3.4 Practical Implementation

In a more practical form, the computational procedure of Procrustes analysis may be summarized in three steps where, given point sets A and B defined as above:

1. Configurations A &B are centered (B translated into A for a registration of B into A) to produce and .
2. is rotated into by (from SVD of ).
3. The global scale is optionally applied.

One must bear in mind that the resulting transformation results in an analytical least squares fit of the two configurations exactly like the resultant fit produced with the more familiar linear least squares techniques for a set of data points with linear tendency. In the analytical fitting of the function to a set of points by the method of maximum likelihood, the optimal transformation of a straight line along the independent axis, discretely evaluated at points which correspond to the raw data's independent variables, which minimizes the discrepancies between the data and the function is similarly found by solving for the coefficients and . The goodness of fit of the straight line function to the raw data may be evaluated by inspecting the residual or root mean square (rms) distances between corresponding points in analogy to the Procrustes statistic. The difference between the two is that an rms difference of zero in the line fitting uniquely implies that there is an exact registration of the two, whereas it merely implies that the configurations of A and B are exactly the same when using the Procrustes algorithm but the anatomical homology may yet be imperfect. For example, if transaxial images of B are displayed with a left-right orientation opposite to similar images of A and homologous points are chosen with exactly matching configurations, but where left and right handedness is wrongly assumed to be correct, then use of the Procrustes algorithm will yield a zero residual but results in an incorrectly oriented registration (by angle about anterior-posterior axes in the transaxial planes). Evaluation of the residual is thus inadequate for the validation of the accuracy of the registration because the exact correspondence of configurations A and B does not necessarily imply exact homology and therefore exact registration. These considerations are presented in the next chapter.