Geometrical analysis of registration errors in point-based rigid-body registration using invariants

Point-based rigid registration is the method of choice for aligning medical datasets in diagnostic and image-guided surgery systems. The most clinically relevant localization error measure is the Target Registration Error (TRE), which is the distance between the image-defined target and the corresponding target defined on another image or on the physical anatomy after registration. The TRE directly depends on the Fiducial Localization Error (FLE), which is the discrepancy between the selected and the actual (unknown) fiducial locations. Since the actual locations of targets usually cannot be measured after registration, the TRE is often estimated by the Fiducial Registration Error (FRE), which is the RMS distance between the fiducials in both datasets after registration, or with Fitzpatrick's TRE (FTRE) formula. However, low FRE-TRE and FTRE-TRE correlations have been reported in clinical practice and in theoretical studies. In this article, we show that for realistic FLE classes, the TRE and the FRE are uncorrelated, regardless of the target location and the number of fiducials and their configuration, and regardless of the FLE magnitude distribution. We use a geometrical approach and classical invariant theory to model the FLE and derive its relation to the TRE and FRE values. We show that, for these FLE classes, the FTRE and TRE are also uncorrelated. Finally, we show with simulations on clinical data that the FRE-TRE correlation is low also in the neighborhood of the FLE-FRE invariant classes. Consequently, and contrary to common practice, the FRE and FTRE may not always be used as surrogates for the TRE.