data association and target identification using range profile

Signal Processing 84 (2004) 571–587www.elsevier.com/locate/sigpro

Data association and target identi cation using range pro leJae-Chern Yoo∗, Young-Soo Kim

POSTECH Information Research Laboratory, Electrical and Computer Engineering Division, Pohang University of Science andTechnology, San 31, Hyojadong, Namgu, Pohang, Kyungpook 790-784, South Korea

Received 1 February 2002; received in revised form 2 June 2003

Abstract

We present a new data association algorithm using range-pro le, which has maneuver-following capability. In this approach,targets can be identi ed as a by-product of the data association, not requiring a separate step for target identi cation. Early dataassociation cannot identify targets and thus requires a large amount of computation not to miss tracks when targets maneuverand cross each other. Our method using range pro le mitigates the complexity of data association. And once the classes oftracks are identi ed, the tracks can be more e8ciently tracked and associated even when target maneuvers. Furthermore,our approach can provide the optimum tracking lter gain for tracking maneuvering target and thus will contribute to theimprovement of performance for maneuvering target tracking. Extensive computer simulations have demonstrated that thenew data association is not only more e8cient in terms of the computational complexity without requiring a separate stepfor target identi cation, but also can provide the optimum tracking lter gain for tracking maneuvering target.? 2003 Elsevier B.V. All rights reserved.

Keywords: Data association; Target tracking; Radar signals; Multiple-target tracking

1. Introduction

Multiple-target tracking (MTT) is an essential re-quirement of surveillance systems which consist ofsensor, communication links, as well as computer sub-systems to interpret the measurements of the sensors[3,13]. The objective of MTT is to form observationsinto tracks so that tracking lters can be applied totrack the states of the individual targets. The accu-racy of data association is critically important sincemis-association of data to the targets will lead to track-ing failure. Other than the association accuracy, the

∗ Corresponding author. Tel.: +82-54-279-5624; fax: +82-54-279-5699.

E-mail addresses: [email protected] (J.-C. Yoo),[email protected] (Y.-S. Kim).

computational speed of the data association is alsoessential to MTT system. This is due to the fact thatthe association decisions must be made in real time asthe sensors regularly scan the surveillance region [13].For the last 20 years, many MTT algorithms have

been developed and considered in a wide variety ofliterature. There have been three main approaches tothe problem of data association, which include Globalnearest neighbor (GNN) [6,7], Joint ProbabilisticData Association (JPDA) [3,5,6], multiple hypoth-esis tracking (MHT) method [6,16] and its variants[6,10,17]. They have not only the problems such as a“hard” association and the computational complexity,but also cannot identify targets.Long-Huwi Wang proposed the data association

using range pro le which is eKective when High Res-olution Radar (HRR) range-pro les are available [20].

0165-1684/$ - see front matter ? 2003 Elsevier B.V. All rights reserved.doi:10.1016/j.sigpro.2003.11.020

mailto:[email protected]

mailto:[email protected]

572 J.-C. Yoo, Y.-S. Kim / Signal Processing 84 (2004) 571–587

The method mitigates the complexity of data asso-ciation, but since it uses only the property that thematching score between range pro les of two consec-utive aspects from the same target is greater than thatfrom diKerent targets, the targets cannot be identi edand a wrong association can be made when the targetsmaneuver [20]. In this paper, a new data associationusing range pro le is proposed and then targets canbe identi ed during the data association, not requiringa separate step. Our new data association algorithmbased on the matching score uses the range pro lesstored in database as the reference for the data asso-ciation. In general, the entries of the database rapidlyincrease as the number of target class to be discrimi-nated increases. So it is crucial to reduce the seek timeduring the evaluation of the matching score. To solvethis problem we proposed a local search algorithmcoupled with the tracking lter and introduced newparameters such as Minor-to-Principal Ratio (MPR)Concentration Of Power (COP) for identifying e8-ciently targets.There are four major advantages of our approach:

(1) targets can be identi ed during data association,not requiring a separate step;

(2) once tracks are identi ed, the tracks can be moree8ciently tracked and associated even whentargets maneuver;

(3) our algorithm can provide the optimum track-ing lter gain for tracking maneuvering target,thus will contribute to the improvement ofperformance for maneuvering target tracking;

(4) and also mitigates the complexity of dataassociation.

