Joint Estimation of K and the Tracks

For the estimation of both the number of tracks K and the tracks themselves $\mbox{\boldmath$\tau$ }_n^i$, (8) can be considered a function of unknown K, resulting in the following joint estimation problem

$$\max_{ K, \mbox{\boldmath$\tau$ }_n^i } \; \; \; \; \; p\lef... ...t p\left( K \right) } {p \left( {\cal Z}^n \right)} \; \; \; .$$ (18)

If we assume that K is uniformly distributed from K=0 to some maximum value K = K_max, and that $p\left( \mbox{\boldmath$\tau$ }_n^i \vert K \right) = p\left( \mbox{\boldmath$\tau$ }_n^i \right)$, and that $p\left( {\cal Z}^n \vert K, \mbox{\boldmath$\tau$ }_n^i \right) = p\left( {\cal Z}^n \vert \mbox{\boldmath$\tau$ }_n^i \right)$. This results in the estimator

$${\hat K}, \mbox{\boldmath${\hat \tau}$ }_n^i = \arg \; \; \... ..._{m=1}^{n} \Lambda_m ( \mbox{\boldmath$\tau$ }_n^i ) \right\}$$ (19)

where all values are as defined for (15), and now the cost is searched over all $\mbox{\boldmath$\tau$ }_n^i ; i=1,2, \cdots I_n^{'} (K)$ for each $K=0,1, \cdots , K_{max}$. (Referring to ( 6 ), note that the number of candidate track sets for a given K is a function of K.)