Adaptive Metrics for Adaptive Samples

Nicholas J. Cavanna,Donald R. Sheehy

doi:10.3390/a13080200

Abstract

We generalize the local-feature size definition of adaptive sampling used in surface reconstruction to relate it to an alternative metric on Euclidean space. In the new metric, adaptive samples become uniform samples, making it simpler both to give adaptive sampling versions of homological inference results and to prove topological guarantees using the critical points theory of distance functions. This ultimately leads to an algorithm for homology inference from samples whose spacing depends on their distance to a discrete representation of the complement space.

Highlights

There is one aspect of these theories that directly contradicts this trend: many surface reconstruction algorithms are able to work with an adaptive sample, while most homology inference algorithms require a uniform sample
We present an alternative metric in Euclidean space that connects adaptive sampling and uniform sampling
We show how to apply classical results from the critical point theory of distance functions to infer topological properties of the underlying space from such samples

Summary

From Points to Topology

Both surface reconstruction and homology inference are algorithmic problems that take points as input and produce a topological representation of the underlying space from which the points were drawn. Cazals et al [1] introduced the conformal alpha shape filtration as a way to build triangulations at different scales that have local connectivity related to the local feature size Their stated goal was surface reconstruction, the work employed many of the methods of homology inference. Gave a more direct generalization of methods in surface reconstruction with adaptive samples to homology inference, achieving some guarantees for smooth manifolds assuming both upper and lower bounds on the density. L for the approximation to the complement space We extend these works by providing guaranteed homology inference for a much more general class of samples and spaces; we do not require the space to be a manifold or the sample to adapt to the medial axis

From Surface Reconstruction to Homology Inference

Overview

Methods

Adaptive Sampling

Interleaving

Approximating X with X

Approximating the Induced Metric

Approximating L with b

Putting It All Together

Smooth Adaptive Distance and Homology Inference

Critical Points of Distance Functions

Smoothing the Metric