Multiple Target Tracking With Competitive Hopfield Neural Network Based Data Association

[5] Misra, P., and Enge, P.Global Positioning System: Signals, Measurements, andPerformance.Lincoln, MA: Ganga-Jamuna Press, 2006.

Multiple-Target Tracking with Competitive HopfieldNeural Network Based Data Association

Data association which obtains relationship between radar

measurements and existing tracks plays one important role in

radar multiple-target tracking (MTT) systems. A new approach

to data association based on the competitive Hopfield neural

network (CHNN) is investigated, where the matching between

radar measurements and existing target tracks is used as a

criterion to achieve a global consideration. Embedded within

the CHNN is a competitive learning algorithm that resolves the

dilemma of occasional irrational solutions in traditional Hopfield

neural networks. Additionally, it is also shown that our proposed

CHNN-based network is guaranteed to converge to a stable

state in performing data association and the CHNN-based data

association combined with an MTT system demonstrates target

tracking capability. Computer simulation results indicate that this

approach successfully solves the data association problems.

I. INTRODUCTION

Multiple-target tracking (MTT) is a prerequisitestep for radar surveillance systems. The objective ofa tracking algorithm is to partition sensor data intosets of observations produced by the same target.Once tracks are formed and confirmed, the numberof targets can be estimated and quantified information,such as the target position and velocity, computed foreach track. Generally, MTT tracking problems consistof three parts: track initiation, track maintenance,and track deletion. Data association is one of mostimportant techniques in order to complete the trackmaintenance.In the past, several algorithms have been

investigated in the literature. A well-known dataassociation algorithm, referred to as joint probabilisticdata association (JPDA) method, is suited for a high

Manuscript received April 7, 2006; revised December 18, 2006 andMay 8, 2007; released for publication July 10, 2007.

IEEE Log No. T-AES/43/3/908425.

Refereeing of this contribution was handled by B. La Scala.

This work was supported by the National Science Council underGrant NSC95-2221-E-212-021.

0018-9251/07/$25.00 c 2007 IEEE

false target density environment [1, 2]. Anothermethod is a unifying approach to MTT which wasdeveloped by Emre and Seo [3]. Some other radartracking algorithms were also discussed in severalother papers [46]. However, these techniques arenearest neighbor or all neighbors-based and usuallyconsider relationship between radar measurements andexisting target tracks independently. Other approachesusing neural networks to MTT are also proposedin [7][9]. Currently, neural network approachesare developed based on traditional Hopfield neuralnetwork, which takes weighted objective cost andconstraints into an overall energy function. Then theHopfield neural network evolves its state of eachneuron in a direction that the overall energy functionis decreased. Since the state of each neuron indicateswhether a measurement should be associated witha target, when the network converges, its final stategives us the solution of the data association results.However, this approach has the problem that theweighting values between the objective cost andconstraints in the overall energy function are verydifficult to be properly determined. As such, it is oftenthe case that the solution may not be satisfactory asreported in Zhou [9]. The proposed approach usingcompetitive Hopfield neural network (CHNN) in[10], [11] can update objective function and costmeasurement to resolve this dilemma.In a dense target environment, some targets

can be very close to each other. The measurementsproduced by these close targets can confuse dataassociation computation algorithms and resultin inaccurate target association. Consequently,an effective approach to solving data associationproblem should be considered globally. Based onthis consideration, the work presented here proposesan MTT approach using a CHNN to obtain a globalmatching between radar measurements and existingtracks. The CHNN is an improved Hopfield neuralnetwork wherein a cooperative decision is made basedon simultaneous inputs of a community of neurons.Each neuron receives information from other neuronsand also broadcasts information to other neurons.With this collective information, each neuron issettled to a stable stage with the lowest value of apredefined energy function. As such, the associationbetween radar measurements and existing trackscan be obtained under global optimal considerationswhich in turn, can increase the accuracy of radartracking systems. Furthermore, due to the embeddedcompetitive updating scheme, the CHNN can relievethe burden of weight setting. It is also proved thatthe network is guaranteed to converge to a stable andrational state during network evolution so that thedilemma of falling into irrational solutions such as intraditional Hopfield neural networks can be avoided.Namely, the network converges to a state that theenergy function is minimized. Furthermore the final

1180 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 43, NO. 3 JULY 2007

Fig. 1. Relations between predicted targets and measurementsbased on gating technique.

state of the network is guaranteed to provide a propersolution to data association.The rest of the paper is organized as follows. The

dynamic models for MTT are presented in Section II.Section III develops a CHNN-based data associationtechnique, while Section IV describes a maneuveringcompensator algorithm. The simulation results for theproposed MTT algorithm are presented in Section V.Finally, conclusions are drawn in Section VI.

