Julien Dardennes

Jean Gutt
email: jean.gutt@math.univ-toulouse.fr

Jun Zhang
email: jun.zhang.3@umontreal.ca

SYMPLECTIC NON-CONVEXITY OF TORIC DOMAINS

 via Ruelle invariant and systolic ratio of the boundary of star-shaped toric domains, we provide elementary operations on domains that can kill the symplectic convexity. These operations only result in small perturbations in terms of domains' volume. Moreover, one of the operations is a systematic way to produce examples of dynamically convex but not symplectically convex toric domains. Finally, we are able to provide concrete bounds for the constants that appear in Chaidez-Edtmair's criterion.

Introduction

A toric domain X Ω in R 4 (≃ C 2 ) is a symplectic manifold with boundary, defined via the moment map µ : C 2 R 2 ≥0 as follows, [START_REF] Abbondandolo | Sharp systolic inequalities for Reeb flows on the three-sphere[END_REF] X Ω := µ -1 (Ω) where µ(z 1 , z 2 ) := (π|z

1 | 2 , π|z 2 | 2 )
and Ω ⊂ R 2 ≥0 is a domain, called moment image, containing a neighborhood of the origin of R 2 . A toric domain X Ω is called star-shaped if the radial vector field of R 4 is transverse to the boundary ∂X Ω and intersects ∂X Ω only once. This is equivalent to the condition, in terms of the moment image, that all the rays in R 2 ≥0 starting from the origin are transverse to ∂Ω. The following Figure 1 provides an example (of the moment image Ω) of a toric star-shaped domain X Ω . In this paper, we always assume Ω (1) Moment image Ω is the region enclosed by bold black curve.

(2) X Ω = µ -1

(Ω) and it is star-shaped by the transversality from dotted rays. that ∂Ω is smooth. It is well-known that for a (toric) star-shaped domain X(= X Ω ) of C 2 , its boundary ∂X is a contact manifold with the contact structure given by the kernel of the contact 1-form λ = λ std | ∂X , where

λ std = 1 2 x i dy i -y i dx i .
The study of the convexity of a symplectic toric domain X Ω has a long history. As (geometric) convexity is not invariant under symplectomorphisms of R 4 , various substitutive notions, of symplectic nature, have been introduced. The most fruitful one is dynamical convexity, introduced in [START_REF] Hofer | The dynamics on three-dimensional strictly convex energy surfaces[END_REF], which requires the minimal Conley-Zehnder index of the Reeb orbits on ∂X Ω has to be sufficiently positive. This serves as a dynamical hypothesis for many interesting problems (see [START_REF] Abreu | Dynamical convexity and elliptic periodic orbits for Reeb flows[END_REF][START_REF] Schneider | Global surfaces of section for dynamically convex Reeb flows on lens spaces[END_REF][START_REF] Viktor | Dynamical convexity and closed orbits on symmetric spheres[END_REF][START_REF] Hryniewicz | Global surfaces of section with positive genus for dynamically convex Reeb flows[END_REF]). In this paper, we are mainly interested in the convexity of the toric domain up to symplectomorphism, for brevity, called symplectic convexity. A toric domain X Ω is called symplectically convex if ϕ(X Ω ) is (geometrically) convex for some ϕ ∈ Symp(R 4 ). A fundamental result in [START_REF] Hofer | The dynamics on three-dimensional strictly convex energy surfaces[END_REF] says that any symplectically convex domain is dynamically convex. Interestingly, it has recently been confirmed that [START_REF] Abreu | Dynamical convexity and elliptic periodic orbits for Reeb flows[END_REF] dynamical convexity ̸ = symplectic convexity on S 3 . This is the main result, Theorem 1.8, of ( [START_REF]3D convex contact forms and the Ruelle invariant[END_REF]).

Remark 1.1. Following [START_REF] Hofer | The dynamics on three-dimensional strictly convex energy surfaces[END_REF] rigorously, up to symplectomorphism, only a strictly convex and non-degenerate domain is dynamically convex. Since both strict convexity and non-degeneracy can be obtained by generic perturbations, we will always consider an arbitrary convex domain as a natural extension from strictly convex and nondegenerate domains, more precisely, a geometric limit of a sequence of such domains. In this way, symplectic convexity as above is defined over any convex domain

Verifying that a given toric domain X Ω is symplectically convex is difficult since it requires one to provide an explicit ϕ ∈ Symp(R 4 ) and to confirm that the image ϕ(X Ω ) is indeed convex. In this paper, we work towards a somewhat opposite direction: try to carry out a minimal amount of operations on a toric domain X so that the resulting domain is not symplectically convex. To this end, a good characterization of a necessary condition for a toric domain to be symplectically convex is needed. We use the following one, which applies to any star-shaped domain, not necessarily toric. Proposition 1.1 (Proposition 3.1 in [START_REF]3D convex contact forms and the Ruelle invariant[END_REF]). Given a star-shaped domain X, denote Y := ∂X. If X is symplectically convex, then

(3) c ≤ ru(Y, λ) • sys(Y, λ) 1 2 ≤ C
where c and C are positive constants, independent of the input (Y, λ). The contact form λ is the restriction of the standard contact form on the boundary:

λ = λ std | Y .
The quantities ru(Y, λ) and sys(Y, λ) in Proposition 1.1 are the Ruelle ratio and the systolic ratio of (Y, λ) respectively. They are defined by ru(Y, λ) := Ru(Y, λ)

Vol(Y, λ) 1 2
, and sys(Y, λ) = (min period of a Reeb orbit) 2 Vol(Y, λ) ,

where Ru(Y, λ) is the Ruelle invariant of the contact manifold (Y, λ). For more details (and a complete definition), we refer to Section 2.

It is important to emphasize that even though the ratios ru and sys are defined only on contact manifolds (here, the boundary of a star-shaped domain), they are in fact invariant under symplectomorphisms of the bounding domains. Indeed, by Lemma 3.5 in [START_REF]3D convex contact forms and the Ruelle invariant[END_REF], it is readily verified that if star-shaped domains X and X ′ are symplectomorphic, then ru(∂X, λ) = ru(∂X ′ , λ) and sys(∂, λ) = sys(X ′ , λ). We can, and we shall, therefore talk about the Ruelle ratio and the systolic ratio of a toric domain, that is, ru (X Ω ) := ru (∂X Ω , λ) and sys (X Ω ) := sys (∂X Ω , λ) .

The applications of the Ruelle invariant, originally defined in [START_REF] Ruelle | Rotation numbers for diffeomorphisms and flows[END_REF], in symplectic and contact geometry were initiated by [START_REF] Hutchings | ECH capacities and the Ruelle invariant[END_REF].

Remark 1.2. It is worth mentioning that for star-shaped domains, even though Ru(X), under Definition 2.1, is invariant under symplectomorphisms, it cannot provide symplectic embedding obstructions in general. For instance, the ellipsoid E [START_REF] Abbondandolo | Sharp systolic inequalities for Reeb flows on the three-sphere[END_REF][START_REF]3D convex contact forms and the Ruelle invariant[END_REF] symplectically embeds into the ball B 4 (2), but by Proposition 1.4 we have Ru(E(1, 4)) = 5 which is bigger than Ru(B 4 (2)) = 4. In fact, Ruelle invariant behaves fundamentally differently from symplectic capacities (cf. [START_REF] Hutchings | ECH capacities and the Ruelle invariant[END_REF]Corollary 1.13]).

The main result of this paper is the following: Theorem 1.3. Given any star-shaped toric domain X Ω in R 4 , there exist small perturbations of X Ω , in terms of the volume, such that the resulting domains X Ω are still star-shaped but the product ru(X Ω ) • sys(X Ω ) 1 2 can be arbitrarily small or arbitrarily large. In particular, the resulting domains X Ω are not symplectically convex.

The notation X Ω indicates that the perturbations of X Ω promised in Theorem 1.3 can be carried on directly on the moment image Ω. Indeed, the proof of Theorem 1.3 in Section 3 provides two explicit constructions of such perturbations -strangulation operation and strain operation. A schematic picture below, Figure 2, illustrate these two operations on the level of moment images. More explicitly, given a moment image Ω of a toric starshaped domain X Ω , the strangulation operation removes a small part (blue shaded region) along a ray from Ω, while the strain operation adds a thin triangle (red shaded region) to Ω. Refined pictures of these operations with precise parameters will show up in Figure 3 and Figure 4. The resulting new toric domain is denoted by X Ω with its moment image Ω. We will apply Proposition 1.1 to deduce that X Ω is not symplectically convex by deforming Ω until passing below the lower bound c or over the upper bound C, even though we don't know explicitly how big c and C are in general. As expected, there will be non-trivial estimations of the ratios ru X Ω and sys X Ω .

This is based on results from Section 2 and Section 3. Explicitly, for the Ruelle ratio ru (in fact, the Ruelle invariant Ru), we have the following computational result. Its proof occupies Section 2.

Proposition 1.4. Let X Ω be any 4-dimensional toric star-shaped domain. Then its Ruelle invariant is given by

Ru(X Ω ) := Ru(∂X, λ std | ∂X ) = a(Ω) + b(Ω)
where a(Ω) and b(Ω) are the w 1 -intercept and w 2 -intercept, respectively, of the moment image Ω in R 2 ≥0 , in (w 1 , w 2 )-coordinate. Remark 1.5. Proposition 1.4 generalizes [START_REF] Hutchings | ECH capacities and the Ruelle invariant[END_REF]Proposition 1.11] since there is no hypothesis that the profile curve, as the boundary ∂Ω ∩ R 2 >0 , has slopes everywhere negative (cf. [10, footnote on page 6]).

