Local Uniqueness of the Circular Integral Invariant

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.


Introduction
In many applications one faces the challenge to model objects, or parts of objects, in a mathematical framework. As an example, one important task is to extract an object from a given data set and manipulate it in a post-processing step in order to obtain further information. Typical applications include medical imaging, object tracking in a sequence of images, but also object recognition, where the postprocessing step consists of the comparison of the extracted object with a database of reference objects. Similarly, such a comparison can be necessary in medical imaging in order to distinguish between healthy and diseased organs. To that end, however, one has to be able to decide whether two given objects are similar or not. This requires a representation of the objects that makes the application of standard similarity measures possible.
Finding a suitable representation of the object of interest, depending on the type of application, is crucial as a first step. For simplicity, it is often assumed that the object is a simply connected bounded domain, allowing for the identification of the domain with its boundary. From a mathematical point of view, this assumption reduces the complexity of the representation. In addition, there exists a larger number of descriptions of boundaries than of domains, and, consequently, more mathematical tools to analyze the geometry of the underlying objects.
In 2D a common approach is to encode the contour of an object by the curvature function of its boundary curve. This approach has, for instance, been used in [8], where the authors set up a shape space of planar curves, where the shapes are implicitly encoded by the curvature function. The main advantage of using a differential invariant -the most prominent representative being the curvatureto represent an object is the well investigated mathematical framework of this type of invariants (see [1,10,12]).
Since all kinds of differential invariants are based on derivatives, they suffer from the shortcoming of being sensitive with respect to small perturbations. To bypass this shortcoming Manay et al. [11] proposed to use integral invariants instead of their differential counterparts (see also [6,7,16]). Integral invariants have similar invariance properties as differential invariants, but have proven to be considerably more robust with respect to noise. Their theory, however, is not that well investigated as opposed to the theory of their differential counterparts.
Beside the classical approach of differential invariants and the novel approach of integral invariants, there exist several other concepts for encoding an object. For instance, in [3] the authors use the zero level set of a harmonic function, which is uniquely determined by prescribing two functions on the boundary of an annulus, to encode the boundary of a 2D object (see [4] for a generalization to compact surfaces in 3D). A similar encoding of the object by a function is given in the article of Sharon and Mumford [15]. Here, the authors first map the 2D object, which is supposed to be a smooth and simply closed curve, to the interior of the unit disc in the complex plane via the Riemann mapping theorem. This conformal mapping is composed with a second one, generated out of the exterior of the original object, and the composition is restricted to the boundary of the unit disc. Thus, the final mapping, which the authors call the fingerprint of the object, is a diffeomorphism from the unit circle onto itself.
One of the challenges in object encoding is the question of uniqueness of the encoding. More precisely, in many applications, e.g. object matching, the correspondence between the object and its encoding should be one-to-one. Thus, a thorough investigation of the operator that maps an object to its encoding is needed. In case of the encoding by a harmonic or conformal mapping -if possible -uniqueness is well known. Also for the encoding of an arc length parameterized curve by its curvature function, it is known that one obtains a one-to-one correspondence between the curve and its encoding (up to rigid body motions). One even has a complete characterization of the set of functions that arise as curvature functions of a class of sufficiently regular curves (see [2]). For integral invariants the situation is different; the cone area invariant, first introduced in [6], is an injective mapping independent of the space dimension, but its application is limited to star-shaped objects. In contrast, for the circular integral invariant, which is the integral invariant most common in use, there exists no proof for the uniqueness conjecture so far.
This article is a contribution towards this goal: We first prove the Fréchet differentiability of the integral invariant in a neighborhood of the circle, seen as a mapping from C k+2 to C k , k ≥ 0. Then we derive a local uniqueness result by proving the local injectivity of the Fréchet differential in case k ≥ 1. The proofs of these results are based on the implicit function theorem on Banach spaces and an application of Riesz-Schauder theory.

