[Paper Review] Rotation invariants of two dimensional curves based on iterated integrals

Joscha Diehl|arXiv (Cornell University)|May 29, 2013

Image Processing and 3D Reconstruction33 references15 citations

TL;DR

This paper introduces a novel class of rotation-invariant features for 2D curves using iterated integrals (the curve's signature), providing a complete algebraic framework to compute invariants up to order six. The method leverages the signature's algebraic structure to derive rotationally invariant quantities that are stable, differentiable, and effective for online character recognition.

ABSTRACT

We introduce a novel class of rotation invariants of two dimensional curves based on iterated integrals. The invariants we present are in some sense complete and we describe an algorithm to calculate them, giving explicit computations up to order six. We present an application to online (stroke-trajectory based) character recognition. This seems to be the first time in the literature that the use of iterated integrals of a curve is proposed for (invariant) feature extraction in machine learning applications.

Motivation & Objective

• To develop a complete and systematic method for generating rotation-invariant features from 2D curves.
• To address the challenge of rotational variability in stroke-based data, such as online handwriting or gesture recognition.
• To provide a mathematically rigorous and computationally feasible framework based on iterated integrals (the signature) for feature extraction.
• To demonstrate the superiority of iterated integrals over curvature-based or Fourier-based methods in handling non-smooth or highly oscillatory signals.
• To establish a theoretical foundation for rotation invariance using representation theory and algebraic independence of signature components.

Proposed method

• Define the signature of a 2D curve as the collection of iterated Riemann–Stieltjes integrals of its components.
• Use the algebraic structure of the tensor algebra and the projection onto homogeneous components to isolate invariants.
• Apply representation theory of the rotation group SO(2) to identify which signature components remain invariant under rotation.
• Derive a criterion: a linear combination of signature components is rotation-invariant if and only if its coefficient vanishes unless the difference between the number of components equal to 2 and 1 is zero.
• Construct an algorithm to compute all rotation invariants up to order six by solving a system of linear equations derived from the rotation invariance condition.
• Verify algebraic independence of the resulting invariants using matrix rank arguments on coefficient vectors of signature monomials.

Experimental results

Research questions

• RQ1 Can a complete and computable set of rotation invariants be derived from the iterated integrals (signature) of a 2D curve?
• RQ2 How can the algebraic structure of the signature be exploited to extract features that are invariant under arbitrary rotations?
• RQ3 What is the relationship between the signature’s homogeneous components and their transformation under rotation?
• RQ4 How do these invariants compare in stability and discriminative power to existing methods like curvature, Fourier, or moment-based features?
• RQ5 Can such invariants be effectively used in real-world applications such as online character recognition?

Key findings

• The paper constructs a complete set of rotation invariants for 2D curves using iterated integrals, with all invariants up to order six explicitly computed.
• The invariants are derived from the signature of the curve, which is known to almost completely characterize the curve, ensuring high discriminative power.
• The method is stable under non-smooth or nowhere-differentiable curves (e.g., Brownian motion), unlike curvature-based methods that require second derivatives.
• The invariants are algebraically independent up to order six, ensuring a rich and non-reduundant feature space.
• The framework is proven to be complete: any rotation-invariant functional on the curve can be expressed as a linear combination of these signature-based invariants.
• An application to online character recognition demonstrates the method’s practical viability, showing clear separation between letters like 'M' and 'W' that are indistinguishable as images but differ in stroke direction.