A Batch Update Using Multiplicative Noise Modelling for Extended Object Tracking

TL;DR

A batch update for the recently introduced elliptical-target tracker called MEM-EKF*, which is based on the same likelihood as the well-established random matrix approach but is derived from the multiplicative error model (MEM) and uses an extended Kalman filter (EKF) to update the target state sequentially, i.e., measurement-by-measurement.

Abstract

While the tracking of multiple extended targets demands for sophisticated algorithms to handle the high complexity inherent to the task, it also requires low runtime for online execution in real-world scenarios. In this work, we derive a batch update for the recently introduced elliptical-target tracker called MEM-EKF*. The MEM-EKF* is based on the same likelihood as the well-established random matrix approach but is derived from the multiplicative error model (MEM) and uses an extended Kalman filter (EKF) to update the target state sequentially, i.e., measurement-by-measurement. Our batch variant updates the target state in a single step based on straightforward sums over all measurements and the MEM-specific pseudo-measurements. This drastically reduces the scaling constant for typical implementations and indeed we find a speedup of roughly 100x in our numerical experiments. At the same time, the estimation error which we measure using the Gaussian Wasserstein distance stays significantly below that of the random matrix approach in coordinated turn scenarios while being comparable otherwise.

Figures (5)

• Figure 1: This figure depicts the multiplication error measurement model for extended object tracking. A measurement $\bm{{{{y}}}}_{i}$ is the measurement source $\bm{{{{z}}}}_{i}$ corrupted with sensor noise $\bm{{{{v}}}}_{i}$. The measurement source $\bm{{{{z}}}}_{i}$ relates object orientation $\alpha$ and axes-lengths $(l_1,l_2)^{\text{T}}$ by a multiplicative noise $\bm{{{{h}}}}_{i}$, which are uniformly distributed on a unit circle.
• Figure 2: This figure shows one example run with three trackers: random matrix (blue), MEM-EKF* (red) and MEM-EIF (green). The ground truth object is black ellipses and measurements are colored dots.
• Figure 3: This figure depicts the average error for 100 simulated runs using the simulation shown in Fig. \ref{['fig:example_run']}.
• Figure 4: Estimation Error of the MEM-EIF[${\bar{\bm{{y}}}}_0$] for different batch sizes ($U$ updates is equivalent to a batch size of $L/U$) versus MEM-EKF* and MEM-EIF[$\bm{{{{y}}}}_L$]. We used a Poisson rate $\lambda = 50$ for the simulation, otherwise the simulation is the same as in Fig. \ref{['fig:example_run']}. The results are averaged over 100 runs.
• Figure 5: Comparison of the runtime performance of the sequential MEM-EKF* (purple line) and the batch MEM-EIF[$\bm{{{{y}}}}_L$] (green line) update based on our python implementation. The number of seconds necessary to perform the update (y-axis) is plotted against the number of measurements which are part of the update (x-axis). Due to the large difference in the scaling constant, the green line representing the scaling of the MEM-EIF[$\bm{{{{y}}}}_L$] update is hugging the x-axis. In the inset, the same data is shown such that the linear scaling of the MEM-EIF[$\bm{{{{y}}}}_L$] update in the number of measurements becomes visible as well.