2. Data association employing range pro�lemeasurements

Echoes of a high-resolution radar contain informa-tion about distribution of the scattering centers of atarget. This information can be used for data associa-tion, and target identi cation as well. However, con-ventional radar signal processing fails in extractingan angular resolved range pro le when several targetsreturns fall in the same angular resolution cell. A num-ber of methods to overcome this problem have beenstudied. They involve either a special antenna pattern

Table 1The database {gij}, where the subscripts i; j represent the classand aspect of an object, respectively (M�dB: angular increment)

j

1 2 3i (006�6M�dB) (M�dB¡�62·M�dB) (2·M�dB¡�63·M�dB)1 g11 g12 g132 g21 g22 g233 g31 g32 g33

and con guration or the monopulse radar employinga variety of algorithm [2,4,18]. In this paper, it isassumed that angular resolved range pro les can beobtained by a narrow beam antenna. The data associ-ation using range pro le measurements is the processof comparing incoming measurements with the exist-ing tracks, and deciding on the correct pairs. In thepaper, the measurement-to-track association is mainlyachieved by calculating the normalized correlation co-e8cient (C(·)) between the measurements and theexisting tracks.

2.1. Global and local search

A typical data base which contains the range pro- les of the targets of interest is shown in Table 1. Itis composed of gij indexed with i; j representing tar-get class and aspect angle, respectively. Note that thedatabase given in Table 1 is structured with an angularincrement (M�dB), while the elevation angle is xedfor simplicity.The M�dB to identify targets using range pro les

should satis es M�dB¡ 2�r=D, where �r is the rangeresolution, and D is the maximum size of the target inthe azimuthal plane [14].Since the number of the entries of the database is

usually large, it is necessary to reduce the seek timeduring the evaluation of the matching score. Limitingthe number of comparisons to be made is achievedby considering only the database which lies withina probable aspect boundary, termed as local search,centered around the predicted aspect of each track.However, for the newly initiated targets, globalsearches are required.

J.-C. Yoo, Y.-S. Kim / Signal Processing 84 (2004) 571–587 573

2.1.1. Global searchLet us assume that ‘I observations at the rst scan

are found inside an initial gate. The global search forthe ‘I initiated targets is then given by

C(gij; mn :∀i; j)¿THmc for n= 1; : : : ; ‘I; (1)

where C(·) is the normalized correlation coe8cientbetween gij andmn,mn the initiated measurements andgij the range pro le in the database indexed with thesu8x i and j.The THmc is a constant that controls the false-alarm

rate, and there is a trade-oK between the complexityof data association and the missing target (or falsealarm) rate.Let us denote by Trackn the set of all possible

index pairs (i; j) such that C(gij; mn :∀i; j)¿THmc fora given n, and by T the set of all Trackn. Each elementof the set T is used as the track-reference, denoted byGt(k), at the next scan.So, we have

T= {Track1;Track2; : : : ;Track‘I};

Gt(1) = Trackt for 16 t6 ‘I: (2)

2.1.2. Local searchFor non-initiated targets (k¿ 2), the local search

is conducted as follows. Here let us assume that ‘observations are found inside a validated gate centeredat a track t.(i) First, nd the target candidates satisfying the

following criterion:

C(gij; m(t)n (k) : i; j∈Gt(k − 1))¿THmc

for n= 1; : : : ; ‘; (3)

where Gt(k − 1) is the track-reference to the track t,given at scan k − 1. m(t)

n (k) the measurements withinthe validated region centered at the track t at scan k(the su8x n is the index of measurements within thevalidated region).Let us denote by rnt (k) the set of all possible in-

dex pairs (i; j) such that C(gij; m(t)n (k) : i; j∈Gt(k −

1))¿THmc for a given n and t, and by Rt(k) the setof all rnt (k).So, we have

rnt (k) = {(i1; j1); (i2; j2); : : :} for a given n and t;

Rt(k) = {r1t (k); r2t (k); : : : ; r‘t (k)} for a given t: (4)

(ii) Next, nd pairing-candidates.When several detections are made within the

validated gate, ambiguous situations may arise:multiple tracks may compete for a singlemeasurement or multiple measurements within thevalidated gate can be associated with the same track.In this case, the data association is performed by con-structing ameasurement-track distance matrix with theset Rt(k). Most of the ambiguities are then resolvedand the best overall selection of measurement-trackpairings can be made.Problems associated with multiple measurements

and tracks at scan k are illustrated in the exampleshown in Fig. 1.(a) First, a matrix of distances with the set Rt(k)

between the measurements and tracks is formed asshown in Table 2.(b) Next, each validated gate is then examined to

nd out which gate has the smallest number of mea-surements by comparing the number of the elementsof the set Rt(k) for each track.(c) Finally, nd the K-nearest neighbors that are

closest to the estimated track among the elements ofthe set Rt(k) corresponding to the gate subject to (b).The K-nearest neighbors are then paired with the

track t, and the associated measurements are allowedto be reused for the remaining gates.Let us denote the K-nearest neighbors by Et(k)

