Hypothesized Track Sets

A single hypothesized track is characterized by a measurement vector or a missed detection for each measurement time. At time n let one such track, the l^th track, be denoted by the measurement-to-track association vector

$$\mbox{\boldmath$\theta$ }_n^l = \{ j_1 (l), j_2 (l) , \cdots , j_n (l) \}$$ (1)

where the subscript j_m (l) is the measurement/missed-detection index at time m for the l^th track. Note that we account for the possibility of a missed detection by letting j_m (l) range from 1 to J_m +1, with j_m (l) = J_m +1 indicating a missed detection. The vector of measurements for the l^th track is

$${\cal Z} ( \mbox{\boldmath$\theta$ }_n^l ) = \{ {\bf z}_{1,j_... ..., {\bf z}_{2,j_2 (l)}, \cdots , {\bf z}_{n,j_n (l)} \} \; \; .$$ (2)

${\bf z}_{m,J_m +1}$ is not an actual measurement, but indicates a missed detection at time m, and is a notational convenience. There are $L_n' = \prod_{m=1}^{n} (J_m + 1)$ of these hypothesized tracks: $\mbox{\boldmath$\theta$ }_n^l ; l=1,2, \cdots, L_n^{'}$. Association of the measurements into a single track is the problem of selecting the best $\mbox{\boldmath$\theta$ }_n^l$, which is equivalent to estimating the discrete parameter $l \in \{ 1,2, ... , L_n^{'} \}$. For a given hypothesized track $\mbox{\boldmath$\theta$ }_n^l$, measurement noise is assumed additive, Gaussian and temporally white. The trajectory and measurements are assumed to evolve in time according to the state/measurement equations¹

$${\bf x}_{m+1}^l = {\bf\Phi}_m {\bf x}_m^l + {\bf w}_m^l$$ (3)

$${\bf z}_{m,j_m (l)} = {\bf H}_m {\bf x}_m^l + {\bf u}_m^l$$ (4)

where at time m and for hypothesized track $\mbox{\boldmath$\theta$ }_n^l$, ${\bf x}_m^l$ is the state vector, and ${\bf\Phi}_m$ and ${\bf H}_m$ are the state transition and output matrices respectively. The ${\bf w}_m^l$and ${\bf u}_m^l$ vectors are zero mean, mutually independent, white and Gaussian with known covariance matrices ${\bf Q}_m^l$ and ${\bf R}_m^l$ respectively. A Kalman filter can be applied to this track to smooth the measurements, to provide minimum variance state vector estimates, and to compute the innovations for the measurement sequence ${\cal Z} ( \mbox{\boldmath$\theta$ }_n^l )$. For each hypothesis, the measurement innovations sequence $\{ {\bf v}_{1,j_1 (l)} , {\bf v}_{2,j_2 (l)}, \cdots , {\bf v}_{n,j_n (l)} \}$ and corresponding covariance matrices $\{ {\bf S}_{1,j_1 (l)} , {\bf S}_{2,j_2 (l)}, \cdots , {\bf S}_{n,j_n (l)} \}$ generated using a hypothesis-specific Kalman filter are to be used in optimum track estimation. Note that a ${\bf v}_{m,j_m (l)}$corresponding to ${\bf z}_{m,J_m +1}$ is not an actual innovation, but is a result of a missed detection at time m, and again is a notational convenience. In such a case, the Kalman state is just the predicted state (i.e. no measurement is available to update the state).