Determining the point of minimum error for 6DOF pose uncertainty representation

In many augmented reality applications, in particular in the medical and industrial domains, knowledge about tracking errors is important. Most current approaches characterize tracking errors by 6x6 covariance matrices that describe the uncertainty of a 6DOF pose, where the center of rotational error lies in the origin of a target coordinate system. This origin is assumed to coincide with the geometric centroid of a tracking target. In this paper, we show that, in case of a multi-camera fiducial tracking system, the geometric centroid of a body does not necessarily coincide with the point of minimum error. The latter is not fixed to a particular location, but moves, depending on the individual observations. We describe how to compute this point of minimum error given a covariance matrix and verify the validity of the approach using Monte Carlo simulations on a number of scenarios. Looking at the movement of the point of minimum error, we find that it can be located surprisingly far away from its expected position. This is further validated by an experiment using a real camera system.