(pairing-candidates) and the closest neighbor amongthem by Ec

t (k).The Ec

t (k) is excluded from the closest neighbor forthe remaining gates.(d) Repeat (b) and (c) for the remaining gates.When Ec

t (k) for all gates is made, the track informa-tion can be updated to produce the re ned estimatesof position and target velocity through the existingtracking lter.In this example, track A is updated by measurement

No. 8, track B by measurement No. 9, and track C bymeasurement No. 7 as shown in Table 2.All other unpaired measurements such that

C(·)¡THmc can be declared as the initiated tracks,and then the global search is conducted. They willbe con rmed or ignored by the succeeding scans. Ifall the matching scores (the normalized correlationcoe8cient) within the validated gate are smaller than


Fig. 1. Example of the problem caused by multiple measurements and tracks in close vicinity.

Table 2Measurement—track distance matrix for the example of Fig. 1(distance: Euclidian distance) (X: not associated) (�: paired)

Track (t )

Measurements A B C

1 X X2 X X3 X X4 X X5 X X6 X X7 10.4 13.5 12.38 9.3 5.6 X9 12.1 X

XXXXXX

X

THmc, whether the corresponding track will be deletedor not will also be determined by the succeedingscans. This strategy has been well known as the trackdeletion and initiation logic such as N detections outof M opportunities [9].(iii) Finally, nd the track-reference Gt(k) for the

next scan.The track-reference Gt(k) is a modi ed version of

the Ect (k) by an aspect constraint. The constraint can

be made by considering only the part of the database

Fig. 2. Local search based on the aspect constraint. Several possiblebranches based on the aspect constraint for a given i, in whichjp = j +M�p and J indicates the scope of j for the local search.

which lies within a probable aspect angle boundary(±M�max) centered around the predicted aspect angle(jp = j +M�p) of the track (see Fig. 2). Note that Jindicates the scope of j to be considered during thelocal search. Here, the M�max depends on the targetclass.Based on the aspect constraint, we have

Gt(k) = {j +M�p −M�max6 J6 j +M�p

+M�max | i; j∈Ect (k)} for a given t; (5)


Fig. 3. An example of the network representing the true and short-lived branches for a given i1. Infeasible connection occurs when theaspect angle change between neighboring scans is greater than M�max.

where M�p=�p(k+1)−�p(k) for the track t, M�max=maximum allowable aspect angle change between twoscans,

�p(k + 1) = tan−1(Vpy(k + 1)=Vpx(k + 1));

Vp(k + 1) =√V 2px(k + 1) + V 2

py(k + 1)

= the predicted velocity on the

Cartesian coordinates:

(For simplicity, it is assumed that the dimension ofthe j; M�p and M�max is identical.)(iv) Repeat (i)–(iii) for the next scan.The above procedures ((i)–(iii)) provide the

information about pair (Et(k); r(k); �(k); v(k)) foreach track. These represent the set of pairing-candidates,range, azimuth position and the velocity of a giventrack t at scan k, respectively.

It can be noted that the number of pairing-candidatesfor identifying a track gradually reduces as the numberof radar scans increases, owing (1) to improvementof the tracking accuracy with the increased amountof processed data, (2) to decrease of the gate size and(3) to the accumulated aspect constraints. Therefore,the unknown tracks are usually tuned as the numberof radar scans increases and can nally be identi edwith a dwelling time.

