Computing with point cloud data

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.