Principles of Modern Radar - Volume 2 1891121537

15.2 Multitarget Tracking 683TABLE 15-3Exhaustive List of Measurement HypothesesH1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11M1 φ T1 T2 T3 φ T1 T3 φ T1 T2 T3M2 φ φ φ φ T2 T2 T2 T4 T4 T4 T4is one whose measurement origins are unique. For example, even though M1 gates withboth T1 and T2 in the notional example, the hypothesis that M1 originates from the targetsbeing tracked by both T1 and T2 is incompatible; the two associations are in conflict. Inthis particular example, eleven compatible hypotheses (H1-H11) are possible, as depictedin Table 15-3. The rows correspond to the potential origins of each measurement, and thecolumns correspond to the eleven possible hypotheses. For example, the first hypothesisis that both measurements are false alarms. The second is that M1 originates from T1 andthat M2 is a false alarm. The eleventh hypothesis is that M1 and M2 are both new targets,which spawn T3 and T4, respectively.These measurement hypotheses will be carried forward over the next M measurementcollections (i.e., scans). Each measurement hypothesis spawns additional sets ofmeasurement hypotheses in each of the next M scans, creating and exponentially growingmeasurement hypothesis tree. The best measurement hypothesis at time t k is then chosenat t k+M , based on the best branch in the hypothesis tree at that time. Hence, the HOMHTdefers the data association decision for M scans, allowing data association ambiguities toresolve naturally over time.If computational resources were infinitely available, then the HOMHT could simplycarry this entire list of measurement hypotheses forward for M scans, updating andmaintaining the resulting tracks in each hypothesis separately. However, given real-worldcomputational constraints, the brute force approach is infeasible (at least for any applicationwith sufficiently dense target scenes to require a MHT). Hence, two main strategies areapplied to limit the computations required in the HOMHT. First, the number of measurementhypotheses actually carried forward is typically constrained. This pruning allowsonly the k-best hypotheses to be carried forward to the next scan, hence limiting theexponential growth of the hypothesis tree over time. Second, rather than updating andmaintaining each track separately in each branch of the measurement hypothesis tree, significantsavings are often gained by exploiting commonalities of track assignments acrosshypotheses.This is accomplished by listing the target hypotheses corresponding to the measurementhypotheses, as in Table 15-4. The target hypothesis list is typically much morecompact than the list of compatible measurement hypotheses. Each target (including thenew tentative ones) gets a row in this matrix, and each measurement (including the nullsymbol,to account for track coasting) gets a column. The checks indicate that the trackand measurement are paired in at least one measurement hypothesis. For example, T1is either coasted (e.g., in H1, H3-H5, H7, H8, H10, H11) or updated with M1 (e.g., inH2, H6, and H9). Hence, even though eleven measurement hypotheses could be carriedforward, only two options for T1 must be maintained at this scan: a version where it iscoasted and a version where it is updated with M1. Similarly, T2 may be associated witheither measurement, or it may be coasted, resulting in checks in all boxes in the T2 row.Hence, three versions of T2 must be maintained at this scan to the possibilities from theset of eleven corresponding measurement hypotheses, as in Table 15-4.