On the other hand, for the systolic ratio, or more generally, the (minimal) action of the Reeb orbits, the discussion at the beginning of Section 3 provides a general formula (15), which is a well-known result in symplectic toric geometry. Then one verifies that the strangulation operation yields only the systolic ratio to be arbitrarily small, while the strain operation yields only the Ruelle ratio to be arbitrarily large.

Remark 1.6 (Related to capacity). The operations in Theorem 1.3 do not always result in small perturbations in terms of symplectic capacities. It is possible to verify that X Ω from method two -strain in subsection 3.2 -satisfies X Ω X Ω hX Ω for a rescaling h close to 1, via symplectic folding technique (see [START_REF] Schlenk | Embedding problems in symplectic geometry[END_REF]), where represents a symplectic embedding. However, the method one -strangulation in subsection 3.1 -is believed to change the symplectic capacities dramatically. How to describe such changes quantitatively in terms of capacities will be explored in further work.

If one applies results in [START_REF] Hofer | The dynamics on three-dimensional strictly convex energy surfaces[END_REF], symplectic convexity can be killed by breaking the dynamical convexity. In fact, this can be achieved as well by a small perturbation also in the sense of the volume. More explicitly, one simply modifies the profile curve of Ω in R 2 ≥0 near the intersection points on the two axes so that the Conley-Zehnder indices of the Reeb orbits corresponding to these intersection points are always negative. We leave the details to interested readers.

Here, we emphasize that our operations in Theorem 1.3 are fundamentally different. In particular, method one -strangulation in subsection 3.1 -can be distinguished with the operation elaborated above via symplectic capacities (for instance, the minimal action), even though it always goes beyond the category of dynamically convex toric domains due to [START_REF] Gutt | Examples around the strong Viterbo conjecture[END_REF]Proposition 1.8]. Method two -strain in subsection 3.2 -can be carried out even within the category of dynamically convex domains. In particular, we have the following useful result.

Corollary 1.7. For any dynamically convex toric domain X Ω in R 4 , there exists a small perturbation in terms of the volume such that the resulting domain X Ω is still dynamically convex but not symplectically convex.

Proof. This directly comes from the construction of strain operation in subsection 3.2, Corollary 3.4, and [START_REF] Gutt | Examples around the strong Viterbo conjecture[END_REF]Proposition 1.8]. □

Note that Corollary 1.7 provides a variety of examples that support [START_REF] Abreu | Dynamical convexity and elliptic periodic orbits for Reeb flows[END_REF]. In sharp contrast to the example produced in subsection 1.5 in ( [START_REF]3D convex contact forms and the Ruelle invariant[END_REF]) (which is closely related to the one invented in [START_REF] Abbondandolo | Sharp systolic inequalities for Reeb flows on the three-sphere[END_REF]), Corollary 1.7 above is to our best knowledge the first systematic way to produce toric examples to confirm (2) and it is more direct and much simpler than the one in [START_REF]3D convex contact forms and the Ruelle invariant[END_REF].

As the proof of Theorem 1.3 is essentially fighting against the constant c and C appearing in (3), one may be curious about how small or large these constants are. In general, due to the complexity of the proof of (3) in [START_REF]3D convex contact forms and the Ruelle invariant[END_REF], it seems difficult to read off any bounds for c and C directly. However, for a special family of star-shaped domains, called monotone toric domains (introduced in [START_REF] Gutt | Examples around the strong Viterbo conjecture[END_REF]), we are able to estimate the product ru(X Ω ) • sys(X Ω ) 1 2 by the following result, which yields concrete bounds for c and C in (3) (cf [START_REF]3D convex contact forms and the Ruelle invariant[END_REF]Remark 1.11]).

Theorem 1.8. For any monotone toric domain X Ω , we have

ru(X Ω ) • sys(X Ω ) 1 2 ≥ 1 2 . If, furthermore, X Ω is (geometrically) convex in R 4 , then ru(X Ω ) • sys(X Ω ) 1 2 ≤ 3.
In particular, the constant c and C in the criterion (3) satisfies c ≤ 1 2 and C ≥ 3, respectively.

Here, monotone means that the outward normal vectors ν = (ν 1 , ν 2 ) at any point of the boundary component

∂ + Ω := ∂Ω ∩ R 2
>0 have non-negative components (ν 1 ≥ 0, ν 2 ≥ 0). This is more general than the classical notion of convex or concave toric domains and admit many nice properties. For instance, by [6, Proposition 1.8], the category of (strictly) monotone toric domains coincides with the category of domains that are both toric and dynamically convex. For another instance, due to [6, Theorem 1.7], all normalized symplectic capacities agree on strictly monotone toric domains. Remark 1.9. In the proof of Theorem 1.3, strain operation in subsection 3.2, shows that within the category of strict monotone toric domains, there is no upper bound for the constant C in [START_REF] Chaidez | Convexity and the Ruelle invariant in higher dimensions[END_REF], see Corollary 3.4. The constants in the conclusion of Theorem 1.8) are sharp. The upper bound is realised by the toric domain whose moment image is given by the blue curve of Figure 6. Moreover, the lower bound 1 2 can be arbitrarily approximated by the family of polydisks P (a, b) (its definition given by ( 23)) with b ∞, we have

ru(P (a, b)) • sys(P (a, b)) 1 2 = (a + b) √ 2ab • a √ 2ab = (a + b) 2b - b +∞ 1 2
For more details, see subsection 4.

Remark 1.10. Soon after our paper was completed, Chaidez-Edtmair uploaded a new preprint [START_REF] Chaidez | Convexity and the Ruelle invariant in higher dimensions[END_REF], where it generalizes the criterion (3) to higher-dimensional domains. Moreover, inspired by our strain operation (see Figure 4), they provide higherdimensional examples that are dynamically convex but not symplectically convex.

Acknowledgement This work was completed while the third author held a CRM-ISM Postdoctoral Research Fellowship at the Centre de recherches mathématiques in Montréal. He thanks this Institute for its warm hospitality. Part of Section 2 is derived from the project [START_REF] Hind | Hamiltonian shape invariant and coarse symplectic Banach-Mazur distance[END_REF], so he thanks Richard Hind for fruitful discussions. The second author was partially supported by the ANR LabEx CIMI (grant ANR-11-LABX-0040) within the French State Programme "Investissements d'Avenir" and by the ANR COSY (ANR-21-CE40-0002) grant. We thank Julian Chaidez for his comments on the initial version of the paper, in particular for the statement of Corollary 1.7. We are also grateful to an anonymous referee for a thorough and thoughtful report.

Ruelle invariant

For simplicity, let us assume our closed contact 3-manifold (Y, λ) is a homology 3sphere. To the contact structure ξ = ker λ, we can associate a real number Ru(Y, λ) as follows. Observe that the Reeb flow ϕ t Rα preserves the contact form λ. In particular, it preserves the contact structure ξ. Then, for any time t and any fixed point y ∈ Y , the pushforward, or linearization, (ϕ t Rα ) * brings the contact 2-plane ξ y to the contact 2plane ξ ϕ t Rα (y) . Under a given trivialization τ of the contact structure, this linearization can be regarded as a linear transformation of R 2 , denoted by Φ τ y,t . For any real T ≥ 0, the path Φ = {Φ τ y,t } t∈[0,T ] defines an element of the universal cover of the symplectic group Sp [START_REF] Abreu | Dynamical convexity and elliptic periodic orbits for Reeb flows[END_REF]. Together with the rotation number, ρ : Sp(2) R (see subsection 1.2 in [START_REF] Hutchings | ECH capacities and the Ruelle invariant[END_REF] or subsection 2.1 in [START_REF]3D convex contact forms and the Ruelle invariant[END_REF]), this yields a real number ρ(y, T, τ ) := ρ({Φ τ y,t } t∈[0,T ] ) and the following limit . For any point p = (w 1 , w 2 ) ∈ ∂ + Ω, consider the polar coordinate (w 1 , θ 1 , w 2 , θ 2 ). Then (recall that µ is the moment map) for any z ∈ µ -1 (p), one can verify that the Reeb vector field R is

(5) R(z) = 2π ν 1 (p)w 1 + ν 2 (p)w 2 ν 1 (p) ∂ ∂θ 1 + ν 2 (p) ∂ ∂θ 2
where (ν 1 (p), ν 2 (p)) is the unit normal vector of ∂ + Ω at point p, pointing outward of Ω. Note that ν 1 (p)w 1 + ν 2 (p)w 2 > 0 for any p ∈ ∂ + Ω due to our hypothesis that X Ω is star-shaped. Moreover, the contact 2-plane at z is given by, ( 6)

ξ z = a 1 ∂ ∂w 1 + b 1 ∂ ∂θ 1 + a 2 ∂ ∂w 2 + b 2 ∂ ∂θ 2 ν 1 (p)a 1 + ν 2 (p)a 2 = 0 w 1 b 1 + w 2 b 2 = 0 ,
and one can choose a basis of ξ z as follows,

(7) e 1 (p) = -ν 2 (p) ∂ ∂w 1 + ν 1 (p) ∂ ∂w 2 and e 2 (p) = -w 2 ∂ ∂θ 1 + w 1 ∂ ∂θ 2 .
Note that (e 1 (p), e 2 (p)) is an ordered basis in that (ω std ) z (e 1 (p), e 2 (p)) > 0. Using this basis, along any Reeb trajectory γ = (γ(t)) t∈[0,T ] , one can chose a trivialization τ : γ * ξ γ × R 2 explicitly defined as follows. For any (z, v) ∈ (γ * ξ) z where z ∈ γ and v ∈ ξ z , ( 8)

τ (p)((z, v)) = (z, (v R , v θ )) where v = v R e 1 (p) + v θ e 2 (p).
Moreover, under this trivialization, the differentials of the Reeb flow along the trajectory γ form a path in Sp(2), denoted by Φ. The following lemma is crucial.

