  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.
