Some Thoughts on Sampling

Topology, Geometry, and Data Analysis Conference at Ohio State University, May 16, 2016

In this talk, I address two new ideas in sampling geometric objects.
The first is a new take on adaptive sampling with respect to the local feature size, i.e., the distance to the medial axis.
We recently proved that such samples can be viewed as uniform samples with respect to an alternative metric on the Euclidean space.
The second is a generalization of Voronoi refinement sampling.
There, one also achieves an adaptive sample while simultaneously "discovering" the underlying sizing function.
We show how to construct such samples that are spaced uniformly with respect to the $k$th nearest neighbor distance function.