The Moving-Frame Method for the Iterated-Integrals Signature: Orthogonal Invariants

Abstract

Geometric, robust-to-noise features of curves in Euclidean space are of great interest for various applications such as machine learning and image analysis. We apply 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 $${\mathbb {R}}^d$$ R d from the iterated-integrals 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 $${\mathbb {R}}^d$$ R d under rigid motions (and tree-like extensions) and an explicit method to compare curves up to these transformations.