Adaptive Metrics for Adaptive Samples

TL;DR

This paper introduces an adaptive metric based on local feature size for surface reconstruction, establishing a duality with Euclidean space and enabling homology inference from samples.

Contribution

It develops a new adaptive metric framework related to landmark sets and proves a duality with Euclidean space, facilitating topological analysis and homology inference.

Findings

Established a near-duality between adaptive and Euclidean metrics.

Proved topological interleavings between offset spaces.

Provided a computable homology inference scheme.

Abstract

In this paper we consider adaptive sampling's local-feature size, used in surface reconstruction and geometric inference, with respect to an arbitrary landmark set rather than the medial axis and relate it to a path-based adaptive metric on Euclidean space. We prove a near-duality between adaptive samples in the Euclidean metric space and uniform samples in this alternate metric space which results in topological interleavings between the offsets generated by this metric and those generated by an linear approximation of it. After smoothing the distance function associated to the adaptive metric, we apply a result from the theory of critical points of distance functions to the interleaved spaces which yields a computable homology inference scheme assuming one has Hausdorff-close samples of the domain and the landmark set.