Setting
Let Emb be the space of all continuous embeddings from S 1 to R 2 . Then every curve γ ∈ Emb has a unique interior, denoted by Int(γ). Following [6,11], this allows us to introduce the circular integral invariant: Definition 2.1. For given r > 0 we define the circular integral invariant where B r (p) denotes the ball of radius r centered at p ∈ R 2 .
The circular integral invariant behaves well under several group actions: • I r is invariant with respect to Euclidean motions: For A ∈ SE [2] we have • I r is equivariant with respect to reparametrizations: For every homeomorphism Φ : S 1 → S 1 we have • For every scalar t > 0 we have The observations above suggest to consider the integral invariant on the space C of all curves modulo Euclidean motions and reparametrizations. Moreover, we assume as an additional smoothness property that the considered curves are of class C k , k ≥ 1. Then it makes sense to use the following representation of C, as it avoids working with equivalence classes of curves.
For the proof of our main theorem we need the following result from differential geometry concerning the manifold structure of C k : Theorem 2.3. For k ≥ 1 the space C k is a smooth submanifold of the Banach space of all C k -curves from S 1 to R 2 . Its tangent space T γ C k at a curve γ ∈ C k consists of all C k -curves σ with σ(ϕ),γ(ϕ) = C for some C ∈ R, σ(0) = (0, 0) and σ(0), γ(0) = 0 .
Proof. The proof of the submanifold result is similar to [14,Thm. 2.2]. In our case the situation is less complicated, as we only deal with C k -curves instead of Sobolev curves. The constant speed parameterization yields the condition The remaining constraints follow directly from the initial conditions.
For k ≥ 2 we obtain the following characterization of the tangent space T γ C k .
Proof. Theorem 2.3 and the fact that γ(ϕ),γ(ϕ) = 0 imply that there exists a constant C ∈ R such that Using the initial conditions for σ, we obtain the initial conditions for a and b and the value of C = c 2 γḃ (0).
We are now able to formulate the main result of this article: Theorem 2.5. The circular integral invariant I r : C k+2 → C k (S 1 ; R), k ≥ 1, is Fréchet differentiable on a neighborhood U ⊂ C k+2 of the circle of radius R > r/2 and its tangential mapping is injective on this neighborhood. In particular, it follows that I r is locally unique on U in the sense that each γ ∈ U has a neighborhood V γ with I r [γ] = I r [γ] for everyγ ∈ V γ . Remark 1. The condition on r to be smaller than 2R is necessary, because otherwise the circular integral invariant in each point ϕ is constant equal to R 2 π, the area of the circle, and the same holds for any sufficiently small deformation of the circle which preserves the area.