II. PROBLEM DEFINITION AND GATINGTECHNIQUE

According to the tracking situation, the movingtargets model can be defined as state variableequations which can be described as follows:

X(k+1) = F(k)X(k) +G(k)W(k)+U(k) (1)

Y(k) =H(k)X(k) +V(k) (2)

whereX(k) is the state vector of the target,Y(k) is the measurement vector of the target,W(k) is the system noise associated with the target,

assumed to be normally distributed with zero meanand variance Q(k),U(k) is the forced input,V(k) is the measurement error associated with

the target, assumed to be normally distributedwith zero mean and variance R(k), and uncorrelatedwith W(k),H(k) is the measurement matrix of the target,F(k) is the transition matrix of the target,G(k) is the noise gain matrix of the target,

The initial state of the target is assumed to beGaussian with known mean vector X(0 j 0) and knowncovariance matrix P(0 j 0).In real situations, we are given a large number

of close measurements from the sensors in orderto determine the trajectory estimates for any targetthat might be presented. It is difficult to preciselydetermine which target corresponds to which ofthe closely spaced measurements. In other words, adata association problem of MTT or associating themeasurement vector Y(k) to the existing track modelfor each step k, must be solved.

In a dense target environment, gating is the firststep in order to solve the problem of associatingobservations with tracks. Additional logic is requiredwhen an observation falls within the gates of multipletarget tracks, or when multiple observations fall withinthe gates of a target track. Fig. 1 illustrates a typicalsituation of a gate diagram consisting of three targettracks P1, P2, and P3. In this figure, there are threetargets and seven measurements. First of all, weapply the gating technique to eliminate less probablemeasurements such as O6 and O7. Then, the CHNNalgorithm will be applied to perform the associationrelationship between the remaining measurements andthe targets.

III. COMPETITIVE HOPFIELD NEURAL NETWORKBASED DATA ASSOCIATION

In this section, the CHNN is applied to thepotential-target measurements to obtain the solutionof the data association problems and is described inthe following.

A. Hopfield Neural Network

The Hopfield neural network is a two-dimensionalbinary neural network. Assume that the networkconsists of n m mutually interconnected neurons,where Vx,i denotes the binary state of the (x, i)thneuron and Tx,i;y,j denotes the interconnection strengthbetween neuron (x, i) and neuron (y,j). A neuron (x, i)in this network receives weighted inputs Tx,i;y,jVy,jfrom each neuron (y,j) and a bias input Ix,i fromoutside. Thus, the total input to neuron (x, i) iscomputed as

Ux,i =nXy=1

mXj=1

Tx,i;y,jVy,j + Ix,i (3)

whereVx,i =

12(sign(Ux,i) +1): (4)

It is proven that for asynchronous update, theLyapunov function of the two-dimensional Hopfieldnetwork [1214] given by

E =nXx=1

nXy=1

mXi=1

mXj=1

Tx,i;y,jVx,iVy,j 2nXx=1

mXi=1

Ix,iVx,i

(5)is decreased during each update. The updating isiterated until the network converges.

B. Mapping Data Association to Competitive NeuralNetwork

In applying the network to data association, letthe state of Vx,i denote an association status between

CORRESPONDENCE 1181

Fig. 2. Diagram of interconnection between predicted targets andmeasurements.

the xth radar measurement and the ith target, with1 and 0 indicating associated and not associated,respectively. The diagram of interconnections betweenpredicted targets and measurements is shown inFig. 2. Then the objective function used for obtainingmeasurements and radar targets association with thebest decision is given by

E = AnXx=1

mXi=1

dx,iVx,i+BnXx=1

nXy=1

mXi=1

mXj=1

Vx,iVy,jx,y

+CmXi=1

nXx=1

Vx,i 1!2

(6)

where the x,y is the Kronecker delta function.The first term is the sum of the distances between

