Mahalanobis association metric

The metric is defined via gaussian function of the Mahalanobis distance $D$

\[A_{rt} = \exp(-f D_{rt}^2 / (2 N_z)),\]

where the pre-factor $f$ is merely a tuning parameter with default value $1$, $N_z$ is the number of variables in the detection vectors. The (square) of Mahalanobis distance makes use a covariance matrix $\Sigma$

\(D^2_{rt} = (\mathbf{z}_r - H\tilde{\mathbf{x}}_t) \Sigma^{-1} (\mathbf{z}_r - H\tilde{\mathbf{x}}_t),\) where $H$ is so-called measurement matrix used in Kalman filters. The choice of covariance matrix $\Sigma$ makes possible to fine-tune the association eventually capturing the correlations between variables in the measurement space. A popular choice of this covariance is so-called innovation covariance defined as sum of the measurement covariance $R$ and process covariance $P$ in measurement space $HPH^T$

\[\Sigma_{rt} = R + H P_t H^T.\]

The matrices $H$, $P_t$ and $R$ are defined ($H$ and $R$) and calculated ($P_t$) in Kalman filters. $^T$ denotes the matrix transpose. Because the measurement covariance is set the same for all measurement vectors, we drop the report index from the measurement covariance $R$.

The Mahalanobis metric has a number of advantages:

• Simple matrix algebra operations involved.
• Definition generalizes straightforwardly to any composition of the state vectors.
• Allows a dynamical adjustment of the values relaxing the association condition at the start of tracking.

A number of disadvantages to mention here:

• The result association quality depends on the choice of the Kalman-filter covariances which affect both the predictions and association likelihoods simultaneously.
• Involves solution of a batch of linear equations which affects the performance.