Mahalanobis size-modulated

This is an attempt to learn from the successful IoU-based metrics. Namely, we observe that the IoU metric makes good use of geometry completely disregarding the accuracy with which the states are currently known. Indeed, no covariance matrix enters the definition of IoU metrics.

On other side, the Mahalanobis distance is much more adjustable, easy to define and based in probability theory governing the whole tracking methodology. In order to acknowledge the importance of geometry within the Mahalanobis-distance approach, we add a size-modulated part to the association covariance. Instead of using the innovation covariance, we suggest

\[\Sigma_{rt} = R_{r} + HP_{t}H^T + S_{rt},\]

where the matrices $R$, $H$ and $P$ forming the usual innovation covariance, while the matrix $S$ is a diagonal matrix scaling with square average sizes of the probed cuboids. For example, to track 2D boxes, the size-modulating matrix reads

\[S_{rt} = \frac{1}{4} \begin{pmatrix} (s_{xr} + s_{xt})^2 & 0 & 0 & 0 \\ 0 & (s_{yr} + s_{yt})^2 & 0 & 0 \\ 0 & 0 & (s_{xr} + s_{xt})^2 & 0 \\ 0 & 0 & 0 & (s_{yr} + s_{yt})^2 \end{pmatrix}.\]

The size-modulated Mahalanobis metric can be used in 2D or 3D tracking. It shows promising results, albeit it’s difficult to quantify its advantages and disadvantages at this point.