MAP Based Track Estimation

Using a MAP formulation which incorporates prior track set probabilities, at time n we have the following optimization problem for determining the measurement-to-track associations. For $\mbox{\boldmath$\tau$ }_n^i ; i=1,2, \cdots I_n^{'}$

$$\max_{ \mbox{\boldmath$\tau$ }_n^i } \; \; \; \; \; p\left( ... ...$\tau$ }_n^i \right)} {p \left( {\cal Z}^n \right)} \; \; \; .$$ (6)

$p \left( {\cal Z}^n \right)$ 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 a priori probability of the set of tracks, and is derived from knowledge of the sensors, targets, noise, clutter, jammers and preprocessors, as manifested for example as the probability of missed detections and the false alarm rate. $p\left( {\cal Z}^n \vert \mbox{\boldmath$\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

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

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

$$p\left( \overline{\cal Z} ( \mbox{\boldmath$\tau$ }_n^i ) \v... ...t) = \left( \frac{1}{Y} \right)^{\sum_{m=1}^{n} F_m^i} \; \; ,$$ (8)

where F_mⁱ is the number of false alarms at time m given hypothesized track set $\mbox{\boldmath$\tau$ }_n^i$. In terms of the Kalman filter generated measurement innovations, for the track measurements,

$$p\left( {\cal Z} ( \mbox{\boldmath$\tau$ }_n^i ) \vert \mbox... ...d_{p=1}^K \prod_{m=2}^{n} \; p ( {\bf v}_{m,j_m ( l_p (i) )} )$$ (9)

$$= d_n^i \; \mbox{exp} \left\{ -\frac{1}{2} \sum_{p=1}^K \sum... ...j_m ( l_p (i) )}^{-1} {\bf v}_{m,j_m ( l_p (i) )} \right\} \\$$ (10)

where for each track p the product (or sum) over m does not include any missed detections since as noted earlier measurements and innovations for missed detections do not exist. Keeping this in mind,

$$d_n^i = \prod_{p=1}^{K} \prod_{m=2}^{n} \frac{1}{{(2 \pi )}^... ... \cdot \sqrt{\det ({\bf S}_{m,j_m ( l_p (i) )} ) }} \; \; \; .$$ (11)

The MAP cost for estimating the best K tracks is described by (6), along with (8), (10), (11) and a specific form of $p\left( \mbox{\boldmath$\tau$ }_n^i \right)$.