Continuous Procrustes distance between two surfaces. Issue 6 (15th February 2013) - Al‐Aifari, Reema; Daubechies, Ingrid; Lipman, Yaron
Abstract 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 <![CDATA[$\inf_{R}\sum_{j=1}^N \|Rx_j-x'_j\|^2$]]> <inline-graphic xlink:href="ark:/27927/pgg1x52v1wg" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />, where the minimization is over all euclidean transformations, and the correspondences <![CDATA[$x_j \leftrightarrow x'_j$]]> <inline-graphic xlink:href="ark:/27927/pgg1x52v1qq" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> 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.