Foundations of Computational Mathematics

, Volume 5, Issue 3, pp 313–347 | Cite as

A Theoretical and Computational Framework for Isometry Invariant Recognition of Point Cloud Data

  • Facundo Mémoli
  • Guillermo Sapiro


Point clouds are one of the most primitive and fundamental manifold representations. Popular sources of point clouds are three-dimensional shape acquisition devices such as laser range scanners. Another important field where point clouds are found is in the representation of high-dimensional manifolds by samples. With the increasing popularity and very broad applications of this source of data, it is natural and important to work directly with this representation, without having to go through the intermediate and sometimes impossible and distorting steps of surface reconstruction. A geometric framework for comparing manifolds given by point clouds is presented in this paper. The underlying theory is based on Gromov-Hausdorff distances, leading to isometry invariant and completely geometric comparisons. This theory is embedded in a probabilistic setting as derived from random sampling of manifolds, and then combined with results on matrices of pairwise geodesic distances to lead to a computational implementation of the framework. The theoretical and computational results presented here are complemented with experiments for real three-dimensional shapes.

Point clouds Gromov-Hausdorff distance Shape comparison High-dimensional data Manifolds Isometrics 


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Copyright information

© Society for the Foundations of Computational Mathematics 2005

Authors and Affiliations

  1. 1.Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455, USA and Instituto de Ingenieria Electrica, Universidad de la Republica, Montevideo Uruguay
  2. 2.Electrical and Computer Engineering and Digital Technology Center, University of Minnesota, Minneapolis, MN 55455USA

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