3.3.1 Analytical Basis

Procrustes analysis is a technique used for assessing the goodness of fit between two points sets with a preassigned correspondence between them (i.e. the images from two different modalities are of the brain of the same person) after construction of an optimal matching with 9 degrees of freedom. The ILM technique's use of Procrustes analysis is not so much concerned with the measure of the goodness of fit as given by the Procrustes statistic or residual, in effect simply a measure of how well the configurations to be matched correspond, as it is interested with the construction of the optimal matching. In fact, the use of the Procrustes statistic as a measure of the goodness of fit does not necessarily yield a quantity that is useful for measuring the error in the resulting brain volume registration because it is essentially only a measure of the degree of configuration correspondence and not necessarily of their homology. A null Procrustes statistic may be produced even if points were incorrectly chosen by for example (please see chapter 4, figure 4.1). The development of the theory presented here closely follows the review given by Sibson [Sib78] and Golub &Van Loan [GL83].

Given two mp data matrices, where m = number of points and p = number of dimensions, consisting of point sets A and B from the MR and SPECT which are selected to be homologous according to the description in the previous section, the construction of the optimal matching under translation is first required before the computation of the matching under rotation and reflection. This is done by first finding the centroids of the points sets. Since the configurations A and B consist of m point coordinates in 3-D space (p = 3), the centroids or centers of mass are found by simple position averaging with the assumption of unit mass densities. For a translation of B into the configuration of A, the difference between the calculated centers of mass of each of the configurations is found, , and added to B. The origin of A and the shifted B are standardized to the centroid of A to produce the centered points sets and . The optimal matching under rotation or reflection (simply rotations of about coordinate axes for images) may be calculated for and by finding the pp orthogonal matrix Q such that

subject to , from which

It is interesting to note that the orthogonality of the transformation Q implies that the operation is the same as a change in the basis vectors of for the best fit rotation into . This may appear to be similar to a principal axes transformation but all that can be said is that if and can be well matched, and if the principal variances are well distinguished, then the principal axes will themselves correspond reasonably closely after fitting under rotation/reflection [Sib79]. Inspection reveals that the minimization problem is reversed into a problem of maximization, where the optimum Q in may be found by calculating the singular value decomposition (SVD) of . The SVD is based on one of the fundamental theorems of linear algebra which states that any mp matrix K whose number of rows m is greater than or equal to its number of columns p, can be written as the product of an mp column-orthogonal matrix U, a pp diagonal matrix with positive or zero elements and the transpose of a pp orthogonal matrix V such that [PFT88]. Application of the SVD to yields unitary matrices U and V such that

If we let and substitute so that

then it is made more clear that the upper bound is easily obtained when , or when . This is the rotation necessary to minimize the squared distances between the standardized point configurations and . If the error in the calibration of pixel dimensions is large, then the optional global dilation may be computed as

The resulting value of is the Procrustes statistic. It may be standardized on a per-point basis by normalizing by m, or more robustly by

to help account for the non-linearity of the sum of squares decomposition introduced in the rotations about the standardized centroid at the origin coordinate system. It has been shown theoretically [Sib79] and experimentally [EMT+91][NCH+92] from simulation studies that G is approximately constant when the perturbations or homology error between the point sets A and B is fairly small, regardless of m or configuration of A (or, symmetrically, B). Although the homology error is generally different for each homologous point pair chosen, it may easily be shown that for a constant homology error there is a clear dependence of G(m), as well as a direct relationship between the magnitudes of G and the homology error [EMT+91][NCH+92].