The Continuous Procrustes Distance between Two Surfaces

The Procrustes distance is used to quantify the similarity or dissimilarity of (3-dimensional) shapes, and extensively used in biological morphometrics. Typically each (normalized) shape is represented by N landmark points, chosen to be homologous (i.e. corresponding to each other), as much as possible, and the Procrustes distance is then computed as infR ∑N j=1 ‖Rxj − xj‖, where the minimization is over all Euclidean transformations, and the correspondences xj ↔ xj are picked in an optimal way. This (discrete) Procrustes distance is easy to compute but has drawbacks – representeding a shape by only a finite number of points, which may fail to capture all the geometric aspects of interest; a need has been expressed for alternatives that are still computationally tractable. We propose in this paper the concept of continuous Procrustes distance, and prove that it provides a true metric for two-dimensional surfaces embedded in three dimensions. The continuous Procrustes distance leads to a hard optimization problem over the group of area-preserving diffeomorphisms. One of the core observations of our paper is that for small continuous Procrustes distances, the global optimum of the Procrustes distance can be uniformly approximated by a conformal map. This observation leads to an efficient algorithm to calculate approximations to this new distance.