R Émi Carles

ON NONLINEAR EFFECTS IN MULTIPHASE WKB ANALYSIS FOR THE NONLINEAR SCHR ÖDINGER EQUATION

We consider the Schrödinger equation with an external potential and a cubic nonlinearity, in the semiclassical limit. The initial data are sums of WKB states, with smooth phases and smooth, compactly supported initial amplitudes, with disjoint supports. We show that like in the linear case, a superposition principle holds on some time interval independent of the semiclassical parameter, in several régimes in term of the size of initial data with respect to the semiclassical parameter. For large data, we invoke properties of compressible Euler equations. For smaller data, we show that there may be no nonlinear interferences on some time interval independent of the semiclassical parameter, and interferences for later time, thanks to explicit computations available for particular phases.

iε∂ t ψ ε + ε 2 2 ∆ψ ε = V ψ ε + |ψ ε | 2 ψ ε .
The potential V = V (x) is supposed real-valued, smooth, and at most quadratic:

(1.1) V ∈ C ∞ (R d , R), ∂ α V ∈ L ∞ (R d ), ∀α ∈ N d , |α| 2.
Typical examples are V = 0, V linear (V (x) = E • x), V harmonic (V (x) = ω 2 |x| 2 ), V ∈ S(R d ), or any sum of such potentials. As initial data, we consider the sum of WKB states:

ψ ε (0, x) = ε γ/2 N j=1
α j (x)e iϕj (x)/ε , with γ 0. The value of γ measures the size of the initial data, and thus the importance of nonlinear effects in the semiclassical limit ε → 0. The case γ = 0 is supercritical in terms of WKB analysis: the evolution of the phase describing the rapid oscillation is given by an eikonal equation which involves the leading order amplitude, and a standard application of the WKB asymptotic expansion leads to systems which are not closed, no matter how many correcting terms are considered (see e.g. [START_REF] Carles | Semi-classical analysis for nonlinear Schrödinger equations: WKB analysis, focal points, coherent states[END_REF]Chapter 1] or [START_REF] Gérard | Remarques sur l'analyse semi-classique de l'équation de Schrödinger non linéaire[END_REF]).

This work was supported by Centre Henri Lebesgue, program ANR-11-LABX-0020-0. A CC-BY public copyright license has been applied by the author to the present document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission.

In order to consider solutions of size O(1) in terms of ε, we consider the unknown function u ε = ε -γ/2 ψ ε instead of ψ ε , where u ε thus solves

(1.2) iε∂ t u ε + ε 2 2 ∆u ε = V u ε + ε γ |u ε | 2 u ε , with initial data (1.3) u ε (0, x) = u ε 0 (x) := N j=1
α j (x)e iϕj (x)/ε .

The assumptions regarding the initial data are listed below:

Assumption 1.1. The phases are smooth and real-valued, ϕ j ∈ C ∞ (R d , R). The initial amplitudes are smooth and compactly supported:

α j ∈ C ∞ 0 (R d , C
), with pairwise disjoint supports, supp α j1 ∩ supp α j2 = ∅, j 1 = j 2 .

The case N = 1, referred to as monokinetic case, is well understood for short time, as we recall below, in the sense that the asymptotic behavior of u ε as ε → 0 is described precisely, locally in time on some interval independent of ε. The large time behavior is, in general, unknown; the one-dimensional case, with V = 0, is an exception, since it is completely integrable, see e.g. [START_REF] Jin | The semiclassical limit of the defocusing NLS hierarchy[END_REF][START_REF] Miller | On the generation of dispersive shock waves[END_REF]. Consider the case γ = 0. When V ≡ 0, the leading order asymptotic description involves the compressible Euler equation (1.4) ∂ t ρ + div (ρv) = 0,

∂ t v + v • ∇v + ∇ρ = 0.
This equation is quasilinear, while (1.2) is semilinear (the nonlinear term is viewed as a perturbation when solving the Cauchy problem). In Section 2, we recall how to justify, in this case, the existence of a WKB approximation of the form u ε (t, x) = a(t, x) + εa 1 (t, x) + . . . + ε k a k (t, x) e iφ(t,x)/ε

+ O ε k+1 , in L ∞ ([0, T ], L 2 ∩ L ∞ (R d ))
, for all k 0, for some T > 0 independent of ε. We choose to measure errors in L 2 ∩ L ∞ in the spatial norm, in order to avoid to introduce ε-dependent norms when derivatives are involved, due to rapid oscillations. This time T can be taken as the lifespan of the smooth solution to the Euler equation (1.4) with suitable initial data. When N 2, the new question arising is the nonlinear interaction of the WKB states. As the problem is supercritical, even a formal computation is a delicate issue: if we plug an approximate solution of the form

u ε app (t, x) = M j=1
b j (t, x)e iφj (t,x)/ε into (1.2), how do we choose M (possibly infinite), and which equations must be satisfied by the amplitudes b j and the phases φ j ? Surprisingly enough, it turns out that as long as the solutions of the Euler equations, involved in the description of each individual initial WKB state, are smooth, there is no interaction, at arbitrary order in terms of powers of ε.

Remark 1.2 (Infinitely many states). The case N = ∞ may also be addressed, under suitable assumptions on the growth in space of the phases φ j compared to the size of the support of α j , as j → ∞. More precisely, as will be clear from the proof of the main result, we can consider the case N = ∞ provided that we may find cutoff functions χ j so that

φ 0 = ∞ j=1 ϕ j χ j ∈ H ∞ (R d ) := ∩ s>0 H s (R d ),
or at least in a weaker form if φ 0 ∈ H s (R d ) for some s > 2+d/2. Another constraint, in this case, is that we have to find a common lower bound for the lifespan of all the approximate solutions (φ j , a j ) considered below, an aspect which is obvious when N is finite, since we consider the minimum of a finite set.

1.2. Main results. The nonlinear evolution of each initial WKB state will play a crucial role:

(1.5)

iε∂ t u ε j + ε 2 2 ∆u ε j = V u ε j + ε γ |u ε j | 2 u ε j ; u j|t=0 = α j e iϕj /ε .
Under our assumptions, for fixed initial data, we know that:

• If d 3, the equation is energy-subcritical, and for fixed ε > 0, there exists a unique solution

u ε ∈ L ∞ (R; H 1 (R d ))
, and it is smooth. See e.g. [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF].

• If d = 4
, the equation is energy-critical: the above conclusion is known to remain when V = 0 ( [START_REF] Ryckman | Global well-posedness and scattering for the defocusing energycritical nonlinear Schrödinger equation in R 1+4[END_REF]), when V is an isotropic quadratic potential ( [START_REF] Killip | Energy-critical NLS with quadratic potentials[END_REF]), or when V is harmonic at infinity ( [START_REF] Jao | Energy-critical NLS with potentials of quadratic growth[END_REF]). • If d 5, the equation is energy-supercritical: only a local in time smooth solution is known to exist by classical theory. In the cases where the global existence of a smooth solution is not known, the local existence time might go to zero as ε → 0, so the existence of a smooth solution on a time interval independent of ε > 0 is already a nontrivial step. The description of the solutions u ε j as ε → 0 on some time interval [0, T j ] independent of ε was evoked above, and is recalled in Sections 2 (case V = 0) and 3 (V satisfying (1.1)). Our main result is the following nonlinear superposition principle:

Theorem 1.3. Let d 1, V satisfying (1.1), γ 0

, and initial data satisfying Assumption 1.1. There exists

T * > 0 independent of ε ∈]0, 1] such that (1.2)-(1.3) has a unique solution u ε ∈ C([0, T * ], H ∞ (R d )). In addition, sup t∈[0,T * ] u ε (t) - N j=1 u ε j (t) L 2 ∩L ∞ = O ε k , ∀k > 0,
where u ε j is the solution of (1.5). Let us discuss this result in the supercritical case γ = 0, as it is the case where Theorem 1.3 may be the most surprising. The result follows from a detailed WKB analysis, as well as a property of finite speed of propagation for the compressible Euler equation, discovered initially in [START_REF] Makino | Sur la solution à support compact de l'équations d'Euler compressible[END_REF]. The key feature of our setting is the compact, disjoint supports of the initial amplitudes α j . In the case V = 0, as long as WKB analysis is valid for each u ε j in (1.5), u ε j remains supported in (essentially) supp α j up to O(ε ∞ ): all the amplitude terms of the WKB expansion (at leading order, as well as correctors at arbitrary order) remain compactly supported, and amplitudes associated with u ε j1 and u ε j2 , respectively, with j 1 = j 2 , do not interact. In the case V ≡ 0, u ε j remains supported in supp α j evolving according to the classical flow generated by V , up to O(ε ∞ ). In other words, we recover the same phenomenon, regarding the evolution of supports, as in the linear case (see e.g. [START_REF] Maslov | Semi-classical approximation in quantum mechanics[END_REF][START_REF] Robert | Autour de l'approximation semi-classique[END_REF]), even though the régime associated to (1.2) is strongly nonlinear (of course Theorem 1.3 is trivial in the linear case, as u ε ≡ u ε j ). In particular, the initial modes cannot interact at a "visible" order before WKB analysis for at least one of the u ε j 's ceases to be valid, that is, before the solution of the corresponding Euler equation (1.4) breaks down (see however Section 5 for a discussion on the influence of our proof strategy on this statement). Recent progress on this precise question, [START_REF] Merle | On the implosion of a compressible fluid I: Smooth self-similar inviscid profiles[END_REF][START_REF] Merle | On the implosion of a compressible fluid II: Singularity formation[END_REF][START_REF] Buckmaster | Smooth imploding solutions for 3D compressible fluids[END_REF] (see also [START_REF] Merle | On blow up for the energy super critical defocusing nonlinear Schrödinger equations[END_REF] for a relation with the nonlinear Schrödinger equation), suggests that the expected scenario is rather that of an implosion: the conclusion of Theorem 1.3 might remain valid even after WKB has ceased to be valid. Remark 1.4 (Wigner measures). Since the proof of Theorem 1.3 relies on WKB analysis, it also implies the characterization of Wigner measures. Recall that the Wigner transform of u ε is defined by

w ε (t, x, ξ) = (2π) -d R d u ε t, x -ε η 2 u ε t, x + ε η 2 e iη•ξ dη.
The position and current densities can be recovered from w ε , by

|u ε (t, x)| 2 = R d w ε (t, x, ξ)dξ, Im (εu ε ∇u ε ) (t, x) = R d ξw ε (t, x, ξ)dξ.
A measure µ is a Wigner measure associated to u ε (there is no uniqueness in general) if, up to extracting a subsequence, w ε converges to µ as ε → 0 (see e.g. [START_REF] Gérard | Homogenization limits and Wigner transforms[END_REF][START_REF] Lions | Sur les mesures de Wigner[END_REF]). In the context of Theorem 1.3, each wave function u ε j has a unique Wigner measure, and the sum of these Wigner measures is the Wigner measure of u ε . For instance, if V = γ = 0,

µ(t, dx, dξ) = N j=1 ρ j (t, x)dx ⊗ δ ξ=vj (t,x) ,
where (ρ j , v j ) solves (1.4) with initial data (ρ j , v j ) |t=0 = (|α j | 2 , ∇(χ j ϕ j )), and

χ j ∈ C ∞ 0 (R d , [0, 1]
)) is (any function) such that χ j ≡ 1 on the support of α j . See Section 5 for the dependence of this statement upon χ j .