Lemma 2.2. With respect to the trivialization given in [START_REF] Hofer | The dynamics on three-dimensional strictly convex energy surfaces[END_REF], along the Reeb trajectory γ = (γ(t)) t∈[0,T ] the resulting path Φ in Sp(2) from the differentials of the Reeb flow is

Φ = 1 0 f (t) 1 t ∈ [0, T ]
where f (t) is a linear function of t depending only on γ(0) and e 1 (µ(γ(0)) in [START_REF] Hind | Hamiltonian shape invariant and coarse symplectic Banach-Mazur distance[END_REF]. In particular, rot τ (γ(0), T ) = 0.

Proof. Suppose γ(0) ∈ µ -1 (p) for some p ∈ ∂ + Ω. For v ∈ ξ γ(0) and any t ∈ [0, T ], to compute the differential (dϕ t R | ξ γ(0) )(v), we need to take a locally defined smooth path r(s) : (-ϵ, ϵ)

∂X Ω for ϵ sufficiently small such that r(0) = γ(0) and r ′ (0) = v. Denote for brevity r(s) = (w 1 (s), θ 1 (s), w 2 (s), θ 2 (s)) where w 1 (0) = w 1 and w 2 (0) = w 2 . For any s ∈ (-ϵ, ϵ), by ( 7) and ( 8),

r(s) = r(0) + sv + o(s) = (w 1 -sν 2 (p)v R , θ 1 -sw 2 v θ , w 2 + sν 1 (p)v R , θ 2 + sw 1 v θ ) + o(s).
Note that the approximation term o(s) exist to guarantee that r(s) ∈ ∂X Ω . Then, by (5), we have

ϕ t R (r(s)) = ϕ t Rα (w 1 (s), θ 1 (s), w 2 (s), θ 2 (s)) = (w 1 (s), θ 1 (s) + Θ 1 (s) • t, w 2 (s), θ 2 (s) + Θ 2 (s) • t) where Θ 1 (s) = 2πν 1 (p(s)) ν 1 (p(s))w 1 (s) + ν 2 (p(s))w 2 (s) and Θ 2 (s) = 2πν 2 (p(s)) ν 1 (p(s))w 1 (s) + ν 2 (p(s))w 2 (s) ,
where by notation p(s) = (w 1 (s), w 2 (s)). Observe that the denominator of Θ 1 (s) and Θ 2 (s) can be simplified as follows,

ν 1 (p(s))w 1 (s) + ν 2 (p(s))w 2 (s) = ν 1 (p(s))w 1 + ν 2 (p(s))w 2 + o(s).
In particular, it converges to ν 1 (p)w 1 + ν 2 (p)w 2 as s 0. Then, by the definition of a differential and computations above,

(dϕ t R )(v) = lim s 0 ϕ t R (r(s)) -ϕ t R (r(0)) s = (-ν 2 (p)v R , -w 2 v θ , ν 1 (p)v R , w 1 v θ ) +   0, lim s 0 Θ 1 (s) - 2πν 1 (p) ν 1 (p)w 1 +ν 2 (p)w 2 s • t, 0, lim s 0 Θ 2 (s) - 2πν 2 (p) ν 1 (p)w 1 +ν 2 (p)w 2 s • t   .
Meanwhile, further simplifications yield (9) lim

s 0 Θ 1 (s) - 2πν 1 (p) ν 1 (p)w 1 +ν 2 (p)w 2 s = 2π • (ν 1 (p)ν 2 (p(s)) ′ | s=0 -ν 1 (p(s)) ′ | s=0 ν 2 (p))(-w 2 ) (ν 1 (p)w 1 + ν 2 (p)w 2 ) 2 ,
and similarly, [START_REF] Hutchings | ECH capacities and the Ruelle invariant[END_REF]

lim s 0 Θ 2 (s) - 2πν 2 (p) ν 1 (p)w 1 +ν 2 (p)w 2 s = 2π • (ν 1 (p)ν 2 (p(s)) ′ | s=0 -ν 1 (p(s)) ′ | s=0 ν 2 (p))w 1 (ν 1 (p)w 1 + ν 2 (p)w 2 ) 2 ,
where the ν i (p(s)) ′ | s=0 denotes the derivative with respect to the variable s and then evaluated at s = 0. For brevity, denote by

A(p; v) := 2π • ν 1 (p)ν 2 (p(s)) ′ | s=0 -ν 1 (p(s)) ′ | s=0 ν 2 (p) (ν 1 (p)w 1 + ν 2 (p)w 2 ) 2 ,
the common factor in ( 9) and [START_REF] Hutchings | ECH capacities and the Ruelle invariant[END_REF]. Then

(11) (dϕ t Rα )(v) = (-ν 2 (p)v R , (A(p; v) + v θ )(-w 2 )t, ν 1 (p)v R , (A(p; v) + v θ )w 1 t).
In particular,

dϕ t R (e 1 (p)) = e 1 (p) + (A(p; e 1 (p))t)e 2 (p) and dϕ t R (v) = e 2 (p).
Representing this by a matrix with respect to the basis (e 1 (p), e 2 (p)), one gets that (12)

dϕ t R | ξ γ(0) = 1 0 A(p; e 1 (p))t 1 .
Thus we prove the first conclusion by setting f (t) := A(p; e 1 (p))t. Moreover, the second conclusion is straightforward, since each matrix representation of the differential dϕ t R | ξz as in ( 12) is similar to a shear matrix, which does not contribute any rotations. □ 2.2. Proof of Proposition 1.4. Note that the trivialization in [START_REF] Hofer | The dynamics on three-dimensional strictly convex energy surfaces[END_REF] does not extend to the entire ∂X Ω (since the polar coordinate is not well-defined for the points where w 1 = 0 or w 2 = 0). For any globally defined trivialization τ , compared with the trivialization via the polar coordinate, the only difference of the rotation number at point z ∈ µ -1 (p) for p ∈ ∂ + Ω comes from how much the function θ 1 + θ 2 changes along the Reeb flow. Indeed, as explained in the proof of Lemma 2.2 in [START_REF] Hutchings | ECH capacities and the Ruelle invariant[END_REF], moving along the circle given by a rotation of either θ 1 or θ 2 results in the desired change of the factor v R in [START_REF] Hofer | The dynamics on three-dimensional strictly convex energy surfaces[END_REF], with respect to the trivialization τ . Then Lemma 2.2 yields

ρ(z) = lim T ∞ rot τ (z, T ) T = 0 + (dθ 1 + dθ 2 )(R α (z)) 2π = ν 1 (p) + ν 2 (p) ν 1 (p)w 1 + ν 2 (p)w 2
where p = (w 1 , w 2 ). Note that ρ(z) is in fact a function of p ∈ ∂ + Ω. Then by the definition of Ruelle invariant,

Ru(X Ω ) = ∂X Ω ρ(z)α ∧ dα = ∂ + Ω ν 1 (p) + ν 2 (p) ν 1 (p)w 1 + ν 2 (p)w 2 (w 1 dw 2 -w 2 dw 1 )
where the second equality comes from a change of variable via the moment map µ defined in (1) (and restricted to ∂X Ω ). Suppose the profile curve

∂ + Ω is parametrized by {(w 1 (s), w 2 (s))} s∈[0,1] such that w 1 (0) = a(Ω), w 1 (1) = 0 and w 2 (0) = 0, w 2 (1) = b(Ω),
where a(Ω) and b(Ω) are the w 1 -intercept and w 2 -intercept. We may assume after a change of parametrization that

w 1 dw 2 -w 2 dw 1 = (w 1 (s)w ′ 2 (s) -w 2 (s)w ′ 1 (s))ds. Meanwhile, observe that (13) (ν 1 (p), ν 2 (p)) = (ν 1 (s), ν 2 (s)) =   -w ′ 2 (s) |w ′ 1 (s)| 2 + |w ′ 2 (s)| 2 , w ′ 1 (s) |w ′ 1 (s)| 2 + |w ′ 2 (s)| 2   .
Therefore, by [START_REF] Schneider | Global surfaces of section for dynamically convex Reeb flows on lens spaces[END_REF],

Ru(X Ω ) = 1 0 ν 1 (s) + ν 2 (s) ν 1 (s)w 1 (s) + ν 2 (s)w 2 (s) (w 1 (s)w ′ 2 (s) -w 2 (s)w ′ 1 (s))ds = 1 0 -w ′ 2 (s) + w ′ 1 (s) -w ′ 2 (s)w 1 (s) + w ′ 1 (s)w 2 (s) (w 1 (s)w ′ 2 (s) -w 2 (s)w ′ 1 (s))ds = 1 0 w ′ 2 (s) -w ′ 1 (s)ds = (w 2 (1) -w 2 (0)) -(w 1 (1) -w 1 (0)) = b(Ω) + a(Ω).
Therefore, we obtain the desired conclusion.

