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6.2 Previous Works

Other studies of existing registration methods have specified the difference between transformations as the basis for error measures [HDaYC+93][BDC+93][ABKC90][FL93]. The results here are different from, and may not be directly compared to, previous studies of the ILM technique because of general case specificity [NGD+94], the measure of error used and the dependence of the errors on the configuration of the points sets has been explicitly isolated. Hill et al [HHH+92] only studied the error's dependence on configuration by contrasting the resultant mean distance measurements at specific points from points generated dispersely on a grid and bound on a single spherical surface. The use of the mean distance to evaluate error may sufficiently specify the error at specific points, but it inadequately provides academic insight into the factors which confound the translation and rotation separately. As opposed to what had been done here, a difference in transformations was not calculated. The mean distance measure between specific points calculated there did not allow the description of the errors in all of the image space. This may only be done if the knowledge of the translation errors in each dimension and the rotation errors in each plane about a specific center of rotation are specified. Additionally, their use of a simple phantom which attempts to model the grid simulation inadequately provided for which is present in dealing with complex structures. Neelin et al [NCH+92] provided much of the insight in this study by specifying the standard deviation as the measure of the distributions of the translation and rotation errors, but their simulations involved variable configurations from sample to sample and therefore produce rotation errors which are approximately equal in all planes because they have been averaged. This is contrary to what has been discovered in the present work because this study has allowed the evaluation of the radial dependence of the rotation error by using the same dimensions for the configurations. It has been shown in the last chapter that the rotation errors decrease as the radial dispersion of the points from the centroid increase. In Neelin's study, the translation errors which are constant across dimensions may not be expected in SPECT/MR registrations because of the anisotropy of the resolution of SPECT.

Similar to what has been done here, Turkington et al [TJP+93] used external fiducials on the Hoffman brain phantom as a basis for error measurements from exact registrations, but they measured error for a single registration as the distance between the fiducial locations averaged over the number of fiducials used. Their measure was an average over space where the error is generally radially dependent and not over N registrations. In general, a repetition of any registration technique will effect a distribution of translation and rotation errors because of finite spatial resolution. and from the finite may manifest itself from the variability of surface or volume mask generation with voxel threshold or the existence of multiple solutions in global search methods.

In Chen's investigation [Che93] of a landmark based registration technique involving outlier stripping and parameter accumulation algorithms, he defined an equivalent error angle, f, in terms of the Frobenius norm of the difference between real and estimated rotation matrices. This is essentially a measure of the mean error angle in 3-space and therefore does not describe the distribution of the dimensionally dependent rotation errors. It is sufficient though in the context of that study where f was evaluated in terms of its variation with a signal to noise ratio measurement in simulations on a unit sphere.

From the non-homogeneous results of the error distributions which were found in the previous chapter, it is recommended that, regardless of the registration technique, certain factors be provided when specifying registration errors. Briefly:

The last point is also necessary because it allows the determination of the approximate errors in all space if the rotation errors are given. Recall though that the external fiduciaries were provided on the brain phantom to explicitly provide test points for exact registration, minute imperfect marker location gave non-zero mean translations and rotations (see table 5.2). It might be incorrectly thought that since and are specified, any set of test points may in fact be used to provide the basis for true registrations - as long as the exact same test points are used for N registrations of dimensionally similar configurations. This is not necessarily true because anatomical homology is necessary as a basis for the error measurement. At the least, real scan results must be compared to point simulation results. That the real scan results here were within about 0.3 mm of the point simulation results is witness to the degree of error that may be expected in error measurements based on supposed true registrations with non-zero anatomical homology (compare table 5.2 with figure 5.6 ). There is another constraint which validates the use of the fiducials in this study. The centroid of the test points should be approximately coincident with the centroid of the clinical registration points. This allows: (a) the measure of from the same centroid, and (b) direct calibration with the point simulation studies.



Next: 7 Conclusion Up: 6 Discussion Previous: 6.1 Introduction


lukban@pet.mni.mcgill.ca
Wed Jan 18 14:28:16 EST 1995