. The first sensor is assumed to be more accurate than the second sensor, i.e. D1 ðjÞoD2 ð jÞ8j ¼ 1; 2; . . . ; na , where na is the total number of attributes. The features may be range, bearing, and speed with corresponding resolutions. It is required to decide whether the two reports represent the same target or not. The idea of the proposed method is to convert all the attribute’s differences to one degree of membership. This degree of membership is compared with a threshold (another degree of membership). The threshold value represents the known physical limitations or specifications of the sensors. In the case of a radar (our case), it is based on bearing resolution, range resolution, and speed error. For automated-dependent surveillance (ADS), it is the relative accuracy of the measurements based on the type of the global positioning system (GPS) used. Thus, the threshold value for a given sensor is a single degree of membership that represents all its attribute resolutions. We consider this problem as a binary hypothesistesting problem for two local sensors. The two hypotheses are 1

the two tracks are the same;

0

the two tracks are different:

kR2 R1 k2 kD2 k2

! ¼

d 11

d 12

d 21

d 22

! , (8)

(6)

kD1 k2 kR2 R1 k2

Dc ¼ 8i,

where c is the number of clusters and n is the total number of measurements. For given observations and prototype (initial) values, the optimum degrees of membership are given by Eq. (5). Thus, the optimum degrees of membership are determined from the following matrix (assume n ¼ c ¼ 2): ! ! d 11 d 12 kx1 v1 k2 kx2 v1 k2 ¼ . DFCM ¼ d 21 d 22 kx1 v2 k2 kx2 v2 k2

H¼

We have two choices of comparison to select: (1) with respect to sensor 1, compare D1 with jR2 R1 j or (2) with respect to sensor 2, compare D2 with jR1 R2 j. We can write a matrix Dc as

(7)

where ( d ik ¼

kRk Ri j2

if iak;

2

if i ¼ k:

kDi k

(9)

The optimum degrees of membership are then given by (Eq. (5)) m11 ¼

m12 ¼

m21 ¼

m22 ¼

ðD01 D1 Þ1=ð1mÞ

, ðD01 D1 Þ1=ð1mÞ þ ððR1 R2 Þ0 ðR1 R2 ÞÞ1=ð1mÞ (10) ððR1 R2 Þ0 ðR1 R2 ÞÞ1=ð1mÞ

, ðD02 D2 Þ1=ð1mÞ þ ððR2 R1 Þ0 ðR2 R1 ÞÞ1=ð1mÞ (11) ððR2 R1 Þ0 ðR2 R1 ÞÞ1=ð1mÞ

, ðD02 D2 Þ1=ð1mÞ þ ððR1 R2 Þ0 ðR1 R2 ÞÞ1=ð1mÞ (12) ðD02 D2 Þ1=ð1mÞ

ðD02 D2 Þ1=ð1mÞ

. þ ððR2 R1 Þ0 ðR2 R1 ÞÞ1=ð1mÞ (13)

Thus, we obtain a similarity matrix U¼

m11

m12

m21

m22

! ,

(14)

where mii represents the degree of membership of the resolution of sensor i and mij represents the degree of membership of the difference between the two reports Ri and Rj with respect to sensor j (the degree of similarity between the pairs of tracks). The fuzzy decision Di based on its resolution Di can be determined based on the most accurate sensor ðD1 Þ


or on the least accurate sensor ðD2 Þ where ( 1 if m21 4m11 ; D1 ¼ 0 if m21 om11 ; ( D2 ¼

with elements (15)

( d ik ¼

kRk Ri k2 kDi k

2

if iak if i ¼ k

;

i; k ¼ 1; 2; . . . ; nr . (21)

1 0

if m12 4m22 ; if m12 om22 :

(16)

With the diversity in the relative sensor resolutions, the global decision of the fusion center ðDg Þ is always based on the least accurate sensor [23,25,66]. In this case the association decision of the fusion center will be Dg ¼ D2 .

