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Continuous Procrustes Distance Between Two Surfaces
Author(s) -
AlAifari Reema,
Daubechies Ingrid,
Lipman Yaron
Publication year - 2013
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21444
Subject(s) - procrustes analysis , mathematics , metric (unit) , similarity (geometry) , euclidean distance , euclidean geometry , distance matrices in phylogeny , algorithm , combinatorics , geometry , artificial intelligence , computer science , image (mathematics) , operations management , economics
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.