The next result shows that in the weakly nonlinear case γ = 1, some explicit information is available, in the sense that indeed, nonlinear interferences are negligible on some time interval [0, T 0 ] with T 0 > 0 independent of ε, while nonlinear interferences occur later.

Proposition 1.5. Let d 1. There exist k 1 , k 2 ∈ R d , and α 1 , α 2 ∈ C ∞ 0 (R d
) with disjoint supports, such that the following holds. There exist T 1 > 0 and T 0 ∈ (0, T 1 ) independent of ε, such that the solution to

(1.6) iε∂ t u ε + ε 2 2 ∆u ε = ε|u ε | 2 u ε ; u ε 0 (x) = α 1 (x)e ik1•x/ε + α 2 (x)e ik2•x/ε , satisfies sup t∈[0,T0] u ε (t) - 2 j=1 u ε j (t) L p = O ε k , ∀k > 0, ∀p ∈ [2, ∞],
and

lim inf ε→0 sup t∈[0,T1] u ε (t) - 2 j=1 u ε j (t) L p > 0, ∀p ∈ [2, ∞].
The proof of Proposition 1.5 relies on explicit computations available in this weakly nonlinear case, and the fact that for linear oscillations, no caustic appears in the case of a single WKB state: the nature of nonlinear interferences is shown in Section 6, and consists of nonlinear phase modulations. In an appendix, we give an alternative argument illustrating another type of nonlinear interferences at leading order, consisting of the creation of a new mode (when d 2): starting from three WKB states, a fourth one, associated with a new phase, may appear by resonant interaction.

1.3. Content. In Section 2, we recall the WKB construction introduced in [START_REF] Grenier | Semiclassical limit of the nonlinear Schrödinger equation in small time[END_REF] for the case γ = 0, and emphasize the finite speed of propagation which appears in our framework. In Section 3, we explain how to adapt the previous approach to the case where V satisfies (1.1), and address the case γ > 0. In Section 4, we complete the proof of Theorem 1.3. Section 5 clarifies the role of the cutoff functions used in the proof of Theorem 1.3. Propositions 1.5 is established in Section 6. In an appendix, we propose an alternative proof of Proposition 1.5, in the case d 2 with N = 3, showing that there are several sorts of nonlinear interferences in the weakly nonlinear case.

The monokinetic case without potential

In this section, we consider (1.2)-(1.3) in the monokinetic N = 1, and in the supercritical case γ = 0, with slightly different notations for future reference:

(2.1)

iε∂ t u ε + ε 2 2 ∆u ε = |u ε | 2 u ε ; u ε |t=0 = a 0 e iφ0/ε .
In view of the setting of this paper, we assume

a 0 , φ 0 ∈ C ∞ 0 (R d ). In particular, a 0 , φ 0 ∈ H ∞ (R d ).
We first consider the case V ≡ 0, then introduce the main ideas that make it possible to incorporate a subquadratic potential V .

We recall the main steps to the construction introduced in [START_REF] Grenier | Semiclassical limit of the nonlinear Schrödinger equation in small time[END_REF] (see also [START_REF] Carles | Semi-classical analysis for nonlinear Schrödinger equations: WKB analysis, focal points, coherent states[END_REF]Section 4.2]). The idea introduced in [START_REF] Grenier | Semiclassical limit of the nonlinear Schrödinger equation in small time[END_REF] consists in writing the solution to (2.1) as

(2.2) u ε (t, x) = a ε (t, x)e iφ ε (t,x)/ε ,
with a ε complex-valued and φ ε real-valued, solving

(2.3)      ∂ t φ ε + 1 2 |∇φ ε | 2 + |a ε | 2 = 0, φ ε |t=0 = φ 0 , ∂ t a ε + ∇φ ε • ∇a ε + 1 2 a ε ∆φ ε = i ε 2 ∆a ε , a ε |t=0 = a 0 .
The key remark is that this leads to a symmetric hyperbolic system, perturbed be a skew-symmetric term. The hyperbolic system appears when considering the unknown

U ε =   Re a ε Im a ε ∇φ ε   =   Re a ε Im a ε v ε   .
Considering the gradient of the first equation in (2.3), the system can be written

(2.4) ∂ t U ε + d j=1 A j (U ε )∂ j U ε = εLU ε , with A(U ε , ξ) = d j=1 A j (U ε )ξ j =   v ε • ξ 0 1 2 Re a ε t ξ 0 v ε • ξ 1 2 Im a ε t ξ 2 Re a ε ξ 2 Im a ε ξ v ε • ξI d   , and
L =   0 -∆ 0 . . . 0 ∆ 0 0 . . . 0 0 0 0 d×d   .
To be precise, the system is made symmetric thanks to the constant symmetrizer

S = I 2 0 0 1 4 I d .
Once v ε is known, one recovers φ ε by integrating in time the first equation in (2.3),

φ ε (t, x) = φ 0 (x) - 1 2 t 0 |v ε (s, x)| 2 ds - t 0 |a ε (s, x)| 2 ds,
and since ∂ t (v ε -∇φ ε ) = 0, v ε = ∇φ ε .
Assuming that a 0 , ∇φ 0 ∈ H s (R d ) for s large (we will always assume a 0 , φ 0 ∈ C ∞ 0 (R d ) in the forthcoming applications), the limit ε → 0 leads to an asymptotic expansion of the form

φ ε ∼ φ + εφ (1) + ε 2 φ (2) + . . . , a ε ∼ a + εa (1) + ε 2 a (2) + . . .
The leading order term is obtained by simply setting ε = 0 in (2.4):

(2.5)

     ∂ t φ + 1 2 |∇φ| 2 + |a| 2 = 0, φ |t=0 = φ 0 , ∂ t a + ∇φ • ∇a + 1 2 a∆φ = 0, a |t=0 = a 0 .
Working with the intermediary unknown v = ∇φ, we get a system of the form

∂ t U + d j=1 A j (U )∂ j U = 0,
and we infer the following result from [START_REF] Makino | Sur la solution à support compact de l'équations d'Euler compressible[END_REF]: (2.5). Moreover, (φ, a) remains compactly supported for t ∈ [0, T * ], and

Proposition 2.1. Let a 0 , φ 0 ∈ C ∞ 0 (R d ), with supp a 0 , supp φ 0 ⊂ K. There exists T * > 0 and a unique solution (φ, a) ∈ C([0, T * ], H ∞ (R d )) 2 to
supp φ(t, •), supp a(t, •) ⊂ K.
The first part of the statement is a consequence of classical theory for symmetric hyperbolic systems (see e.g. [START_REF] Alinhac | Pseudo-differential operators and the Nash-Moser theorem[END_REF][START_REF] Majda | Compressible fluid flow and systems of conservation laws in several space variables[END_REF]). The property stated that initial compactly supported condition lead to a zero speed of propagation is due to the structure of this hyperbolic system, and is well understood from the simplest model of the Burgers equation

∂ t u + u∂ x u = 0, u |t=0 = u 0 ∈ C ∞ 0 (R).
Suppose we have a smooth solution on some time interval [0, T * ]. In particular,

T * 0 ∂ x u(t) L ∞ dt < ∞.
We have directly, for all (t, x)

∈ [0, T * ] × R, |∂ t u(t, x)| ∂ x u(t) L ∞ |u(t, x)|.
Gronwall lemma then shows that if u 0 (x 0 ) = 0, then u(t, x 0 ) = 0 for all t ∈ [0, T * ], hence the zero speed of propagation for smooth solutions. As the matrix A(U, ξ) is linear in U , the result follows in the setting of (2.5). Note that to prove this zero speed of propagation, we do not invoke the symmetry of A: it was used in order to get Sobolev estimates (which ensure that U ∈ L 1 ([0, T * ], W 1,∞ )), but only the fact that it is (at least) linear in U is used at this stage. We then have, for the same T * as in Proposition 2.1:

Proposition 2.2. Let a 0 , φ 0 ∈ C ∞ 0 (R d ). There exists T * > 0 independent of ε ∈]0, 1] such that for all s 0, there exists C = C(s) such that φ ε -φ L ∞ ([0,T * ],H s (R d )) + a ε -a L ∞ ([0,T * ],H s (R d )) Cε.
To infer the pointwise description of u ε at leading order, we must in addition know φ ε up to o(ε), which is achieved by considering the linearization of (2.5). At the next step of the WKB expansion, we find that

φ ε -φ -εφ (1) L ∞ ([0,T * ],H s (R d )) + a ε -a -εa (1) L ∞ ([0,T * ],H s (R d )) Cε 2 ,
where the first corrector (φ (1) , a (1) ) solves the system:

           ∂ t φ (1)
+ ∇φ • ∇φ (1) + 2 Re aa (1) = 0, ∂ t a (1) + ∇φ • ∇a (1) + ∇φ (1) • ∇a

+ 1 2 a (1) ∆φ + 1 2 a∆φ (1) = i 2 ∆a, φ (1)
|t=0 = 0 ; a (1)
|t=0 = 0. At higher order k 2, the corrector (φ (k) , a (k) ) is given by:                        ∂ t φ (k) + ∇φ • ∇φ (k) + 2 Re aa (k) = F k ∇φ (ℓ) 1 ℓ k-1 , a (ℓ) 1 ℓ k-1 , ∂ t a (k) + ∇φ • ∇a (k) + ∇φ (k) • ∇a + 1 2 a (k) ∆φ + 1 2 a∆φ (k) = i 2 ∆a (k-1)
-

1 ℓ k-1 ∇φ (ℓ) • ∇a (k-ℓ) - 1 2 1 ℓ k-1 a (k-ℓ) ∆φ (ℓ) , φ (k) |t=0 = 0 ; a (k)
|t=0 = 0, for some function F k which is a polynomial in its arguments, without constant term, and whose precise expression is unimportant here. The left hand side is always the linearization of the left hand side of (2.5) about (φ, a), and the right hand side depends on previous correctors. We infer (see [START_REF] Grenier | Semiclassical limit of the nonlinear Schrödinger equation in small time[END_REF][START_REF] Carles | Semi-classical analysis for nonlinear Schrödinger equations: WKB analysis, focal points, coherent states[END_REF]), by induction:

Proposition 2.3. Let a 0 , φ 0 ∈ C ∞ 0 (R d ).
Let T * > 0 given by Proposition 2.1. For all k 1, there exists a unique solution (φ

(k) , a (k) ) ∈ C([0, T * ], H ∞ (R d ))
2 to the above system, and for all s 0, there exists C = C(k, s) such that