(18) The previous track-to-track fuzzy association rule can easily be extended to the case of nr reports obtained from more than two sensors (nr represents the number of reported tracks before fusion). In this case, all the comparison terms can be defined in a matrix as 2

kD1 k B kD k2 B 2 B B B B @ kDnr k2

Thus, we obtain the following distance matrix: 0

d 11 Bd B 21 B Dc ¼ B B B @ d nr 1

d 12 d 22

d 13 d 23

d nr 2

d nr 3

1 d 1nr d 2nr C C C C. C C A d nr nr

(22)

(17)

The correlation between the two reports R1 and R2 can be defined as ( 1 if Dg ¼ 1 ðsame tracksÞ; CORRð1; 2Þ ¼ 0 if Dg ¼ 0 ðdifferent tracksÞ:

0

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2

kR2 R1 k2

kRnr R1 k2

kR1 R2 k

kR1 Rnr k

2

1

kR2 Rnr k2 C C C C. C C A 2 kRnr Rnr1 k

kRnr R1 k2

kR1 R2 k2 kD2 k2 ... ...

... ...

kRnr R2 k2

m11 Bm B 21 B U¼B B B @ mnr 1

m12 m22

m13 m23

mnr 2

mnr 3

1 m1nr m2nr C C C C. C C A mnr nr

(23)

The correlation between any two tracks h and p ðp is less accurate) is defined as (

1 0

if mhp 4mpp ðsame tracksÞ; if mhp ompp ðdifferent tracksÞ; (24)

By putting the resolutions elements in the diagonal, we can write the matrix Dc as kD1 k2 B kR R k2 2 1 B B Dc ¼ B B B @

0

CORRðh; pÞ ¼

(19)

0

By determining the similarity measures as the optimum degrees of membership using Eq. (5), we obtain the following similarity matrix:

1 kR1 Rnr k2 kR2 Rnr k2 C C C C, C C A kDnr k2

(20)

where in general, CORRðh; pÞ is the association decision based on the least accurate sensor ðminðmpp ; mhh ÞÞ. 3.3. Fuzzy track fusion (TF) Once a pair (or more) of track estimates has been aligned, i.e. the common origin hypothesis is accepted; the next step is to fuse the two estimates into a unique global estimate. One approach to obtain the global track is to adopt the superior of the tracks. The second approach is to fuse the tracks according to some score (weights). It will be shown that under certain conditions the performance of the fused track may perform worse than


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the performance of the superior track. In this case, it is recommended to adopt the superior track and TF is not recommended. The superior track can be chosen according to the resolutions of the sensors. If the sensors have the same resolution, the superior track is chosen according to the operating performance and the relative distance to the target [4,16–18,28]. The smaller the distance to the target the more accurate is the sensor estimate. In our approach, the superior track is determined automatically from the data. The superior track is the track that has maximum degree of membership in the diagonal elements of the similarity matrix. Suppose it is decided that s tracks are the same (represent the same target i), i.e. CORRðk1 ; iÞ ¼ CORRðk2 ; iÞ ¼ ¼ CORRðks ; iÞ ¼ 1.

ð25Þ

The superior track is Rsup ¼ Rksup ,

(26)

where ksup ¼ Maxk fmkk g;

k ¼ k1 ; k2 ; . . . ; ks .

(27)

In this case, the superior track is determined according to the sensor resolutions as well as the similarity between all the estimated tracks. The same tracks can also be fused as a weighted sum of the sensor estimates. The weights are the corresponding degrees of membership. A simple fuzzy average can be used to yield an overall association score for the tracks which represent the same target. The fused estimate can be defined as Rf ¼

Rik1 mk1 k1 þ Rik2 mk2 k2 þ þ Riks mks ks , mk1 k1 þ mk2 k2 þ þ mks ks

(28)

which can be rewritten as Pk s k¼k1 Rik mkk . Rf ¼ P ks k¼k1 mkk

The proposed fuzzy TTTA and TF approach is depicted in the block diagram of Fig. 2. 4. Numerical results Two examples are considered to evaluate the performance of the proposed clustering approach. The first example considers the case of four emitters with different radio frequencies (RF) and pulse repetition intervals (PRI) [63]. Fig. 3 depicts the four basic data points ðRFi0 ; PRIi0 ; i ¼ 1; 2; 3; 4Þ where 0 denotes the original values of RF and PRI in a clear environment (unknown values). The measured RF and PRI are the original values plus noise. The noise in both RF and PRI directions is assumed to be a zero mean Gaussian noise with a common standard deviation s (sigma). Figs. 4 and 5 shows the scenario in the case of random values in both RF and PRI directions for small and large values of s, respectively. The resolution of each sensor is 3 times the noise standard deviation s, i.e. ! 3sRFi Di ¼ ; i ¼ 1; 2; 3; 4, (30) 3sPRIi where sRFi ¼ sPRIi ¼ s.

