Extended Neural Network Solution of Radar Tracking Problems

Dend-Jyi Juang, Kuo-Chang Hu, Mang-Liang Lee, Chih-Yung Wang, and Yi-Nung Chung Department of Electrical Engineering,

Da-Yeh University Chang-Hua 515, Taiwan, R.O.C. E-mail: chung@mail.dyu.edu.tw

Corresponding author: Yi-Nung Chung, Email: chung@mail.dyu.edu.tw Abstract- Data association is the key technique to solve

the multiple-target tracking problems. A new approach to tracking algorithm for radar system based on the neural network is investigated in this paper. The proposed algorithm will solve both the data association and the target tracking problems simultaneously. With this approach, the matching between radar measurements and existing target tracks can achieve a global consideration. Moreover, one simple detection and estimation algorithm denoted adaptive procedure will be applied when tracking targets with maneuvering situations. Keywords: Data association, neural network, adaptive procedure

I. Introduction Data association is the major technique in

multiple-target tracking (MTT) systems. Once the measurements are received by the radar sensor, such a technique is applied to compute the relation of measurements and existing targets. Therefore, the best estimates for the tracking targets can be obtained. The techniques of tracking algorithms for radar system have been investigated in many papers. A multiple adaptive estimation algorithm for the tracking system was developed by Fisher [2]. A new approach about multiple sensor fusion work done in this area was presented by Chung and Okello in [1,3]. A well-known data association algorithm denoted as the Joint Probabilistic Data Association (JPDA) method which is suited for a high false target density environment was also addressed by Bar-Shalom [4]. Sengupta [5] developed an MTT algorithm with Neural Network approach. However, these techniques of data association may be nearest neighbor or all neighbors-based, usually consider the relations between radar measurements and existed target tracks independently. Nevertheless, in a dense target environment, some targets can be very close to each other. The measurements produced by these close targets can confuse the data association computation algorithm and result to wrong relations. However, the data association problems should be considered globally. Based on this consideration, this paper proposes a multiple-target tracking approach using a neural network technique to solve the data association problem and to obtain a global matching between radar measurements and each existed target. Chung [6] developed a new approach dented the Competitive Hopfield Neural Network (CHNN) to improve the Hopfield Neural Network wherein a cooperative decision is made based on the simultaneous

input of a community of neurons. Each neuron receives information from other neurons and also gives information to other neurons. With this collective information, each neuron settles to a stable stage with the lowest value of a predefined energy function. Based on the CHNN approach, the association between radar measurements and each existed target can be obtained under global considerations which in turn, can increase the accuracy of radar tracking systems.

II. Tracking Models The dynamic model of tracking system can be

defined to be state variable equations which are as follows:

( ) ( ) ( ) ( ) ( )kWkGkXkFkX +=+1 (1)

( ) ( ) ( ) ( )kVkXkHkY += (2) In order to solve the tracking problem, a well

known technique denoted as Kalman filter is applied. Based on the dynamic system defined as above, the Kalman filter for this signal model is applied to the tracking process. In a dense target environment, gating is the first step in order to solve the problem of associating observations with tracks. To describe the gating technique, we first define the variables in the gate. According to the Kalman filter equations, the observation of the combined data in time k is

( ) ( ) ( ) ( )kVkXkHkY += (3) The vector difference between the measurement and the predicted quantities,

( ) ( ) ( ) ( )1ˆ −−= kkXkHkYkτ (4)

is defined to be the residual vector with residual covariance matrix S(k). Assume that the measurement is of dimension m. Hence, if we define 2 1( ) ( ) ( )T

kd k S k kτ τ∆

−= , the m-dimensional Gaussian probability density for the residual is

2

2

2( ( ))

(2 )

dk

M

k

ef kS

τπ

−

= (5)

An observation satisfies the gates of a given track if all the elements, ( )kτ of the residual vector satisfy the relationship:

ˆ( ) ( ) ( ) G gY k Y k k Kτ σ− = ≤ (6)

where GK is a constant and gσ is the residual standard

deviation as defined in terms of the measurement ( 2rσ )

and the prediction ( 2pσ ) variances:

2 2g r pσ σ σ= + (7)

In this approach, a neural network technique is applied to obtain the solution of the MTT problems and is described as follows. For each step k, once a measurement vector is received, the corresponding probability of each hypothesis, can be obtained by using the following formula.