φ ε -φ -εφ (1) -. . . -ε k φ (k) L ∞ ([0,T * ],H s (R d )) + a ε -a -εa (1) -. . . -ε k a (k) L ∞ ([0,T * ],H s (R d )) Cε k+1 .
In addition, if supp a 0 , supp φ 0 ⊂ K, then (φ (k) , a (k) ) remains compactly supported for t ∈ [0, T * ], and

supp φ (k) (t, •), supp a (k) (t, •) ⊂ K.
The support property is a consequence of the same argument as in the proof of Proposition 2.1. Using the embedding

H s (R d ) ⊂ L ∞ (R d ) for s > d/2,
we also deduce from the above error estimate the bound, for k 1:

(2.6) u ε - k-1 ℓ=0 ε ℓ a (ℓ) exp i ε k ℓ=0 ε ℓ φ (ℓ) L ∞ ([0,T * ],L 2 ∩L ∞ (R d )) = O ε k ,
with the convention (φ (0) , a (0 ) = (φ, a). The standard form of WKB expansions,

u ε (t, x) = a(t, x) + εa 1 (t, x) + . . . + ε k a k (t, x) e iφ(t,x)/ε + O L ∞ T * (L 2 ∩L ∞ ) ε k+1
, is then obtained by setting a = ae iφ (1) , a 1 = a (1) e iφ (1) + iaφ (2) e iφ (1) , etc. Remark 2.4 (Higher order nonlinearities). If instead of (2.1), one considers

iε∂ t u ε + ε 2 2 ∆u ε = |u ε | 2σ u ε ; u ε |t=0 = a 0 e iφ0/ε ,
with σ 2 an integer, then the justification of WKB analysis requires a different approach. We refer to [START_REF] Alazard | Supercritical geometric optics for nonlinear Schrödinger equations[END_REF][START_REF] Chiron | Geometric optics and boundary layers for nonlinear Schrödinger equations[END_REF] for two different proofs, which show that the conclusions of the propositions stated in this section remain valid.

Remark 2.5 (Focusing nonlinearity). If instead of (2.1), one considers a cubic focusing nonlinearity,

iε∂ t u ε + ε 2 2 ∆u ε = -|u ε | 2 u ε ; u ε |t=0 = a 0 e iφ0/ε ,
then the analogue of (2.5) is no longer hyperbolic, but elliptic. Working with analytic initial data (φ 0 , a 0 ) is then necessary in order to solve (2.5) ( [START_REF] Lerner | The onset of instability in first-order systems[END_REF][START_REF] Métivier | Remarks on the well-posedness of the nonlinear Cauchy problem[END_REF]), and this is a framework where nonlinear WKB analysis is fully justified ( [START_REF] Gérard | Remarques sur l'analyse semi-classique de l'équation de Schrödinger non linéaire[END_REF][START_REF] Thomann | Instabilities for supercritical Schrödinger equations in analytic manifolds[END_REF]). However, analyticity is incompatible with an initial compact support. On the other hand, in the weakly nonlinear case γ = 1 (and more generally if γ 1), it is possible to justify WKB analysis with a focusing nonlinearity and compactly supported initial data (see e.g. [START_REF] Carles | WKB analysis for nonlinear Schrödinger equations with potential[END_REF] or [START_REF] Carles | Semi-classical analysis for nonlinear Schrödinger equations: WKB analysis, focal points, coherent states[END_REF]Chapter 2]).

The monokinetic case with a potential

In this section, we first recall some elements of WKB analysis in the linear case. We then show how this case can be merged with the analysis presented in the previous section, when γ = 0. We sketch how the case of a weaker nonlinearity, 0 < γ < 1. To conclude, we briefly discuss the weakly nonlinear régime γ = 1, and more generally the situation γ 1.

3.1. Linear case. The eikonal equation associated to

(3.1) iε∂ t u ε + ε 2 2 ∆u ε = V u ε ; u ε |t=0 = a 0 e iϕ0/ε
, that is, without initial rapid oscillation, reads:

(3.2) ∂ t φ eik + 1 2 |∇φ eik | 2 + V = 0 ; φ eik|t=0 = ϕ 0 .
In this subsection, we assume that ϕ 0 is smooth and at most quadratic, in the same sense as in (1.1). This eikonal equation is solved by introducing the classical trajectories, solving (3.3) ẋ(t, y) = ξ(t, y), x(0, y) = y ; ξ(t, y) = -∇V (x(t, y)) , ξ(0, y) = ∇ϕ 0 (y).

As V is at most quadratic, from (1.1), the above system has a unique, global, smooth solution, and in addition

∇ y x(t, y) = I d + O(t),
uniformly in y ∈ R d , for any matricial norm on R d×d . Therefore, the Jacobi determinant J t (y) = det ∇ y x(t, y), remains non-zero and bounded on some time interval [0, T ] with T > 0. Since we also have, by uniqueness in ordinary differential equations, ∇φ eik (t, x(t, y)) = ξ(t, y), for any smooth solutions to (3.2), the global inversion theorem implies the following result (see also [5, Proposition 1.9]): Lemma 3.1. Let V satisfying (1.1), and ϕ 0 satisfying the same properties. There exists T > 0 and a unique solution

φ eik ∈ C ∞ [0, T ] × R d to (3.

2). In addition, this solution is at most quadratic in space:

∂ α x φ eik ∈ L ∞ ([0, T ] × R d )
as soon as |α| 2. There exists C > 1 such that the Jacobi determinant satisfies:

1 C J t (y) C, ∀(t, y) ∈ [0, T ] × R d .
Since the above relations imply ẋ(t, y) = ∇φ eik (t, x(t, y)) ,

we infer the classical formula (3.4) ∂ t J t (y) = J t (y)∆φ eik (t, x(t, y)) .

In the linear case, the leading order amplitude is given by the linear transport equation

∂ t a + ∇φ eik • ∇a + 1 2 a∆φ eik = 0 ; a |t=0 = a 0 .
Following the classical trajectories, this transport equation becomes trivial, since A(t, y) := J t (y)a (t, x(t, y)) satisfies ∂ t A = 0.

3.2. Supercritical case: γ = 0. We consider the same framework as in the previous section, now with a potential:

(3.5) iε∂ t u ε + ε 2 2 ∆u ε = V u ε + |u ε | 2 u ε ; u ε |t=0 = a 0 e iφ0/ε .
As noticed in [START_REF] Carles | WKB analysis for nonlinear Schrödinger equations with potential[END_REF], it is possible to adapt the above WKB analysis in the presence of an external potential satisfying (1.1) by simply mixing the standard approach followed in the linear case (see e.g. [START_REF] Robert | Autour de l'approximation semi-classique[END_REF]) and Grenier's method.

3.2.1. Introducing the nonlinearity. As noticed in [START_REF] Carles | WKB analysis for nonlinear Schrödinger equations with potential[END_REF], the approach presented in the case V = 0 for the nonlinear case can be adapted by changing the representation (2.2) to u ε (t, x) = a ε (t, x)e iφ eik (t,x)/ε+iφ ε (t,x)/ε , where φ eik solves (3.2) with ϕ 0 ≡ 0, and requiring

(3.6)            ∂ t φ ε + ∇φ eik • ∇φ ε + 1 2 |∇φ ε | 2 + |a ε | 2 = 0, ∂ t a ε + ∇φ eik • ∇a ε + ∇φ ε • ∇a ε + 1 2 a ε ∆φ eik + 1 2 a ε ∆φ ε = i ε 2 ∆a ε , φ ε |t=0 = φ 0 ; a ε |t=0 = a 0 .
The new terms compared to (2.3) involve φ eik , and since φ eik is at most quadratic in space, it turns out that they can be estimated like (semilinear) perturbative terms (using commutator estimates for the transport part). The natural limit for (3.9) when ε → 0 is given by

(3.7)            ∂ t φ + ∇φ eik • ∇φ + 1 2 |∇φ| 2 + |a| 2 = 0, ∂ t a + ∇φ eik • ∇a + ∇φ • ∇a + 1 2 a∆φ eik + 1 2 a∆φ = 0, φ |t=0 = φ 0 ; a |t=0 = a 0 .
The following result is a consequence of [START_REF] Carles | WKB analysis for nonlinear Schrödinger equations with potential[END_REF]: 2 , and for all s 0, there exists C = C(s) such that

Proposition 3.2. Let a 0 , φ 0 ∈ C ∞ 0 (R d ), V satisfying (1.

1), and T , φ eik given by Lemma 3.1. There exists

0 < T * T independent of ε ∈]0, 1] such that (3.9) has a unique solution (φ ε , a ε ) ∈ C([0, T * ], H ∞ (R d )) 2 , (3.7) has a unique solution (φ, a) ∈ C([0, T * ], H ∞ (R d ))
φ ε -φ L ∞ ([0,T ],H s (R d )) + a ε -a L ∞ ([0,T ],H s (R d )) Cε.
The correctors φ (j) , a (j) j 1 as obtained in the same fashion as in Section 2. The only difference is that the operator ∂ t is replaced by

∂ t + ∇φ eik • ∇ + 1 2 ∆φ eik .

Finite speed of propagation: following the classical trajectories.

In order to prove that if a 0 ∈ C ∞ 0 (R d ), the solution to (3.1) remains compactly supported in the support of a 0 transported by the classical flow (3.3), it is standard to introduce the following change of unknown function (e.g. [START_REF] Robert | Autour de l'approximation semi-classique[END_REF][START_REF] Carles | Semi-classical analysis for nonlinear Schrödinger equations: WKB analysis, focal points, coherent states[END_REF]):

A(t, y) := J t (y)a (t, x(t, y)) , where a solves the transport equation

∂ t a + ∇φ eik • ∇a + 1 2 a∆φ eik = 0 ; a |t=0 = a 0 .
as given by WKB analysis. Indeed, using (3.4), we easily check that A is constant in time, ∂ t A = 0. Correctors (a (k) ) k 1 in the (linear) WKB analysis solve the equation

∂ t a (k) + ∇φ eik • ∇a (k) + 1 2 a (k) ∆φ eik = i 2 ∆a (k-1) ; a (k) |t=0 = 0,
with the convention a (0) = a. Setting

A (k) (t, y) := J t (y)a (k) (t, x(t, y)) ,
we infer that supp A (k) (t, •) ⊂ supp a 0 , ∀t ∈ [0, T ], ∀k 0, where T given by Lemma 3.1. Thus, for t ∈ [0, T ], up to O(ε ∞ ), u ε remains compactly supported, in the support of a 0 transported by the classical flow.