Proof of Theorem 1.3

In this section, we will give the proof of the main result, Theorem 1.3 in the introduction. Let us start from the following elementary observation on the closed Reeb orbit on a star-shaped domain (∂X, λ). By ( 5), for a point p = (w 1 , w 2 )

∈ ∂ + Ω = ∂Ω ∩ R 2 >0 , we know that (14) a Reeb trajectory at µ -1 (p) is closed if and only if ν 2 (p) ν 1 (p) ∈ Q. Assume that ν 2 (p) ν 1 (p) ∈ Q. Denote by h p ∈ R >0 the unique non-zero positive scalar such that (i) (h p ν 1 (p), h p ν 2 (p)) ∈ Z 2 ; (ii) h p ν 1 (p), h p ν 2 (p) are coprime.
For brevity, denote (m p , n p ) := (h p ν 1 (p), h p ν 2 (p)). The second condition (ii) above guarantees that the corresponding closed Reeb orbit, denoted by γ (mp,np) is primitive (that is, not a multiple cover of another closed Reeb orbit). Note that the period of γ (mp,np) is, by definition, the action A(γ (mp,np) ). Hence, (15)

A(γ (mp,np) ) = (ν 1 (p)w 1 + ν 2 (p)w 2 ) • h p = m p w 1 + n p w 2 .
This can be viewed as the inner product of the (integer-normalized) normal vector (m p , n p ) and the position vector (w 1 , w 2 ) (for point p). Due to our hypothesis that X Ω is star-shaped, the action A(γ (mp,np) ) is always positive, even though the vector (m p , n p ) does not have both of its components positive.

3.1. Method one: strangulation. Since X Ω is a star-shaped domain, we can assume, for its moment image Ω, without loss of generality, that the diagonal of R 2

>0

intersects ∂Ω at point (w * , w * ) such that a neighborhood within R 2 >0 of the subset {(w, w) ∈ R 2 >0 | 0 ≤ w < w * } lies in the interior of Ω. In general, there always exists some ray in R 2 ≥0 satisfying this condition. By our assumption, since ∂ + Ω is smooth, for any given ϵ > 0, there exists some angle θ(ϵ) such that the unbounded sector with vertex (ϵ, ϵ), divided in half by the diagonal, and angle equal to 2θ(ϵ), intersects Ω in a closed region S(ϵ) with points (w 1 , w 2 ) ∈ ∂Ω ∩ S satisfying

|w 1 -w * | ≤ ϵ and |w 2 -w * | ≤ ϵ.
Now, carry on the following strangulation operation on Ω, that is, define

Ω := Ω\(int(S(ϵ)) ∪ int(∂Ω ∩ S(ϵ))).
For a picture of this operation, see Figure 3. After smoothing all singularities of Ω, we have that the resulting domain, still denoted by Ω, is again a closed domain in R 2 ≥0 with its pre-image under the moment map µ -1 ( Ω) =: X Ω being a star-shaped domain. Moreover, X Ω satisfies the following quantitative properties.

w 1 = w 2 Ω w 1 = w 2 (
Lemma 3.1. The strangulation operation on X Ω results in a star-shaped domain X Ω which satisfies

(1) Vol(X Ω ) = Vol(X Ω ) + O(ϵ). (2) sys X Ω ≤ O(ϵ). (3) Ru(X Ω ) = Ru(X Ω ).
Here, O(ϵ) represents a constant, proportional to ϵ in Figure 3, that can be arbitrarily small.

Assuming Lemma 3.1, we continue the proof of Theorem 1.3 via the strangulation operation. First, we have the following computations on volumes from Stokes' theorem,

Vol(∂X Ω , λ) = ∂X Ω λ ∧ dλ = X Ω d(λ ∧ dλ) = X Ω dλ ∧ dλ = 2Vol(X Ω ).
Second, suppose ϵ is sufficiently small so that O(ϵ) < Vol(X Ω ) 2 . Then (1) and ( 2

) in Lemma 3.1 imply that ru X Ω 2 = Ru(X Ω ) 2 2Vol(X Ω ) = Ru(X Ω ) 2 2Vol(X Ω ) -2O(ϵ) ≤ 2 • ru(X Ω ) 2 ,
where, in particular, the upper bound 2

• ru(X Ω ) 2 is finite. Third, (3) in Lemma 3.1 implies that ru X Ω • sys X Ω 1 2 ≤ √ 2 • ru(X Ω ) • O(ϵ) 0 as ϵ 0.
Therefore, the product ru X Ω •sys X Ω 1 2 will be lower than the constant c appearing in criterion (3), whenever ϵ is sufficiently small. In conclusion, the domain X Ω is not symplectically convex.

Proof of Lemma 3.1. When ϵ < w * , the 4-dimensional volume of X Ω and X Ω (with respect to the standard symplectic structure on R 4 ) satisfy ( 16)

|Vol(X Ω ) -Vol(X Ω )| ≤ π • 2(w * + ϵ) 2 • θ(ϵ) π = 8w 2 * • θ(ϵ)
which goes to 0 as ϵ goes to 0 (since θ(ϵ) goes to 0). Therefore, X Ω is indeed a small perturbation in terms of the volume of X Ω . This proves [START_REF] Abbondandolo | Sharp systolic inequalities for Reeb flows on the three-sphere[END_REF]. By the discussion above on the closed Reeb orbits in [START_REF] Usher | Symplectic Banach-Mazur distances between subsets of C n[END_REF], applied to the new domain Ω, there exists a closed Reeb orbit at p = (ϵ, ϵ) corresponding to the normal vector [START_REF] Abbondandolo | Sharp systolic inequalities for Reeb flows on the three-sphere[END_REF][START_REF] Abbondandolo | Sharp systolic inequalities for Reeb flows on the three-sphere[END_REF]. In particular, by (15) its action is 1 • ϵ + 1 • ϵ = 2ϵ. Denoting by T min the minimal period of a closed Reeb orbit of ∂X Ω , we have

T min ≤ 2ϵ Therefore, sys X Ω ≤ 4ϵ 2 Vol(∂X Ω , λ) = 4ϵ 2 2Vol(X Ω ) ≤ 4ϵ 2 2Vol(X Ω ) -16w 2 * • θ(ϵ)
where the second inequality comes from (16). Hence, when ϵ is sufficiently small so that w 2 * • θ(ϵ) < Vol(X Ω ) 16

, we have

(17) sys X Ω ≤ 4ϵ 2 Vol(X Ω ) 0 as ϵ 0.
This proves [START_REF] Abreu | Dynamical convexity and elliptic periodic orbits for Reeb flows[END_REF]. Finally, since the strangulation operation does not change the w 1 or w 2 -intercepts of the original domain Ω, due to Proposition 1.4, the Ruelle invariant does not change, that is, Ru(X Ω ) = Ru(X Ω ). This proves [START_REF] Chaidez | Convexity and the Ruelle invariant in higher dimensions[END_REF]. □ Remark 3.2. It is not necessary to carry out the strangulation operation along the diagonal, as we did above. In general, most rays starting from the origin work in a similar way. An extreme case is to carry our such an operation along the w 1 -axis or w 2 -axis. The only difference is that the Ruelle invariant will change but gets smaller (so we still obtain the result that the product of ratios ru • sys 1 2 will be eventually smaller than the constant c in the criterion in [START_REF] Chaidez | Convexity and the Ruelle invariant in higher dimensions[END_REF]. In fact, such an operation has been investigated in [START_REF] Usher | Symplectic Banach-Mazur distances between subsets of C n[END_REF] on ellipsoids, called truncated ellipsoid.

3.2. Method two: strain. Given a star-shaped domain X Ω , suppose that the w 1intercept of ∂ + Ω is a > 0. Consider a generic small perturbation of Ω near (a, 0) but with the w 1 -intercept a fixed, which also results in a small perturbation of X Ω in terms of the volume, such that in a neighborhood N of (a, 0), the boundary ∂ + Ω has a constant slope k, either positive or negative (but not equal to ±∞). This can be achieved due to our hypothesis that ∂ + Ω is smooth, and we can consider N sufficiently small so that the minimal period of the Reeb orbit of ∂X Ω changes in an arbitrarily small way. For brevity, we still denote the domain after this perturbation by Ω.

Next, for any ϵ > 0, sufficiently small so that the (unique) point (w * (ϵ), ϵ) ∈ ∂ + Ω for some w * > 0 lies in the neighborhood N above, we have ϵ-0 w * (ϵ)-a = k, that is,

(18) w * (ϵ) = ϵ k + a.
Consider the following triangle T (ϵ) := the triangle determined by vertices (0, 0), (w * (ϵ), ϵ), and

1 √ ϵ , 0
where ϵ is sufficiently small so that

(19) -ϵ 1 √ ϵ -w * (ϵ) > k if k < 0.
This can be achieved since ( 19) is equal to k(a -1 √ ϵ ) > 0, so when ϵ 0, we have a -1 √ ϵ < 0 (since k < 0). Then consider the following strain operation on Ω, that is,

Ω := Ω ∪ T (ϵ).
For a picture of this operation, see Figure 4. Observe that condition (19) together with the hypothesis that X Ω is star-shaped, implies that Ω ⊂ Ω and the pre-image X Ω = µ -1 ( Ω) is again star-shaped. In particular, X Ω being star-shaped is used to deal with the case when k > 0. Similarly to Lemma 3.1, we have the following quantitative result for X Ω . Lemma 3.3. The strain operation on X Ω results in a star-shaped domain X Ω that satisfies [START_REF] Abbondandolo | Sharp systolic inequalities for Reeb flows on the three-sphere[END_REF]

Ω Ω = Ω ∪ {red triangle} (w * ( ), ) ( 1 √
Vol(X Ω ) = Vol(X Ω ) + O(ϵ). (2) sys X Ω ≥ A, where A is a constant independent of ϵ. (3) Ru(X Ω ) ≥ 1 O(ϵ) .
Here, O(ϵ) represents a constant, proportional to ϵ in Figure 3, that can be chosen arbitrarily small. Assuming Lemma 3.3, we continue the proof of Theorem 1.3 via the strain operation. The conclusions (1) and (3) in Lemma 3.3 imply that, if O(ϵ) < Vol(X Ω ), we have

ru X Ω 2 = Ru(X Ω ) 2 2Vol(X Ω ) = Ru(X Ω ) 2 2Vol(X Ω ) + 2O(ϵ) ≥ 1 O(ϵ) 2 • 1 4Vol(X Ω ) . Then we have (20) ru X Ω • sys X Ω 1 2 ≥ 1 O(ϵ) 2 • A 4Vol(X Ω ) +∞ as ϵ 0.
Hence, the product of the ratios will be larger than the constant C appearing in criterion (3). Therefore, the domain X Ω is not symplectically convex.

