Derivation of Expected Registration Error for Point-based Rigid-body Registration

This paper presents two new expressions for estimating registration accuracy in point-based rigid-body registration and points out a danger in using the traditional measure of registration accuracy. Rigid-body registration is appropriate for applications in which relatively rigid parts of the body are involved. In some such applications rigid-body registration is accomplished by aligning two sets of discrete points. In neurosurgical guidance, for example, the points are found by localizing the centroids of fiducial markers. This paper provides two new fundamental theoretical results on the relationship between localization error and registration error in rigidbody, point-based registration. Rigid-body, point-based registration is accomplished by finding a rigid-body transformation that aligns pairs of homologous "fiducial" points. The imprecision in locating a fiducial point is known as the "fiducial localization error" (FLE). Fiducial points may be centroids of attached markers, which tend to have small, equal FLEs, or salient points in segmented anatomic features, whose FLEs tend to be larger and more varied. Alignment is achieved by minimizing the "fiducial registration error" (FRE) , which is the root mean square distance between homologous fiducials after registration. Closed form solutions for the rigid transformation that minimizes FRE have been known since 1966. The expected value (FRE2) depends on the number N of fiducials and expected squared value of their fiducial localization error (FLE2), but in 1979 it was shown that (FRE2) is approximately independent of the fiducial configuration C. The importance of this surprising result seems not yet to have been appreciated by the registration community: Poor registrations caused by poor fiducial configurations may appear to be good. A more critical and direct measure of registration error is the "target registration error" (TRE) ,which is the distance between homologous points other than the centroids of fiducials. Efforts to characterize its behavior have been made since 1993 or earlier; published simulations have shown that (TRE2) is roughly proportional to (FLE2)/N and, unlike K FRE2), does depend in some way on C. In this work we present two major results: we first derive an approximate expression for (TRE2), and we then use this to calculate the expected squared alignment error for individual fiducials. This leads to a surprising conclusion the registration accuracy is generally worst near the fiducials which are most closely aligned! This should demonstrate the danger of relying on the traditional accuracy measure, namely fiducial registration error, to judge the quality of a registration.