the associated measurements and the radar targets,where dx,i is the distance between the ith predictedtarget and the xth measurement. The dx,i here needsspecial design to achieve the task of data associationas described in the following. If there are not anymeasurements in one gate then the previous targetinformation should be chosen as the next targetinformation. To achieve this destination, the m targetdata is included as part of the measurement. Assumethat there are m targets and nm newly obtainedradar measurements. Assume an index vector withthe first m entries indexing the m targets, followedby the indices of the m n measurements. Letdx,i given by (7) define the Mahalanobis distance(with regards to measurement innovation andcovariance matrix, respectively) between targets andmeasurements. If there are no measurements (m= n),introduce a dummy measurement with the targetindex; the distance between a target and this dummymeasurement is chosen to be the radius of the gatingwindow. After these arrangements, the distance dx,iwhere x= i is defined as dx,i = , where is theradius of the gate. Hence, if there are measurementsinside the gate, then one of the measurements shouldbe chosen. But if there are no measurements insidethe gate, then the target itself should be chosen.Another constraint is that if x 6= i and 1 xm,then dx,i =1. This constraint prevents one target fromchoosing another target as its measurement. Basedon the above discussions, the distance dx,i is then

defined as

dx,i =

8>:[T(k)S(k)1(k)]1=2 if x 6= i and x > m1 if x 6= i and 1 xm if x= i

(7)

where (k) = Y(k)HX(k j k 1), and S(k) is thecovariance matrix of the innovation ( (k)).The second term in (6) attempts to ensure that

each measurement can be associated with onlyone target. The third term forces the conditionthat each target has one and only one associatedmeasurement. The parameters A, B, and C specifythe important factors in the object function. Howeverit is very difficult to determine proper values for theseparameters A, B, and C, which are highly dependenton the number of targets, target-measurementdistances, and the radius of the gate. Because of this,it has been reported that irrational solutions may resultfrom the use of traditional Hopfield approaches [7, 8]if the weighting factors are not properly determined.In order to reduce the burden of determining

the values of the weighting factors, a competitivewinner-take-all updating is proposed as follows:

Vx,i =1, if Ux,i =maxfU1,i Un,ig0, otherwise

: (8)

With this modified updating rule, the hard constraintthat each target should be associated with oneand only one measurement will be automaticallyembedded inside the network evolution results. Assuch, the third term can be subsequently removedfrom the objective function. Thus, the objectivefunction can be further simplified as follows:

E = AnXx=1

mXi=1

dx,iVx,i+BnXx=1

nXy=1

mXi=1

mXj=1

Vx,iVy,jx,y:

(9)

It is also worth noting that once the competitivewinner-take-all updating is applied, with A set to be1, B can be easily set to be greater than the radiusof gate , a relatively constant value. By so doing,the network would be avoided from trapping intoirrational solutions.Comparing the resultant objective function with the

Lyapunov function of the two-dimensional Hopfieldnetwork in (5), we can obtain

Ix,i =A

2dx,i (10)

Tx,i;y,j =Bx,y: (11)From (10) and (11) we can see that the CHNN is notfully interconnected; instead, it is locally connectedwith the neurons in the same column. Applyingthese two equations to (3) and then it will be

1182 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 43, NO. 3 JULY 2007

obtained as

Ux,i =mXj=1

BVx,j A

2dx,i: (12)

The proposed algorithm is found to be convergentduring network evolutions. Detail proof of theconvergent property is shown in the Appendix.

IV. MANEUVERING ESTIMATION AND ADAPTIVEPROCEDURE

If target maneuvers occur, maneuver detectionand acceleration estimation algorithm is applied tomodify the parameters of the tracking filter. This willallow the maneuvers to be tracked without divergingor severely distorting the estimate. Such an adaptiveprocedure which modifies the Kalman filter equationsis described as follows. Let

(k) = Y(k)H(k)X(k j k1) (13)(k) =H(k)P(k j k 1)HT(k) (14)S(k) = (k)+R(k) (15)

where (k) is the measurement innovation and S(k)is the innovation covariance matrix. One detectioncriterion is defined as

f(k) =kX

j=kN+1T(j)S1(j) (j)

H1?H0

" (16)

where H0 is the hypothesis that the system behavioris normal, and H1 is the hypothesis that the targetis moving with maneuvers. Since the noises V(k)and W(k) have been assumed to be zero mean whiteGaussian noises, f(k) is a chi-squared random variablewith N m degrees of freedom, where N is theresidual window size and m is the dimension of themeasurement vector. The criterion " can be chosenfrom standard chi-squared tables.Based on the detection results, if the target

initiates and sustains a sudden maneuver, then thisalgorithm will detect this situation based on statisticalcalculations. Under this situation, another algorithmis also applied to increase the Kalman gain and thus,the tracking filter will have faster responses for thesudden maneuvering situations. Therefore, betterestimation accuracy will be obtained. This algorithmevaluates the innovations based on the following test

ji(k)j DpSii(k)