Proof of Lemma 3.3. Comparing the difference of the volume in R 4 , we have

(21) |Vol(X Ω ) -Vol(X Ω )| ≤ ϵ • 1 √ ϵ 2 =
√ ϵ 2 which goes to 0 as ϵ goes to 0. This proves [START_REF] Abbondandolo | Sharp systolic inequalities for Reeb flows on the three-sphere[END_REF].

This operation possibly introduces various new closed Reeb orbits. Besides the one corresponding to the w 1 -intercept point ( 1 √ ϵ , 0) with large action, others will concentrate only near the point p = (w * (ϵ), ϵ), after smoothing Ω at p. By ( 14), these closed Reeb orbits correspond to the pairs of integers,

(m p , n p ) ∈ Z >0 × Z with min - 1 k , 0 ≤ n p m p ≤ √ ϵ -w * (ϵ) ϵ .
Concerning their action, we have by (15),

A(γ (mp,np) ) = m p w * (ϵ) + n p ϵ = m p w * (ϵ) + n p m p ϵ ≥ m p ϵ k + a + min - ϵ k , 0 ≥ a 2 ,
when ϵ is sufficiently small. We denote as above, T min the minimal period of a closed Reeb orbit on ∂X Ω and T min the minimal period of a closed Reeb orbit on ∂X Ω . If T min < a 2 , then obviously T min = T min . If, on the other hand, T min ≥ a 2 , then T min ≥ a 2 . Meanwhile, by assumption, T min ≤ a. Therefore, in either case, we have

T min ≥ T min 2 .
In particular, the lower bound T min 2 is independent of the parameter ϵ. Now, for the ratios in discussion, by (21),

sys X Ω ≥ T 2 min 4Vol(∂X Ω , λ) = T 2 min 8Vol(X Ω ) ≥ T 2 min 8Vol(X Ω ) + 4 √ ϵ
When ϵ is sufficiently small, say √ ϵ < Vol(X Ω ), we have This proves [START_REF] Abreu | Dynamical convexity and elliptic periodic orbits for Reeb flows[END_REF].

Finally, the strain operation results in an essential change of the Ruelle invariant. By Proposition 1.4, Ru(X Ω ) = (w 2 -intercept of

∂ + Ω) + 1 √ ϵ ≥ 1 √ ϵ .
This proves [START_REF] Chaidez | Convexity and the Ruelle invariant in higher dimensions[END_REF]. □

As an immediate consequence from the strain operation, we have the following.

Corollary 3.4. The product of ratios ru • sys 1 2 is unbounded on the category of monotone toric domains.

Proof. Note that the strain operation is closed within the category of strictly monotone toric domains. Hence, it is closed within the category of dynamically convex toric domains by [START_REF] Gutt | Examples around the strong Viterbo conjecture[END_REF]Proposition 1.8], since by the definition of a monotone toric domain, near the w 1 -intercept the corresponding slope k is always negative. Then the desired conclusion follows from the computation (20). □

Estimate of constants

Recall that a strictly monotone toric domain is a star-shaped domain such that the outward normal vectors along the boundary component ∂ + Ω = ∂Ω ∩ R 2 >0 all have both components positive. In contrast with Corollary 3.4, in this section, we give a proof of Theorem 1.8, which provides a uniform bounds of the product of ratios ru • sys Proof. By the proof of [6, Theorem 1.11], we know that c Gr (X Ω ) is equal to the largest L > 0 such that the right triangle with vertices (0, L), (L, 0) and (0, 0) is contained in Ω. Denote by (s, t) one of these intersection points. Then observe that X Ω being monotone implies that X Ω ⊂ P (s, b) ∪ P (a, t). Therefore, we have

f c (w 1 ) =          √ b -b-c c w 1 2 if 0 ≤ w 1 ≤ c b (b -c) c -w 1 if c b (b -c) ≤ w 1 ≤ c 2

π|z 1 | 2 π|z 2 | 2 Figure 1 .

 2221 Figure 1. An example of a toric star-shaped domain.

1

 1

Figure 2 .

 2 Figure 2. Strangulation operation and strain operation.

Definition 2 . 1 . 2 . 1 .

 2121 , T, τ) T is well-defined. In particular, rot(y) is independent of the trivialization τ , since here, Y has a unique trivialization up to homotopy. In general, rot(y) only depends on the homotopy class of a trivialization. As elaborated by Hutchings in[START_REF] Hutchings | ECH capacities and the Ruelle invariant[END_REF], subsection 1.2, right above Definition 1.3 (and which traces back to Ruelle in[START_REF] Ruelle | Rotation numbers for diffeomorphisms and flows[END_REF]), we have that rot(y) is an integrable function. Proposition 2.13 in[START_REF]3D convex contact forms and the Ruelle invariant[END_REF] also proves these properties. Suppose the closed contact 3-manifold (Y, λ) is a homology 3-sphere, then its Ruelle invariant is defined byRu(Y, λ) := Y rot(y) λ ∧ dλ.In particular, if X is a star-shaped domain in R 4 , then we define Ru(X) := Ru(∂X, λ std | ∂X ) Linearized Reeb flow. Denote by ∂ + Ω := ∂Ω ∩ R 2 >0

Figure 3 .

 3 Figure 3. Strangulation operation.

Figure 4 .

 4 Figure 4. Strain operation.

2 min

 2 Vol(X Ω ) (:= A) which is independent of ϵ.

1 2Lemma 4 . 1 .

 141 for such domains (when they are geometrically convex in R 4 ). In terms of the notation, P (a, b) denotes the polydisk in C 2 defined as(23) P (a, b) = (z 1 , z 2 ) ∈ C 2 π|z 1 | 2 ≤ a, π|z 2 | 2 ≤ b .Let us start from the following useful result. Denote by c Gr (X Ω ) the Gromov width of a toric domain X Ω , measuring the largest B 4 (a) that can be symplectically embedded into X Ω . Let X Ω be a monotone toric domain where the w 1 -intercept and w 2intercept of ∂ + Ω are (a, 0) and (0, b) respectively. Suppose that b ≥ a, then Vol(X Ω ) ≤ b • c Gr (X Ω ).

Figure 5 .

 5 Figure 5. Graph of g c in red.

if c 2 ≤ w 1 ≤ 1 ,

 21 also see Figure6.

Figure 6 .

 6 Figure 6. Graph of f c in red.

Proof of Theorem 1.8. For a monotone toric domain X Ω , [START_REF] Gutt | Examples around the strong Viterbo conjecture[END_REF]Theorem 1.7] shows that all normalized symplectic capacities coincide. In particular, the minimal period of a Reeb orbit is equal to c Gr (X Ω ). Without loss of generality, assume b ≥ a. Then by Proposition 1.4 and Lemma 4.1, we have

Thus, we complete the proof of the first conclusion. Now, suppose furthermore that X Ω is geometrically convex in R 4 . Up to a rescaling, assume the w 1 -intercept of Ω is 1 while the w 2 -intercept of Ω is still b. Up to a reflection between w 1 and w 2 , we can assume that b ≥ 1. Therefore, we have Ru(X Ω ) = 1 + b for any such domain and so

depends only on c Gr (X Ω ) Vol(X Ω ) . We thus aim to bound above this quantity among monotone toric domains which are geometrically convex.

By [6, Proposition 2.3], the following subset

In particular, when restricted to R 2 ≥0 , the boundary ∂ Ω can be written as a decreasing concave function µ 2 = g(µ 1 ). Since g is concave, we have

. Therefore, g is above the broken curve consisting of the two previous curves. Hence, among these g, the one whose domain maximizes c Gr Vol is the one minimizing the volume i.e. the convex hull of this broken curve, see Figure 5. It has the following boundary:

By a change of variables w i = µ 2 i , we know that the boundary ∂ + Ω (minus the components on w 1 -axis and w 2 -axis) is a function w 2 = f c (w 1 ) := g c ( √ w 1 ) 2 given by Denote by X fc the monotone toric domain such that f c is the boundary of its moment map minus the w 1 and w 2 -axis. Then we have

Meanwhile, by integrating along the graph f c (w 1 ), we get that

Moreover, since X Ω is geometrically convex with the w 1 -intercept and w 2 -intercept being a(Ω) = 1 and b(Ω) = b, respectively, we have b 1+b ≤ c Gr (X Ω ) ≤ 1, since the Gromov width of the domain with boundary w

where the maximum is obtained for c = b 1+b i.e. for the domain whose boundary is given by f (w 1 ) = b(1 -√ w 1 ) 2 . Hence, ru(X Ω ) 2 • sys(X Ω ) = (1 + b) 2 c Gr (X Ω ) 2 4Vol(X Ω ) 2 ≤ 9.