3. Target identi�cation

A network structure with the set {Et(k) | i = i1}can be made as shown in Fig. 3, which illustrates theidenti cation mechanism for a given track t. Here,each node (point) represents the elements of the set{Et(k) | i1} with the index of aspect angle versusradar scans. The connected nodes form branches to


connect between the nodes. The problem is to ndthe true branch among probable branches formulatedon the network structure. Provided that a given targeti1 moves with a linear aspect angle change, the con-nected nodes by {Et(k) | i1} will form a linear-likebranch on the network and most of the elements ofthe set {Et(k) | i1} will be around the linear branch.As stated previously, since the elements of the set{Et(k) | i1} are usually tuned as the number of radarscans increases, more and more points will be aroundthe linear branch.The best- tting line (the linear branch) to the set

{Et(k) | i1} can be obtained by “eigenvector line t-ting” and is in the direction of the principal eigenvec-tor of the set {Et(k) | i1}.Our prescription for the best- tting line is the fol-

lowing:(i) Standardize the points by subtracting the mean of

the set from each point and let us denote it’s correlationby the matrix S

S = [{Et(k) | i1} − {Et(k) | i1}] · [{Et(k)|i1}−{Et(k)|i1}]′;

in which the superscript ′ denotes the transpose of amatrix.(ii) Find the principal eigenvector of the matrix S

of the set of standardized points, and denote it by(xeigen ; yeigen).Based on the above prescription, the proposed iden-

ti cation algorithm is as follows:First, let us assume that a target is measured by

a radar device and moves with a linear aspect anglechange, but not yet identi ed.

Step 1: Calculate the eigenvector for the set{Et(k) | i = i1}.To identify the target, we rst think of the target

as i1 and nd it’s principal axis on the network bycalculating the eigenvector (xeigen ; yeigen) for the set{Et(k) | i1}, which is extracted from the set {Et(k)}.If the assumption that the target is i1 is true, then

most of the points of the set {Et(k) | i1} will be onthe principal axis. Otherwise, the points will be ran-dom distributed regardless of the principal axis. Theprincipal axis having the random distributed points isreferred to as “apparent branch” in this paper. Figs.4(a) and (b) are examples representing the points(circle dots) spread along the 45◦ line of a true

branch while Fig. 4(c) shows an apparent branch.Since the elements of the set {Et(k)|i1} are usuallyuntuned in the vicinity of target’s initiation, it is nec-essary to have an initial delay kdelay for calculatingtheir eigenvector with robustness. Comparing withFig. 4(a) and (b) shows that the true branch can bebetter found in Fig. 4(b), the principal axis of whichis obtained by calculating the eigenvector for the set{Et(k)|i1; kdelay6 k6 kdwell}. The dwell time (kdwell)represents the minimum scan duration required toobtain a reliable eigenvector after the initiation kinit .

Step 2: Calculate the minimum to maximum ratio(MPR).To discriminate the true branch from apparent

branches, calculate the minimum to maximum ratiofor eigenvalue, termed as minor-to-principal ratio(MPR):

MPR =min(eigen value for {Et(k) | i1})max(eigen value for {Et(k) | i1})

for kdelay6 k6 kdwell: (6)

The MPR is generally very small when most of thepoints are on the principal axis [11,15].It can be noted, from a distribution for the MPR

feature shown in Fig. 13, that the MPR for a truebranch is smaller than that for apparent branches, anda discrimination between them can be made by simplysetting a threshold THMPR.However, as shown in Fig. 13, since there is an over-

lap between the true and apparent branches, no mat-ter how THMPR is chosen, it is impossible to perfectlyseparate the true branch from the apparent brancheswithout false alarms.The typical example for the ambiguous situation is

shown in Fig. 5. Since the apparent branch shown inFig. 5(a) has a small MPR, it may be misconceivedas a true branch, thus occur false alarm.The problem can be usually solved by additionally

considering the sum (d2) of squares of the perpen-dicular distances from the points to the principal axis,which is given by Haykin [8]:

d2 = N ′SN; (7)

where N is the unit normal vector of the principal axis.Since the sum (d2) of a true branch is generally

much smaller than that of apparent branches, it makesa good discriminator. However, this method has veryhigh computational requirements that make them


Fig. 4. An example representing the eigenvector for a true and apparent branch for a given target i1, in which it is assumed that thetarget is moving with a linear aspect change M�=5◦ between the consecutive scans: (a) true branch with kdelay = 0, (b) true branch withkdelay = 10 and (c) apparent branch with kdelay = 10.

impractical, especially for a long dwell time (kdwell).The alternative good discrimination can be made byemploying the THCOP.

Step 3: Compute the concentration of power (COP).The concentration of power (COP) indicates how

many points are within a principal window (seeFig. 6):

COP=thenumberof pointswithinaprincipalwindow

k − kinit

× 100(%);

where the principal window is given by

y =yeigenxeigen

k ±M�max for kinit6 k6 kdwell; (8)

the y-axis represents the aspect angle (j).

Since COP only counts the number of the pointswithin the principal window, it is computationally verye8cient, compared to Eq. (7). Note that COP may begreater than 100%.

