On Isometries And Equivalences Between Point Configurations: Labelled and Unlabeled Data

Neophytos Charalambides, Steven B. Damelin, Michael Werman

Published 2017-05-17Version 1

This paper deals with the Orthogonal Procrustes Problem in R^D by considering either two distinct point configurations or the distribution of distances of two point configurations. The objective is to align two distinct point configurations by first finding a correspondence between the points and then constructing the map which aligns the configurations.This idea is also extended to epsilon-distorted diffeomorphisms which were introduced in [30] by Fefferman and Damelin. Examples are given to show when distributions of distances do not allow alignment if the distributions match, and when we can partition our configurations into polygons in order to construct the maximum possible correspondences between the configurations, considering their areas. Included is also a brief overview of reconstructing configurations, given their distance distributions. Finally, some algorithms are described for configurations with matching points along with examples, where we find a permutation which will give us a relabeling, and also the affine transformation which aligns the configurations.