In the nonlinear case, we check that the same argument remains valid. Consider φ eik solution to (3.2), and (φ, a) solving (3.7). The natural adaptation of the above computation consists in showing that if φ 0 , a 0 ∈ C ∞ 0 (R d ), the new unknown (ψ, A), defined by (3.8)

A(t, y) := J t (y)a (t, x(t, y)) , ψ(t, y) := φ (t, x(t, y)) , enjoys a zero speed of propagation. Note that in view of Proposition 3.2, we already know that φ, a ∈ C([0, T * ], H ∞ (R d )), so it suffices to check that (ψ, A) solves a system for which the argument presented on the toy model of Burgers equation in Section 2 remains valid. Introducing

M (t, y) = ∇ y x(t, y) ∈ R d×d ,
whose determinant is by definition J t (y), we find:

∂ t ψ + 1 2 M -1 ∇ψ, M -1 ∇ψ + 1 J t (y) |A| 2 = 0, ψ |t=0 = φ 0 , ∂ t A = -J t (y) ∇φ • ∇a + 1 2 a∆φ (t, x(t, y)) , A |t=0 = a 0 .
We do not express the right hand side of the last equation in terms of (ψ, A): differentiating the first equation with respect to y, the bounds stated in Proposition 3.2 make it possible to infer an inequality of the form

|∂ t ∇ψ(t, y)| + |∂ t A(t, y)| |∇ψ(t, y)| + |A(t, y)|, (t, y) ∈ [0, T * ] × R d . Therefore, if supp φ 0 , supp a 0 ⊂ K, then supp ∇ψ(t, •), supp A(t, •) ⊂ K for all t ∈ [0, T * ].
Integrating in time the equation solved by ψ, we conclude to the zero speed of propagation for (ψ, A). Arguing like in Section 2 for the correctors, we have:

Proposition 3.3. Let φ 0 , a 0 ∈ C ∞ 0 (R d
) with supp φ 0 , supp a 0 ⊂ K. There for any t ∈ [0, T * ], where T * is given by Proposition 3.2,

supp ψ(t, •), supp A(t, •) ⊂ K,
where ψ and A are related to φ and a through (3.8). The same is true for the correctors (ψ (k) , A (k) ) k 1 corresponding to the next terms (φ (k) , a (k) ) k 1 in the asymptotic expansion in (3.9). Remark 3.4 (Special potentials). In the case where V is linear in x or isotropic quadratic, explicit formulas allow to bypass the above arguments.

If V (x) = E • x for some (constant) E ∈ R d and u ε solves (1.2), then v ε (t, x) = u ε t, x - t 2 2 E e i tE•x-t 3 3 |E| 2 /ε solves (2.1). If V (x) = ω 2 2 |x| 2 , ω > 0, then w ε (t, x) = 1 (1 + (ωt) 2 ) d/4 u ε arctan(ωt) ω , x 1 + (ωt) 2 e i ω 2 t 1+(ωt) 2 |x| 2 2ε solves iε∂ t w ε + ε 2 2 ∆w ε = (1 + t 2 ) d/2-1 |w ε | 2 w ε ; w ε |t=0 = a 0 e iφ0/ε . If d = 2 (the cubic nonlinearity is L 2 -critical)
, we recover exactly (2.1). Otherwise, a (smooth) time dependent factor has appeared, which obviously does not change the conclusion of Propositions 2.1 and 2.3. The case of a potential with the opposite sign is obtained by changing ω to iω in the formulas. See e.g. [START_REF] Carles | Semi-classical analysis for nonlinear Schrödinger equations: WKB analysis, focal points, coherent states[END_REF]Section 11.2] and references therein regarding these changes of unknown functions. For such potentials, the classical trajectories given by (3.3) are computed explicitly, and we can check directly the conclusions of Proposition 3.3.

Weaker nonlinearity.

We now consider the case 0 < γ < 1. This case is still a supercritical case as far as WKB analysis is concerned, in the sense described in the introduction: a "natural" asymptotic expansion of the solution u ε still involves a system of equations which is not closed. As noticed in [START_REF] Carles | WKB analysis for nonlinear Schrödinger equations with potential[END_REF], this intermediary case can be handled like the case γ = 0, by replacing (3.9) with (3.9)

           ∂ t φ ε + ∇φ eik • ∇φ ε + 1 2 |∇φ ε | 2 + ε γ |a ε | 2 = 0, ∂ t a ε + ∇φ eik • ∇a ε + ∇φ ε • ∇a ε + 1 2 a ε ∆φ eik + 1 2 a ε ∆φ ε = i ε 2 ∆a ε , φ ε |t=0 = φ 0 ; a ε |t=0 = a 0 .
The matrices A j and S now depend on ε, in an explicit way, and the asymptotic expansion of (φ ε , a ε ) involves more terms. Let N = [1/γ], where [r] is the largest integer not larger than r > 0: N new intermediary terms appear compared to the case γ = 0,

φ ε = ε γ φ + ε 2γ φ(1) + • • • + ε N γ φ(N) + O(ε), a ε = a + ε γ a (1) + • • • + ε N γ a (N ) + O(ε),
where the estimate holds in L ∞ ([0, T ], H s ) for any s > 0. This can be seen by setting φε = ε -γ φ ε : the leading order term is given by   

∂ t φ + ∇φ eik • ∇ φ + |a| 2 = 0 ; φ |t=0 = 0, ∂ t a + ∇φ eik • ∇a + 1 2 a∆φ eik = 0 ; a |t=0 = a 0 .
The leading order amplitude solves the same transport equation as in the linear case, and it is readily observed that the analogue of Proposition 3.3 remains valid, up to adapting the hierarchy of equations.

3.4. Weakly nonlinear and linearizable cases. We now assume γ 1. As in [START_REF] Carles | WKB analysis for nonlinear Schrödinger equations with potential[END_REF] (or [5, Chapter 2]), we present a strategy for any γ 1, and emphasize the fact that the value γ = 1 is specific. In this setting, the coupling between phase and amplitude changes dramatically: rapid oscillations are described by φ eik only, and the analysis consists in expanding the amplitude a ε = u ε e -iφ eik /ε in powers of ε:

∂ a ε + ∇φ eik • ∇a ε + 1 2 a ε ∆φ eik = i ε 2 ∆a ε -iε γ-1 |a ε | 2 a ε ; a ε |t=0 = a 0 .
Like above, the case when γ > 1 is not an integer requires a special asymptotic expansion, and we do not discuss this case. When γ > 1, the leading order amplitude satisfies the same transport equation as in the linear case. When γ = 1, it satisfies

∂ t a + ∇φ eik • ∇a + 1 2 a∆φ eik = -i|a| 2 a ; a |t=0 = a 0 .
Following the classical trajectories, that is resuming the change of unknown function (3.8), this equation reads

∂ t A = -iJ t (y) -1 |A| 2 A.
In particular, ∂ t |A| 2 = 0, and the nonlinear effect in a consists of a phase selfmodulation. In particular, the support of A(t, •) is independent of t ∈ [0, T ]. The same is true for all correctors in the asymptotic expansion, as can be checked easily.

Separation of states

We complete the proof of Theorem 1.3, by proving the nonlinear superposition.

For 1 j N , let χ j ∈ C ∞ 0 (R d , R), 0 χ j 1, with χ j ≡ 1 on supp α j , supp χ j1 ∩ supp χ j2 = ∅ if j 1 = j 2 .
We set

a 0 (x) = N j=1 α j (x), φ 0 (x) = N j=1 ϕ j (x)χ j (x).
Then a 0 , φ 0 ∈ C ∞ 0 (R d ), φ 0 is real-valued, and

u ε 0 (x) = a 0 (x)e iφ0(x)/ε .
We can then resume the analysis from the monokinetic case as presented in Sections 2 and 3, with the same notations. Let φ eik be given by Lemma 3.1 (it does not depend on the initial data, but only on V ).

4.1. Supercritical case. When γ = 0, the WKB analysis for each u ε j , solution to (1.5), involves the following system:

(4.1)      ∂ t φ j + ∇φ eik • ∇φ j + 1 2 |∇φ j | 2 + |a j | 2 = 0, φ j|t=0 = ϕ j χ j , ∂ t a j + ∇φ eik • ∇a + 1 2 a∆φ eik + ∇φ j • ∇a j + 1 2 a j ∆φ j = 0, a j|t=0 = α j .
To simplify the discussion, suppose first that V = 0, hence φ eik = 0. Each solution to (4.1) remains smooth on some time interval [0, T j ] for some 0 < T j T , and, on this time interval, enjoys a zero speed of propagation. As a consequence of Proposition 2.1, we have

φ = N j=1 φ j , a = N =1 a j ,
since nonlinear terms containing two indices j 1 = j 2 involve two functions whose supports are disjoint. Also, for all k 1, the correctors satisfy

φ (k) = N j=1 φ (k) j , a = N =1 a (k) j . Set T * = min (T * , T 1 , . . . , T N ) . As we have, in view of (2.6), in L ∞ ([0, T * ], L 2 ∩ L ∞ ), for any k > 0, u ε - k-1 ℓ=0 ε ℓ a (ℓ) exp i ε k ℓ=0 ε ℓ φ (ℓ) = O ε k , u ε j - k-1 ℓ=0 ε ℓ a (ℓ) j exp i ε k ℓ=0 ε ℓ φ (ℓ) j = O ε k , j = 1, . . . , N,
we obtain Theorem 1.3 in the case V = 0. In the case where V is not trivial, we just have to resume the above arguments by replacing the functions (φ, a) (possibly with indices and/or superscripts) with (ψ, A), as defined by the change of unknown function (3.8), which involves only V (see (3.3)), and not the initial data.

4.2. Other cases. When γ > 0, we have seen that the leading order amplitude is the same as in the linear case, up to a phase modulation. Leading order oscillations are given by φ eik , where we now set ϕ 0 = φ 0 in (3.2). The features used in the supercritical case then remain, regarding the evolution of the support of the terms involved in WKB analysis.

iε∂ t u ε + ε 2 2 ∆u ε = V u ε u ε |t=0 = a 0 e iφ0/ε , with a 0 ∈ C ∞ 0 (R d ) and φ 0 ∈ C ∞ (R d , R). Like in Section 4, consider χ ∈ C ∞ 0 (R d , [0, 1]), with χ ≡ 1 on supp a 0 .
For any such function χ, we have u ε |t=0 = a 0 e iχφ0/ε . However, the eikonal equation now depends on χ, as (3.2) becomes

∂ t φ eik + 1 2 |∇φ eik | 2 + V = 0 ; φ eik|t=0 = χφ 0 .
As recalled in Section 3.1 (in the case φ 0 = 0), the solution is constructed, locally in time, via the classical trajectories, or, equivalently, through characteristic curves.

As V is smooth, the slope of characteristic curves at time t = 0 is uniformly bounded on the support of a 0 . By finite speed of propagation, there exists T (χ) > 0 such that φ eik does not depend on χ for t ∈ [0, T (χ)]. In practice, the introduction of χ may shorten the time interval of validity of WKB analysis, as we now illustrate. Let d = 1, V = 0, and φ 0 (x) = x 2 /2. The solution to the eikonal equation (without cutoff χ) is given explicitly by

φ eik (t, x) = x 2 2(1 + t) .
This is a case where there is no singularity for t 0 (but a caustic reduced to one point at t = -1). Indeed, the classical trajectories, solving ẋ(t, y) = ξ(t, y), x(0, y) = y ; ξ(t, y) = 0, ξ(0, y) = φ ′ 0 (y) = y, are given by x(t, y) = (1 + t)y, obviously inverted, for all t 0, as

y(t, x) = x 1 + t ,
and the leading order amplitude in WKB analysis is given by

a(t, x) = 1 √ 1 + t a 0 x 1 + t .
For χ a (usual) cutoff function as above, χφ 0 has two humps: in the presence of χ, y → x(t, y) ceases to be invertible for all t 0 (φ ′ eik solves the Burgers equation), but for short time (independent of ε, but depending on χ), a(t)e iφ eik (t)/ε does not depend on χ.

