Continuous procrustes distance between two surfaces

The Procrustes distance is used to quantify the similarity or dissimilarity of (three-dimensional) shapes and extensively used in biological morphometrics. Typically each (normalized) shape is represented by N landmark points, chosen to be homologous, as far as possible, and the Procrustes distance is then computed as $∈f_R∑_j=1^N |Rx_j-x'_j|^2$, where the minimization is over all euclidean transformations, and the correspondences $x_j ≤ftrightarrow x'_j$ 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. We also propose an efficient algorithm to calculate approximations to this new distance. © 2012 Wiley Periodicals, Inc.