The moving frame method for iterated-integrals: orthogonal invariants

TL;DR

This paper develops a method combining the moving frame technique with the log-signature transform to create geometric invariants for curves in Euclidean space, aiding in noise-robust shape analysis.

Contribution

It introduces an algorithmic approach to construct orthogonal invariants from iterated-integral signatures, enabling curve comparison under rigid motions.

Findings

Constructs a set of invariants characterizing curves under orthogonal transformations.

Provides an explicit method for comparing curves up to rigid motions.

Enables robust geometric feature extraction for applications in machine learning and image analysis.

Abstract

Geometric features, robust to noise, of curves in Euclidean space are of great interest for various applications such as machine learning and image analysis. We apply the Fels-Olver's moving frame method (for geometric features) paired with the log-signature transform (for robust features) to construct a set of integral invariants under rigid motions for curves in $\mathbb{R}^d$ from the iterated-integral signature. In particular we show that one can algorithmically construct a set of invariants that characterize the equivalence class of the truncated iterated-integrals signature under orthogonal transformations which yields a characterization of a curve in $\mathbb{R}^d$ under rigid motions (and tree-like extensions) and an explicit method to compare curves up to these transformations.