Fréchet Differentiability of the Circular Integral Invariant
In the following, we show that I r is Fréchet differentiable and derive an analytic formula for I r and its derivative I r valid in a neighborhood of the circle. As a first step, we recall the following result on the differentiability of the composition mapping.
is in addition continuous for the operator norm on the image space.
According to [5,13], the mapping Comp : A straightforward calculation shows that a weak C 2 -mapping is Fréchet differentiable, which concludes the proof.
Remark 2. We will need two different types of derivatives for the formulation of our results: First, Fréchet derivatives in the function space C k , and, second, derivatives of functions f ∈ C k (S 1 ) with respect to their argument ϕ ∈ S 1 . In order to highlight this difference, we use the following notation: For a function F : C k → C j , we denote by F : C k → L(C k , C j ) its Fréchet derivative. In contrast, if f ∈ C k , thenḟ denotes its derivative in the parameter space.
In order to make the notation less cumbersome, we omit the argument ϕ in γ(ϕ), σ(ϕ) and similar expressions if the argument is clear from the context.
of the constant speed parameterized circle of radius R > r/2 such that the following hold: (1) For each γ ∈ V and ϕ ∈ S 1 the circle B r γ(ϕ) intersects the curve γ in exactly two points, denoted by γ p γ (ϕ) and γ m γ (ϕ) . Here m γ (ϕ) denotes the previous intersection parameter and p γ (ϕ) the next one (see Figure 1).
are Fréchet differentiable. The derivatives in direction σ ∈ C k+2 (S 1 ; R 2 ) are given by Here Diff k (S 1 ) denotes the group of C k -diffeomorphisms on the unit circle.
Proof. Denote by γ 0 : S 1 → R 2 the constant speed parameterized circle, that is, γ 0 (ϕ) = R cos(ϕ), R sin(ϕ) . Then, for every ϕ ∈ S 1 , the circle B r γ 0 (ϕ) intersects γ 0 precisely at the two points R cos(ϕ ± ϑ), R sin(ϕ ± ϑ) , where Obviously, the mappings p γ and m γ we are searching for satisfy F (γ, p γ ) = 0 and F (γ, m γ ) = 0. In particular, the equation F (γ 0 , d) = 0 has the two solutions p γ0 (ϕ) := ϕ + ϑ and m γ0 (ϕ) := ϕ − ϑ. Now, Lemma 3.1 implies that the mapping Comp : C k+2 (S 1 ; R 2 )×Diff k (S 1 ) → C k (S 1 ; R 2 ), and consequently also F , is Fréchet differentiable. Moreover, it is easy to see that the derivative of F at (γ 0 , p γ0 ) in direction (0, τ ) ∈ C k+2 (S 1 ; R 2 ) × C k (S 1 ; R) is given as which is obviously an isomorphism of C k (S 1 ; R), as the assumption R > r/2 > 0 implies that sin(ϑ) = 0. Thus the implicit function theorem on Banach spaces (see [9, Sec. I.5]) implies the existence of a neighborhood V ⊂ C k+2 (S 1 ; R 2 ) of γ 0 and unique Fréchet differentiable mappings m, p : V → Diff k (S 1 ) satisfying the equations F (γ, p γ ) = 0 = F (γ, m γ ). The formula for the directional derivative of p at γ in direction σ now follows from the fact that which is a simple application of the chain rule. The formula for m γ (σ) can be derived analogously. Theorem 3.3. For each k ≥ 0 there exists a neighborhood V ⊂ C k+2 (S 1 ; R 2 ) of the circle of radius R > r/2 such that the following hold: (1) For each γ ∈ V the circular integral invariant I r [γ] can be written as (2) The circular integral invariant is Fréchet differentiable. Its derivative in direction σ ∈ C k+2 (S 1 ; R 2 ) is given by Proof. Under the given assumptions for R, r and V, Formula (1) can be easily deduced from Figure 1. A term by term investigation of Formula (1), using Lemma 3.1 and Lemma 3.2, shows the Fréchet differentiability of I r on V. Figure 1. Sketch of the derivation of the analytical formula for the circular integral invariant assuming two points of intersection.
To calculate the differential of I r we treat the two terms of Formula (1) separately. For the first term we obtain A simple application of the chain rule yields for the second term Using the formulas for the intersection parameters, we obtain the desired result.
In the special case where γ equals the unit circle the lemma above reduces to: Lemma 3.4. Let γ ∈ C k+2 (S 1 ; R 2 ), k ≥ 0, be the constant speed parameterized unit circle, that is, γ(ϕ) = cos(ϕ), sin(ϕ) , and let r < 2. Then the derivative of is given by The proof of this lemma is postponed to the appendix.

