Department of Electrical Engineering and Computer Science, Vanderbilt University, Nashville, TN 37235, USA.
IEEE Trans Med Imaging. 2011 Mar;30(3):679-93. doi: 10.1109/TMI.2010.2091513. Epub 2010 Nov 11.
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.
提出了一种刚性点配准中误差的一阶分析的通用方法，该方法可以容纳可能不均匀（随点变化）和各向异性（随方向变化）的基准定位误差（FLE），并且还可以容纳可能不均匀和各向异性的任意加权。根据 FLE 的协方差，推导出目标配准误差（TRE）和加权基准配准误差（FRE）的协方差，最终得到一个简单的实现，包含了所有加权和各向异性的组合。此外，还表明对于理想加权，其中每个基准的加权矩阵等于其两个空间 FLE 的交叉协方差的平方根的倒数，FRE 和 TRE 的波动是相互独立的。这些结果通过与先前发表的表达式和模拟进行比较得到验证。此外，还提出了针对随机生成的基准位置和 FLE 的模拟结果，表明在理想和均匀加权（即无加权）的精确情况下，相关性可以忽略不计（相关系数 < 0.1），后者在商业手术引导系统中使用。从这些结果中我们得出结论，对于这些加权方案，虽然存在将 FRE 的协方差与 TRE 的协方差相关联的有效表达式，但对于给定的配准，没有任何衡量基准拟合度的指标可以首先提供关于 TRE 从其期望值的波动的任何信息，也没有任何指标可以在精确情况下提供有用的信息。因此，作为配准精度的估计量，这些措施都应该由引导系统的供应商和使用它们的从业者谨慎对待。