Step 4: Identify the track t as follows:(i) If MPR6THMPR and COP¿THCOP, then

declare the track t as i1, go to Step 6 (the THCOP is adecision boundary of COP between true and apparentbranches).(ii) Otherwise, go to Step 5.Step 5: Repeat steps 1–4 for the other i (i =

i2; i3; : : :).Step 6: Repeat steps 1–5 for the next track t.Note that once tracks through the above steps are

identi ed, even though the tracks maneuver theycan be more e8ciently tracked and associated than


Fig. 5. An example of an apparent branch having a small MPR(M�max = 15◦): (a) apparent branch: M� = 2◦, SNR = −3 dB,kdelay=10, kdwell=40, MPR=0:0153, COP=5:0%; (b) true branch:M�= 2◦, SNR =−3 dB, kdelay = 10, kdwell = 40, MPR = 0:0045,COP = 125:0%.

when they are not identi ed. Our algorithm requiresthe assumption that targets move with a tendencyto a linear aspect angle change within M�max. How-ever, since most of the targets usually keep movingbetween maneuvering and stationary state, the targetscan be identi ed at the instant that it moves with thestationary state.

4. Data association for maneuvering target

The Kalman lter performs almost perfect trackingin case where target model ts the real target trajec-

Fig. 6. Principal clusters within the principal window(M�max = 15◦).

tory, and when the statistical characteristics of the tar-get maneuver and measurement noise such as meanand variance are known [12]. However, in practice,it is di8cult to know the statistical characteristics ofthe target in advance when the target maneuvers. Toovercome the problems, the two-stage Kalman lter,turn detector which consists of two validated gatesor maneuver adaptive ltering methods can be used[1,6,19]. However, all of them have the problem thatthe optimization of the maneuver detector is quitedi8cult and complicate.The tracking index (target maneuvering index: �)

is the parameter which indicates the value of target’sTuctuation as follows [3]:

�= T 2 �!�n; (9)

where �2! and �2n denote variance of target’s maneuver

and measurement noise, respectively.It is well known that the optimum lter gain can be

obtained uniquely by variance of target’s maneuverand measurement noise [3]. While variance of mea-surement noise can be obtained, variance of target’smaneuver is unknown because there is in advanceno way to know the target’s trajectory from radarsite. However, if it is possible to measure the degreeof target’s maneuver or the tracking index (�), theoptimum lter gain for tracking maneuvering targetcan be obtained.


Fig. 7. The data association using the gate-growing process, in which the growing direction is decided by the predicted velocity,

Vp =√V 2px + V 2

py .

4.1. Validated gate and the tracking index

The estimation of the tracking index is based onanalytical results of the relationships between the val-idated gate size and tracking index. The validated gatesize is given by Farina and Studer [9]

Gr = �r"(1 +

2(2k + 1)k(k + 1)

+�2r

4

)1=2(10)

and

G� = ��"(1 +

2(2k + 1)k(k + 1)

+�2�

4

)1=2;

where �2r ; �2� is the variances of measurement errors

on the polar coordinates, �r ; �� the tracking index onthe polar coordinates, " the parameter " de nes thevalidation region is chi-square distributed with numberof degrees of freedom equal to the dimension of themeasurement and probability which the true target isfall in the gate [3] and k the number of radar scan.Here, it can be noted that the gate size reduces as

the number of radar scans increases (gate-decreasingprocess). The tracking index, on the polar coordi-nates, corresponding to a given gate size can be easily

obtained from Eq. (10):

�r = 2

√(Gr

�r"

)2− 1− 2(2k + 1)

k(k + 1)

and

�� = 2

√(G��"

)2− 1− 2(2k + 1)

k(k + 1): (11)

Miss track will occur when the measured value is farfrom the predicted value. In this case, the larger vali-dated gate is required to associate with the track. Thatis, when a pairing is not made during association, theassociation is continued with a growing gate until apairing conforming to the following criterion (12) ismade (see Fig. 7: the gate-growing process):

C(gij; m | i; j∈ J)¿THmc; (12)

where i is the class of an identi ed track, m the rangepro le of a new measurement introduced into the val-idated region during the gate-growing process, J thescope of j to be considered during the local search.There is the allowable maximum gate size that de-

pends on the target’s attribute such as the maximumspeed and turn-rate. From the nal gate size, it ispossible to determine not only the degree of target’smaneuver (tracking index: �) using Eq. (11), but also


