Comparing Point Clouds
Guillermo Sapiro,
University of Minnesota
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 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 talk. 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
leading to a computational implementation of the framework. The
theoretical and computational results here presented are complemented with
experiments for real three dimensional shapes. Joint work with F. Memoli.