  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 far as possible, and
the Procrustes distance is then computed as the infimum, over all Euclidean
transformations R, of the sum of the squared Euclidean distances between R x_j
and x'_j, and the correspondences x_j <-> x'_j are picked in an optimal way.
The discrete Procrustes distance has the drawback that each shape is
represented by only a finite number of points, which may not capture all the
geometric aspects of interest; a need has been expressed for alternatives that
are still easy to compute. 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.