, for all i (17)

where the subscript i means the ith component of theinnovation vector, and D is a constant related to theGaussian probability density function. The probabilityassociated with that interval, which can be evaluatedform the normal distribution table, is 0.682 for D = 1,0.954 for D = 2, 0.998 for D = 3, and so on. If thetargets have maneuvering situations, the sudden

changes may make ji(k)j greater thanDpSii(k)

for some elements. In order to hold the equality in(17), the variance of the rejected innovation should bemodified as

D2 = 2i (k)fai(k)ii(k) +Rii(k)g1 (18)and then (k) will exist on the boundaries of theacceptable region defined by (17). Thus, the parameterai(k) can be computed as follows:

ai(k) =[ii(k)=D]

2Rii(k)ii(k)

: (19)

In order to keep the target on track, the covarianceof the prediction error P(k j k 1) is modified to[am(k) P(k j k 1)], where am(k) is the largest valueof all the ai(k). With this approach, the Kalman gain isincreased and thus, the tracking filter will have fasterresponses for the sudden maneuvering situations.Therefore, a better performance will be obtained.

V. SIMULATIONS

A simulation program using the proposed dataassociation technique is developed. In our simulations,the measurement noise and clutter points were createdusing random number generators. The measurementdata were obtained using a discretized version of thetrue target motion in addition to the measurementerrors. The clutter points were assumed to beuniformly located in the measurement space withan average of about two clutter points per validationgate. Kalman filters were used to estimate the statevector X(k j k) recursively. As soon as measurementdata were received, the corresponding correlationswere calculated based on each hypothesis. The resultsof tracking multiple targets in the planar case weresimulated under several different situations.In the simulation examples two targets were

chosen with the initial conditions as listed in Table I.The maneuvering situations for all targets wereshown in Table II. The standard deviation of systemposition noise was 50 m for both X and Y axis. Thestandard deviation of measurement position noisewas assumed to be 200 m for both axis. Moreover,the standard deviation of the velocity noise was atleast 0.05 times the mean velocity. We assume allthe noise to be uncorrelated. In the simulation, weapplied two different data association techniquesnamely, the CHNN and the one-step conditionalmaximum likelihood for comparison. The valuesA= 1 and B = +1, where was the radius of thegate, were chosen for CHNN. For tacking targets withmaneuvering situations, we also employed two kindsof approaches which were adaptive procedure andinteracting multiple model (IMM) algorithm [15, 16]for comparison. The number of returns for each stepwas assumed to be five times the number of targets.

CORRESPONDENCE 1183

Fig. 3. Tracking two targets using CHNN and adaptive procedure.

TABLE IInitial Conditions for Simulation Examples

x (m) _x (m/s) y (m) _y (m/s)

Target 1 100 120 100 100Target 2 100 150 600 250

The simulation results of tracking two maneuveringtargets are shown in Figs. 3 and 4 when CHNN withadaptive procedure and one-step conditional maximumlikelihood with adaptive procedure, respectively, areapplied. After fifty Monte Carlo runs, their trackingrms errors of positions and velocities are shown

TABLE IITarget Maneuvering Situations

Step 1020 Step 2535 Step Other Step

Acceleration a(x) a(y) a(x) a(y) a(x) a(y)(m/s2) (m/s2) (m/s2) (m/s2) (m/s2) (m/s2)

30 30 30 30 0 0

in Tables III and IV. From Tables III and IV, wecan see that the CHNN with adaptive proceduredemonstrates better performance, with smalleraveraged position errors and velocity errors, thanthe one-step conditional maximum likelihood with

1184 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 43, NO. 3 JULY 2007

Fig. 4. Tracking two targets using one-step conditional maximum likelihood and adaptive procedure.

Fig. 5. Tracking two targets using CHNN and IMM algorithm.

TABLE IIISimulation Results of Tracking Two Targets using CHNN and

Adaptive Procedure

Position Errors Velocity Errors

Target 1 113.53 25.87Target 2 116.15 25.82

adaptive procedure. The results of an experimentusing CHNN with IMM and one-step conditionalmaximum likelihood with IMM are also shown inFigs. 5 and 6, respectively. Their tracking rms errorsof positions and velocities are shown in Tables Vand VI. By comparing the results in Tables V and

TABLE IVSimulation Results of Tracking Two Targets using One-StepConditional Maximum Likelihood and Adaptive Procedure

Position Errors Velocity Errors

Target 1 134.04 30.64Target 2 126.83 26.82

VI, we can see that the CHNN with IMM is betterthan the one-step conditional maximum likelihoodwith IMM. This experiment again demonstrates thatthe CHNN used as data association can give betterperformance for target tracking. By comparing theresults in Tables III and V, and comparing the results

CORRESPONDENCE 1185

Fig. 6. Tracking two targets using one-step conditional maximum likelihood and IMM algorithm.

