5.3.1 Point Simulations

From the point simulations within spheres and shells, for different (FWHM) is given in figure 5.4, 5.5, 5.6 and 5.7 for and r ranging from 15 to 90 mm. As would be expected, it shows that the translation errors are independent of configuration dimension for the range of homology errors that may be expected clinically because the translation transformations necessary for registration are simply calculated as the difference between centroid positions (refer to chapter on Procrustes analysis). The results given are for the x dimension, but it sufficiently describes the behavior in all three dimensions through symmetry arguments because the simulations were performed within spherical geometries.

The same behavior is obtained from simulations within the shells. It is of interest to point out that the results from configurations within shells or spheres are essentially the same. In general, the delineation of the data's dependence on radius is simply less scattered for the simulations within shells. This may be seen for in figures 5.9, 5.10, 5.11 and 5.12 for both spheres and shells. The curve gives a Simplex fit to (where k is a constant that is proportional to and inversely proportional to plus a constant). The bottom plot in figure 5.8 give the values for the constant k resulting from the fit. The change in rotation error with radius for spheres and shells is shown to be the same from the correspondence of the k values resulting from the fits. Although the development of an exact functional dependence is not the purpose here, this correspondence clearly demonstrates the decrease of with increase of r. The deviation of the data points to the curve may be seen to be generally greater for spheres than for shells because the shells constrain the radial dependence more tightly than the spheres. This corresponds to the standard deviation of the difference between the fits and the simulation data being larger for the spheres than for the shells. This may be seen in the upper plot of figure 5.8. For small perturbations where mm, both and are fairly independent of point set configuration. This has previously been shown theoretically in terms of the Procrustes statistic [Sib79].

The dependence of the errors on the number of points used is well established. Studies show that they decrease by [NCH+92][HHH+92]. It is clear from the figures that the uncertainty in the rotation error depends very greatly on the radial dispersion of the points from their centroid. is greater for smaller radii. This is due to the larger lever introduced by the same homology error when the points are nearer the centroid (see figure 5.13). The independence of on the boundary used to constrain the radial dependence, either shell or sphere, is of interest. This suggests that there is no significant advantage to limiting the selection of point pairs to shell-like configurations, corresponding to points selected on the surface of the brain (refer to constants, k, in the bottom part of figure 5.8). This is contrary to what has been suggested elsewhere [HHH+92]. The results from points simulations within spheres and shells indicate that the homology error does affect and dramatically. Changes in both and are most dependent on the range of the expected variation of as well as by the differences and possible range of r and m.