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Computing with Point Cloud Data

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Statistics and Analysis of Shapes

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Abstract

Point clouds are one of the most primitive and fundamental manifold representations. A popular source of point clouds are three-dimensional shape acquisition devices such as laser range scanners. Another important field where point clouds are found is 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. Under the assumption that the underlying object is a submanifold of Euclidean space, we first discuss how to approximately compute geodesic distances by using only the point cloud by which the object is represented. We give probabilistic error bounds under a random model for the sampling process. Later in the chapter we present a geometric framework for comparing manifolds given by point clouds. 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.

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Author information

Authors and Affiliations

  1. Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN, 55455

    Facundo Mémoli

  2. Instituto de Ingeniería Eléctrica, Universidad de la República, Montevideo, Uruguay

    Facundo Mémoli

  3. Electrical and Computer Engineering and Digital Technology Center, University of Minnesota, Minneapolis, MN, 55455, USA

    Guillermo Sapiro

Authors
  1. Facundo Mémoli
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  2. Guillermo Sapiro
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Editors and Affiliations

  1. Department of Electrical and Computer Engineering, North Carolina State University, Raleigh, NC, 27606-7914, USA

    Hamid Krim

  2. School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA, 30332-0250, USA

    Anthony Yezzi Jr.

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Mémoli, F., Sapiro, G. (2006). Computing with Point Cloud Data. In: Krim, H., Yezzi, A. (eds) Statistics and Analysis of Shapes. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4481-4_8

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