Fig. 8. The Towchart showing the relationship between the validated gate and the tracking index.

the optimum lter gain. Here note that the associa-tion(Eq. (12)) based on the identi ed range pro lemakes the gate-growing process very reliable.Fig. 8 is the Towchart showing the relationship

between the validated gate and the tracking index. Todetect the degree of the target’s maneuver when misstrack occurs, the data association is continued with a

growing gate until a pairing is made and then a ma-neuver is declared. Next, by using the nal validatedgate size, determine the tracking index � from Eq.(11), and thus we can nd the optimal lter gain.Finally, the tracking information is recursively up-dated to produce re ned estimates of the position andthe velocity.


Fig. 9. Target generation for simulation: (a) six groups generated by scrambling the original 58 range pro les, (b) target 1 generated byconcatenating the six groups and (c) 10 targets generated by scrambling the range pro les of target 1, each of which consists of 328 rangepro les.

5. Experimental results

Simulations were conducted to test the associationand identi cation algorithm proposed in this paper.In the simulation, it was assumed that (i) all the tar-gets have the property of azimuthal symmetry in theaspect angle ranges from 0◦ to 180◦, (ii) the targetaspect angle has been changed by M� between theconsecutive scans, (iii) two targets always exist insidea validated gate, 2-nearnest neighbor rule was usedand (iv) the angular resolved range pro les can beobtained.

5.1. Target generation and database

Ten kinds of targets using a model airplane coatedwith silver were generated for the simulation. Thephysical size of the plane is approximately 100 cm inlength and 20 cm in height. Hp 8720C network an-alyzer was used as the transmitter and the receiver,and separate transmit and receive antennas were usedat VV polarization. A bandwidth of 2 GHz at X-band(9–11 GHz) was used in the experiment to have therange resolution of 7:5 cm. This ne resolution wouldbe able to justify the use of small model targets. For


Fig. 10. An example of C(gij ; gij | i) over � for diKerent SNR.

the model, 58 range pro les having 0:55◦ angular in-crement (M�dB = 0:55◦), where the elevation anglewas xed at 25◦, were measured.The measured range pro les were stored into the

hard disk on Personal Computer from the networkanalyzer through GPIB interface.These 58 range pro les were scrambled and con-

catenated to generate the 10 targets with aspect an-gles ranging from 0o to 180o. To this end, as shownin Figs. 9(a) and (b), six groups denoted by #1–#6,each of which consists of the 58 range pro les, weregenerated by random scrambling the original rangepro les (1–58). And then they were concatenated for328 range pro les of target 1. Note that the target 1 hasat least ve identical range pro les at diKerent aspects.As shown in Fig. 9(c), the other targets were gener-ated by random scrambling the 328 range pro les oftarget 1. Each of the 10 targets has 328 range pro- les, and thereby the database (10×328) explained inTable 1 was established. Since all the 10 targets weremade by scrambling and concatenating the original 58range pro les, all of them are ambiguous and confus-ing enough to test the performance of the data associ-ation and identi cation algorithm. That is, the fact that

each target contains at least ve identical range pro- les at its diKerent aspects and shares the same rangepro les with the other will make the data associationand identi cation very hard.

5.2. The choice of THmc

Fig. 10 shows the examples of C(gij; gij) for a giveni over �, where gij = gij +AWGN.The gij is used for the test data set. It can be seen

that the normalized correlation coe8cient decreasesas the signal to noise ratio (SNR) decreases. Fig. 11provides information on how to choose the suitablethreshold THmc, which is given by

THmc = C(gij; gij)− �ij; (13)

where C(gij; gij) and �ij are the mean and standarddeviation of the normalized correlation coe8cientbetween gij and gij for all i; j.The lower the SNR is, the lower the THmc is. The

values of THmc versus the SNR are stored in the formof look-up table and are used when Eqs. (1), (3) and(12) are evaluated.


Fig. 11. THmc versus SNR.

5.3. Target dynamic model and measurement model

It was assumed that the target motion is modeled as[position : x(k + 1)

velocity : v(k + 1)

]

=

[1 T

0 1

][x(k)

v(k)

]+

[12T

2

T

]w(k) (14)

and the corresponding measurement model for posi-tion is given by

z(k) = x(k) + n(k); (15)

where T is the scan period, w(k) the unknown maneu-verability of target and n(k) the radar measurementnoise.The w(k) and n(k) are zero-mean white Gaussian

distributed.The Kalman ltering algorithm has been coded in

the Matlab environment. A tracking system is consid-ered to measure a target position every T = 1 s witherrors of 20 m standard deviation in range, and30 mrad in azimuth direction.And it is also assumed that the target maneuver

standard deviation is 2 gm=s2 in range and 29 mrad=s2

in azimuth direction, and there are always two

measurements within the validated gate. The pa-rameter " was set to “3”, which implies that thevalidated gate is chi-square distributed with 2degrees of freedom and probability of 0.989 inthe gate.

