*3.2. GLMB Multi-Scan Estimator*

The concept of a multi-scan estimator is introduced in Reference [50]. Given a multi-scan GLMB from time step *j* to *k*, the cardinality distribution of the number of trajectories is given as follows:

$$\Pr(|\mathcal{L}(\mathbf{X}\_{j:k})| = n) = \sum\_{\vec{\xi}, I\_{j:k}} \delta\_{\mathbb{N}}[|I\_{j:k}|] w\_{j:k}^{(\vec{\xi})}(I\_{j:k}) \tag{22}$$

One possible form of a multi-scan estimator is to determine the component with the highest weight *w*(*ξ*) *<sup>j</sup>*:*<sup>k</sup>* (*Ij*:*k*) given that it has the most probable cardinality by maximizing Equation (22). The expected trajectory estimate can then be computed from *p* (*ξ*) *<sup>j</sup>*:*<sup>k</sup>* (·, *l*) for each *l* ∈ *Ij*:*k*.

In this work, we proposed modifications to the multi-scan estimator in Reference [50], which can eliminate track fragmentation and improve localization performance. The set of all estimated trajectories is updated at each time step via the most significant hypothesis with the most probable cardinality in the GLMB density. At the time step when state estimation is required, the information of estimated trajectories is passed into the estimator. At this stage, trajectories pruning is applied to eliminate short-term tracks. Subsequently, standard filtering and RTS smoothing techniques are applied on each trajectory to produce smoothed state estimates. The significance of this estimator is that it allows the application of smoothing techniques to improve the tracking accuracy while completely eliminates track fragmentation as the entire trajectory is estimated as a whole. In addition, as the complexity of the estimator depends only on the number of estimated tracks, the additional computational effort of the estimator is negligible compared to GLMB filtering. The detailed implementation of the estimator is given as in following subsections.
