Sebastián Torcida

Since first formulated, orthogonal Procrustes analysis has been mainly related to the biometrical study of shapes (shape analysis) based on landmarks. Essentially, this is a method to superimpose two configurations of points by means of a similarity – a combination of a translation term, an isotropic change of scale and an orthogonal transformation – in order to produce an optimal fit, in a least-squares sense. And among them, the problem of finding the best orthogonal transformation has its own particular interest from a biological point of view and it is known as the Procrustes rotation (PR) problem. In morphometrics, the presence of a particular type of symmetry often constrains the search of the PR to be either a determinant 1 or a determinant -1 orthogonal matrix. While the analytical solution to the first problem is well known, a formal proof of the solution to the latter is still required. A common-analytical proof of the solution for both constrained PR problems is given in this work, based on the standard (and unique) singular-value descomposition.