The distribution of registration error of a fiducial marker in rigid-body point-based registration

Many image-guidance surgical systems rely on rigid-body, point-based registration of fiducial markers attached to the patient. Marker locations in image space and physical space are used to provide the transformation that maps a point from one space to the other. Target registration error (TRE) is known to depend on the fiducial localization error (FLE), and the fiducial registration error (FRE) of a set of markers, though a poor predictor of TRE, is a useful predictor of FLE. All fiducials are typically weighted equally for registration purposes, but is also a common practice to ignore a marker at position r by zeroing its weight when its individual error, FRE(r), is high in an effort to reduce TRE. The idea is that such markers are likely to have been compromised, i.e., perturbed badly between imaging and surgery. While ignoring a compromised marker may indeed reduce TRE, the expected effect of ignoring an uncompromised marker is to increase TRE. There is unfortunately no established method for deciding whether a given marker is likely to have been compromised. In order to make this decision, it is necessary to know the probability distribution p(FRE(r)), which has not been heretofore determined. With such a distribution, it may be possible to identify a compromised marker and to adjust its weight in order to improve the expected TRE. In this paper we derive an approximate formula for p(FRE(r)) accurate to first order in FLE. We show by means of numerical simulations that the approximation is valid.