Thus we completed the proof of the second conclusion. □


Petitjean Théo

Zongwei Wu

Cédric Demonceaux

Olivier Laligant

OLF : RGB-D Adaptive Late Fusion for Robust 6D Pose Estimation

Keywords: Late fusion, Deep learning, Self Optimized parameter, PSNR and noise study, RGB-D

RGB-D 6D pose estimation has recently gained significant research attention due to the complementary information provided by depth data. However, in real-world scenarios, especially in industrial applications, the depth and color images are often more noisy 1 . 2 Existing methods typically employ fusion designs that equally average RGB and depth features, which may not be optimal. In this paper, we propose a novel fusion design that adaptively merges RGB-D cues. Our approach involves assigning two learnable weight α 1 and α 2 to adjust the RGB and depth contributions with respect to the network depth. This enables us to improve the robustness against low-quality depth input in a simple yet effective manner. We conducted extensive experiments on the 6D pose estimation benchmark and demonstrated the effectiveness of our method. We evaluated our network in conjunction with DenseFusion on two datasets (LineMod 3 and YCB 4 ) using similar noise scenarios to verify the usefulness of reinforcing the fusion with the α 1 and α 2 parameters. Our experiments show that our method outperforms existing methods, particularly in low-quality depth input scenarios. We plan to make our source code publicly available for future research.

INTRODUCTION

6D pose estimation is crucial for implementing computer vision in real-world and industrial problems, such as robotic manipulation 6 and augmented reality. [START_REF] Marder-Eppstein | Project tango[END_REF] Solutions must be robust to object texture and geometry, occlusion, and external constraints. However, expensive RGB-D sensors are currently used in the industry to limit noise, limiting access to cheaper sensors while demand for vision applications grows. Deep learning has led to more modern research in camera pose estimation, with direct regression through the network [START_REF] Wang | Densefusion: 6d object pose estimation by iterative dense fusion[END_REF] being a simple and applicable approach. Some methods use DNNs and the Perspective-n-Point algorithm (PnP), [START_REF] He | Pvn3d: A deep point-wise 3d keypoints voting network for 6dof pose estimation[END_REF] but suffer from problems transitioning from 2D to 3D space and loss of geometric information. Traditional methods [START_REF] Rios-Cabrera | Discriminatively trained templates for 3d object detection: A real time scalable approach[END_REF][START_REF] Wohlhart | Learning descriptors for object recognition and 3d pose estimation[END_REF] based on feature extraction and mapping to model clouds were effective but limited in performance due to changing scene brightness and occlusion.

The development of inexpensive and easily usable RGB-D sensors has enabled the creation of datasets carrying depth information, highlighting the importance of 3D and depth information in pose detection. This modality has improved processes used on 2D, whether in direct regression [START_REF] Wang | Densefusion: 6d object pose estimation by iterative dense fusion[END_REF] or based on keypoints. [START_REF] He | Pvn3d: A deep point-wise 3d keypoints voting network for 6dof pose estimation[END_REF] Methods using multiple modalities have highlighted the importance of RGB and Depth information, leading to input data fusion strategies. However, most fusion methods [START_REF] Wang | Densefusion: 6d object pose estimation by iterative dense fusion[END_REF][START_REF] Xu | Pointfusion: Deep sensor fusion for 3d bounding box estimation[END_REF] simply concatenate the information, potentially propagating noise throughout the network. To address this, we propose a method using an external parameter to enhance the network's resistance to input noise. Our method's advantages are described by addressing two points:

• How to adapt a well-known fusion network (DenseFusion [START_REF] Wang | Densefusion: 6d object pose estimation by iterative dense fusion[END_REF] ) and to make it reliable even under extreme noise conditions.

• How much noise can impact classical networks and how much state-of-the-art metrics can be degraded by noise.

The smart grid will be driven according to the following scenario: the training data will be considered perfect and under controlled laboratory conditions. The test data will be noisy to get closer to potential complex conditions encountered in industrial environments.

RELATED WORK

Given an RGBD image, the goal of 6D pose estimation is to predict the matrix that transforms the object coordinates to the camera coordinates.

Pose with RGB only. The traditional methods were essentially based on point of interest detection techniques. [START_REF] Collet | The moped framework: Object recognition and pose estimation for manipulation[END_REF][START_REF] Rothganger | 3d object modeling and recognition using local affine-invariant image descriptors and multi-view spatial constraints[END_REF] These techniques already used in computer vision allowed to create correspondence between two images. Some methods sought to learn to predict 2D keypoint and determine the position using PnP algorithm. [START_REF] Fischler | Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography[END_REF] The arrival of methods using deep learning allowed to outperform the classical methods particularly sensitive to the quality of the images and to the texture of the objects. These new methods based on CNN have allowed the emergence of networks that directly predict the pose from RGB images. [START_REF] Tulsiani | Viewpoints and keypoints[END_REF] However, despite good results, 3D prediction remains a problem, especially because of the difficulty of back-projecting 2D points in 3D space. Some methods such as Xiang et al. [START_REF] Xiang | Data-driven 3d voxel patterns for object category recognition[END_REF] have tried to tackle this problem by taking advantage of the 3D models of the searched objects by producing 3D features from them. Mousavian et al. [START_REF] Mousavian | 3d bounding box estimation using deep learning and geometry[END_REF] have tried to predict 3D parameters using geometric constraints.

Pose with 3D point cloud. Some studies have also tried to solve the problem of pose estimation with methods based on voxel and point cloud studies. The traditional methods [START_REF] Song | Sliding shapes for 3d object detection in depth images[END_REF] were very time consuming. More recently, methods based on deep learning have allowed to directly predict the pose from networks adapted to 3D point clouds such as frustrumPointNet, [START_REF] Qi | Frustum pointnets for 3d object detection from rgb-d data[END_REF] PointNetLike, 18 VoxelNet 19 etc ... These methods although very robust on urban driving application dataset, remain limited in the problem of 6D pose estimation. The sparsity and lack of texture in the point clouds limit the pose estimation accuracy. Moreover, objects with high reflectivity are an almost inescapable problem : active depth sensors are unable to create clouds on these particular surfaces.

Pose with RGB-D. Similar to the classical RGB methods, the classical RGB-D methods are based on two-step algorithms. The 3D features extracted from the input are therefore used by matching and verification techniques. [START_REF] Brachmann | Learning 6d object pose estimation using 3d object coordinates[END_REF][START_REF] Lowe | Object recognition from local scale-invariant features[END_REF] These methods are nevertheless sensitive to the change of brightness in the image.RGB-D methods can be categorized on three different approaches. Holistic methods based on Deep Neural Network (DNN) like DenseFusion 5 or PoseCNN, [START_REF] Xiang | Posecnn: A convolutional neural network for 6d object pose estimation in cluttered scenes[END_REF] seek to regress the position directly. To circumvent the problem induced by the non-linearity these methods use refining loops to estimate the position. Dense correspondence methods, 23 using a Hough voting scheme to vote for the final results with either random forest [START_REF] Brachmann | Learning 6d object pose estimation using 3d object coordinates[END_REF] and CNN. [START_REF] Doumanoglou | Recovering 6d object pose and predicting next-best-view in the crowd[END_REF] KeyPoint based methods are also very present in RGB-D, works such as PVnet, [START_REF] Peng | Pvnet: Pixel-wise voting network for 6dof pose estimation[END_REF] or PVN3D. [START_REF] He | Pvn3d: A deep point-wise 3d keypoints voting network for 6dof pose estimation[END_REF] Fusion. Many methods based on RGB-D images [START_REF] Qi | Frustum pointnets for 3d object detection from rgb-d data[END_REF] choose to extract features from the point cloud and the RGB image and then merge them to improve the robustness and efficiency of their network. As also shown by DenseFusion, [START_REF] Wang | Densefusion: 6d object pose estimation by iterative dense fusion[END_REF] PointFusion, 10 MoreFusion, 26 late fusion strategies are efficient and allow a better understanding of input data. However the concatenation strategy seems to propagate the noise inside the network. Some more modern methods have underlined the impact that a badly optimized fusion could have on a network : new strategies have been developed like in FFB6D. [START_REF] He | Ffb6d: A full flow bidirectional fusion network for 6d pose estimation[END_REF] With this in mind, and in order to improve the fusion while making it as insensitive to noise as possible, we propose in this work a new late fusion method balanced by two coefficient parameters(α 1 and α 2 ).

PROPOSED METHOD

With an input of type RGB-D, the 6D pose estimation seeks to predict a transformation matrix [R|t] which transforms the object model from these coordinates to the camera space coordinate. The accuracy of pose estimation is highly dependent on the quality of the RGB and Depth data. Thus we would like to take into account the quality of the texture and geometry extracted from the RGB-D camera in order to improve the accuracy of the camera pose estimation.

