Shape Analysis of Euclidean Curves under Frenet-Serret Framework

TL;DR

This paper introduces a Riemannian geometric framework based on generalized curvatures for analyzing Euclidean curves, improving shape comparison and registration by capturing all geometric features.

Contribution

It extends the square root-velocity transform to all dimensions using Frenet-Serret curvatures, enabling more accurate shape analysis that includes torsion and curvature.

Findings

The proposed geometry avoids artefacts common in first-order methods.

Simulated data supports the effectiveness of the approach.

Application to sign language trajectories demonstrates practical relevance.

Abstract

Geometric frameworks for analyzing curves are common in applications as they focus on invariant features and provide visually satisfying solutions to standard problems such as computing invariant distances, averaging curves, or registering curves. We show that for any smooth curve in R^d, d>1, the generalized curvatures associated with the Frenet-Serret equation can be used to define a Riemannian geometry that takes into account all the geometric features of the shape. This geometry is based on a Square Root Curvature Transform that extends the square root-velocity transform for Euclidean curves (in any dimensions) and provides likely geodesics that avoid artefacts encountered by representations using only first-order geometric information. Our analysis is supported by simulated data and is especially relevant for analyzing human motions. We consider trajectories acquired from sign language, and show the interest of considering curvature and also torsion in their analysis, both being physically meaningful.