Two Tracks Feature

(31)

The objective of any clustering approach is to determine the right number of clusters (four emitters in our example). The performance of the proposed clustering approach is determined in terms (corr)

Fuzzification (FCM)

(29)

Fuzzy o/p

Hard Comparison and Decision

Decision No (Different

(Δ)

Differences

Tracks) Yes

Sensors Resolution Limitations (Same Tracks) Priority of Superior Tracks or Track Fusion Fig. 2. Proposed fuzzy TTTA and TF approach.


1481

1550 1500

PRI (msec)

1450 1400 1350 1300 Separatio 1250 1200 1150 850

900

950

1000

1050

1100

1150

1200

1250

1200

1250

RF (MHz)

Fig. 3. Initial values of RF and PRI for clustering analysis.

1550 1500

Random Fixed

1450

PRI (msec)

1400 1350 1300 1250 1200 1150 850

900

950

1000

1050

1100

1150

RF (MHz)

Fig. 4. Random data of both RF and PRI for small sigma.

of correctly clustering the data points into four clusters for different values of separation/sigma. We process 100 samples (four hundred data points). The performance is evaluated as an average value over 1000 Monte Carlo simulations. Fig. 6 compares the

performance of the proposed clustering approach with the performance of Euclidean clustering [76,80]. The percentage of correct clustering using Euclidean clustering varies from 35% to 99.4%, while it varies from 64.9% to 99.8% using the


1482 1550 1500

Random Fixed

1450

PRI (msec)

1400 1350 1300 1250 1200 1150 850

900

950

1000

1050

1100

1150

1200

1250

RF (MHz)

Fig. 5. Random data of both RF and PRI for large sigma.

100

Percentage of Correct Clustering

90

80

70

Fuzzy

60

Euclidean 50

40

30 0

0.5

1

1.5

2

2.5

3

Separation / Sigma

Fig. 6. Comparison of fuzzy and Euclidean clustering.

proposed clustering approach. The performance of the proposed clustering approach is always better than the performance of the Euclidean clustering for all values of separation/sigma. The results show that the proposed clustering approach is much more

efficient than the Euclidean clustering. The proposed approach is also compared to the fuzzy approach presented by Smith [62,63]. The results show that the proposed approach achieves as much as 30% performance improvement as opposed to


target 1

sensor 1

target 2

sensor 2

R22 R11

target 3

sensor 3

R32

target 4

sensor 4

R23 R33

R34

1483

sensor 5

R44 R45

Fusion Center

Fig. 7. Four targets and five sensors in overlapping coverage scenario.

15% in the case of the fuzzy approach of Smith [62,63] in the same simulated example. The second example considers the case of four targets ðnt ¼ 4Þ, moving in straight lines, observed by five sensors ðns ¼ 5Þ in an overlapping coverage scenario. This scenario is depicted in Fig. 7. The lines on the figure indicate which sensors see which targets. The five sensors send eight reports fRij g to the data fusion center ðnr ¼ 8Þ, where Rij represents the report from sensor j due to observing target i. Each report represents the x and y positions of the targets ðna ¼ 2Þ. The targets motion model at scan K is assumed to be

where H is the measurement matrix given by 1 0 0 0 H¼ . 0 0 1 0

xðK þ 1Þ ¼ FxðKÞ þ zðKÞ,

where s2ij represents the variance of the measurement error due to observing target i by sensor j. We assume that the targets are moving in straight lines with constant velocities. The measurement noise is assumed to be Gaussian. The values of the noise uncertainties (in meters) are taken as s11 ¼ 25, s22 ¼ 30, s32 ¼ 33, s23 ¼ 40, s33 ¼ 42, s34 ¼ 45, s44 ¼ 48, and s45 ¼ 50. The measurement error is defined as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ 2 þ ðytrue y^ Þ2 , e ¼ e2x þ e2y ¼ ðxtrue xÞ (38)

(32)

where zðKÞ is the plant noise and F is the state transition matrix given by 0 1 1 d 0 0 B0 1 0 0C B C F¼B (33) C, @0 0 1 dA 0

0 0

1

and d is the sampling interval. The state vector xðKÞ contains the x- and y-target positions and velocities, i.e. 1 0 xðKÞ C B B vx ðKÞ C C (34) xðKÞ ¼ B B yðKÞ C. A @ vy ðkÞ The measurements are the x- and y-target positions, i.e. RðKÞ ¼ HðKÞxðKÞ þ wðKÞ,

(35)

(36)

