3.2.4. Forward Filtering-Backward Smoothing of Trajectories

In this step, by using the measurement association history, the initial birth information (the state and the time at birth) in the estimated trajectories tuples set, and the measurements set, we apply standard single-object filtering and backward RTS smoothing techniques to produce a set of smoothed distributions of the trajectories. In this work, spatial distributions of tracks are assumed to be Gaussian distributed; hence, the estimated spatial distribution of track labeled *l* at time *k* is represented by the mean *m<sup>l</sup> <sup>k</sup>* and the covariance *<sup>P</sup><sup>l</sup> <sup>k</sup>*. The details of the procedure to produce the tracks distributions are given in Algorithm 4. The SingleObjectPrediction and SingleObjectUpdate functions are chosen according to the dynamic model, which can be Kalman prediction and Kalman update or their nonlinear variances. The linearity of the system also determines the SingleObjectSmoothing function, which takes the form of either Algorithm 2 or Algorithm 3 to smooth each individual trajectory. The output of the algorithm is the smoothed spatial distributions of all estimated trajectories, which is {*m*<sup>ˆ</sup> *ln <sup>k</sup>* , *<sup>P</sup>*<sup>ˆ</sup> *ln k* } *k* ˆ *ln <sup>i</sup>* :*k* ˆ *ln e* . From this set of distributions, the mean values can be extracted to be used as the estimated states of the trajectories.
