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 ( ) ( ) FRE p 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 ( ) ( ) FRE r p accurate to first order in FLE. We show by means of numerical simulations that the approximation is valid.