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 $\inf_{R}\sum_{j=1}^N \|Rx_j-x'_j\|^2$ , where the minimization is over all euclidean transformations, and the correspondences $x_j \leftrightarrow x'_j$ 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.