Application of EM algorithm to adaptive filter for

APPLICATION OF EM ALGORITHM TO ADAPTIVE FILTER FOR MULTIPLE TARGET TRACKING Jeong-Ho Park and Hong Jeong Dept. of E.E., POSTECH Pohang 790-784, Korea (pijah,hjeong} Qpostech. edu ABSTRACT Tracking the targets of interest has been one of the major research areas in radar surveillance system. We formulate the tracking problem as an incomplete data problem and apply the E M algorithm to obtain the MAP estimate. The resulting filter has a recursive structure analogous t o the Kalman filter. The difference is that the measurementupdate deals with multiple measurements and the parameterupdate can estimate the system parameters. Through extensive experiments, it turns out that the proposed system is better than PDAF and NNF in tracking the targets. Also, the performance degrades gracefully as the disturbances become stronger.

We include the system parameters, like the gate detection probability a , clutter density &, and measurement error variance crz and U?, t o the parameters t o be estimated, in addition to the states of targets. As a result, the formulation becomes an adaptive filter that consists of a recursive time-update in time and a recursive measurement-update in each fixed time frame. This approach leads t o a more efficient filter that integrates EM and Kalman filtering so that one can recursively predict, independently for each target, the states while incoming measurements improve the states and system parameters.

1. INTRODUCTION

Tracking all the targets of interest within the coverage region of radar has been one of the major research fields in radar surveillance system. One of the pioneering works in this field is the joint probabilistic data association filter (JPDAF) 171. This is an Another approach for obtaining optimal association is the connectionist scheme [13, 181. In this approach, the outputs of the neurons indicate association between targets and measurements. One of the disadvantages of this approach is that the parameters of the neural network are determined by trial-and-error. As yet another approach, there have been emerging interests in applying the EM algorithm [6, 141 to the tracking problem [I, 8, 9, 171. Major works include the papers by Avitzour [l], Gauvrit (91, and hlolnar [17]. All of these schemes focus only on data association and stat,e estimation based on measurements and thus additionally need a good filter, such as the Kalman filter [12], for predicting the target state in the next time frame. These two operations, often denoted as measurement- and timeupdates, are the two key components in a tracking filter as shown in Fig. 1. Integration of measurement- and tirneupdates in an EM approach has been successfully achieved by Molnar [I71 by using a MAP estimate instead of Avitzour’s ML approach. This work h a s been supported by grants from MARC (Microwave Applications Research Center) and ADD (Agency for Defense Development).

0-7 803-6293-4/00/$10.0002000IEEE

Measurement

----j

MeasurementUpdate

Target State

Figure 1: The structure of the proposed filter.

2. PROBLEM FORMULATION

The state &*( I C ) and measurement jj ( k ) of the target t are expressed by

{

+

+,(k) = F + , ( k - 1) Gw(k - l), + v(kL YJ(k)=

(1)

where the noise w ( k - I) and v(k) are mutually independent white Gaussian noise [Ill. We assume the measurement is expressed in rectangular coordinate. If y , ( k ) does not originate from the target t , y, ( k ) is distributed uniformly. We define the combination of measurements y ( k ) and validation vectors w ( k ) as the incomplete data w ( k ) . The association vectors z ( k ) is considered t o be the missing data, and the complete data x ( k ) becomes x(k) = { y ( k ) ,w ( k ) ,z(k)}. The validation vectors and the association vectors are the columns of the validation matrix and the association matrix defined in [2], respectively. The system parameter

3188

0 = {a;,u;, Ld,a} consists of the variances of observation error, the clutter density, the gate detection probability. a! 4 P d P g , where P d is the detection probability and Pg is the probability that the target is in the gate. The posterior pdf is

where V is the area of the gate, and 6(z) is the Kronecker delta function. As can be seen in (3), p[xly,U ,~ $ ( ~ ) ( k l6k'1) ,is needed for the expectation of ( 5 ) . That is P[XlY, U ,#(p)(klkC),

eo]

1 = ZP[YIZ, d P ) @ l k ) , RalP14z, L:lP[zbo1

(6) where xk = {x(k),x(k - l ) , . . .,x(O)}. We omit (k) in the equations for simplicity if there is no possibility of confusion. This posterior pdf enables us to employ the penalized EM [lo]. To get the estimates of # and 0 simultaneously, we use a slightly different method with ECM [16] given by E-step : Q[#J#(P)(klk),6'1 = E{logp[xl#, 6'1lY, U , #(P)(kIk), M-step : q5(P+1)(kJk) = argmax+