TABLE VSimulation Results of Tracking Two Targets using CHNN and

IMM Algorithm

Position Errors Velocity Errors

Target 1 136.41 30.22Target 2 136.62 30.21

TABLE VISimulation Results of Tracking Two Targets using One-StepConditional Maximum Likelihood and IMM Algorithm

Position Errors Velocity Errors

Target 1 144.81 33.04Target 2 144.97 33.01

in Tables IV and VI, we can also see that the adoptedmaneuvering approach adaptive procedure showssmaller averaged position errors and velocity errorsthan the IMM for both targets.The results again demonstrate that the CHNN with

adaptive procedure presents higher performance intarget tracking, in terms of smaller averaged positionerrors and velocity errors, than other combinations.In order to evaluate the speed of the tracking system,the computation time for tracking 2, 4, and 6 targetswhen CHNN with adaptive procedure and CHNNwith IMM algorithm are applied, respectively, arealso listed in Table VII. From the results in Table VII,we can see that since IMM computes the trackingbased on probabilistic weighted estimations whichare obtained using various Kalman filters, it requireshigher computation loads, especially when the numberof tracked targets increases.

TABLE VIIThe Computation Time for Tracking Various Numbers of Targetswhen CHNN with Adaptive Procedure and CHNN with IMM

Algorithm are Applied

Adaptive Procedure IMM Algorithm

Two Targets 0.187 s 0.578 sFour Targets 0.297 s 1.063 sSix Targets 0.531 s 1.671 s

VI. CONCLUSIONS

A new data association algorithm based on aCHNN for tracking multiple targets was presented.This tracking technique has an advantage of choosingthe optimal correlation between radar measurementsand existing target tracks. Because of the use ofCHNN, it is found that the system is relieved ofthe burden of determining the proper weightingfactors as in traditional Hopfield neural networks.As such, the network can always achieve a rationalsolution. Furthermore, we also applied an adaptiveprocedure for tracking maneuvering targets. Basedon the simulation results, this algorithm was capableof obtaining the optimal correlations between truetargets and radar measurements. It was also foundout that the proposed approach performed quite wellwhen tracking both constant velocity and maneuveringtargets.

APPENDIX

In this Appendix, it is proved that our algorithmalways converges to a stable state in the network

1186 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 43, NO. 3 JULY 2007

evolutions. First, consider the energy function

E = AnXx=1

mXi=1

dx,iVx,i+BnXx=1

nXy=1

mXi=1

mXj=1

Vx,iVy,jx,y

= AnXx=1

mXi=1

dx,iVx,i+BnXx=1

mXi=1

mXj=1

Vx,iVx,j : (20)

The energy function can be separated into five termsconsisting of i= k, j = k, and others.

E = AnXx=1

dx,kVx,k +BnXx=1

mXj=1

Vx,kVx,j +AnXx=1

mXi=1i 6=k

dx,iVx,i

+BnXx=1

mXi=1i 6=k

mXj=1j 6=k

Vx,iVx,j +BnXx=1

mXi=1i 6=k

Vx,iVx,k

=

(A

nXx=1

dx,kVx,k +BnXx=1

mXj=1

Vx,kVx,j +BnXx=1

mXi=1i 6=k

Vx,iVx,k

)

+

(A

nXx=1

mXi=1i 6=k

dx,iVx,i +BnXx=1

mXi=1i 6=k

mXj=1j 6=k

Vx,iVx,j

)(21)

where only the first term, bracketed by f g, will beaffected by the state change of a specific column k.Assume the neuron (p,k) is the only active neuron

on the kth column before a specific updating cycle ofcolumn k, i.e., Voldp,k = 1, and V

oldx,k = 0 8i 6= P. Further

assume that after the updating cycle, neuron (q,k)becomes the winning node, i.e., Vnewq,k = 1, and Vx,k = 08x 6= q. According to Eq.(11) and the winner-take-allrule of Eq.(7), we obtain

mXj=1

BVq,j A

2dq,i = max

x=1,2,:::,n

8

mXj=1

BVp,j A

2dp,i: (23)

Therefore,

mXj=1

BVq,j +A

2dq,i