Time Varying Number-of-Track Estimation

For the tracking scenario described in Section 2 and number-of-tracks estimation problems formulated in Section 3, the number of tracks K was assumed fixed over the processing interval. This lead to the following set of hypothesized track-sets:

$$\mbox{\boldmath$\tau$ }_n^{i(K)} \ ; \; K=0,1, \cdots , K_{max}$$ (20)

where the i(K) are the hypothesis indices for the different K. Although this can be a formidable number of hypotheses, as we have described Viterbi MHT can be employed to manage this, though as with any pruning/merging strategy this results in a suboptimal estimator.

Here we generalize the scenario by partitioning the processing interval into measurement blocks of temporal length T, and by allowing K to vary from block to block. For time n=tT, where t is the block index, the set of hypothesized track-sets is now:

$$\mbox{\boldmath$\tau$ }_n^{i(K_1 , K_2 , \cdots , K_t)} \ ; \; K_t =0,1, \cdots , K_{max} \; ; \; \; t=1,2, \cdots , T \; .$$ (21)

Now $i(K_1 , K_2 , \cdots , K_t)$ are the hypothesis indices for the number-of-tracks sequences $K_1 , K_2 , \cdots , K_t$. The number of hypotheses is even more formidable, but again Viterbi MHT can be used to manage this.

The Viterbi MHT algorithm as described above is modified by "truncating the trellis", a technique commonly used in digital communication applications of the Viterbi algorithm. We propose the following. Consider processing the t^th measurement block to estimate K_t. We assume that the estimates for previous blocks, i.e. $\hat{K}_1 , \cdots , \hat{K}_{t-1}$, are correct. For the current block, t, the hypotheses are:

$$\mbox{\boldmath$\tau$ }_n^{i(\hat{K}_1 , \hat{K}_2 , \cdots , \hat{K}_{t-1}, K_t)} \ ; \; K_t =0,1, \cdots , K_{max} \; ,$$ (22)

where $i(\hat{K}_1 , \hat{K}_2 , \cdots , \hat{K}_{t-1}, K_t)$ are the hypothesis indices for $\hat{K}_1 , \hat{K}_2 , \cdots , \hat{K}_{t-1}$ fixed and K_t varying from 0 to K_max. Note that n=tT.

In the Viterbi MHT algorithm, at time (t-1)T, $\hat{K}_{t-1}$ is computed, the costs for the hypotheses $\mbox{\boldmath$\tau$ }_{(t-1)T}^{i(\hat{K}_1 , \ \hat{K}_2 , \cdots , \hat{K}_{t-1})}$ are computed, and the trellis paths (i.e. the hypotheses) are merged and pruned. Starting at time n=(t-1)T+1, at each time n Viterbi MHT is run as described above. That is, costs for hypotheses for all $K_t = 1,2, \cdots , K_{max}$ are computed and trellis paths are merged and pruned. At time n=tT, the trellis is truncated. That is, $\hat{K}_t$ is computed (using either joint estimation or merging) and fixed for the next block.