Point-based rigid registration: clinical validation of theory

This paper presents the first comparison between theoretical estimates and clinically observed values for registration accuracy in point-based rigid-body registration. 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. We have previously provided two fundamental theoretical results on the relationship between localization error and registration error in rigid-body, point-based registration and have justified these results by showing them to be close to those given by numerical simulations. 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. 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. This error measure has been investigated by numerical simulation for many years, and we presented at this meeting in 1998 the first derivation of theoretical estimates of TRE. In that paper we showed that these estimates agree well with our simulations and those of others. We made the simplifying assumption in both the derivations and the simulations that the FLE is isotropic, i.e. that it has the same distribution in each coordinate direction. In the present work, we extend the validation beyond simulated data sets to clinically acquired head images from a set of 86 patients. We use the actual localizations of skull-implanted, visible fiducial markers in the images to compare the observed TRE values with those estimated by theory. This approach provides a clinically relevant estimate of the usefulness of the theoretical predictions. We also make a comparison between the observed TRE values and those given by numerical simulation: this allows us to determine whether the assumptions we use for the derivation of our results are good ones in practice. Although the distributions of observed and theoretical values appear different, ROC analysis shows that the theoretical values are good predictors of the observed ones. This gives some validity to the assumptions we make governing the point- based registration process (e.g., that the FLE is isotropic, and that it is independent and identically distributed at each fiducial point), and shows that our theory has practical use in a clinical setting.