Continuous Procrustes distance between two surfaces. Issue 6 (15th February 2013)

• Record Type: Journal Article
• Authors: Al‐Aifari, Reema; Daubechies, Ingrid; Lipman, Yaron
• Is Part Of: Communications on pure and applied mathematics. Volume 66:Issue 6(2013:Jun.)
• Page Start: 934
• Page End: 964
• Publication Date: 2013-02-15
• Subjects: Mathematics -- Periodicals; Mechanics -- Periodicals; Mathématiques -- Périodiques; Mécanique -- Périodiques; 510.5
• DOI: 10.1002/cpa.21444
• Languages: English
• ISSNs: 0010-3640

Abstract

<abstract abstract-type="main" xml:lang="en"> <title>Abstract</title> <p>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 <italic>N</italic> landmark points, chosen to be homologous, as far as possible, and the Procrustes distance is then computed as <tex-math notation="tex"><![CDATA[$\inf_{R}\sum_{j=1}^N \|Rx_j-x'_j\|^2$]]></tex-math><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 <tex-math notation="tex"><![CDATA[$x_j \leftrightarrow x'_j$]]></tex-math><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.</p> <p>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.</p> </abstract>