Estimation of K Tracks

Using a Bayesian formulation which incorporates prior track set probabilities, at time n we have the following optimization problem for the selection from $\mbox{\boldmath$\tau$ }_n^i ; i=1,2, \cdots I_n^{'}$

$$\max_{ \mbox{\boldmath$\tau$ }_n^i } \; \; \; \; \; p\left( ... ...$\tau$ }_n^i \right)} {p \left( {\cal Z}^n \right)} \; \; \; .$$ (7)

$p \left( {\cal Z}^n \right)$ is a normalizing factor which serves to make the possible values of $p\left( \mbox{\boldmath$\tau$ }_n^i \vert {\cal Z}^n \right)$ sum to unity, and can be ignored since it is independent of $\mbox{\boldmath$\tau$ }_n^i$. $p\left( \mbox{\boldmath$\tau$ }_n^i \right)$ is the apriori probability of the set of tracks, and is derived from the known probability of missed detections and the false alarm rate. Letting J_m = T_mⁱ + F_mⁱ, where T_mⁱ is the number of measurement vectors at time m used in the Ktrack set $\mbox{\boldmath$\tau$ }_n^i$ and F_mⁱ is the number of assumed false alarms, we have that

$$p\left( \mbox{\boldmath$\tau$ }_n^i \right) = \prod_{m=1}^{n} \; P_d^{T_m^i} \; (1-P_d)^{K-T_m^i} \; p( F_m^i ) \; \; ,$$ (8)

where p( F_mⁱ ) is the Poisson distributed probability of F_mⁱ false alarms.

$p\left( {\cal Z}^n \vert \mbox{\boldmath$\tau$ }_n^i \right)$is the joint probability density function of the measurements ${\cal Z}^n$ conditioned on the track set $\mbox{\boldmath$\tau$ }_n^i$. For a given $\mbox{\boldmath$\tau$ }_n^i$, we can partition ${\cal Z}^n$ into the data associated with the tracks, ${\cal Z} ( \mbox{\boldmath$\tau$ }_n^i )$, and data associated with false alarms, $\overline{\cal Z} ( \mbox{\boldmath$\tau$ }_n^i )$. We then have

$$p\left( {\cal Z}^n \vert \mbox{\boldmath$\tau$ }_n^i \right)... ...$ }_n^i ) \vert \mbox{\boldmath$\tau$ }_n^i \right) \; \; \; ,$$ (9)

where, under the assumption that false alarms are uniformly distributed over the surveillance volume Y,

$$p\left( \overline{\cal Z} ( \mbox{\boldmath$\tau$ }_n^i ) \v... ...) = \left( \frac{1}{Y} \right)^{\sum_{m=1}^{n} F_m^i} \; \; .$$ (10)

In terms of the Kalman filter generated measurement innovations, for the track measurements,

$$p\left( {\cal Z} ( \mbox{\boldmath$\tau$ }_n^i ) \vert \mbo... ...d_{p=1}^K \prod_{m=1}^{n} \; p ( {\bf v}_{m,j_m ( l_p (i) )} )$$ (11)

$$= d_n^i \; \mbox{exp} \left\{ -\frac{1}{2} \sum_{p=1}^K \su... ...(i) )}^{-1} {\bf v}_{m,j_m ( l_p (i) )} \right\} \nonumber \\$$

where for each track p the product over m does not include any missed detections since as noted earlier measurements and innovations for missed detections don't exist. Keeping this in mind,

$$d_n^i = \prod_{p=1}^{K} \prod_{m=1}^{n} \frac{1}{{(2 \pi )}^... ...\cdot \sqrt{\det ({\bf S}_{m,j_m ( l_p (i) )} ) }} \; \; \; .$$ (12)

The Bayesian cost for estimating the best K tracks is described by (8), along with (9), (11), (13) and (14).

To derive an equivalent cost from which a time recursive trellis structure representation can be derived, take the negative natural log of (8), ignoring $p \left( {\cal Z}^n \right)$. For the i^th track set, the following equivalent Bayesian optimization problem is obtained:

$$\min_{ \mbox{\boldmath$\tau$ }_n^i } \; \; \; \; \; \Lambda... ...) = \sum_{m=1}^{n} \Lambda_m ( \mbox{\boldmath$\tau$ }_n^i )$$ (13)

where the time incremental cost is

$$\Lambda_m ( \mbox{\boldmath$\tau$ }_n^i ) = \lambda_m ( \mb... ...{p=1}^{K} \lambda_m ( \mbox{\boldmath$\theta$ }_n^{l_p (i)} )$$ (14)

with

 

$$\lambda_m ( \mbox{\boldmath$\tau$ }_n^i ) F_m^i \ln (Y) - T_m^i \ln (P_d )$$ = (15)

$$(K-T_m^i ) \ln (1-P_d ) - \ln (p(F_m^i )) \; \; ,$$ -

and the p^th track incremental cost at time m is

$$\lambda_m ( \mbox{\boldmath$\theta$ }_n^{l_p (i)} ) \frac{1}{2} \ln ( \det ({\bf S}_{m,j_m ( l_p (i) )})) + \frac{D}{2} \ln (2 \pi )$$ = (16)

$$\frac{1}{2} {\bf v}_{m,j_m ( l_p (i) )}^T {\bf S}_{m,j_m ( l_p (i) )}^{-1} {\bf v}_{m,j_m ( l_p (i) )} \; .$$ +

Although from (15), the time recursion

$$\Lambda^n ( \mbox{\boldmath$\tau$ }_n^i ) = \Lambda^{n-1} (... ...$ }_n^i ) + \Lambda_n ( \mbox{\boldmath$\tau$ }_n^i ) \; \;$$ (17)

can be formed, to solve this Bayesian problem directly, at time n I_n^' incremental costs must be computed. In the following sections we develop an algorithmic approach which reduces this computational requirement.