Overview

We therefore propose, in order to limit the propagation of noise from the input to the whole network, a smart fusion allowing the network itself to give a weight to each modality. The model works as follows : first we extract the features for each object location. More concretely, two networks will extract features from each modality in parallel: on the one hand the RGB on which a CNN is applied and on the other hand a network similar to PointNet 18 on the depth. Then these features are merged in a so-called late fusion strategy. This fusion based on DenseFusion 5 assemble the features previously extracted from each network, and output a pixel-wise feature. This fusion is balanced by the network itself with the help of an additional parameter α, one parameter is linked to Rgb (α 1 ) and the other one to the depth (α 2 ). The purpose of this coefficient is to give confidence to information extracted from one of the two data sources according to their quality and/or relevance. Then the features are used by a network that predicts the position of the targeted object. Finally a refining network iteratively improves the results.

Global architecture

In this section we will see the different steps necessary to predict the final pose, from the initial segmentation, to the feature extraction.

Semantic Segmentation. In the first pre-processing step, the objects are segmented. The segmentation network generates N +1 segmentation maps. Each mask then generated is a binary mask whose class corresponds to the N objects of the dataset.

Feature extraction. Feature extraction is a primordial step in a pose estimation algorithm. We extract the features from the color and depth, in order to take advantage of the RGB-D data. As proposed in DenseFusion, we first extract a point cloud from the previously segmented depth map.

Then, with an architecture similar to PointNet, the geometric information is extracted from the generated point cloud. The features extracted from the depth map would best characterize the geometric information of the scene. In the same way a CNN encoder-decoder network will try to extract the information from the RGB image.

Adaptive Late Fusion

Once the features are extracted from each modality, we want to find a method limiting the impact of noise on the quality of the information transmitted in the network afterwards. Based on a late fusion technique, massively used in the state-of-the-art, we modify the commonly used strategy of information concatenation. This fusion strategy is massively used in several applications such as PVN3D [START_REF] He | Pvn3d: A deep point-wise 3d keypoints voting network for 6dof pose estimation[END_REF] and DenseFusion (DF). [START_REF] Wang | Densefusion: 6d object pose estimation by iterative dense fusion[END_REF] Although PVN3D outperforms DenseFusion, we chose to use DenseFusion because its direct regression prediction method in the network allows a better observation of the influence of the fusion on the final result.

As shown in Figure 1, the objective of the fusion is to create balancing α parameters : the network balances the weight of information for each modality on its own. This additional parameter in the PixelWiseFusion stage 5 allows, in addition, better occlusion robustness (induced by the method of Wang et al.), outlier filtering and reducing noise resistance (moreover, the concatenation strategy initially used in the DenseFusion method induces a propagation of the noise potentially present in each input modality). The geometrical and color features are then merged through a pooling strategy balanced by the α 1 and α 2 parameters. As proposed in prior work we keep, afterwards, an average-pooling strategy in order to add in addition to the geometrical and color features a parameter representing the global appearance of self balancing with α.

It is important to note that the merged features are injected into both the pose prediction network and the refinement network equally. The entire original network benefits from the noise resistance brought by our Optimized Late Fusion method.

Pose estimation and refinement

Once the fusion is done, we use a classical scheme of the state of art for position estimation. This estimation is done in two steps. First the network predicts for each point a rotation R and a translation t balanced by confidence C. The specificity of the method, initially proposed by dense fusion, is to minimize the Euclidean distance calculation for all the points and to choose the most efficient prediction, having the highest confidence.

L p i = 1 M j (Rx j + t) -( Ri x j + ti )) (1)
Where x j is the j st randomly selected 3D point wrote as M from the object model. The two values R and t are the prediction p. R and t are the GT values known thanks to the dataset annotations. This distance calculation is well adapted for asymmetrical objects. For symmetrical objects, we minimize the distance to get the closest one with the model of points, removing a part of the problem of the rotation. In order to minimize the distance we replace i from equation ( 1) by i min 0<k<M

The Loss optimized by the network is the one proposed by DenseFusion:

L = 1 N i (L p i c i -ωlog(c i )) ( 2
)
Where N is the randomly sampled fused dense-pixel from the P element of the initial segmentation and ω is the balancing hyper-parameter.

EXPERIMENTS

We simulate scenarios of more or less intense noise in order to verify how the fusion parameterized by α 1 and α 2 allows to minimize the impact of the noise latter on the network. Noise is alternatively (or both) added to RGB images or (and) to depth images. This will allow us to verify which noise has the most impact on the network and which modality has the most impact on the final results. For the RGB image we add a additive Gaussian white noise (GWN) whose strength will be fixed by its standard deviation σ noise . The depth information is corrupted by a multiplicative GWN in order to avoid creating unwanted information on the 0 returned by the physical sensor. To quantify the noise level in our experiments, we use the SNR for each information channel (RGB ou depth). The closer the SNR value will be to 0 (or negative value) is, the stronger the noise is.

Our experiments are based on the DenseFusion algorithm. While this approach is no longer the best approach of the state of the art, it allows to easily demonstrate the impact on the proposed fusion stage. Nevertheless, our approach could be integrated in any other method using the two modalities (RGB and D). All the experiments presented here are obtained with 2 refinement loops.

DataSet

LineMod dataset. The LineMOD dataset proposed by Hinterstoisser et al. [START_REF] Hinterstoisser | Multimodal templates for real-time detection of texture-less objects in heavily cluttered scenes[END_REF] is the main benchmark used to compare 6D pose estimation methods. It consists of a series of videos and indoor images including 13 objects with a low texture and different lighting level.

YCB dataset. The YCB-Video dataset Xiang et al. [START_REF] Xiang | Posecnn: A convolutional neural network for 6d object pose estimation in cluttered scenes[END_REF] characterizes 21 objects from the YCB Calli et al. [START_REF] Calli | The ycb object and model set: Towards common benchmarks for manipulation research[END_REF] database and being of different texture, size, and shape. The dataset contains a total of 92 videos with depth and RGB modality, in staging inside. These videos also include the annotations necessary for the 6D pose estimation work. We follow the previous work [START_REF] Xiang | Posecnn: A convolutional neural network for 6d object pose estimation in cluttered scenes[END_REF] regarding to the test and training set splitting.

Evaluation Metrics

The results of our network are evaluated by two metrics proposed in previous works. [START_REF] Xiang | Posecnn: A convolutional neural network for 6d object pose estimation in cluttered scenes[END_REF] We therefore use the ADD metrics for non-symmetric objects and ADD-S for symmetric objects. ADD is defined as follow :

ADD = 1 m i∈o (R i + t) -( Ri + t)) (3)
where i denotes a vertex in object o, and R, t the predicted pose and R, t the ground truth. And for symmetric object we calculate ADD-S such as:

ADD-S = 1 m i∈o min j∈o (R i + t) -( Rj + t)) (4)
According to the work of, [START_REF] Xiang | Posecnn: A convolutional neural network for 6d object pose estimation in cluttered scenes[END_REF] we evaluate the results of YCB DataSet by reporting the area under the ADD-S curve (AUC) and set the maximum threshold to be 0.1m. We also report the percentage of ADD smaller than 2cm.

LineMOD

LineMOD evaluation. The results are given in several parts. First, as shown in Table 1, we compare our network to classical DenseFusion approach re-trained from scratch for fair comparison. DF Ori is the orginal DenseFusion method, DF rt means that the fusion strategies has been handle by using a simple addition in the fusion bloc and OLF is our method where this fusion bloc takes into account the α parameters. Our method outperforms both DenseFusion strategies.

Robustness toward LineMOD+Noise. This is the core of our contribution. Introducing α 1 and α 2 parameter produces a method less sensitive to noises which could be present during the inference phase. This test is extremely important because in practice the input data are often noisy, especially depth modality in industrial environments. Thus, we consider the network is trained on exact data (LineMod) and the noise is added in the inference images. The results of the different noise scenarios can be seen in Table 1. The objective is both to identify the most problematic noise for the network and to verify that the Optimized Late Fusion is better than the classical fusion method. The noise strength is known through the SNR value estimated for each inputted σ.

• RGB+hN & D+hN: This case is the most noisy of all, here we apply a strong white noise on both modalities at the same time, that give us a SN R d of 9 for Depth and -15 for SN R RGB . We notice that in these conditions, the network is very poorly performing, on each type of objects. Nevertheless, we can observe that the OLF network performs better than the initial network by 3%

• RGB & D+hN: Here we only add a white noise on the depth modality with a equivalent SN R d of 8. We can notice a strong degradation of the network performance. Nevertheless, OLF remains more resistant than DF in this noise condition.

• RGB+hN & D: This particular case is interesting because despite a strong noise applied on RGB with SN R RGB = 1.7, the network keeps a very convincing performance or even almost unchanged. This result can be explained in part by the fact that the network gives more weight to the depth. To degrade the results in a similar way with noise only in RGB images, it would be necessary to apply an extremely strong noise to these images (see figure 2).

• RGB+N & D+N: Here a white noise of slightly reduced strength is applied on the two modalities (SN R d = 25, SN R RGB = 7) . This scenario is probably the closest to reality and the most interesting in our case. The Optimized Late Fusion performs much better than the original network. The resistance to noise brought by this new fusion method is felt on almost all objects, and globally the average success is improved by 4% on all test images.

Evolution of average on LineMod. Figure 2 shows the behavior of the average success of the DenseFusion network and the OLF network with growing noises. The noise is added with a growing value, allowing us to see the evolution of the percentage of success of the network as a function of the noise. The noise is, for this experiment, added either on RGB or on D, but never simultaneously.