5.4. Scenarios

Two kinds of target trajectory were taken into con-sideration, which are:

• radial trajectory and• trajectory with centripetal acceleration.

Each trajectory model includes both stationary andmaneuvering trajectory. Fig. 12(a) consider the radialtrajectory of target, whose position is sampled everyT = 1 s. It is non-maneuvering moving with constantvelocity of 40 m=s until k = 50 (A6 k6B).A constant acceleration with 20 m=s2 starts at k=51,

and is completed at k = 64 (B¡k6C).Non-maneuvering moving continues again during

another 137 scan (C¡k6D). Fig. 12(b) considerthe trajectory of target with centripetal accelera-tion, whose position is sampled every T = 1 s. It isnon-maneuvering moving with constant velocity of40 m=s until k = 50 (A6 k6B). A turn starts atk = 51 with turn gravity of 2:0 G, and is completedat k = 151 (B¡k6C). Non-maneuvering movingcontinues again during another 50 scan (C¡k6D).Also, Fig. 12 shows their validated gate and trajec-

tories. Note that the validated gate size varies as thedegree of target’s maneuver and has its peaks in thevicinity of maneuvering area.The database (gij) and the test data set (gij) were

used in a series of identi cation experiments. Letus denote the classes of the database by i1; i2; : : : ; i10and while those of the test data set by i1; i2; : : : ; i9and i10.The simulation was conducted for a given ia (a =

1; 2; : : : ; 10) and was assumed that there are two mea-surements (ia; ib for a �= b) within the validated gate.One (ia) of the two measurements is moving with a xed M� while the other (ib) with a random aspect an-gle change regardless of M�. It emulates well the sit-uation that many diKerent targets (ib) frequently crossthe track of the ia.


Fig. 12. Validated gate and its size along various trajectories: (a) validated gate along the radial trajectory and (b) validated gate alongthe centripetal trajectory.


Table 3The average MPR and COP for true and apparent branches

SNR M�

True branch Apparent branch

2◦ 5◦ 7◦ 10◦ 2◦ 5◦ 7◦ 10◦

Average MPR kdelay = 10, kdwell = 40−3 0.01835 0.01206 0.00906 0.00612 0.03487 0.05821 0.06957 0.05928−2 0.03017 0.00880 0.00745 0.00689 0.03386 0.05950 0.08671 0.05888−1 0.06116 0.00692 0.00740 0.01080 0.03278 0.06631 0.07709 0.052900 0.00543 0.00089 0.00094 0.00374 0.03161 0.07062 0.08088 0.060551 0.00543 0.00089 0.00083 0.00347 0.03171 0.06101 0.08021 0.052872 0.00543 0.00089 0.00083 0.00347 0.03171 0.06177 0.08223 0.055863 0.00543 0.00089 0.00083 0.00347 0.03171 0.06177 0.08021 0.055864 0.00543 0.00089 0.00083 0.00347 0.03171 0.06177 0.08021 0.055865 0.00543 0.00089 0.00083 0.00347 0.03171 0.06177 0.08021 0.05586

Average COP kdelay = 10, kdwell = 40−3 121.25 124.00 122.25 92.50 9.75 11.00 12.75 11.50−2 115.50 121.75 115.75 91.75 8.55 11.00 12.00 27.75−1 106.00 118.25 114.75 81.25 7.75 12.50 11.25 12.250 122.50 132.20 126.50 100.2 10.5 13.75 13.50 11.501 122.25 132.25 127.50 100.75 10.5 13.25 13.50 13.202 122.25 132.25 127.50 100.50 10.5 13.25 13.50 13.503 122.25 132.25 127.50 100.50 10.5 13.25 13.50 13.504 122.25 132.25 127.50 100.50 10.5 13.25 13.50 13.505 122.25 132.25 127.50 100.50 10.5 13.25 13.50 13.50

The simulation procedure is as follows:

for ia = i1; i2; : : : ; i10(i) The global and local search are con-ducted until the kdwell and then nd the setEt(k|kinit6 k6 kdwell; k ∈ stationary trajectory).(ii) To identify the target at the kdwell, we rst thinkof the target as i1 and nd it’s MPR and COP withthe set Et(k|i = i1; kinit6 k6 kdwell).(iii) If the target was not identi ed, repeat the abovestep (ii) with thinking of the target as the otheri(i2; : : : ; i10).

endFour kinds of linear aspect change trajectory (M� =2◦; 5◦; 7◦; 10◦) for each target trajectory with variousS=N ratios were simulated.