Supercritical WKB analysis for the nonlinear Schrödinger equation.

In the case addressed in Section 2, the above eikonal equation is replaced by (2.5). By considering the gradient of the phase instead of the phase, the Burgers equation (in the case of WKB analysis for the linear Schrödinger equation without potential) is replaced by the symmetrization of the Euler equation. Like above, finite speed of propagation implies that the introduction of a cutoff function in the initial phase does not alter the solution to (2.5) on some time interval [0, T (χ)], for some T (χ) > 0 possibly depending on χ. This time is of course independent of ε, as ε is absent from (2.5). This is why in Remark 1.4, the Wigner measure does not depend on the χ j 's, even though its construction seems to depend on these cutoff functions: the time of validity that we can prove may, on the other hand, depend on the choice of these cutoff functions.

We conclude this discussion by an illustration similar to the one given in the previous subsection. Let a 0 ∈ C ∞ 0 (R d ), and assume that for s > d/2+1, a 0 H s (R d ) is sufficiently small. Suppose also that v 0 = ∇φ 0 satisfies:

∇ 2 v 0 ∈ H s-1 (R d ), ∇v 0 ∈ L ∞ (R d )
, and there exists δ > 0 such that for all x ∈ R d , dist(Sp(∇v 0 (x)), R -) δ, where we denote by Sp(M ) the spectrum of a matrix M . Then it follows from the main result in [START_REF] Grassin | Global smooth solutions to Euler equations for a perfect gas[END_REF] that (2.5) has a global (in the future) solution

a, v -v ∈ C j ([0, ∞), H s-j (R d )), j = 0, 1,
where v is the unique, global smooth solution to the (multidimensional) Burgers equation

∂ t v + v • ∇v = 0, v |t=0 = ∇φ 0 .
We may for instance consider φ 0 (x) = |x| 2 /2 (see the previous subsection), and then v(t, x) = x 1 + t .

On the other hand, if φ 0 is multiplied by a cutoff function χ, then the initial data in (2.5) belong to C ∞ 0 (R d ): it follows from [START_REF] Makino | Sur la solution à support compact de l'équations d'Euler compressible[END_REF] that the corresponding solution develops a singularity in finite time. Like in the previous subsection, the introduction of the cutoff χ reduces the lifespan of the solution involved in WKB analysis but, for short time, does not alter the asymptotic description of the solution u ε .

Weakly nonlinear case

In this section, we prove Proposition 1.5. Instead of (1.2)-(1.3), we consider the weakly nonlinear case,

iε∂ t u ε + ε 2 2 ∆u ε = ε|u ε | 2 u ε ; u ε 0 (x) = N j=1 α j (x)e iϕj (x)/ε .
When d 2, the creation of new WKB terms is possible by resonant interactions, provided that N 3, as recalled in the appendix. The one-dimensional cubic case is special, as there are no nontrivial resonances, see [START_REF] Carles | Multiphase weakly nonlinear geometric optics for Schrödinger equations[END_REF]. In order to present an argument including the cubic one-dimensional case, we propose a proof which does not use the creation of, e.g., a fourth term out of three.

Consider linear phases,

ϕ j (x) = k j • x.
The first part of Proposition 1.5 is simply a restatement of Theorem 1.3 in this case. To prove the appearance of nonlinear interferences, we will not consider cutoff functions, and rely on explicit computations. WKB analysis in the monokinetic case N = 1 leads to the hierarchy

∂ t φ + 1 2 |∇φ| 2 = 0 ; φ(0, x) = k • x, ∂ t a + ∇φ • ∇a + 1 2 a∆φ = -i|a| 2 a ; a(0, x) = α(x).
The eikonal equation is solved explicitly,

φ(t, x) = k • x - |k| 2 2 t.
As ∆φ = 0, the initial amplitude α is transported along the vector k with a phase self-modulation: a(t, x) = α(xtk)e -it|α(x-tk)| 2 .

In the case N = 2, no new WKB term is created, but interactions between the two modes lead to a modification of the phase modulation. As computed in [6, Section 3], we find

a 1 (t, x) = α 1 (x -tk 1 )e -i(2 t 0 |α2(x+(τ -t)k1-τ k2)| 2 dτ +t|α1(x-tk1)| 2 ) , a 2 (t, x) = α 2 (x -tk 2 )e -i(2 t 0 |α1(x+(τ -t)k2-τ k1)| 2 dτ +t|α2(x-tk2)| 2 ) .
In addition, we have

sup t∈[0,T ] u ε (t) -a 1 (t)e iφ1(t)/ε -a 2 (t)e iφ2(t)/ε L 2 ∩L ∞ = O(ε),
for any T > 0 (independent of ε); see Corollary 5.13 and Theorem 6.5 in [START_REF] Carles | Multiphase weakly nonlinear geometric optics for Schrödinger equations[END_REF].

The leading order nonlinear interactions between the two modes correspond to the integrals in time in the exponentials. For small time though, the integrals are zero on the support of the transported amplitudes: in other words, there exists T 0 > 0 independent of ε such that for t ∈ [0, T 0 ],

a 1 (t, x) = α 1 (x -tk 1 )e -it|α1(x-tk1)| 2 , a 2 (t, x) = α 2 (x -tk 2 )e -it|α2(x-tk2)| 2 ,
in agreement with the conclusion of Theorem 1.3 (up to the order of precision). The conclusion in Proposition 1.5 then follows from the property:

sup t∈[0,T1] α 1 (• -tk 1 ) e -2i t 0 |α2(•+(τ -t)k1-τ k2)| 2 dτ -1 L p > 0, or, equivalently, sup t∈[0,T1] α 1 sin t 0 |α 2 (• + τ (k 1 -k 2 ))| 2 dτ L p > 0.
This is possible as soon as the transport of the support of α 1 meets the support of α 2 , as transported in the above integral. Let α ∈ C ∞ 0 (R d ) supported in the ball centered at the origin, of radius 1, and set

α 1 (x) = α(x), α 2 (x) = α(x + 3e 1 ),
where (e 1 , . . . , e d ) is the canonical basis of R d . Setting k 2k 1 = λe 1 for λ > 0, we see that the above property is satisfied for some T 1 > 0. We also remark that T 1 → 0 as λ → ∞.

Appendix A. Weakly nonlinear case and creation of a new term

In this appendix, we prove that in the weakly nonlinear case, if d 2, then nonlinear interactions may lead to the creation of a new WKB term, which is a stronger phenomenon than that used in the proof of Proposition 1.5. Consider

iε∂ t u ε + ε 2 2 ∆u ε = ε|u ε | 2 u ε ; u ε 0 (x) = N j=1
α j (x)e iϕj (x)/ε , with now N = 3, and d 2. The one-dimensional cubic case is special, as there are no nontrivial resonances, see [START_REF] Carles | Multiphase weakly nonlinear geometric optics for Schrödinger equations[END_REF]. Again, we consider linear phases,

ϕ j (x) = k j • x.
Recall ( [START_REF] Colliander | Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation[END_REF], see also [START_REF] Carles | Multiphase weakly nonlinear geometric optics for Schrödinger equations[END_REF]Lemma 2.2]) that the resonant set is defined by

Res(n) = {(j, ℓ, m), k j -k ℓ + k m = k n , |k j | 2 -|k ℓ | 2 + |k m | 2 = |k n | 2 }
is characterized as follows: (k j , k ℓ , k m ) ∈ Res(n) when the endpoints of the vectors k j , k ℓ , k m , k n form four corners of a nondegenerate rectangle with k ℓ and k n opposing each other, or when this quadruplet corresponds to one of the following two degenerate cases: (k j = k n , k m = k ℓ ) or (k j = k ℓ , k m = k n ). Note that we always have (A.1) {(j, j, n), ((n, j, j), a j ≡ 0} ⊂ Res(n), where a j is the amplitude associated with the phase

φ j (t, x) = k j • x - |k j | 2 2 t.
In order for the nonlinearity to create a term associated with a phase φ 4 , out of three phases associated with wave numbers k 1 , k 2 and k 3 , we must have

k 4 := k 2 -k 1 + k 3 , |k 4 | 2 = |k 2 | 2 -|k 1 | 2 + |k 3 | 2 .
This resonant condition is equivalent to the following conditions:

(k 1 -k 2 ) • (k 1 -k 3 ) = 0,
and the endpoints of k 1 , k 2 and k 3 are not aligned (the case of alignment corresponds to the set on the left in (A.1)); this is possible with pairwise different k 1 , k 2 , k 3 and k 4 ∈ {k 1 , k 2 , k 3 } provided that d 2, see [START_REF] Carles | Multiphase weakly nonlinear geometric optics for Schrödinger equations[END_REF] (or [5, Section 2.6]). For instance if d = 2, we can choose, for λ > 0, k 1 = λ(1, 1), k 2 = λ(1, 0), k 3 = λ(0, 1), hence k 4 = (0, 0).

In higher dimension, we simply complete each vector by zero coordinates. Then a new term, associated with the phase φ 4 may be created by nonlinear resonance. Because of the geometric characterization of resonances, no other term can be created apart from this one, since we have completed a rectangle. The creation is effective only if the associated amplitude does not remain zero. The equation for the corresponding amplitude is If we assume that the mode 4 is not effectively created, that is a 4 ≡ 0, then the inclusion (A.1) is actually an equality, and

∂ t a 4 + k 4 • ∇a 4 = -i
∂ t a j + k j • ∇a j = -2i 3 k=1
|a k | 2 a j + i|a j | 2 a j , j = 1, 2, 3. hence a j (t, x) = α j (xtk j )e -iSj (t,x) , j = 1, 2, 3, for some explicit real-valued phase, whose expression is irrelevant here (see [START_REF] Carles | Multiphase weakly nonlinear geometric optics for Schrödinger equations[END_REF]Section 3.1] for the formula). Given any T > 0, we may choose α 1 , α 2 , α 3 compactly supported, with disjoint supports, k 1 , k 2 , k 3 like above, so that a 2 ā1 a 3|t=T /2 ≡ 0.

This shows that the term a 4 is actually created, in the sense that a 4 does not remain trivial on [0, T ]. The error estimate proved in [START_REF] Carles | Geometric optics and instability for NLS and Davey-Stewartson models[END_REF] (see also [5,

1 . Introduction 1 . 1 .

 111 Setting. We consider the cubic defocusing Schrödinger equation on R d , d 1, in the semiclassical régime

5 . 5 . 1 .

