Labelled OSPA metric for fixed and known number of targets

Multitarget tracking systems should solve two basic problems. The first one is to estimate the number of targets and their states at the current time. The second one is to connect target state estimates that belong to the same target along time to form tracks. The conventional way of building tracks in the random finite set framework (RFS) [1] is to attach a label to the individual target states [2], [3]. Labels have two important properties: they are unique (no two targets can have the same label) and they are fixed over time. Labels were used for track formation in [4], [5] using a vector-based formulation and in [2], [3] using the RFS framework. The approaches of [4], [5] and [2], [3] are equivalent due to the bijection between the labelled RFS state and the hybrid labelled multitarget state vector [6, Appendix B]. For the same reason, for fixed and known number of targets, representing the multitarget state as a vector is equivalent to a labelled set. One way to evaluate performance of tracking algorithms based on labelled set is using the labelled optimal subpattern assignment (LOSPA) metric [7]. In some cases, it is convenient to assume that the number of targets is fixed and known [8], [9]. This way we can study some properties of tracking algorithms more easily. In these cases, it is usually useful to use vector notation, in which labels are implicit in the ordering of the components of a vector, to denote a labelled set. The problem is that the LOSPA metric in [7] is defined with explicit labels. In this paper, we fill this gap and provide an expression for this metric when the number of targets is fixed and known and vector notation is used. This paper is organised as follows. In Section II, we introduce the two equivalent representations of the multitarget state based on a labelled set and a vector. We provide the