Principles of Modern Radar - Volume 2 1891121537

684 CHAPTER 15 Multitarget, Multisensor TrackingTABLE 15-4List of Target HypothesesM1 M2 φT1 ✓ ✗ ✓T2 ✓ ✓ ✓T3 ✓ ✗ ✓T4 ✗ ✓ ✓TABLE 15-5Hypothesis Relationship MatrixM1 M2 φT1 H2, H6, H9 H1, H3-H5, H7, H8, H10, H11T2 H3, H10 H5-H7 H1, H2, H4, H8, H9, H11T3 H4, H7, H11 H1-H3, H5, H6, H8-H10T4 H8-H11 H1-H7A hypothesis relationship matrix is then maintained in the HOMHT to map the versionsof each track to each hypothesis. Table 15-5 shows the hypothesis relationship matrix forthis example.Before continuing to the next scan, each measurement hypothesis must be scored.Reid proposes using the Bayesian probability of each hypothesis. Note that (15.20) doesnot match the formulation in the original paper by Reid, although the intent is the same;Bar-Shalom et al. later extended Reid’s original formulation to produce (15.20) [4].Two main tricks are often used in practice to limit the number of computations requiredto execute this step. First, a mathematically-equivalent, computationally-preferable, andmore numerically-stable log-likelihood ratio (LLR) score is commonly used as a shortcutto computing the hypothesis probabilities. The LLR derivation is explained in depth innumerous sources [8, 9, 42], so will not be repeated here. In short, the LLR takes thenatural logarithm of the ratio that the track is valid (i.e., based on target measurements)to the ratio that the track is based entirely on false alarms. The hypothesis score is foundby summing the LLRs for each track assignment in the hypothesis. This hypothesis scoreis then converted to a probability through the approach outlined in [4]. Second, the LLR(which is already more numerically-stable and computationally-convenient than the fullhypothesis probability) can be computed as a recursion for each track, minimizing thenumber of new computations required each time. More details on implementation of thisrecursion are readily available in [8, 9].Having scored the measurement hypotheses, the scores can then be used to prune thelist, so that only the best are carried forward to the next scan. For example, suppose that H6had the highest probability, followed by H7 and H5. Further suppose that the remainingeight hypotheses probabilities fell below some allowable threshold. This pruning stepwould immediately get rid of these eight low-scoring hypotheses; only H5, H6, and H7would be carried forward to subsequent scans. Note that this can substantially simplify theassociation problem. In this example, the only three hypotheses remaining after pruning(H5-H7) all assign M2 to T2 and fail to initiate T4. In the next scan, it is thus unnecessaryto include T4 or to consider branches from other association alternatives at t k for T2.Once the hypothesis scores have all been updated and the low-scoring ones havebeen pruned at t k , the next step is retro-active pruning. The best hypothesis (e.g., H6)is identified, and all branches in the tree M scans back (at t k−M ) that don’t lead to it are