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A theoretical and computational framework for isometry invariant recognition of point cloud data. (English) Zbl 1101.53022

Summary: 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.

MSC:

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
54E35 Metric spaces, metrizability
60D05 Geometric probability and stochastic geometry
62P30 Applications of statistics in engineering and industry; control charts
68T10 Pattern recognition, speech recognition

Software:

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