 551 On the role of the cutoff function(s) WKB analysis for the linear Schrödinger equation. Consider (3.1) in the presence of rapid initial oscillations,(5.1)

  (j,ℓ,m)∈Res[START_REF] Carles | WKB analysis for nonlinear Schrödinger equations with potential[END_REF] a j āℓ a m , a 4|t=0 = 0.More generally, the term a n solves∂ t a n + k n • ∇a n = -i (j,ℓ,m)∈Res(n) a j āℓ a m , a n|t=0 = α n .

  hence again the conclusion of Proposition 1.5. The proof also implies that for t ∈ [0, T ], u ε has a unique Wigner measure, given by µ(t, dx, dξ) =

							Section 2.6]) yields
			4
	sup t∈[0,T ]	u ε (t) -	j=1	a j (t)e iφj (t)/ε	L 2 ∩L ∞	= O(ε),
				4
				j=1	|a

j (t, x)| 2 dx ⊗ δ ξ=kj .

Acknowledgments. The author wishes to thank Patrick Gérard for his remarks which helped improve the consistency of the paper.


Clément Cazorla

Nathanaël Munier

Renaud Morin

Pierre Weiss

Sketchpose: Learning to Segment Cells with Partial Annotations

Keywords:

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

Image segmentation plays a fundamental role in the analysis of biological images. It enables the extraction of quantitative information on diverse objects ranging from molecules, droplets, membranes, nuclei, cells, vessels or other structures. In modern biological research, accurate segmentation is often pivotal to better understand the mechanisms of life. The increasing availability of highthroughput imaging technologies has led to a surge in the quantity and complexity of image data, raising significant challenges and opportunities. Manual annotation of the resulting images is labor-intensive, time-consuming, and often impractical for large-scale datasets. Automated segmentation is therefore widely accepted as a critical step in biological research.

A simplified history of cell segmentation Image segmentation has long been dominated by handcrafted algorithms. The processing pipelines typically combine popular tools such as linear filtering, thresholding [START_REF] Otsu | A threshold selection method from gray-level histograms[END_REF], morphological operations [START_REF] Serra | Mathematical morphology and its applications to image processing[END_REF][START_REF] Legland | Morpholibj: integrated library and plugins for mathematical morphology with imagej[END_REF], active contour models (Snake) [START_REF] Kass | Snakes: Active contour models[END_REF] or watershed [START_REF] Vincent | Watersheds in digital spaces: an efficient algorithm based on immersion simulations[END_REF]. A significant issue with handcrafted approaches is that they are usually imagespecific and rely on the manual tuning of a few complicated hyper-parameters. Although excellent performance can be achieved, it is often the work of a handful of talented people and these techniques are not broadly applicable.

The introduction of machine learning and especially random forests made image segmentation accessible to a much larger range of researchers. These techniques automatically combine and tune elementary image processing bricks. They are driven by a few easily interpretable user annotations. Embedded in well conceived software such as Ilastik [START_REF] Berg | Ilastik: interactive machine learning for (bio) image analysis[END_REF] or Labkit [START_REF] Arzt | Labkit: labeling and segmentation toolkit for big image data[END_REF], these techniques heavily contributed to democratize image segmentation and classification.

"Deep learning" and convolutional neural networks played an important role in improving the segmentation performance around 2015. For instance, the popular U-Net architecture [START_REF] Ronneberger | U-net: Convolutional networks for biomedical image segmentation[END_REF] increased the accuracy on some cell segmentation challenges by more than 10%, which can be considered as a small revolution. This type of neural network architecture seems to be a good prior for segmenting "natural" images, as suggested by the so-called Deep Image Prior principle [START_REF] Lempitsky | Deep image prior[END_REF]. However, it can sometimes demonstrate limited effectiveness when it comes to separating nearby or touching objects. Many applications in biology involve densely packed objects (e.g. cells, nuclei) and a vanilla U-Net is often insufficient to perform a satisfactory analysis. To address this issue, new architectures coming from computer vision such as Mask R-CNN [START_REF] He | Mask r-cnn[END_REF] have been developed and continued improving the performance.

Roughly at the same time, a few approaches (Deep Regression of the Distance Map [START_REF] Naylor | Segmentation of nuclei in histopathology images by deep regression of the distance map[END_REF], StarDist [START_REF] Schmidt | Cell detection with star-convex polygons[END_REF], Cellpose [START_REF] Stringer | Cellpose: a generalist algorithm for cellular segmentation[END_REF], Omnipose [START_REF] Cutler | Omnipose: a high-precision morphology-independent solution for bacterial cell segmentation[END_REF]) have been developed and generated results with an unprecedented quality. Despite certain differences, they all share a common underlying principle. The idea is to make a regression with respect to some distance function. Given a set of annotated objects, a distance function to the objects centers or boundaries is computed. A convolutional neural network is then trained to predict the distance function rather than a binary map of the objects. This principle created a new gap in the segmentation accuracy, especially for objects with touching boundaries. Indeed, the gradient of this distance function points in opposite directions on each side of the boundary, which makes it possible to determine them with much greater precision.

Finally, let us mention that a current trend consists in involving the user in the training procedure. This "human in the loop" principle was recently incorporated in Cell-Pose 2.0 [START_REF] Pachitariu | Cellpose 2.0: how to train your own model[END_REF].

It would be hazardous to call these approaches the current "state-of-the-art", since this field is expanding extremely quickly. However -as of 2023 -we can safely claim that algorithms based on the distance map are at the basis of some of the most popular and efficient cell segmentation methods. illustrated in Figure 1. Hence, it is a priori unclear how partial annotations can be used in this framework. It is however much easier to draw a few strokes and boundaries, than to segment all the objects in a complex image.

In this paper, we introduce an idea that allows us to use the distance function even with partial annotations. To assess its potential, we develop a Napari plugin [START_REF] Chiu | napari: a python multi-dimensional image viewer platform for the research community[END_REF] named Sketchpose, that relies on a modified version of the Omnipose [START_REF] Cutler | Omnipose: a high-precision morphology-independent solution for bacterial cell segmentation[END_REF] algorithm. After drawing just a few regions and boundaries, the user can train a task-aware neural network. This approach capitalizes on the generalization capacity of neural networks, reducing the overall annotation effort without sacrificing accuracy. We explore the performance of the proposed architecture in 3 different settings:

• Frugal learning: starting from random weights, we show that just a few annotations are already enough to quickly realize complex cell segmentation analyses. This is interesting when faced with a problem for which no close pre-trained model exists.

• Transfer learning: starting from Omnipose's optimized weights, we show that just a few clicks at locations where the segmentation is inaccurate lead to improved weights and fast adaptation to out-ofdistribution images.

• Large databases: finally, we show that large, but partially annotated sets can also be used to optimize neural networks in a robust way.

This comprehensive evaluation on both small and largescale datasets, overall showcases the advantages of our approach in terms of time and resource savings.

Methodology

Preliminary definitions and notation

In all the paper X refers to the image domain, which can be understood as a discrete set of coordinates, or as a continuous domain depending on the context. In the discrete setting, we let |X| denote the number of pixels of X.

Definition 1. For an arbitrary set S ⊂ X, we let 𝜕S denote its boundary. We use the 4-connectivity in the discrete setting.

Definition 2 (Point to set distance). The distance from a point x ∈ X to a set S ⊆ X is defined by

dist(x, S) def = inf x ′ ∈ S ∥x -x ′ ∥ 2 .
(1)

Omnipose

Our work is based on the Omnipose cell segmentation architecture [START_REF] Cutler | Omnipose: a high-precision morphology-independent solution for bacterial cell segmentation[END_REF]. In this section, we justify this choice, explain its founding principles and then demonstrate how they can be adapted to deal with partial annotations.

Why Omnipose

Cellpose [START_REF] Stringer | Cellpose: a generalist algorithm for cellular segmentation[END_REF] has now become a standard in cell segmentation. Its excellent perfomance, processing speed, and ergonomic graphical interface make it a near ideal tool for every day cell biology image analysis. However, it occasionally fails in scenarios involving complex and elongated objects. In such cases, it tends to produce over-segmentation, where neighboring objects are split in smaller fragments.

The Omnipose [START_REF] Cutler | Omnipose: a high-precision morphology-independent solution for bacterial cell segmentation[END_REF] algorithm was initially conceived in order to address this limitation but is currently not as popular as Cellpose. Among the explanations for this phenomenon, we may think of the following facts i) Cellpose was developed earlier, ii) Cellpose is trained continuously with new data coming on a daily basis, while Omnipose is trained on a much smaller and fixed dataset, iii) as a consequence, Omnipose currently produces less accurate results in our experience, except for elongated bacteria datasets.

Omnipose can be seen as an evolution of Cellpose, as some features of the latter architecture have been upgraded. For instance, the distance map is defined as the distance to the cell boundaries in Omnipose, while it is defined as a distance to an arbitrary cell centroid in Cellpose. The problem is that there is no canonical choice to define this center, hence Omnipose's choice seems preferable.

With more developers, training data and further methodological advances, the features of Omnipose might therefore turn it into a serious competitor. This explains our decision to choose and base our work on its architecture.

The main principles

Fig. 2 summarizes the main ideas behind the Omnipose architecture and its training. Omnipose is based on a regular convolutional neural network (CNN), with a U-Net like architecture [START_REF] Ronneberger | U-net: Convolutional networks for biomedical image segmentation[END_REF]. Given an input 2D image with 𝑁 pixels, the CNN can be seen as a mapping 𝑁 𝑤 of the form

𝑁 𝑤 : R 𝑁 → R 𝑁 × R 𝑁 × R 2𝑁 𝑢 ↦ → (𝑁 𝑏 𝑤 (𝑢) , 𝑁 𝑑 𝑤 (𝑢) , 𝑁 v 𝑤 (𝑢))
.

(2) It depends on weights 𝑤 that should be optimized during a training stage. It returns 3 different outputs (illustrated on the top of Fig. 2):

• 𝑁 𝑏 𝑤 (𝑢) ≡ boundary probability: at every pixel, the value of this image can be interpreted as a probability of being a boundary between the objects to segment.

• 𝑁 𝑑 𝑤 (𝑢) ≡ distance map: at a given pixel, the value of this map is equal to:

-The distance of the pixel to the closest object boundary, if the pixel is inside an object.

-0 (or a fixed negative value) elsewhere.

• 𝑁 v 𝑤 (𝑢) ≡ flow field: can be interpreted as the gradient of the distance map. It is an essential feature of the Cellpose and Omnipose architectures. Ultimately, the flow is used through a procedure called Euler integration to generate a segmentation mask. This is illustrated on the top right of Fig. 2.

The training procedure

The original training stage involves a collection of 𝐾 ∈ N images (𝑢 𝑘 ) 1≤ 𝑘 ≤𝐾 together with their exhaustive segmentation masks. For every image 𝑢 𝑘 in the dataset, an algorithm creates the gold standard boundary probability 𝑏 ★ 𝑘 , distance map 𝑑 ★ 𝑘 and flow field v ★ 𝑘 . This is illustrated on the bottom of Fig. 2.