Q[#,@ol#(p)(klk),61'

e",

- K ( # , e'),

(3)

MAP : 8 = arg maxe

Q[#(klk), W ( J 4 k ) ,01 ' - K(#(klk), e),

+

where S ( k l k ) = H P ( k J k ) H T Ro(k)and y y ) = yj H # ( P ) ( k l k ) .In this equation, we replace 6' with the individual parameters, and delete independent variables. Because we can not use P ( k l k ) to calculate S ( k l k ) , we use P ( k l k - 1) instead.

where the penalty function is given by

K(#l

='2{

e> - logP[#le, #(klk log[(Zn)dIP(klk

+ [#(klk

- 111 - logp[@I@(k - 1)1

- 1)1]

- 1) - #]ITP-'(klk

- 1)[#(klk - 1) - #I}>

4. E- AND M-STEP SOLUTIONS

(4) where d is the dimension of

#.

In this section, we derive the explicit solutions for E- and M- steps. Substituting (5) and ( 7 ) into (3), we have Aft

3. PARAMETRIC MODELS

&[#,el#(P'(~lk),eO1= - 5

~~P)log[(2.rr)nlRo(k)ll 3=1

We assume the N targets are statistically independent and thus a,derivation for only one target will suffice. In this subsection we derive probability distributions to be used to get Q-function. The complete data log-likelihood function is decomposed into

- -1 Mi

Z;P)b$

- H+]TRo(k)-Ily3 - H # ]

J=1

(8)

Mt

- Lo + ( M t -

z;p))

log Lo

+ zt?"'log(1 - a')

3=1

+

Mi ZIP) j=1

log

-,a'

Mt

otherwise.

(9) MI

M.

3=1

3=1

+ b ( ~ z 3 ) l o g ( l - a ) + 6 ( ~ -1)log-, z, a Mt

(5 )

where s;3 = yi - W # ( p ) ( k l k ) .Substituting (8) into (3) and applying the necessary condition for the maximizer, one

3 189

obtains

where

and (a) Track maintenance ratio

In (12), if we use +,(klk) and +,(klk) instead of cb ,Z and G9, these estimates are biased. This bias can be corrected by rearranging (12) to use +,(klk) instead of +z as follows:

(b) RMS position error

where P,, is the variance of 4,. Here, we assume that the measurement and estimate errors are mutually independent. In order to get more stable estimates, we add one more step for time averaging with a window of size D. The original EM method mentioned nothing about the covariance matrix of the estimate. Meng and Rubin I151 introduced an algorithm called SEM to resolve this issue. With the aid of their method, we calculate the state estimation error covariance matrix.

Figure 2: The performance with respect to Ld, given cr2 = 0.0225 km2 and Pd = 0.8.

5. EXPERIMENTAL RESULTS

We do not apply the process noise to the target movement ~ 0 - 5 1 is used for the filter pageneration, but Q = 1 . 2 1 0 6 1 rameter. the gate probability Pg is 0.99. We set the initial values according to [2]. The initial states of the crossing targets are&(O) = (-16.0 km, 0.20 k m K ’ , 4.0 km, -0.05 km. s - ’ ) ~and G(0) = (-16.0 km,0.20 km.s-’, -4.0 km,0.05 km. S-1)T.

Monte Carlo simulation was carried out for 500 runs with u2 = 0.0225 km2 and P d = 0.8 for the first target only. Figs. 2 show the track maintenance ratio (TMR) and RMS position error with those of PDAF and the nearest neighbor filter (NNF). For TMR, PDAF is the best of all, NNF is the poorest, and the proposed method is in-between. This fact can be expected since PDAF controls the gate size according to the known operating environment [2]. However, the proposed method, together with NNF, are the best in term of the RMS performance. It is obvious that the optimal condition of high TMR and low RMS error can not be

achieved simultaneously and thus must be compromised as in the case of the new filter. We examine the parameter-update performance of the proposed filter for the crossing target case. The ensemble averages of a Monte Carlo simulation of 500 runs are depicted in Figs. 3, 4, and 5. There is a tendency for the average of u z ( k ) to increase slightly after the targets are cross at 802’. This observation is natural since the parameter estimation must be influenced by nearby targets. Incidentally, only ot(k)tends to rise because the two targets are separated in y axis. Fig. 5 shows that the mean of the estimate & ( k ) increases when targets are crossing, that is, the filter can not discern the nearby target from the false alarms. Also, we can observe the effect of adjacent targets when they are approaching each other in space. 6. CONCLUSION