Measurements are affected by noise which is modeled as Gaussian, zero mean, with a certain standard deviation. The noise sequence wðKÞ has a covariance matrix ! s2ij 0 CK ¼ , (37) 0 s2ij

where xtrue and ytrue are the true (actual) target trajectories while x^ and y^ are the estimated target trajectories. The fusion center has to process all the reported tracks and fuse the redundant tracks into a unique set of tracks. The actual target trajectories are shown in Fig. 8. The displayed tracks before fusion are shown in Fig. 9 ðnr ¼ 8Þ. The displayed tracks after fusion (after applying the clustering approach) are shown in Fig. 10. The proposed approach successfully associates all the reported


1484

7400 7200 target 3

y−position (m)

7000 6800 6600 target 2 target 4

6400 6200 6000

target 1 ns=5 nt=4 na=2

Initial positions

5800 5600 6000

6500

7000

7500

8000

8500

9000

x−position (m)

Fig. 8. Actual targets trajectories.

7400 7200 target 3

y−position (m)

7000 6800 6600 target 2

target 4

6400 6200 target 1

ns=5 nt=4 na=2 nr=8

6000 5800 5600 6000

6500

7000

7500

8000

8500

9000

x−position (m)

Fig. 9. Displayed tracks before fusion.

tracks and displays nrf number of tracks, where nrf represents the number of displayed tracks after fusion ðnrf ¼ 4Þ. All the redundant tracks are fused and all the superior tracks are correctly determined.

The performance of the fused track is also compared to the performance of the superior track of target 3 in Fig. 7. Target 3 is detected by three different sensors; sensors 2, 3, and 4. Three tracks, R32 , R33 , and R34 are reported to the fusion center.


1485

7400 7200 target 3

y−position (m)

7000 6800 6600 target 2 6400

target 4

6200 target 1

ns=5 nt=4 na=2 nrf=4

6000 5800 5600 6000

6500

7000

7500

8000

8500

9000

x−position (m)

Fig. 10. Displayed tracks after fusion. 140 Superior Track Fused Track

Mean Measurements Errors

120

100

80

SD2 = 100

60

SD3 = 100 SD4 = 100

40

20

0 0

5

10

15

Time

Fig. 11. Comparison of fused and superior track in case of same sensor resolutions.

The data fusion center can either adopt the superior of the three reported tracks (using Eq. (26)) or fuse them into a global one. The fused track is defined from Eq. (29). The sensor resolutions are defined in terms of the noise standard deviation for each sensor assuming a

common standard deviation in both x- and y-positions, i.e. SDxi ¼ SDyi ¼ SDi ;

i ¼ 1; 2; 3; 4.

(39)

The performances of the superior track and the fused track are compared in terms of the mean


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Fig. 12. Comparison of fused and superior track in case of comparable sensor resolutions.

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Fig. 13. Comparison in case of large differences in sensor resolutions.

measurements errors for different values of sensor resolutions assuming independent estimation errors. The results are depicted in Figs. 11–14. The results

show that the performance of the fused track may be worse than the performance of the superior track. As shown in Figs. 11 and 12, the performance


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200 SD2 = 100 SD3 = 500

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Fig. 14. Comparison in case of very large differences in sensor resolutions.

of the fused track is better than the performance of the superior track when the sensors have similar or comparable resolutions. As shown in Fig. 11, the best performance of the fused track occurs when the sensors are similar (having the same resolutions). Figs. 13 and 14 show that the performance of the fused track is worse than the performance of the superior track when the sensor resolutions vary widely. In this case, fusion of sensor tracks is not recommended and adopting the superior track is recommended. These results match the results of the optimum (algorithmic) techniques [17,28,30–35]. We also compare the performance of the fused track using the proposed method with TF using the Bayesian minimum mean square error (MMSE) criterion (see [18,20] for details). We consider the same scenario of target 3 in Fig. 7 when target 3 is a maneuvering target. The dominant acceleration in deterministic target maneuvers are coordinated turns. The reasons are: (1) turns generate higher accelerations (up to 9g for an aircraft turn versus 1g for thrust), (2) targets prefer to maintain a high speed when in danger, turning rather than slowing down to avoid danger. Hence, turning motion models are the dominant models for target maneuver in tracking systems [7,29,30,83]. The target motion model has the form of (32) with a plant

noise zðKÞ with a covariance Q given by [83] 1 0 D3 D2 0 0 C B 2 C B 3 C B 2 C BD C B D 0 0 C B 2 C Q ¼ q2 B 3 2 C, B D D C B 0 0 C B 3 2 C B C B A @ D2 D 0 0 2

