The Subset Problem: Costs for Two-Sensor Assignment Using a Model-Order Penalty

We desire to fuse a list of observations from sensor 1 to one at sensor 2. The data are unlabeled (i.e., permuted), such that it must be decided which data from sensor 1 should be fused with which data at sensor 2. The challenge, perhaps more important than the lack of label, is that the underlying object lists at the two sensors can be different. With respect to the former, this is a two-list assignment: It has been solved. The second is the "subset problem": The lists of true objects observed at sensor 1 and at sensor 2 differ, and the issue is that fusion "costs" of fused pairs and singletons are not comparable. In this paper we treat the problem as one of model-order selection, since while a hypothesis that fuses two observations will necessarily have a (negative) log-likelihood "cost" that is greater than that for a hypothesis that leaves them unfused, the unfused hypothesis relates to a model of greater parametric complexity, and hence, to penalize a more complex model that may be "fitting the noise" should be discouraged. We find that the familiar Akaike Information Criterion (AIC) model-order penalty offers an excellent approach to this problem.