where FLE is the error for each point and N is the number of points in three dimensions (a different adjustment is used for two-dimensional registration).

Using perturbation analysis, Fitzpatrick et al. [1] have shown that the error at some arbitrary target location can be approximated using FLE, N, the distance from the target location to each of the three principal axes of the fiducials (dx!, dy, and dz,), and the RMS distance of the fiducials to each of the three principal axes of the fiducials (f^, jy, and fzi). The equation relating these quantities is



Note that the error should be smallest near the centroid of the fiducials. Errors grow larger with increasing distance in any direction. For any given distance from the centroid, the errors are worst when the displacement corresponds to the axis along which the fiducials are least dispersed from one another and best when it corresponds to the axis along which the fiducials are most dispersed.

This equation provides guidance for selecting fiducial points in a way that will optimize accuracy. First of all, it is always preferable to use points that are widely dispersed as far from their centroid as possible in all directions. This will make the f values large. Secondly, errors will be inversely proportional to the square root of the number of points, so quadrupling the number of fiducials identified will reduce the errors by half, so long as subsequent fiducials are identified with the same accuracy as the previous fiducials. A point of diminishing returns may be reached if FRE begins to increase more rapidly than VN — 2 as a result of adding inaccurately identified points. Finally, if all the individual fiducials can be localized twice as accurately, this will double the accuracy of the registration derived from the fiducials at any location.

The dependence of accuracy on location serves to emphasize the earlier point that registration accuracy should not generally be assumed to be uniform throughout the images. In this particular case, the underlying geometry makes it fairly easy to summarize the spatial dependency of the errors. It is likely that other rigid-body registration methods may also have error isocontours that are approximately ellipsoidal in space, making their spatial dependency amenable to characterization by a reasonably small number of parameters.

While fiducial-based registration has classically depended on anatomic expertise, computerized identification of landmarks based on differential geometry is a relatively new approach based on similar principles. This approach is discussed in the chapter entitled "Landmark-Based Registration Using Features Identified Through Differential Geometry". In this case, error estimation becomes more complicated since information about landmark orientation often supplements the traditional information about landmark location [5].

When registration accuracy is extremely important in a clinical context (e.g., during neurosurgical procedures as discussed in the chapters "Clinical Applications of Image Registration" and "Registration for Image-Guided Surgery"), external fiducials may be attached directly to bone. So long as the structure of interest cannot move with respect to the bone, this should improve registration accuracy over anatomically identified fiducials for three reasons: (1) The fiducials are always located farther from their centroid than the anatomic regions of interest, thereby minimizing the ¿//ratios in the TRE estimate; (2) the external fiducials are deliberately designed to be easily and accurately identified, making it possible to have a large N and a small FRE; (3) the fiducials are often constructed as part of a stereotactic frame that is designed in a way that allows the relationships among the fiducials (e.g., the fact that several are known to lie on the same line or within parallel planes) to further reduce FRE (Fig. 1). Particularly when frames are used instead of simple fiducials, these theoretical advantages translate into a substantial improvement in accuracy [6]. As a result, images obtained with external fiducial frames in place can reasonably be used to derive gold standard transformations for validating other intermodality registration methods [9,11]. In one large validation study based on this methodology, West et al. [9] estimated that the TRE using external fiducials was approximately 0.39 millimeters for registration of MRI to CT images and approximately 1.65 millimeters for MRI to PET registration. These results were then used as gold standards to validate a large number of other independent techniques. This study illustrates the importance of having highly accurate gold standards, since some registration methods produced results consistent with the conclusion they were as accurate as the gold standards used to test them. Additional comments on this comparative study are included in the chapter "Across-Modality Registration Using Intensity-Based Cost Functions."

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