Joint Estimation of K and the Tracks

We begin with a MAP formulation, which incorporates prior track set probabilities, for the estimation of both the number of tracks K and the tracks themselves (i.e. the track set associations $\mbox{\boldmath$\tau$ }_n^{i(K)}$). At time n we have the following optimization problem. For $K=0,1, \cdots , K_{max}$ and $\mbox{\boldmath$\tau$ }_n^{i(K)} ; i(K)=1,2, \cdots I_n^{'} (K); K=0,1, \cdots , K_{max}$

$$\max_{ K, \mbox{\boldmath$\tau$ }_n^{i(K)} } \; \; \; \; \; ... ... p \left( K \right) } {p \left( {\cal Z}^n \right)} \; \; \; .$$ (8)

$p \left( {\cal Z}^n \right)$ is a normalizing factor which serves to make the possible values of $p\left( \mbox{\boldmath$\tau$ }_n^{i(K)} \vert {\cal Z}^n \right)$ sum to unity, and can be ignored since it is independent of $\mbox{\boldmath$\tau$ }_n^{i(K)}$ and K. Concerning p(K), any prior distribution can be incorporated. However, here we will assume that K is uniformly distributed from K=0 to some maximum value K = K_max. The term $p\left( \mbox{\boldmath$\tau$ }_n^{i(K)} \vert K \right)$ is the a priori probability of the i(K)^th 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. In Section 5 a specific $p\left( \mbox{\boldmath$\tau$ }_n^{i(K)} \vert K \right)$will be used for illustration purposes. $p\left( {\cal Z}^n \vert K, \mbox{\boldmath$\tau$ }_n^{i(K)} \right)$is the joint probability density function of the measurements ${\cal Z}^n$ conditioned on the number of tracks K and the track set $\mbox{\boldmath$\tau$ }_n^{i(K)}$. For a given K and $\mbox{\boldmath$\tau$ }_n^{i(K)}$, we can partition ${\cal Z}^n$ into the data associated with the tracks, ${\cal Z} ( \mbox{\boldmath$\tau$ }_n^{i(K)} )$, and data associated with false alarms, $\overline{\cal Z} ( \mbox{\boldmath$\tau$ }_n^{i(K)} )$. We then have

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

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

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

where F_m^i(K) is the number of false alarms at time m given hypothesized track set $\mbox{\boldmath$\tau$ }_n^{i(K)}$. In terms of the Kalman filter generated measurement innovations, for the track measurements⁴,

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

$$= d_n^{i(K)} \; \mbox{exp} \left\{ -\frac{1}{2} \sum_{p=1}^K... ...m ( l_p (i(K)) )}^{-1} {\bf v}_{m,j_m ( l_p (i(K)) )} \right\}$$ (12)

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,

$$d_n^{i(K)} = \prod_{p=1}^{K} \prod_{m=1}^{n} \frac{1}{{(2 \p... ...dot \sqrt{\det ({\bf S}_{m,j_m ( l_p (i(K)) )} ) }} \; \; \; .$$ (13)

Note that, for the K=0 hypothesis, there are no measurements ${\cal Z} ( \mbox{\boldmath$\tau$ }_n^i )$associated with tracks. Then the first term on the right of (9) is not present. All measurements are considered false alarms, resulting in a simple MAP probability computation composed of (10) and a single $p\left( \mbox{\boldmath$\tau$ }_n^{i(K)} \vert K=0 \right)$ (i.e. for i(0)=1).

The resulting joint MAP estimator of K and $\mbox{\boldmath$\tau$ }_n^{i(K)}$ is

$${\hat K}, \mbox{\boldmath${\hat \tau}$ }_n^{i(K)} = \arg \; ... ...ft( \mbox{\boldmath$\tau$ }_n^{i(K)} \vert K \right) \right\}$$ (14)

where the propability is searched over all $\mbox{\boldmath$\tau$ }_n^{i(K)} ; i=1,2, \cdots I_n^{'} (K)$ for each $K=0,1, \cdots , K_{max}$. (Referring to (5), note that the number of hypothesized track sets for a given K is a function of K.)