General approach to first-order error prediction in rigid point registration

Abstract

A general approach to the first-order analysis of error in rigid point registration is presented that accommodates fiducial localization error (FLE) that may be inhomogeneous (varying from point to point) and anisotropic (varying with direction) and also accommodates arbitrary weighting that may also be inhomogeneous and anisotropic. Covariances are derived for target registration error (TRE) and for weighted fiducial registration error (FRE) in terms of covariances of FLE, culminating in a simple implementation that encompasses all combinations of weightings and anisotropy. Furthermore, it is shown that for ideal weighting, in which the weighting matrix for each fiducial equals the inverse of the square root of the cross covariance of its two-space FLE, fluctuations of FRE and TRE are mutually independent. These results are validated by comparison with previously published expressions and by simulation. Furthermore, simulations for randomly generated fiducial positions and FLEs are presented that show that correlation is negligible (correlation coefficient < 0.1) in the exact case for both ideal and uniform weighting (i.e., no weighting), the latter of which is employed in commercial surgical guidance systems. From these results we conclude that for these weighting schemes, while valid expressions exist relating the covariance of FRE to the covariance of TRE, there are no measures of the goodness of fit of the fiducials for a given registration that give to first order any information about the fluctuation of TRE from its expected value and none that give useful information in the exact case. Therefore, as estimators of registration accuracy, such measures should be approached with extreme caution both by the purveyors of guidance systems and by the practitioners who use them.

Fig. 1

FRE and TRE for two example registrations (a) Image space showing patient (dashed outline), three fiducials (dotted circles), and an anatomical target (dotted cross). For simplicity localization errors are zero (b) Physical space showing a set of fiducial localization errors. Arrows show the displacements from true positions (dotted outlines) to localized positions (solid outlines). The same anatomical target is shown (solid cross). This set of localization errors can be duplicated exactly by a rigid transformation that comprises a clockwise rotation R about the “bull’s eye” that is located just to the right of the nose (c) Point registration has been applied to the image to register the localized positions in (a) with those in (b) The transformation is incorrect by the same rotational error R (dashed arrow) about the bull’s eye, but it achieves an FRE of zero. TRE (arrow) is large, however, because R is large (d) Physical space showing a second set of fiducial localization errors. This set of localization errors can be duplicated by expansion from the target point but cannot be approximated by any rigid transformation (e) Point registration has been applied to the image in (a), and the resulting rigid transformation is perfect. Since the transformation is perfect, TRE is zero, but since no rigid transformation can approximate the localization errors, the fiducial fit is poor, as can be seen from the relatively large size of the individual FREs (distance between circles with dotted and solid outlines, so the root-mean square FRE is large.

Fig. 2

FRE-TRE correlation coefficient CC for ideal weighting (a) FLE < 1 mm with varying number of fiducials (b) Varying FLE with four fiducials. In each case, the solid line is the mean of 15 sets of 10,000 registrations for randomly selected fiducial and target positions (see text). The dashed lines are mean ± one standard deviation. Correlation is insignificant at p < 0.05 for all of (a) and for FLE < 10 mm in (b). Statistically significant but negligible (<0.1) correlation occurs for FLE ≥ 20 mm, showing that, while FRE has some predictive power when higher order terms are included, the power is negligible.

Fig. 3

FRE-TRE statistics for FRE < 1 mm when uniform weighting (i.e., no weighting) is employed (a) Correlation coefficient CC versus number of fiducials. A statistically significant but negligible (<0.1), correlation is apparent that decreases with increasing number of fiducials (b) The chi-square statistic q for dependence versus number of fiducials. The dashed lines are mean ± one standard deviation. The horizontal solid line in (b) is the critical level above which values of q indicate dependence. The slight dependence implied by the small correlation in (a) is not detected by q, whose mean remains below the critical level.

Fig. 4

Computer code to implement the derived formulas. The code is a function written in Matlab (The Mathworks, Natick, MA). Its input and output parameters are described in comments (lines beginning with a percent sign) and in the text.