1 1

1/ 2 1/ 2

( ( ) | , )1 1exp ( ) ( ) ( )

(2 ) | ( ) | 2

i k kx

Tm

p Y k Y

k S k kS k

β

τ τπ

− −

− = − (8)

where m is the dimension of the measurement vector, ( )kτ is the innovation due to Y(k), and S(k) is the

innovation covariance matrix. These equations can be obtained from the Kalman filter equations as

ˆ( ) ( ) ( )k Y k Y kτ = − (9) ( ) ( ) ( ) ( ) ( )KRkHkkPkHkS T +−= 1 (10)

For each measurement, we can obtain one weighting probability i

xp corresponding to the correlated target. Moreover, in order to have clear probability value between each pair of measurement and target, we define

∑ =

= n

viv

ix

ixp

pP1

, (11)

which is the normalized probability for the x-th measurement and the i-th target.

An approach denoted as the CHNN is then applied to obtain the solution of the data association problems. The CHNN is a two-dimensional binary Hopfield neural network. Assume that the network consists of n*m mutually interconnected neurons, where

,x iV denotes the binary state of the ( , )x i -th neuron and

, ; ,x i y jT denotes the interconnection strength between neuron ( , )x i and neuron ( , )y j . A neuron ( , )x i in this network receives weighted inputs , ; ,x i y jT ,x iV from each neuron ( , )y j and a bias input ,x iI from outside. Thus, the total input to neuron ( , )x i is computed as

, , ; , , ,1 1

n m

x i x i y j y j x iy j

U T V I= =

= +∑∑ (12)

Then the state of ,x iV is determined by ,x iV =1 if 0U ≥ , and 0 if otherwise. This way of neuron updating rule is proven to decrease the Lyapunov function of the two-dimensional Hopfield network given by

, ; , , , , ,1 1 1 1 1 1

2n n m m n m

x i y j x i y j x i x ix y i j x i

E T V V I V= = = = = =

= − −∑∑∑∑ ∑∑

(13) In applying the network in data association, let the

state of ,x iV indicate an association status between the x-th radar measurement and the i-th target, with “1” and “0” indicating associated and not associated, respectively. Then the objective function used for obtaining

measurements and radar targets association with the best decision is given by

∑ ∑∑∑ ∑ ∑∑∑= = = = = ===

−++=m

i

n

x

n

y

m

i

m

i

n

xix

m

jyxjyix

n

xixix VCVVBVAE

1 1 1 1 1

2

1,

1,,,

1,, 1δη

(14)

where ix

ix P ,,

1=η

The first term is the sum total of the related parameter between the associated measurements and the radar targets. In data association, it is unavoidable to encounter a situation wherein none of the newly obtained measurements is fit for some specific targets. In this case, the previous target information will be chosen as the next target information. Assume that if there are measurements inside the gate, then one of the measurements should be chosen. But if there are no measurements inside the gate, then the target itself should be chosen. Another constraint will prevent one target from choosing another target as its measurement.

The second term in Eq.(14) attempts to ensure that each measurement can be associated with only one target. The third term forces the condition that each target has one and only one associated measurement. The constants A, B, and C specify the important factors of the three terms. In order to reduce the burden of determining the values of the constant factors, a competitive winner-take-all updating is proposed as follows:

{ }, 1, ,,

1 , max

0 ,x i i n i

x i

if U U UV

otherwise

== (15)

With this modified updating rule, the hard constraint that each target should be associated with one and only one measurement will be automatically embedded inside the network evolution results. As such, the third term can be naturally removed from the objective function.

Comparing the resultant objective function with the Lyapunov function of the two-dimensional Hopfield network in Eq. (13), we can obtain

ixixAI ,, 2

η−= (16)

yxjyix BT ,,;, δ−= (17)

III. Maneuvering Estimation and Adaptive Procedure

In the tracking process, target maneuvers usually will be happened. In this paper, one acceleration estimation algorithm is applied to modify the parameters of the tracking filter. Such an adaptive procedure which modifies the Kalman filter equations is described as follows. Let

)1(ˆ)()()( −−= kkXkHkYkτ (18)

)()1()()( kHkkPkHk T−=β (19)

)()()( kRkkS += β (20) where )(kτ is the measurement innovation and S(k) is the innovation covariance matrix. If the target has a sudden maneuver, then this algorithm will detect this situation based on statistical calculations. In this algorithm, the components which have jumps are first detected using the following test

iallforkSDk iii ,)()( ≤τ (21)

where the subscript i means the i-th component of a vector, and D is a constant related to the Gaussian probability density function. The variance of the rejected innovation can be modified as

122 )}()()(){( −+= kRkkakD iiiiii βτ (22)

so that )(kτ exists on the boundaries of the acceptable region defined by Eq. (22). Thus, the parameter

)(kai can be computed as follows:

)(

)(]/)([)(

2

kkRDk

kaii

iiiii β

τ −= (23)

In order to keep the target in track, the covariance of the prediction error )1( −kkP is modified to

)]1()([ −⋅ kkPkam , where )(kam is the largest value of all the )(kai . With this approach, the Kalman gain is increased and as such, the tracking filter will have faster responses for the sudden maneuvering situations. Therefore, a better performance will be obtained.