Proof of the Main Theorem
Proof. Without loss of generality we may assume that R = 1. Let V be the neighborhood of the unit circle defined in Theorem 3.3. The formula for I r implies that for every γ ∈ V ⊂ C k+2 (S 1 ; R 2 ) the mapping I r [γ] is bounded as a mapping from C k (S 1 ; R 2 ) to C k (S 1 ; R). Thus, I r [γ] has a unique bounded extension R). Moreover, the mapping J r is continuous with respect to the C k+2 -topology seen as a mapping from V to L C k (S 1 ; R 2 ), C k (S 1 ; R) . In addition, we denote for γ ∈ V byJ r [γ] the restriction of J r [γ] to T γ C k . Denote now by γ 0 : S 1 → R 2 the constant speed parameterized unit circle. Define for given σ ∈ T γ0 C k the function Aσ : S 1 → R by Because γ 0 is a C ∞ -curve, it follows that Aσ is C k . Using Lemma 2.4 it follows that Aσ(0) = 0 and ∂ ϕ (Aσ)(0) = 0. Denoting by C k 0 (S 1 ; R) the space of all C kfunctions a on the circle satisfying a(0) =ȧ(0) = 0, it follows that A is a bounded linear mapping from T γ0 C k to C k 0 (S 1 ; R). In addition, it follows from Lemma 2.4 that A is boundedly invertible with A −1 given by The expression forḃ(0) is due to the periodicity of b, which implies that Therefore A is in fact an isomorphism between T γ0 C k and C k 0 (S 1 ; R). According to Lemma 3.4, the mappingJ r [γ 0 ] evaluated at σ = aγ ⊥ 0 +bγ 0 ∈ T γ0 C k can be written asJ where the operator B : C k (S 1 ; R) → C k (S 1 ; R) is given by and ı is the embedding from C k 0 (S 1 ; R) into C k (S 1 ; R). Lemma 6.1 (see Appendix) implies that the mapping σ → χ [−ϑ,ϑ] * a is compact and thus B is a compact perturbation of the identity. Therefore the Riesz-Schauder theory (see [17, Chap. X.5]) implies that B has a closed range.
Next we compute the kernel of B. To that end we consider the mapping in the Fourier basis. A short calculation shows that in this basis the operator B is the diagonal operator that maps a sequence of (complex) Fourier coefficients (c j ) j∈Z to the sequence (d j c j ) j∈Z , where Because sin(ϑ) = 1 and sin(jϑ) = j sin(ϑ) whenever j ∈ Z \ {−1, 0, 1} (see Lemma 6.2 in the Appendix), it follows that the kernel of B consists of the functions a of the form a(ϕ) = c −1 exp(−iϕ) + c 1 exp(iϕ) for some c −1 , c 1 ∈ C.
In the next step we show that the kernel ofJ Because a(0) = 0, it follows that c −1 + c 1 = 0; becauseȧ(0) = 0, it follows that −c −1 + c 1 = 0. Together, this shows that c −1 = c 1 = 0, implying that the intersection of Ker B with C k 0 (S 1 ; R) is trivial. Since A is an isomorphism this proves the injectivity of We have thus shown thatJ r [γ 0 ] = B • ı • A is strongly closed and injective. Now note that the set of strongly closed and injective, bounded linear functionals between two Banach spaces X and Y is open with respect to the norm topology on L(X, Y ). Because of the continuity of J r and thereforeJ r , this proves the existence of a neighborhood γ 0 ∈ U ⊂ C k+2 such thatJ r [γ] is injective for every γ ∈ U. In particular, this proves the injectivity of I r [γ] for every γ ∈ U.
The local uniqueness follows directly from the Fréchet differentiability of I r and the injectivity of I r on U.

Conclusion
In this article, we have shown a local uniqueness result for the circular integral invariant on a neighborhood U of the circle. Note, however, that the derived result does not prove the local injectivity of the invariant on U. We still believe that also the injectivity holds, but the following difficulties arise using our line of argumentation: For an application of the inverse function theorem, we would require that the mapping I r is Fréchet differentiable and its derivative I r is an isomorphism of the corresponding tangent spaces. Although our results prove that I r can be extended to an isomorphism J r of C k , we cannot use the inverse function theorem, as the mapping I r is not Fréchet differentiable (and not even Gâteaux differentiable) from C k to C k -in fact, our results do not even prove that I r maps C k curves into C k integral invariants. Conversely, seen as a mapping from C k+2 to C k , the tangent mapping, though injective, is not a surjection near the circle. The main problem regarding the differentiability of I r is that our formula involves a composition mapping (see Lemma 3.1 and Theorem 3.3). thank the referee for the careful reading of the article and for pointing out a mistake in the proof of the original main theorem.

Appendix
Lemma 6.1. For every k ≥ 0, the mapping a → K ϑ a := χ [−ϑ,ϑ] * a is compact as a mapping from C k (S 1 ; R) to C k (S 1 ; R).
Because γ is the unit circle, there exists ϑ ∈ S 1 such that Obviously, all assumptions of Lemma 3.3 are satisfied. It remains to calculate all the terms that appear in the expression of the Fréchet derivative in Lemma 3.3 for the special case of the unit circle. In particular, we obtain γ, γ(p γ ) − γ(m γ ) = 2 sin(ϑ) .