Local Uniqueness of the Circular Integral Invariant

TL;DR

This paper proves that the circular integral invariant uniquely represents curves near a circle and provides stability estimates, addressing a key gap in understanding its injectivity and robustness.

Contribution

It establishes local injectivity and stability of the circular integral invariant for curves close to a circle, using advanced functional analysis tools.

Findings

Proves injectivity of the circular integral invariant near a circle.

Provides a stability estimate for the invariant in this neighborhood.

Uses Riesz-Schauder theory and the implicit function theorem in the proof.

Abstract

This article is concerned with the representation of curves by means of integral invariants. In contrast to the classical differential invariants they have the advantage of being less sensitive with respect to noise. The integral invariant most common in use is the circular integral invariant. A major drawback of this curve descriptor, however, is the absence of any uniqueness result for this representation. This article serves as a contribution towards closing this gap by showing that the circular integral invariant is injective in a neighbourhood of the circle. In addition, we provide a stability estimate valid on this neighbourhood. The proof is an application of Riesz-Schauder theory and the implicit function theorem in a Banach space setting.