IV. Simulations The simulation results of tracking multiple targets

under several different situations are obtained in this section. Table 1 shows the target initial conditions for the simulation example of tracking four targets. We assume all the noise to be uncorrelated. The targets’ maneuvering situations are shown in Table 2.The simulation results are shown in Figure 1 and the position and velocity RMS errors are shown in Figure 2. The average RMS errors are shown in Table3.

In order to compare the performance of the proposed algorithm with other technique, another simulation with same conditions is also presented. In this simulation example, we apply a different data association techniques denoted the one-step conditional maximum likelihood. The standard deviations of both system and measurement noise are same as previous experiment. The simulation results are shown in Figure3 and the position and velocity RMS errors are shown in Figure 4. The average RMS errors are shown in Table4. According to the simulation results based on several different situations, we know that the performance of the proposed algorithm is quite well.

V. Conclusion A radar tracking algorithm based on a Competitive

Hopfield Neural Network for multiple targets was

accomplished in this paper. This tracking technique has the advantage of choosing the optimal correlation between radar measurements and existing target tracks. Furthermore, we also applied the adaptive procedure to solve the target maneuvering problems for tracking system. Based on the simulation results, the proposed algorithm performed quite well when tracking various situations’ targets.

Acknowledgement The work was supported by the National Science

Council under Grant NSC 95-2221-E-212-021

References [1] Yi-Nung Chung, Hsin-Ta Chen, Pao-Hua Chou, and Maw-Rong

Yang, “An Improved Estimator Using Multiple Sensor Data Fusion for Radar Maneuvering Target Tracking Systems,” Journal of The Chinese Institute of Engineers 2007.

[2] K.A. Fisher and P.S. Maybeck, "Multiple Adaptive Estimation with Filter Spawning," IEEE Trans. Aerosp. Electron. Syst., Vol.38, No. 3, pp.755-768, 2002.

[3] N. Okello and B. Ristic, "Maximum Likelihood Registration for Multiple Dissimilar Sensors," IEEE Trans. Aerosp. Electron. Syst., Vol. 39, No.3, pp.1074-1083, 2003.

[4] Y. Bar-Shalom and T.E. Fortmann," Tracking and Data Association," Academic Press ,Inc., 1989.

[5] D. Sengupta and R. A. Iltis, "Neural solution to the multitarget tracking data association problem", IEEE Trans. Aerosp. Electron. Syst., 25, 86-108, 1989.

[6] P.C. Chung, C.T. Tsai, E.L. Chen and Y.N. Sun, "Polygonal Approximation Using A Competitive Hopfield Neural Network," Pattern Recognition, Vol. 27, No. 11, pp.1505-1512, 1994.

Table 1: Initial Conditions of four Targets

( )x m ( / )x m s ( )y m ( / )y m s

Target1 1500 400 3500 560

Target2 1000 600 4000 440

Target3 0 550 8000 80

Target4 2000 540 17000 -100

Table2: Target maneuvering situations

Step 20~30 step 50~60 step other step

Acceleration a(x)

( 2/m s )

a(y)

( 2/m s )

a(x)

( 2/m s )

a(y)

( 2/m s )

a(x)

( 2/m s )

a(y)

( 2/m s )

Target1 25 25 15 -15 0 0

Target2 15 25 15 -5 0 0

Target3 10 5 10 -5 0 0

Target4 10 -25 10 5 0 0

Table 3: Simulation Results of Tracking Four Targets (Using the proposed technique)

Position errors Velocity errors

Target1 134.71 30.12

Target2 132.93 28.54

Target3 125.22 25.72

Target4 133.91 25.83

Table 4: Simulation Results of Tracking Four Targets (Using one-step conditional Maximum Likelihood)

Position errors Velocity errors

Target1 144.71 33.12

Target2 142.93 32.54

Target3 135.22 35.72

Target4 138.91 35.83

Figure 1: Simulation of Tracking Four Targets (Using the

proposed technique)

Figure 2: The RMS Position and Velocity Errors {Using

the proposed technique)

Figure3: Simulation of Tracking Four Targets (Using

one-step conditional Maximum Likelihood)

Figure4: The RMS Position and Velocity Errors (Using

one-step conditional Maximum Likelihood)