5.5. Results

Table 3 summarizes the average MPR and the aver-age COP on the test data set, respectively. The tables

show that the average MPR for true branches is usu-ally lower than that for apparent branches while theaverage COP for true branches is much greater thanthat for apparent branches. Since there is an overlapbetween the true and apparent branches in the MPRdistribution as shown Fig. 13, the THMPR is not enoughto discriminate the true branches. The good discrimi-nation can be made by considering the THCOP togetherwith THMPR.Fig. 14(a) shows the average COP versus the SNR

for various kdwell. It can be noted that true branches canbe eKectively discriminated from apparent branches bysetting the THCOP to such as 40, 60 and 80. Fig. 14(b)shows the average identi cation rate versus THMPR

for various kdwell when we set the THCOP to 80. Wealso obtained the same results as Fig. 14(b) for settingthe THCOP to 40 or 60. Note that the choice of THCOP

as discriminator is not rigorous. The identi cation rateis de ned as the percentage of successful recogni-tions for all trials and was obtained by averaging overdiKerent S=N ratios and M�. This simulation result


Fig. 13. Distribution of the MPR for true and apparent branches.

performed for the stationary areas (kdwell6 k6B)shows that the identi cation (data association) rate isabout 96% with the false alarm rate of 0% when it isassumed that M�max =15◦, kdelay =10, THMPR =0:03.Table 4 summarizes the rate of correct-association

in cases of radial and centripetal trajectory, divid-ing the trajectory into the stationary and maneuveringareas.The high accuracy in maneuvering areas is mainly

due to the fact that the tracks during passing throughthe stationary area have already identi ed. It is there-fore considered that the proposed scheme has muchbetter maneuver-following capability. The limited ex-perimental results presented in this paper show thattargets can be identi ed with a practical performanceas a by-product of the data association, not requiringa separate step for target identi cation.

6. Conclusions

In this paper, a new data association algorithm us-ing range-pro le is proposed for maneuver-followingcapability. In this approach, targets can be identi edas a by-product of the data association, not requiringa separate step. Our algorithm requires the assump-tion that the targets move with a tendency to a linearaspect angle change within M�max. However, sincethe targets usually keep moving between maneuver-ing and stationary state, the targets can be identi ed

Fig. 14. Simulation results (kdelay = 10): (a) average COP versusS=N ratio for various kdwell and (b) correct identi cation rateversus THMPR for various kdwell.

at the instant that moves with a linear aspect anglechange. And once the classes of tracks are identi ed,the tracks can be more e8ciently tracked and asso-ciated even when the targets maneuver. Our methodusing range pro le can considerably mitigate thecomplexity of data association since the data associ-ation is performed by comparing the matching scorebetween candidates. Furthermore, our approach usingthe gate-growing process can provide the optimumtracking lter gain for tracking maneuvering target.Therefore, it will contribute to the improvement ofperformance for maneuvering target tracking. Thelimited experimental results presented in this paper


Table 4The rate of correct-association averaged over diKerent SNR (kdelay = 10, kdwell = 40, M�max = 15◦, THMPR = 0:03, THCOP = 80)

Radial trajectory Centripetal trajectory

kdwell-B B-C C-D kdwell-B B-C C-D

The rate of correct-association (%) 2◦ 98.2 97.3 98.3 96.8 93.3 90.15◦ 97.0 94.5 98.5 95.6 93.7 93.27◦ 97.3 95.6 93.7 96.1 92.1 90.010◦ 92.2 93.7 94.4 95.2 90.8 88.5

show that targets can be identi ed with a practicalperformance as a by-product of the data association,not requiring a separate step for target identi cation.

Acknowledgements

This work was supported in part by the Agencyfor Defense Development, Korea through MicrowaveApplication Research Center and by the Ministryof Education, Korea through its BK21 Program andPOSTECH Information Research Laboratory (PIRL).

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data association and target identification using range profile

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