The distribution of target registration error in rigid-body point-based registration

Guidance systems designed for neurosurgery, hip surgery, spine surgery and for approaches to other anatomy that is relatively rigid can use rigid-body transformations to accomplish image registration. These systems often rely on point-based registration to determine the transformation and many such systems use attached fiducial markers to establish accurate fiducial points for the registration, the points being established by some fiducial localization process. Accuracy is important to these systems, as is knowledge of the level of that accuracy. An advantage of marker-based systems, particularly those in which the markers are bone-implanted, is that registration error depends only on the fiducial localization and is, thus, to a large extent independent of the particular object being registered. Thus, it should be possible to predict the clinical accuracy of marker-based systems on the basis of experimental measurements made with phantoms or previous patients. For most registration tasks, the most important error measure is target registration error (TRE), which is the distance after registration between corresponding points not used in calculating the registration transform. In this paper, we derive an approximation to the distribution of TRE; this is an extension of previous work that gave the expected squared value of TRE. We show the distribution of the squared magnitude of TRE and that of the component of TRE in an arbitrary direction. Using numerical simulations, we show that our theoretical results are a close match to the simulated ones.