The two curves (OLF + DN and DF + DN ) present the evolution of the average results obtained when the depth images are noisy. Our method dealing with noisy depth (OLF + DN ) is more efficient than the original method whatever the intensity of the noise added. We can see that the methods is very relevant for Depth and less for RGB, it's simply due to the methods itself, much more sensible to depth variation than to RGB variation. Inded when RGB is completly destroyed we can still perfom 6d pose estimation in 40 % of the cases.

OLF + RGBN and DF + RGBN show in an identical way, the evolution of the average precision according to the intensity of the noise applied on the RGB images. For higly noisy RGB images (SNR -20) the curve stabilises around 38%, which means that once the RGB is fully unusable, depth achieves to recover 38% of the poses. The OLF fusion allows to increase the resistance to noise in all the tested scenarios. We can conclude from these series of tests on the LineMod, that our fusion method has well strengthened the original methods against the noise.

YCB dataset

In order to more illustrate the robustness of our method, we tested its performances on the YCB datest with high levels of noise. We choose two different white noise values. The SNR are different from LineMod because the image ranges are not the same. noise is added in YCB images jointly on both D and RGB. Table 2 presents some results. The OLF method is slightly better than the DenseFusion in both ADD-s 2cm and AUC metrics. In presence of white noise, the OLF method strongly outperforms the original method. For SN R = 39 for example, a gap of 7% between the two methods is observed. These experiments confirm the robustness of our approach compared to DenseFusion.

Experiments under different noise Scenarios

We conducted experiments to evaluate the effectiveness of our fusion method in reducing the impact of noise on the network. We added noise to either RGB images, depth images, or both, and measured the impact of each type of noise on the final results. Gaussian white noise (GWN) was added to the RGB images, and multiplicative GWN was added to the depth information to avoid unwanted information on the 0 returned by the physical sensor. We used the signal-to-noise ratio (SNR) to quantify the noise level in our experiments, with a lower SNR indicating stronger noise.

The two curves (OLF + DN and DF + DN ) present the evolution of the average results obtained when the depth images are noisy. Our method dealing with noisy depth (OLF + DN ) is more efficient than the original method whatever the intensity of the noise added.

OLF + RGBN and DF + RGBN show, in an identical way, the evolution of the average precision according to the intensity of the noise applied on the RGB images.

CONCLUSION

In this paper, we have proposed an optimized fusion method based on DenseFusion. Even if DenseFusion is working well, we have shown that this method is very sensitive to noise which makes our method more efficient for real application where data (especially depth data) are often noisy. We have shown here that a simple modification of the fusion module which takes into account the accuracy of the modalities improves the results and the robustness. These revisited DenseFusion-like methods are then more efficient and usable in real scenario. Many neural network methods of the state of the art as (PVN3D, 7 FFB6D, 27 ...) use a fusion module which concatenate RGB and Depth information and are good candidates for this new fusion methods. As our Optimized Late Fusion module is generic, it can be also added in these neural network methods to improve their robustness toward noise.

Figure 1 .

 1 Figure 1. Global overview of our Optimized late fusion Our method is based on the DenseFusion 5 Architecture : after segmentation, Features are extracted from RGB and Depth and are then fused by our adaptive Fusion Module. The features generated by this fusion are then used by a pose predictor and a refiner. The pose Predictor predicts pose for each pixel and the predictions are voted to generate the final pose.

Figure 2 .

 2 Figure 2. DenseFusion and OLF comparison related to noise evolution in RGB and depth

Table 1 .

 1 Results of LineMOD under different noise scenario, with OLF(our)+(Refinement) and DF+(Refinement), hN mean high noise and N noise.

		RGB+hN & D+hN	RGB & D+hN	RGB+hN & D	RGB+N & D+N	NoNoise
	Network Type Df+R	OLF+R	Df+R OLF+R Df+R OLF+R Df+R OLF+R	DF	OLF
	ape	00,9	00,7	01,0	00,6	90,1	93,6	63,4	78,6	92,3	94,0
	bench vi.	11,4	22,9	11,3	21,7	91,4	92,3	90,6	91,0	93,2	92,5
	camera	02,6	02,9	02,3	03,2	94,7	94,0	88,9	91,6	94,4	94,4
	can	04,6	13,6	04,2	12,8	92,4	96,2	90,7	94,9	93,1	96,6
	cat	01,3	03,8	01,2	04,9	95,8	96,6	91,2	94,7	96,5	97,0
	driller	01,3	08,4	01,9	08,1	85,0	92,4	83,0	91,1	87,0	92,2
	duck	00,9	01,0	00,5	00,4	91,4	95,9	76,3	85,4	92,3	96,2
	eggbox	35,3	48,3	36,4	45,9	99,8	99,6	99,8	99,8	99,8	99,9
	glue	41,4	26,9	40,1	28,5	99,8	99,5	99,6	99,4	100,0 99,8
	hole p.	01,2	03,3	01,9	03,9	88,9	88,3	78,8	82,3	92,1	88,2
	iron	18,5	19,2	18,1	18,6	97,5	96,9	96,9	96,1	97,0	97,0
	lamp	01,6	08,4	01,7	09,5	95,6	96,2	94,3	95,5	95,3	96,0
	phone	12,0	11,7	11,4	13,3	92,5	95,7	90,8	94,3	92,8	95,8
	AVERAGE	10,3	13,3	10,2	13,2	93,4	95,2	88,0	92,0	94,3	95,3

Table 2 .

 2 Quantitative evaluation of 6D pose (AUC and ADD(2cm)) on the YCB data + Noise.

		DF ori	OLF(ours)
	nonoise	AUC < 2cm AUC < 2cm
		93.1	96.8	93.8	97.2
	SN R D = 39 AUC < 2cm AUC < 2cm
	SN R rgb = 6 57.4	60.2	63.4	67.1
	SN R D = 33 AUC < 2cm AUC < 2cm
	SN R rgb = 3 37.6	40.1	44.6	51.1


Joachim Claudet

Cassandra M Brooks

HAL is

Making protected areas in the high seas count

More than 15 years in the making, the High Seas Treaty-the legally binding instrument under the United Nations Convention on the Law of the Sea on the conservation and sustainable use of marine biological diversity of areas beyond national jurisdiction-was agreed upon by UN member states in March. Vast in scope, the treaty applies to about two-thirds of the ocean and includes a provision to conserve biodiversity by using legal tools to design, implement, and manage marine protected areas (MPAs) in areas beyond national jurisdiction. MPAs can be a valuable step toward conservation goals, but their effectiveness depends on their ability to limit human activities within their borders. Because the High Seas Treaty could not directly address the legal fragmentation of ocean governance, participating states will have to work together through multiple regulatory frameworks to implement MPAs coherently.

The main drivers of marine biodiversity erosion are fishing and sea use change ( 1), but the High Seas Treaty does not guarantee that future MPAs will directly regulate high seas fisheries or mining of the international seabed. Article 4 stipulates that the Treaty should not undermine the legal frameworks already put in place by global, regional, subregional or sectoral bodies (2). Hence, a siloed status quo is a troubling possibility: Regional Fisheries Management Organizations (RFMOs), as recognized by the UN Food and Agriculture Organization, could continue to manage high seas fisheries, and mining of the international seabed could remain within the scope of the UN International Seabed Authority.

The power to ensure that the High Seas Treaty realizes its full potential and becomes a tool of coherence rather than fragmentation rests with states. Poorly designed MPAs undermined by sectoral squabbling would be a frustration to conservationists and states alike. A multisectoral approach that builds on the best elements of multilateralism in the High Seas Treaty negotiations will require that countries seek coherence across the relevant regional and sectoral bodies to ensure that MPAs in the high seas effectively conserve biodiversity (3). For instance, countries conducting fishing in an area of the high seas proposed to be part of an MPA should agree to implement and enforce, through the relevant RFMOs, fishing regulations consistent with the MPA objective of biodiversity conservation. Not doing so carries the risk that MPAs will not protect against existing threats to biodiversity or that MPAs will only be designated in areas with limited conservation value (4). Aquat. Conserv. Mar. Freshw. Ecosyst. 25, 480 (2014).

Joachim

10.1126/science.adh4924

  sciencemag.org SCIENCE VOL. xxx • galley printed 30 April, 2023 • • For Issue Date: ???? 1

Claudet 1 *, Cassandra M. Brooks 2 , Robert Blasiak 3,4

  Draft agreement under the United Nations Convention on the Law of the Sea on the conservation and sustainable use of marine biological diversity of areas beyond national jurisdiction" (2023). 3.K. Grorud-Colvert et al., Science 373, eabf0861 (2021). 4.R. Devillers et al.,

	1 National Center for Scientific Research (CNRS),
	Université Paris Sciences & Lettres, Centre de
	Recherches Insulaires et Observatoire de
	l'Environnement, Paris, France. 2 Department of
	Environmental Studies, University of Colorado Boulder,
	Boulder, CO, USA. 3 Stockholm Resilience Centre,
	Stockholm University, Stockholm, Sweden. 4 Graduate
	School of Agricultural and Life Sciences, The University
	of Tokyo, Tokyo, Japan.
	*Corresponding author. Email:
	joachim.claudet@cnrs.fr
	REFERENCES AND NOTES
	1.Intergovernmental Science-Policy Platform on Biodi-
	versity and Ecosystem Services, "Summary for poli-
	cymakers of the global assessment report on biodi-
	versity and ecosystem services of the
	Intergovernmental Science-Policy Platform on Bio-
	diversity and Ecosystem Services" (2019).
	2.UN General Assembly, "