Analytical solution for both orthogonal procrustes rotation problems with determinant constraints
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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.
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