Multi-Hypothesis Tracking

where1 ˆ xi k|k = N_H X h=1 βhxˆi,h_k|k (3.14) Pi_k|k = N_H X h=1 βh n

Pi,h_k|k+ ˆxi_{k|k− ˆx}i,h_k|k
xˆi_{k|k− ˆx}i,h_k|k
To (3.15) are the mean and covariance matrix of the single Gaussian.

The JPDA algorithm is often described in an alternative, but equivalent, fashion [20], where the computation of the state estimates includes the cal- culation of a weighted measurement residual, which is used in an ordinary Kalman filter update.

Both the PDA and the JPDA filter have a tendency to merge tracks when the targets are closely spaced. This is referred to as the track coalescence effect [25]. In Paper II, an adjusted version of the JPDA filter is presented, which does not experience track coalescence. Another adjustment of the JPDA filter to avoid track coalescence is the JPDA* algorithm [26].

3.1.4

Multi-Hypothesis Tracking

The two previously discussed approaches to data association are either to as- sign the (globally) best measurement to each target, or to assign a weighted sum of measurements to each target, where a weight depends on the proba- bility of the corresponding data association hypothesis. A third alternative is to wait with the assignment, and instead keep several data association

1_{Although the sums in (3.14) and (3.15) can be done as written, in practice there would}

hypotheses, and aggregate them over time. Then, a hypothesis describes a possible sequence of assignments for each target. This is the foundation of Multi-Hypothesis Tracking (MHT) [20]. MHT is not a single algorithm, but a family of algorithms that maintain multiple data association hypotheses.

Multi-hypothesis tracking is a deferred decision logic in the sense that the data association at a time k is not set until at a later time step k+N (N > 0), at which time the availability of more data increases the probability that the true hypothesis at time k is retained. In each time step, MHT forms a set of plausible association hypotheses. Over time, the hypotheses will build up a hypothesis tree for each target, in the same way as for the conceptual solution to data association in Section 3.1.1. This makes MHT different to the single-hypothesis algorithms, as it propagates more than one hypothesis in time. Since the hypothesis tree grows with time, a reduction of branches is performed by discarding the least likely, or least probable, hypotheses at each time step. The original formulation of MHT was given in [27], and since then many different formulations of MHT has been given, e.g., in [20, 28–35]. Compared to a conventional single-hypothesis tracking algorithm, MHT can handle 10-100 times higher false-alarm densities [20]. MHT is also better suited than conventional algorithms when the target density is high. If the MHT algorithm is probability-based, another advantage with MHT is that the algorithm can tell the user how certain it is about the existence of currently presented tracks.

There are two fundamentally different approaches to MHT. The first one is hypothesis-oriented MHT, which is the one described in the original MHT formulation [27]. The second one is track-oriented, and it is described, e.g., in [20, 28–31]. We will give a short introduction to both approaches, starting with track-oriented MHT.

Track-Oriented MHT

In the track-oriented approach, as the name implies, we work on a per- track perspective. In our description, a track constitutes the description of a possible target, and each track is represented by a hypothesis tree. A branch running through the tree from top to bottom is referred to as a track hypothesis. In other descriptions, e.g., in [20] a track is a sequence of detections associated to a target, i.e., what we call a track hypothesis.

At each time step, all n possible associations of measurements to a certain track hypothesis are used to create n new branches of that track hypothesis. Consider the example in Fig. 3.1, where we have a single-target scenario, for which the target is described by a single validated track. To the left in the figure, we have plotted the track hypotheses predictions at time k, for each of

3.1 Tracking with Target Identity 3 0 1 0 1 2 0 0 2 5 0 4 0 1 0 2 3 1 0 2 5 0 4 0 2

Figure 3.2: Example of N-scan pruning (N = 2) for a single-target scenario. To the left we have the hypothesis tree before N-scan pruning, and to the right we have the corresponding tree after pruning. The best track hypothesis (in terms of track score or probability) at time k is marked with a circle. All track hypotheses with the same root node N = 2 steps back (detection 1) are to be kept, whereas the rest are pruned. In the example, the hypotheses which have detection 0 as root node N = 2 steps back are thus removed.

the four track hypotheses at time k_{−1, and the gate of each such hypothesis.} In the figure, we have also plotted the five received measurements at time k. To the right, the hypothesis tree is depicted, where the marked nodes of the tree represent the four hypotheses at k_{− 1. Each hypothesis prediction} is associated with all measurements within its gate. For each association, a new branch of the tree is created. The numbers next to the branches declare the measurement number. Recall that the number 0 branches represent the hypothesis that the target is not detected at time k.

Since the number of hypotheses grows exponentially over time, the num- ber of hypotheses must be reduced to obtain a practicable approach. This can be done by merging similar branches together, or by pruning low-probability branches, or by performing both merging and pruning. One way of perform- ing pruning is N-scan pruning [36], which is illustrated in Fig. 3.2. The pruning is either based on track hypothesis probability or likelihood.

In the multi-target case, the track-oriented MHT is more complicated, since we also need to consider possible conflicts between targets, where a conflict occurs if two tracks are associated with the same detection. To treat conflicts, the concept of a global hypothesis is introduced. Consider the two-target example of Fig. 3.3. A few of the conflicts between the hypothe- sis trees are marked with circles, and the globally possible data association

1 0 1 0 0 0 1 2 2 5 4 0 1 0 2 2 0 2 0 2 5 0 0 2 1 0 1 4 H1 H2 3 H3

Figure 3.3: Example of hypothesis trees for a two-target scenario. Each track represents one target. Three of the global hypotheses are marked asH1 to H3, and some conflicts are marked with circles.

hypotheses are the ones where there are no conflicts between the targets. Those hypotheses are referred to as feasible global hypotheses. In Fig. 3.3, three of the feasible global hypotheses are marked with arrows. As seen, a global hypothesis contains one track hypothesis from each track. Note that missed-detection hypotheses are never in conflict. If there are L tracks, a global hypothesis Hk is described as

Hk={h1i1k, h2i2k, . . . , hLiLk} , (3.16) where hlijk represents the track hypothesis ij of hypothesis tree l, under the global hypothesis k.

Just as for the single-target case, the hypothesis trees need to be reduced for the algorithm to be feasible. The N-scan pruning approach is applicable also to the multi-target case, where the global hypotheses are used instead of the local ones. The pruning algorithm then finds the best global hypothesis, H∗_{, and prunes the hypothesis trees according to that hypothesis.}

Hypothesis-Oriented MHT

The original MHT formulation [27] was hypothesis-oriented. This means that it works directly on the feasible global hypotheses. So at time k, the set of global hypotheses from time k_{−1 are expanded to consider the measurements} received at time k. Each global hypothesis from the previous time instant is hence expanded into a set of global hypotheses, which are feasible with

3.1 Tracking with Target Identity

respect to conflicts between the targets that are part of the hypothesis. Just as for track-oriented MHT, pruning is used to reduce the number of global hypotheses.

According to [20], a problem with the original MHT formulation is that a lot of low-probability hypotheses are created, and then immediately discarded due to their low probability. This wastes computational resources. However, the problem is relieved in [32] through the use of an extended version of Murty’s method [37].