(40)

where q2 is a scalar given by [83,84] q2 ¼ a2 D,

(41)

and a is the acceleration. The initial state estimates and the corresponding initial covariance matrix are obtained from the first two measurements by a method described by BarShalom [83]. The target motion is initially in a straight line with constant velocity. The measurements are taken every 0:1 s. After generating 250 measurements ð25 sÞ, the target institutes a 10g right turn (g ¼ 9:8 m2 =sÞ and holds the turn for 100 measurements and then returns to straight line motion for an additional 250 measurements (producing 600 measurements in all). The performance is evaluated based on 500-run Monte Carlo simulations. The results of TF are shown in Figs. 15–17.


1488 9000

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Fig. 15. Actual target trajectory and fused trajectory (proposed method).

8000

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Fig. 16. Actual target trajectory and fused trajectory (MMSE).

The actual target trajectory and the fused trajectory, using the proposed fuzzy method, are shown in Fig. 15. The results indicate the ability of the fused track, using the proposed method, to track the target correctly. Fig. 16 shows the same plots in case of MMSE criterion. Fig. 17 compares the mean

measurement errors of the proposed method with the MMSE criterion. The performance of the MMSE criterion is slightly better than the performance of the proposed method. The reason is that the proposed method does not depend on any statistical knowledge. The proposed fuzzy method is


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80 70 proposed method 60 50 MMSE 40 30 20 10 0 0

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Fig. 17. Comparison of mean measurement errors.

suboptimal (compared to MMSE), but our objective here is to show the feasibility of using a fuzzy method (approximate solution) in TTTA and TF techniques. The advantages of the proposed method are listed in the following section. The results are promising. 5. Advantages of the proposed approach The proposed fuzzy TTTA and TF approach has the following advantages: 1. The optimal membership functions are generated from the data using the FCM algorithm. They are not chosen heuristically (they are not fixed a priori). 2. The degrees of membership of the sensor resolutions are affected by the received measurements. This means that the values of the membership functions are changed according to the relative positions of the targets with respect to the sensors (adaptation to the environment). 3. The similarity between tracks are obtained by treating all the tracks at once (not pairwise). Thus, it avoids the conflict situation when track A is associated with track B, track B is associated with track C, but track A is not associated with track C.

Fig. 18. Comparison of hard and soft decisions.

4. It reduces the computational complexity with a factor of na , where na is the total number of attributes (since the proposed approach assigns only one degree of membership to each report rather than assigning one degree of membership for each attribute, the number of comparisons does not grow with the number of attributes). This also reduces the sensitivity of the final decision to individual attribute fluctuations and has the advantage of the soft decision over the hard decision (see Fig. 18). 5. The superior track is determined by the values of the sensor resolutions as well as the measurements.


A.M. Aziz / Signal Processing 87 (2007) 1474–1492

6. Using the proposed method, observations containing kinematics and non-kinematics data can be combined and updated estimates thereby formed. 6. Conclusions The problems of TTTA and TF in a MSMT environment have been considered. Fuzzy approaches to both these processes have been proposed. Performance evaluation of the proposed techniques using Monte Carlo simulations in case of Gaussian noise has been provided. Performance evaluation has been done and compared to other correlation techniques reported in the literature. Comparison of the performances of the fused track and the superior track has been also provided. The proposed approach performed correctly under all simulated scenarios. It has been shown that the performance of the fused track may perform worse than the performance of the superior track. The best performance of the fused track occurs when the sensors have the same resolutions. The performance of the fused track is worse than the performance of the superior track when the sensor resolutions vary widely. In this case, adopting the superior track is recommended and TF is not recommended. It has been shown that the performance of the proposed method significantly outperforms the Euclidean clustering and is reasonably close to the performance of the MMSE criterion. The proposed approach has many advantages including reduction of the computational complexity and performance improvement. Unlike, all fuzzy TTTA and TF approaches in which the membership functions are chosen heuristically, the optimal membership functions, using the proposed approach, are generated in unsupervised mode (automatically) from the data using the FCM algorithm. Acknowledgments The author sincerely thank Professors M.H. ElAyadi and Eweda. They played a big part in my development during my study in the Military Technical College. They have been true professionals and very supportive. References [1] E. Waltz, J. Llinas, Multisensor Data Fusion, Artech House, Norwood, MA, 1990.

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