Recovery of the metric structure of a pattern of points using minimal information

Abstract

A new method is proposed in order to reconstruct the geometrical configuration of a large points set using minimal information. The paper develops algorithms based on graph and kinematics theories to determine the minimum number of distances, needed to uniquely represent n points in d-dimensional Euclidean space. Therefore, it is found that this theoretical minimum is d(n-2)+1 interpoint distances. The method is evaluated, on the basis of basic parameters, by means of Monte Carlo simulation using genetic algorithms for better optimization procedures. This evaluation takes into account the real case where the metric informations are interpoint dissimilarities instead of exact Euclidean distances. Two applications on real data successfully illustrate the efficiency of the method. Finally, on the basis of Monte Carlo results, the authors provide some practical recommendations to experimenters who wish to use the method in order to scale a many-objects set.