The Chordal Distance Transform of Geometric Loops and its Persistent Homology

TL;DR

This paper introduces the chordal distance transform as an invariant for circle embeddings, analyzing its persistent homology and critical points to understand geometric features in a stable, continuous manner.

Contribution

It defines a new invariant called the chordal distance transform, proves its stability and invariance properties, and characterizes its critical points for generic embeddings.

Findings

The transform is invariant under isometries and parametrizations.

Persistent homology of the transform is finite-dimensional for generic $C^2$ embeddings.

Critical points are finite and non-degenerate for generic $C^2$ and piecewise linear embeddings.

Abstract

We present an isometry and parametrisation invariant of embeddings of $S^1$ into Euclidean space. We do so by representing the distance between pairs of points on the embedded circle as a function on a M\"obius band, the two-point finite subset space of $S^1$. We call this function the chordal distance transform of the embedding. We show that the sublevel set persistent homology of the chordal distance transform satisfies the desired isometry and parametrisation invariance, and is a continuous transform with respect to the Whitney topology on the space of circle embeddings and the bottleneck distance in the space of persistence diagrams. We then considered the generic behaviour of the chordal distance transform for $C^2$ and finite piecewise linear embeddings. In the $C^2$-case, we show that non-boundary critical points of the chordal distance transform are finite and non-degenerate on an open and dense subset of circle embeddings. Consequently, its persistent homology is pointwise finite dimensional for generic $C^2$-embeddings. In the finite piecewise linear case, we also find piecewise-continuous analogues of non-degenerate critical points, and give generic conditions for the homological critical points of the chordal distance transform to be non-degenerate. In order to gain a geometric interpretation of the chordal distance transform and its persistent homology, we give a geometric characterisation of the $C^2$ and finite piecewise linear non-degenerate critical points. Finally, we consider how the chordal distance transform can be generalised to capture geometric features involving $n\geq 2$ points on an embedded shape, as a function on the $n$-point finite subset space.