The weights 𝑤 of the neural network are then optimized so as to minimize a loss function that compares the output of the CNN with the gold standard:

inf 𝑤 loss(w) def = 1 𝐾 𝐾 ∑ 𝑘=1 ℓ B (𝑏 𝑘 , 𝑏 ★ 𝑘 )+ℓ D (𝑑 𝑘 , 𝑑 ★ 𝑘 )+ℓ V (v 𝑘 , v ★ 𝑘 ), (3)
where

𝑏 𝑘 = 𝑁 𝑏 𝑤 (𝑢 𝑘 ), 𝑑 𝑘 = 𝑁 𝑑 𝑤 (𝑢 𝑘 ), v 𝑘 = 𝑁 v 𝑤 (𝑢 𝑘 ).
The optimization is performed using standard stochastic gradient procedures with the RAdam algorithm [START_REF] Liu | On the variance of the adaptive learning rate and beyond[END_REF]. The different losses are defined as follows:

• Boundary loss ℓ B : This loss simply compares the predictions 𝑏 to 𝑏 ★ using a binary cross-entropy loss. It is defined as

ℓ B (𝑏, 𝑏 ★ ) def = 𝜆 B |X| ∑ x∈ X 𝑔(𝑏[x], 𝑏 ★ [x]), (4)
where 𝑔 : R × R → R combines a sigmoid and a binary cross entropy loss. It corresponds to the function BCEWithLogitsLoss in the package torch.nn.

• Distance loss ℓ D : This loss calculates a weighted mean squared error between the predicted distance fields and the ground truth distance fields. It is defined as

ℓ D (𝑑, 𝑑 ★ ) def = 𝜆 D |X| ∑ x∈ X (𝑑 [x] -𝑑 ★ [x]) 2 • 𝜌[x],
where 𝜌 ∈ R 𝑁 is a weight image with higher values around the gold standard boundaries. We will use 𝜌 = 𝛽 + exp(-𝑑 ★ /𝛼) in our numerical experiments.

The parameter 𝛼 > 0 can be interpreted as a characteristic distance, measuring how far from the boundary we want 𝑑 to match 𝑑 ★ closely. The parameter 𝛽 allows the used to balance the relative importance of the boundaries. We pick 𝛼 = 5 and 𝛽 = 2 in our implementation.

• Flow loss ℓ V :

This loss is defined as a weighted sum of three losses

ℓ V = ℓ 1 V + ℓ 2 V + ℓ 3 V .
The first one is a mean squared error loss:

ℓ 1 V (v, v ★ ) def = 𝜆 V,1 |X| ∑ x∈ X v[x] -v ★ [x] 2 2 • 𝜌[x]. ( 5
)
The second one compares the norms of the vector fields:

ℓ 2 V (v, v ★ ) def = 𝜆 V,2 |X| ∑ x∈ X (∥v[x] ∥ 2 -∥v ★ [x] ∥) 2 • 𝜌[x]. (6)
The third one aims to minimize the distance between trajectories generated through the ground truth and predicted flows. Trajectories starting from an initial point z can be generated by simple explicit Euler discretization:

x 0 (z) def = z x 𝑙+1 (z) def = x 𝑙 (z) + Δ𝑡 • v[x 𝑙 (z)] x ★ 0 (z) def = z x ★ 𝑙+1 (z) def = x ★ 𝑙 (z) + Δ𝑡 • v ★ [x ★ 𝑙 (z)].
where Δ𝑡 is a step-size. Letting 𝐿 ∈ N denote an integration time, the "Euler" loss then becomes:

ℓ 3 V (v, v ★ ) def = 𝜆 V,3 |X| ∑ z∈ X 𝐿 ∑ 𝑙=1 ∥x 𝑙 (z) -x ★ 𝑙 (z) ∥ 2 2 . ( 7
)
It measures how two trajectories generated by Euler integration using the ground truth and predicted vector fields deviate. This loss is implemented in the torchVF library by [START_REF] Peters | Torchvf: Vector fields for instance segmentation[END_REF]. For more information, we refer the reader to the related report.

An inspection of the code reveals that the different weights have been set empirically as:

𝜆 B = 10, 𝜆 D = 2, 𝜆 V,1 = 2, 𝜆 V,2 = 2, 𝜆 V,3 = 1.
Remark 1. The different losses probably have been combined by trial and error to produce the best possible results. It seems however that some simplifications might be good to consider. For instance, the flow field can be obtained from the distance map directly. Hence, the network could likely be simplified to return the boundary probability and flow only. The loss ℓ 1

V is an upper-bound of ℓ 2 V . The loss ℓ 1

V is also present already in the first term of ℓ 3 V . We decided to stick to these choices to perform some numerical comparisons.

Adapting to partial annotations

All the principles described above heavily depend on an exhaustive segmentation of the cells. Indeed, the distance functions and gradient flows -which are instrumental to define the loss functions -are global properties which do change heavily if the objects boundaries are incomplete. In this section, we describe the main methodological contribution of this paper, which will allow us to handle partial boundaries.

The gold standard

We assume that the domain X = X 0 ⊔ X 1 is partitioned with the background set X 0 and the foreground set X 1 . A difficulty in instance segmentation is that multiple objects may exist within the connected components of a region X 𝑖 . To differentiate them, we let (X 𝑖, 𝑗 ) 1≤ 𝑗 ≤ 𝐽 𝑖 denote a partition of the set X 𝑖 as different objects within a similar class. For instance in Fig. 3a, the foreground set

X 1 is split in 13 components. A connected component of X 1 can be split as X 1,2 ∪ X 1,3 . The background set X 0 is split in a single component X 0,1 .
We let

E def = 𝑖 ∈ {0,1} 𝐽 𝑖 𝑗=1 𝜕X 𝑖, 𝑗 (8)
denote the set of all edges (or object boundaries) within the image. It is depicted in red in Fig. 3a.

The annotation set

The input of our neural network is a set of "strokes" drawn by the user. We let S 0 and S 1 denote those strokes describing the background and foreground respectively. They are depicted in brown and blue respectively in Fig. 3b. The intersection of the brown and blue strokes define natural boundaries. We can indeed construct the touching boundaries B 0,1 between different strokes as

B 0,1 def = S 0 ∩ S 1 ,
where X is the closure of X in the continuous setting and the interface between neighboring pixels in the discrete setting.

In addition, the user can delineate other boundaries, denoted B manual , to separate touching objects within a class. We can concatenate all the boundaries to obtain a complete boundary set B defined as

B = B manual B 0,1 . (9)
For the algorithm to work properly, we require the following set of assumptions.

Assumption 1 (Assumptions on the strokes).

• The strokes correctly separate the background and foreground: S 0 ⊆ X 0 and S 1 ⊆ X 1 .

• The strokes do not overlap: S 0 ∩ S 1 = ∅. This is actually forced by our Napari interface.

• The boundaries B are a subset of the exact boundaries E, that is:

B ⊆ E. (10)
• If the stroke S 𝑖 contains multiple objects, then the boundaries between the objects need to be completely drawn with B manual (see Fig. 4). Letting S def = S 0 ∪ S 1 denote the complete stroke set drawn by the user, this condition reads:

E ∩ S ⊆ B. (11)

The main observation

The main result we will use to define and certify our algorithm is summarized in the following theorem.

Theorem 1 (The valid distance set). Let CB def = 𝜕S 0 ∪ 𝜕S 1 ∪B denote the complete set of annotation boundaries and define the valid distance set D as

D def = {𝑥 ∈ S, dist(𝑥, B) ≤ dist (𝑥, CB)} . ( 12
)
The following relationships hold:

For all 𝑥 ∈ D, dist(𝑥, E) = dist(𝑥, B). For all 𝑥 ∈ S 0 ∪ S 1 , dist(𝑥, CB) ≤ dist(𝑥, E).
The proof of this theorem is given in Appendix 5. This theorem should be understood as follows. The first identity informs us that we can compute the exact distance map dist(𝑥, E) to the set of exact boundaries E on the valid set D. This set can be computed using only the partial annotations of the boundaries B ⊆ E and the different semantic regions S 𝑖 ⊆ X 𝑖 . The second inequality tells us that we have an information everywhere on the strokes S 0 and S 1 . Moreover, in the case of total annotations, we get D = X and the proposed idea will lead to a training equivalent to the one in Omnipose. Figure 5 schematically summarizes Theorem 1.

Adapting the training

Different summation sets Equipped with the valid distance set D, we are ready to adapt the losses to cope with partial annotation. In Omnipose, the losses ℓ B , ℓ D , ℓ 1 V and ℓ 2

V are defined by summation over the set X (see paragraph 2.2.3). With partial annotation, the gold standard is not properly defined on this set and we therefore need to change the summation sets.

First, we propose to modify the weight vector 𝑤 by defining it as:

𝜌 = exp(-𝛼𝑑 C B ) + 𝛽, (13)
where 𝑑 C B is the distance to the complete boundary set.

For the loss ℓ B , we know the partial boundaries B and that there should be no boundaries in the strokes S 𝑖 , hence we can sum over S 0 ∪ S 1 ∪ B:

ℓ partial B (𝑏, 𝑏 ★ ) = 𝜆 B |S 0 ∪ S 1 ∪ B| ∑ x∈ S 0 ∪S 1 ∪B 𝑔(𝑏[x], 𝑏 ★ [x]). (14)
Based on Theorem 1, the losses related to the distance set and to the flows become:

ℓ partial D (𝑑, 𝑑 ★ ) def = 𝜆 D |D | ∑ x∈ D (𝑑 [x] -𝑑 ★ [x]) 2 • 𝜌[x], ℓ 1,partial V (v, v ★ ) def = 𝜆 V,1 |D | ∑ x∈ D v[x] -v ★ [x] 2 2 • 𝜌[x], ℓ 2,partial V (v, v ★ ) def = 𝜆 V,2 |D | ∑ x∈ D (∥v[x] ∥ 2 -∥v ★ [x] ∥ 2 ) 2 • 𝜌[x].
We simply replaced x∈ X by x∈ D in section 2.2.3. This means that we compare the ground truth and prediction only where it makes sense to do so. Since X = D in the case of exhaustive annotation, the training of Omnipose and ours coincide and we can see our setting as a generalization of Omnipose.

