Principles of Modern Radar - Volume 2 1891121537

15.2 Multitarget Tracking 685removed. In doing so, the HOMHT produces a definitive association answer M scans afterthe data are collected, using all the subsequent hypotheses (that exceeded some minimumprobability, in practice) created since. The entire process repeats with every new set ofmeasurements that are collected, so that at t k+M , the association answer for t k will bedefinitively determined. By deferring the association decision by M scans, the HOMHTallows changes in target motion over time to resolve the previous association ambiguity.Track-Oriented MHT Recall that the number of possible hypotheses is typically ordersof magnitude larger than the number of targets. The TOMHT capitalizes on this fact bycarrying tracks (not hypotheses) forward from one scan to the next. As a result of thesignificant computational savings, the TOMHT is more commonly used and cited in theliterature.When a new set of measurements is collected, the TOMHT begins by independentlyforming possible associations to tracks. Uniqueness isn’t imposed yet in the algorithm.Rather, multiple tracks can be updated with the same measurement at this stage.The LLR is then computed for each track, using the recursion from [8]. The LLRsfor all potential tracks (which need not form compatible sets, or hypotheses yet) are thencompared with a threshold. Those that fail this threshold are immediately pruned.The remaining association candidates are clustered into groups linked by commonobservations. For example, if measurement 1 could potentially associate with tracks 1,2, or 3, then these three tracks would be grouped together in a cluster. Furthermore, notall tracks in the cluster have to share a common observation; they could be linked byanother track that shares common observations with both. Bear in mind that the intent ofthis clustering step is not to form hypotheses, or compatible sets of associations—Thathappens in the next step. Rather, the intent is to divide the association candidates intoseparable groups so that a series of small association problems can be solved in lieu ofone large problem. Since the computational complexity eventually required to solve theproblems grows exponentially as a function of the number of tracks in the cluster, it isadvantageous to break the associations into as many small clusters as possible.Finally, once poor association candidates have been pruned out based on their LLRsand once the remaining association candidates have been segregated into separable clusters,the TOMHT forms hypotheses. The candidate associations, along with their updatedLLRs and compatibility constraints, serve as inputs for this multi-frame association (MFA)problem, where the frames correspond to the previous scans of data that have been collected.The most common technique for efficiently solving the MFA problem is Lagrangianrelaxation [35, 36, 37], the goal of which is to maximize the hypothesis score under theconstraint that no tracks in the hypothesis share observations (i.e., are incompatible witheach other). Hence, the hypothesis score serves as the objective function, and the constraintsare replaced with well-chosen Lagrangian multipliers, allowing efficient creationof hypotheses.Pruning then occurs again, in the interest of limiting computational complexity. Therecently-created hypothesis scores are equal to the sum of their contributing track scores.Any hypotheses or tracks that score too low at time t k are dropped, to reduce the computationalburden. Associations M scans back are also finalized based on the currenthypotheses. Only the t k−M associations in the same branch as the current best hypothesisare maintained. Hence, the TOMHT also allows associations to be deferred untiladditional data is available, so that subsequent target motion has a chance to resolve theambiguities.