trees, and automata produced trees with delay constraints of 0 and 2 hops greater than the shortest path lengths. The automata routed trees produce dynamic costs that are lower than the shortest path trees. Even the automata which are constrained to choose from all shortest paths (i.e. +O delay constraint), minimise the dynamic cost over the shortest path tree approach. Finally, we let the automata converge as before, let all the potential receiver nodes join the tree in a random order and measured the resulting static cost of the tree. In Fig. 3, we plot the static cost against number of receivers for the above automata schemes, shortest path trees and the trees produced by the KMB algorithm [5], which is a centralised Steiner heuristic known to produce trees with costs within a few percent of the minimum achievable [2]. Again, the automata based trees produce costs that are significantly less than the shortest path trees, while the automata with a (+2) delay constraint produce costs comparable to those of the KMB algorithm for this network scenario. Steiner trees minimise cost at the expense of delay, since some receiver-source pairs take longer paths to maximise sharing potential. In both Figs. 2 and 3, the (+O) delay constrained automata produce higher costs since they are constrained to use the shorter paths, which limits the amount of sharing that can take place. In summary, we have shown that learning automata, which use minimal state information and require only local connectivity knowledge, are suitable for creating minimum cost, delay bounded multicast trees in dynamic environments. The automata learn to minimise the number of hops taken to join the tree, thus minimising the overall resources consumed by the tree. Previous studies have largely focused on centralised Steiner tree heuristics operating in static multicast environments. Future work should focus on distributed algorithms applied to dynamic multicast environments.
Our research is confined to the association matrix and the related energy function for optimal constraints. By adopting the constraints on missing targets and false alarms, the new energy function is more general and natural.
0 IEE 1999 2 November 1998 Electronics Letters Online No: 19990019 J. Reeve and P. Mars (Centre for Telecommunication Networks, School of Engineering, University of Durham, South Road, Durham DHI 3LE, United Kingdom) E-mail:
[email protected] T. Hodgkinson (BT Laboratories, Martlesham Heath, Ipswich, Suffolk IPS 7RE, United Kingdom)
where w(k) and v(k) are mutually independent Gaussian noise sources with zero mean. The prediction part assigns a Kalman fdter to each target. A predicted state is used to produce the gate centre given by
References
MTT system structure: The MTT system consists of three parts, as shown in Fig. 1. The track initiation part detects a new target. The detected target is then tracked by the joint co-operation of the
track initiation
Multiple target tracking using constrained MAP data association
For multiple target tracking, data association between measurement data and targets is needed, but is difficult to solve because of measurement error, false alarms, and missing targets. Important current MTT schemes are joint probabilistic data association [l], the expectation maximisation approach [6], and the neural net approach [A. Introduction:
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it(k,k)
The state x,(k) and measurement z,(k) of the target t are represented by
t E [l,n]
g t ( k ) = H t ( k ) X t ( k l k - 1)
(2)
The distance between a measurementj and the centre of gate t is given by [{Y,@) - gt(k)ITS,l(k.){y,(k)
-gt(k)}T]1/2
(3) where S,(k) is a measurement prediction covariance matrix. The distance from the gate centre is directly related to the probability that the measurement is generated from the target. Therefore, we define a modified validation matrix as 00 = { w$ E [1, m],t E [1 , nl} > and (4)
where m and n are the number of measurements and targets, respectively, and U(.) is the unit step function. The association part calculates an association matrix o = { q , b E [l, m], t E [ l , n]}, and generates the measured position of the target t by z,(k) = y,,(k), wherej’ = argmaxp,,. MAP estimate f o r association matrix: An MAP estimate for the association matrix is given by w * = argmin(E(w1wO) W
A new scheme is introduced for data association for multiple target tracking. The scheme is formulated using the MAP estimation method and a new energy function. The natural constraints between targets and measurements are reflected in the energy function.
