Neural data association methods

The central feature of the PDAF and JPDAF is the determination of association prob- abilities

t i

. Since the computation of association probabilities is complex, especially with the presence of clutter, the neural network approach is suggested to improve the computational efficiency of

t i .

The main motivation for using a neural network is the claim that the complexity of computing the association probabilities is reduced. As far as we know, literature survey reviewed that there are 2 types of neural networks used for data association, namely:

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Boltzmann machine networks [107, 108].

Hopfield neural networks [205, 139].

In [107], Iltis and Ting show that the Boltzmann machine network, with simple binary computing elements, can estimate the association probabilities. The results are similar to the JPDAF, however with less computational load. Iltis and Ting have also shown in [108] that by using sufficient parallel Boltzmann machines the associ- ation probabilities can be computed with arbitrarily small errors.

The Hopfield’s neural network for computing the association probabilities is viewed as a constrained optimization problem. The constraints are obtained by a careful evaluation of the properties of the JPDA association rule. The problem is formulated in the same manner as the Hopfield’s travelling salesman problem (TSP). That is, the energy function of the data association problem is similar to the energy function of the TSP. In the data association, by minimizing the energy function, it is hoped that the association probabilities

t i

from the likelihoods could be computed accurately with minimum computational load.

The use of the Hopfield neural network to compute t i

(k)was first demonstrated in [205]. The distance is measured from track to measurement. The energy function of the data association problem (DAP), using the Hopfield neural network, is given as follows [205]: E DAP = A 2 m(k ) X i=0 T X t=1 T X r6=t r=1 V t i V r i + B 2 m(k ) X l=0 T X t=1 T X j6=l j=1 V t i V t j + C 2 T X t=1 f m(k ) X i=0 V t i 1g 2 + D 2 m(k ) X i=0 T X t=1 (V t i t i ) 2 + E 2 m(k ) X i=0 T X t=1 T X r6=t 0 @ V t i m(k ) X j6=l j=1 r j 1 A 2 where t i = p t i (k ) P m(k ) j=0 p t j (k )

is the normalized version of the likelihood functionp t i

(k). Taking the derivative ofE

DAP with respect to V

t i

and obtaining the neuron dynam- ics: du t i dt = au t i A T X r6=tr=1 V r i B m(k ) X j6=i j=0 V t j C 0 @ m(k ) X j=0 V t j 1 1 A [D+E(T 1)]V t i +(D+E) t i +E T 1 T X r i ! : (6.16)

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The solution from equation (6.16) is calculated using the neuron input-output transfer function to obtain the neuron output voltage (association probability) [55]:

t i (k)=V t i =f(u t i )= 1 2 (1+tanh(u t i )) (6.17) t i

is approximated from the output voltage ofV t i

of a neuron in an(m+1)n array of neurons, wheremis the number of measurements andnis the number of targets.

The connection weights are given as:

T xi;yi = AÆ xy (1 Æ ij ) BÆ ij (1 Æ xy ) C Dd xy (Æ jj+1 +Æ jj+1 ) I xi = Cn

In [139], a modified Hopfield network is used to approximately compute t i . This modified Hopfield network uses the Runge-Kutta method and the Aiyer net- work’s structure [2]. Using the Aiyer network structure, the new connection weights are given as:

T xi;yi = AÆ xy (1 Æ ij ) BÆ ij (1 Æ xy ) 2A 1 Æ xy Æ ij C+2(An A+A i )=n 2 D(1 Æ ij )(1 Æ xy )(d xi +d yj ) I xi = Cn

and with the following recommended choice of choosing the parameters(A;B;C ;D;A 1 ): B=AandA 1 = 31 32A andC= A 10 .

The performance of the neural data association depends mainly upon the selec- tion of these five parameters. The modified Hopfield neural network is shown to have a much better performance in the sense that it follows the correct path closer, especially in the manoeuvre region.

It is claimed that the advantage of using the Hopfield neural network is that it reduces the complexity of computing the association probabilities. For a large number of targets, it only requires a larger array of neurons as opposed to the digital approach, which requires an exponential increase of computer resources.

The Hopfield neural network is also used to reduce a likelihood matrix, properly defined, to the assignment matrix [209]. The neural network processes the elements of the likelihood matrix to produce an assignment matrix.