Metrics and Uniqueness Criteria on the Signatures of Closed Curves

TL;DR

This paper advances the understanding of differential signatures of closed curves by establishing new criteria for their uniqueness and robustness, with implications for applications like handwriting verification and medical imaging.

Contribution

It introduces new criteria for the uniqueness of curve signatures and extends the signature concept to include higher order derivatives, enhancing robustness analysis.

Findings

New criteria for signature-curve correspondence uniqueness.

Extension of signatures to higher derivatives of curvature.

Robustness results linking signature proximity to curve equivalence classes.

Abstract

This paper explores the paradigm of the differential signature introduced in 1996 by Calabi et al. This methodology has vast implications in fields such as computer vision, where these techniques can potentially be used to verify a person's handwriting is consistent with prior documents, or in medical imaging, to name a few examples. Motivated by examples provided by Hickman in 2011 and Musso and Nicolodi in 2009 regarding key failures in this invariant, we provide new criteria for the correspondence between a curve and its signature to be unique in a general setting. To show this result, we introduce new methods regarding the signature, particularly through the lens of differential equations, and the extension of the signature to include information on higher order derivatives of the curvature function corresponding to the curve and desired group action. We additionally show results regarding the robustness of the signature, showing that under a suitable metric on the space of subsets of $\mathbb{R}^n$, if two signatures are sufficiently close then so too will the corresponding equivalence classes of curves they correspond to, given certain conditions on these signatures.