7th January 1999
i
~
Fig. 1 Multiple target tracking system
Hong Jeong and Jeong-Ho Park
ELECTRONICS LETTERS
. j
target state prediction
T J t ( k )=
‘Steiner problem in networks : a survey’, IEEE Netw., 1987, 17, (2), pp. 129-167 SALAMA, H.F., REEVES, D.s., and VINIOTIS, Y.: ‘Evaluation of multicast routing algorithms for real-time communication on high-speed networks’, IEEE J. Sel. Areas Commun., 1997, SAC-15, (3), pp. 332-345 REEVE, J., MARS, P., and HODGKINSON, T.: ‘Learning algorithms for quality of service multicast routing’, Electron. Lett., 1998, 34, (12), pp. 1195-1 I97 NARENDRA, K.A., and THATHACHAR, M.A.L.: ‘Learning automata An introduction’, (Prentice-Hall, 1989) KOU, L., MARKOWSKY, G., and BERMAN, L.: ‘A fast algorithm for Steiner trees’, Acta Informatica, 1981, 15, pp. 141-145
WINTER, P.:
Y(0)
;
parameter estimation
I I
+E(wo)}
(5)
Here, we assume that p(wolw) and p ( w ) are Gibbsian, given by
In a parameter-observation model, we can assume that R - Ro is a Gaussian noise process and therefore , n
No. I
m
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To model the prior R, we require all the constraints involved in this quantity. A target must be associated with at most one measurement, i.e. CEl a,,5 1. If no measurement is assigned for a target, then it means that a missed detection has occurred. A measurement is assigned for at most one target, i.e. C ;=I 9, 5 1. If a measurement is not assigned then that is a false alarm. These constraints are different from those in [3] that dealt with equality constraints. The final constraint is 0 5 qt. All the above constraints are integrated into eqn. 5 using the three Lagrangian multipliers h, E, and p [5] to obtain the Lagrangian L(w): n
m
where p 0, h,, E,, p,, 2 0. We apply the generalised Kuhn-Tucker theorem to obtain a set of necessary conditions for minimisation of the Lagrangian. As a result, there are three sets of nonlinear simultaneous equations for the minimiser:
(9)
We use the Gauss-Seidel method to solve eqn. 9. After calculation of the Lagrange multipliers, the minimum point is given by WJt
=
P j t - At - € 3
+ Pw,”,
P
(10)
Experimental results: Simulations were carried out for two crossing targets with initial positions ( 4 . O k m , 1 km) and (4.0km, -l.Okm), and velocities (0.20km/s, 4 . 0 5 k d s ) and (0.20km/s, O.OSkm/s). F,(k) and G,(k) in eqn. 1 were the same as those of [7] and the sampling interval was 1s. This model was for constant speed targets with acceleration noise. The process noise covariance and the measurement noise covariance were diagonal matrices with diagonal elements of 1.2106 x
1 0 5
and 0.0225, respectively. The proba-
bility of detection P, was 0.7, and the probability of validation Pc was 0.99. Monte Carlo simulation was performed for N = 50 runs. Fig. 2 shows one of the many samples that used the clutter density C = O.2km2. Extensive experimentation shows that the proposed method tends to successfully separate two crossing targets in most clutter cases.
Clutter km2
0.2 0.4 0.6
Track maintenance rate RMS position error Proposedl PDA NNF Proposedl PDA NNF
I I
1
!A>
86 88 86
I
I
(Yo
I
98 100 100
I
I
I
1 I
1
1
1%
86 80 86
k
m
k
m
km
1 0.1393 I 0.1695 1 0.1592 I 0.1755 I 0.1608 I 0.2276 I 0.2703 I 0.1776 I 0.2889 I
The track maintenance rate and RMS position error were observed and are listed in Table 1. The nearest neighbour filter (”F) [l, 41 and the probabilistic data association (PDA) [2, 13 techniques were also used for the same target model. We considered a track to have been lost when there was no measurement in the gate of the track for at least the last five sampling times. Although PDA is generally the best, the proposed method performs comparably. One of the advantages of the proposed method is that, like NNF, it does not need to know the probability of detection P, and the clutter density C. However, these parameters of a priori knowledge are mandatory for PDA, and the reason for its better performance. Generally the proposed method displays better performance than NNF, especially in terms of the RMS error rate, which is improved by 13.9% on average. Conclusion: We have introduced a new scheme for data association, that computes the data association by an MAP estimation method and uses a new energy function. Unlike PDA, this algorithm does not need any additional information such as the probability of detection and the clutter density. Also our algorithm does not require any parameters such as the balance control coefficients used in the neural network approach. As a result, there is no need for trial and error procedures to select values for the necessary coefficients. These properties are important for adaption to unknown environments with good performance and stability. Acknowledgment: This work has been supported by grants from
the MARC (Microwave Applications Research Center) and ADD (Agency for Defense Development).
0 IEE 1999 Electronics Letters Online No: 19990002
14 October 1998
Hong Jeong and Jeong-Ho Park (Department of Electronic and Electrical Engineering, Pohang University of Science and Technology, Poliang, Kyungbuk 790-784. Korea)
References 1.0
I
1,
1
BAR-SHALOM, Y., and FORTMANN, T.E.: ‘Tracking association’ (Academic Press, Inc., 1988)
2
BAR-SHALOM, Y., and TSE, E.: ‘Tracking in a cluttered environment with probabilistic data association’, Automation, 1975, 11, pp. 451460
3
YANG-WEON LEE,, and HONG JEONG, : ‘A mdp estimate O f Optimal data association for multi-target tracking’. Proc. ICSPAT, 1997
4
RONG LI, x.,and BAR-SHALOM, Y.: ‘Tracking in clutter with nearest neighbor filters: Analysis and performance’, IEEE Trans., 1996, AES-32, (3), pp. 995-1010
5
LUENBERGER. D.G.: ‘Optimization by vector space methods’ (John Wiley & Sons, Inc., 1969)
6
MOLNAR, K.J., and MODESTINO, J.w.: ‘Application of the EM algorithm for the multitarget/multisensor tracking problem’, IEEE Trans., 1998, SP-46, (l), pp. 115-129
7
SENGUPTA, D., and ILTIS. R A . : ‘Neural solution to the multitarget tracking data association problem’, IEEE Trans., 1989, AES-25, pp. 96-108
0.5 E Y ai c
O
E -0.5 0
8
r-1.0 -1.5
-2.0 x co-ordinate, km
m
Fig. 2 Tracking crossing targets using proposed method true track _ _ - - measurement ........... estimate ~
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