We introduced an adaptive tracking filter following an approach based on the EM method. We improve the measurementupdate by introducing validation vectors as an additional observation in EM estimation. As a result, the measurementtarget association is optimally decided under the given conditions of the environment. The other contribution is the introduction of parameter-update that relieves us of the

3 190

Figure 3: The update of

0% and U:.

Figure 5: The update of the false alarm density Ld.

[8] L. F'renkel and M. Feder. Recursive expectationniaximization (EM) algorithms for time-varying parameters with applications to multiple target tracking. IEEE Trans. Signal Processing, 47(2):306-320, Feb. 1999. (91 H. Gauvrit, J. P. Le Cadre, and C. Jauffret. A formulation of multitarget tracking as an incomplete data problem. IEEE Trans. Aerosp. Electron. Syst., 33(4):1242-1257, Oct. 1997.

[lo] P.

Green. On use of the EM algorithm for penalized likelihood estimation. J . Roy. Statist. Soc. Ser. B, 52(3):443-452, 1990.

Figure 4: The update of the gate detection probability N.

[ll] H. Jeong and J. Park. A constrained optimal data as-

sociation for multiple target tracking. In IEEE International Conference on Acoustics, Speech, and Signal Processing, volume V, pages 2977-2980, March 1999.

need t o manually decide some critical values. Based on MAP, estimates of the parameters have been derived. In this manner, the filter can track multiple targets that move in a complicated manner and in variable environments.

[12] R. E. Kalman. A new approach to linear filtering and prediction problems. Trans: ASME, ( J . Basic Eng.,), 82:34-45, hlar 1960.

7. REFERENCES

[13] Henry Leung. Neural network data association with application t o multiple-target tracking. Opt. Eng., 35(3):693-700, Mar. 1996.

D. Avitzour. A maximum likelihood approach to data association. IEEE Trans. Aerosp. Electron. Syst., 28(2):560-565, Apr. 1992.

[14] G. J. McLachlan and T. Krishnan. The EM Algorithm and Extensions. A Wiley-Interscience Publication, 1996.

Y. Bar-Shalom and T. E. Fortmann. Tracking and Data Association. Academic Press, Inc, 1988.

[15] X. L. Meng and D. B. Rubin. Using EM to obtain asymptotic variance-covariance matrices: the SEM algorithm. Journal of the American Statistical Association, 86:899-909, 1991.

Y . Bar-Shalom and X. R. Li. Estimation and tracking: principles, techniques, and software. Artech House, Inc, 1993.

Y . Bar-Shalom and X. R. Li. Multitarget-Multisensor Tracking: Principles and Techniques. YBS Publishing, Storrs, CT, 1995.

[16] X. L. Meng and D. B. Rubin. Maximum likelihood estimation via the ECM algorithm: A general framework. Boimetrika, 80:267-278, 1993.

Y . Bar-Shalom and E. Tse. Tracking in a cluttered environment with probabilistic data association. Automation, 11:451-460, Sep. 1975.

[17] K. 3. Molnar and J. W. Modestino. Application of the EM algorithm for the multitarget/multisensor tracking problem. IEEE Trans. Signal Processing, 46(1):115129, Jan 1998.

A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data via the EM algorithm. J . Roy. Statist. Soc. Ser. B , 39(1):1-38, 1997.

[18] D. Sengupta and R. A. Iltis. Neural solution t o the multitarget tracking data association problem. ZEEE Trans. Aerosp. Electron. Syst., AES-25:96-108, Jan. 1989.

T. E. Fortmann, Y. Bar-Shalom, and M. Scheffe. Sonar tracking of multiple targets using joint probabilistic data association. IEEE Journal of Oceanic Engineering, 03-8(3):173-183, Jul 1983.

3191