The Euler loss A specific care needs to be taken for the Euler integration loss ℓ 3 V . In that case, we need to replace the summation over X by a summation over D, but this is insufficient since trajectories might escape the valid distance set during integration. To avoid this, we mask the ground truth v ★ and predicted v vector fields outside of the valid set D. Therefore, the trajectories generated from points in D by Euler integration will at stop at the boundary of D. In equations, this yields:

x 0 (z) = z x 𝑙+1 (z) def = x 𝑙 (z) + Δ𝑡 • v[x 𝑙 (z)] if x 𝑙 (z) ∈ D x 𝑙 (z) otherwise
with a similar modification for x ★ 𝑙 . The loss then becomes

ℓ 3,partial V (v, v ★ ) def = 𝜆 V,3 |D | ∑ z∈ D 𝐿 ∑ 𝑙=1 ∥x 𝑙 (z) -x ★ 𝑙 (z) ∥ 2 2 . ( 15
)
Dealing with inequalities Until now, we just used the first identity in Theorem 1, but the second inequality brings some additional information. We propose to integrate it in the training through the additional asymmetric loss:

ℓ ineq D (𝑑, 𝑑 ★ ) def = 𝜆 D,ineq |S 1 ∪ S 2 | ∑ z∈ S 1 ∪S 2 ReLU 2 𝑑 [z] -𝑑 ★ [z] 𝜌[z]. (16)

The Sketchpose plugin

A significant part of this work lies in the development of a user-friendly graphical interface to train and use the neural network. It is integrated in Napari [START_REF] Chiu | napari: a python multi-dimensional image viewer platform for the research community[END_REF], which is well suited to embedding the Python/PyTorch codes at the core of our approach. Sketchpose can be easily installed through either the pip package manager or Napari's [START_REF] Chiu | napari: a python multi-dimensional image viewer platform for the research community[END_REF] built-in interface. A detailed documentation can be accessed at: https://sketchpose-doc.readthedocs. io/en/latest/index.html. It offers step-by-step instructions illustrated by short videos, to assist users in effectively testing all the capabilities of the plugin. Two approaches can be used for labeling. In the first one the user directly draws a few strokes for the background, foreground and boundaries. The brush size can be adjusted similarly to usual paint software. An entire stroke boundary can be added to the boundary set by a double right-click. The second approach is similar to the one implemented in Cellpose 2.0 [START_REF] Pachitariu | Cellpose 2.0: how to train your own model[END_REF]. After an initial prediction, the user manually chooses a bounding box and performs a full labeling in the corresponding area. This is more tedious and the first option should be preferred. An overview of the interface is depicted in Fig. 6.

Experiments

In this paragraph, we conduct several experiments to explore three distinct use cases of the method.

• Learning from a limited set of annotations on a single image with randomly initialized neural network weights.

• Learning from a limited set of annotations on a single image, starting from a pre-trained neural network.

• Learning with randomly initialized weights using a large dataset with sparse annotations. We study the impact of the percentage of labeled pixels (10%, 25%, 50% and 100%) on the segmentation quality, when training on thousands of cells.

After describing the metrics used for validation, we will turn to the practical results.

Evaluation metrics

To quantify the predictions quality, we enumerate the true positives (TP), the true negatives (TN) and the false positives (FP). A true positive is an object in the gold standard that can be matched to an object in the prediction with an Intersection over Union (IoU) criterion higher than a Figure 6: The Sketchpose plugin in annotation mode threshold 𝜏. We let 𝑇 𝑃(𝜏) denote the total number of true positives. The total number of estimated objects without matches is denoted 𝐹𝑃(𝜏) (for false positives). The total number of gold standard objects without valid matches is denoted 𝐹 𝑁 (𝜏) (for false negatives). Utilizing these values, we compute the standard average precision metric (𝐴𝑃(𝜏)) for each image using the formula:

𝐴𝑃(𝜏) = 𝑇 𝑃(𝜏) 𝑇 𝑃(𝜏) + 𝐹𝑃(𝜏) + 𝐹 𝑁 (𝜏) .
The reported average precision is then computed as the average over all images in the test set.

This evaluation process corresponds to utilizing the matching_dataset function from the StarDist [START_REF] Schmidt | Cell detection with star-convex polygons[END_REF] package, with the by_image option set to True. Additionally, we computed the average DICE and the Aggregated Jaccard Index defined as follows:

𝐽 aggregated ( 𝐴, 𝐵) = 1 𝑁 𝑁 ∑ 𝑖=1 | 𝐴 𝑖 ∩ 𝐵 𝑖 | | 𝐴 𝑖 ∪ 𝐵 𝑖 | , average DICE( 𝐴, 𝐵) = 1 𝑁 𝑁 ∑ 𝑖=1 2 • | 𝐴 𝑖 ∩ 𝐵 𝑖 | | 𝐴 𝑖 | + |𝐵 𝑖 | ,
where 𝐴 and 𝐵 are a dataset and its groundtruth.

Training from scratch on a single image

In this section, we will showcase several results achieved while training from scratch on a small set of images. All the gold standards in this section and the next were generated by us and validated by biologists. The tests are made on a variety of biological structures (dendritic cells, osteoclasts, bacteria, insect eggs, adipose tissue, artistic image of cells).

Cells (artistic paint)

In the example of Figure 7, we use an artistic cell representation to showcase the possibility to reach satisfactory segmentation results in short annotation and computing times. On this painting, the nuclei are shown in red and the cytoplasmic membrane in green. To facilitate visualization and ensure that two contiguous areas do not have the same color when labeled differently, we used the 4-color theorem library called ncolor developed by K. J. Cutler.

After drawing for less than one minute and training for 100 epochs (2 minutes), we achieve an excellent result with no over segmentation, contrarily to the trained model of Omnipose (see Figure 7). The quality metrics is shown in Figure 7d.

Eggs on a tree leaf

In this section, we picked an image from the Omnipose dataset, which likely represents eggs of an insect on a tree leaf. At first sight, the segmentation task is uneasy, since the objects are tightly connected, with identical textures and blurry boundaries. We first annotated a subset of 5 eggs in Fig. 8b with a minimal amount of background. The segmentation result after training is already surprisingly good in Fig. 8d, but some objects are not detected, and others are merged. We annotated 2 extra eggs in Fig. 8c. With this extra information, retraining the network now produces a near perfect segmentation mask, with a single error (2 pink eggs on the left). This experiment illustrates a unique feature of Sketchpose: it is possible to interactively annotate while training. This offers a possibility to label a minimum amount of regions to reach the desired output. This principle sometimes called "active learning" or "human-in-the-loop" [START_REF] Budd | A survey on active learning and human-inthe-loop deep learning for medical image analysis[END_REF] is significantly enhanced by using partial annotations and the user-friendly Napari interface.

Transfer learning on a single image

In this section, we explore the feasibility of improving pre-trained weights using transfer learning.

Bacteria segmentation

Bacteria are often used as biological models (e.g. in DNA studies). A precise segmentation can be difficult to achieve, because they have elongated shapes and can be clustered.

The Omnipose [START_REF] Cutler | Omnipose: a high-precision morphology-independent solution for bacterial cell segmentation[END_REF] model was initially conceived to address the shortcomings of Cellpose for this task. Figure 9 shows how transfer learning with sparse annotations can improve the Omnipose results by separating touching bacterias. Figure 9d shows a quantitative comparison of both methods. As can be seen, Sketchpose's adapted weights provide much higher performance. A visual inspection indicates that all objects have been correctly separated, apart from the cluster touching the boundary on the bottom left.

Adipocytes segmentation

The image in Fig. 10d shows a crop of a very large image of a skin explant provided by DIVA Expertise. One can see a part of the dermis (in pink) and above it, adipose tissue (large white circular cells). Adipose tissue is the third skin layer after the epidermis and dermis, also known as the hypodermis. Hypodermal cells (adipocytes) secrete specific molecules (e.g. adiponectin, leptin) which have a direct impact on the biology of fibroblasts present in the dermis, and also on keratinocytes present in the epidermis. They are the subject of numerous studies (see [START_REF] Bourdens | Short exposure to cold atmospheric plasma induces senescence in human skin fibroblasts and adipose mesenchymal stromal cells[END_REF] [START_REF] Zhu | Effects of fixation on bacterial cellular dimensions and integrity[END_REF] and [START_REF] Neil S Sadick | The facial adipose system: its role in facial aging and approaches to volume restoration[END_REF] for instance). For most of the studies where skin explants are imaged, we first need to count the adipocytes number in the image, and remove any potential outliers detected in the dermis and epidermis.

While Omnipose cyto2 results in some undersegmentation for this task, the adapted weights provided by Sketchpose yields significantly enhanced results. Annotating 6 cells and a training for 100 epochs (less than 1 minute) were sufficient to significantly improve the quality of the segmentation and to remove the outliers from the dermis (see Figure 10). Figure 10d shows a quantitative comparison between Omnipose and Sketchpose on this example.

Osteoclasts segmentation

Osteoclasts are responsible for bone resorption, and are widely studied (see [START_REF] Labour | Tgf𝛽1-induced recruitment of human bone mesenchymal stem cells is mediated by the primary cilium in a smad3dependent manner[END_REF] for instance) as being responsible for certain pathologies such as osteoporosis when dysfunctional. Their differentiation goes through several stages, culminating in the activated osteoclast. The latter is generally large and contains numerous nuclei. Atlantic Bone Screen (ABS) company is investigating the effect of different drugs in inducing either proliferation or cell death in these activated osteoclasts, in order to regulate their population. To do so, they extract osteoclasts from biopsies, culture them, apply the drugs and image them under a bright-field microscope.

The studied image is a crop of an image containing around 20,000 cells. We can see touching cells presenting a great variety in size, shape color. The image is complex to segment and poses a real challenge. What is more, ABS does not want to count pre osteoclasts (small black nuclei), but only the mature cells (according to specific nuclei criteria). Each study comprises around sixty images, hence manual counting task performed at ABS is costly and laborious.

In Figure 11, we present a qualitative depiction that underscores the enhancement in segmentation accuracy attained through transfer learning with just a few labels. Labeling required approximately 2 minutes, while the training process took about 5 minutes. A quantitative comparison is available in Figure 11d.

Training from scratch on a large dataset

The aim of this experiment is to highlight the possibility to train our model on a large dataset with sparse annotations.

The Cellpose dataset

To compare trainings with full or partial annotations, we used the Cellpose dataset, which was initially conceived with the data science bowl [START_REF] Juan C Caicedo | Nucleus segmentation across imaging experiments: the 2018 data science bowl[END_REF]. This dataset used for training and evaluation is a carefully curated compilation of 540 images, each capturing various cellular morphologies and imaging scenarios. A sample of this dataset is presented in Figure 12. Notice that the latest Cellpose models were trained using a considerably enriched dataset. To the best of our knowledge, this larger dataset is not publicly accessible.

Selecting annotation subsets

In this section, we investigate the model robustness across various annotation levels each characterized by a different percentage of annotated pixels: 10%, 25%, 50%, and 100%. We generate randomly binary masks by threshold- ing white Gaussian noise with a Gaussian filter of variance 𝜎 2 . The resulting Gaussian process is then thresholded to keep only a given proportion of pixels.

While the model is stochastic in nature, the generated data is created once and for all, enabling its deterministic reuse across multiple training sessions. Figure 13 shows an example of training image from which we have kept 25% of the label mask pixels.

Results

After training four models, 500 epochs each (2.5 days per model with a single Nvidia RTX5000) on the whole dataset with the four percentages of annotated pixels we described above, we evaluated the performance through our three metrics (see Figure 14).

Surprisingly, it seems that training with 100%, 50% or 25% of labeled pixels lead to near identical performance. The performance seems to break down for "10"% Figure 12: Some image crops extracted from Cellpose dataset, showing its variability. In addition to cells, we can observe various objects (e.g., fruits, pebbles, scales)

annotation, yet the average precision (56%) decreased by only 16% compared to the best model (72%) with an IoU threshold of 50%. As a conclusion, it seems that training with only 25% of annotated pixels is a good strategy to save labelling time without degrading the segmentation quality. Figure 15 shows an example of how segmentation quality declines as the percentage of annotated pixels decreases.

Discussion & conclusion

We introduced Sketchpose, an open-source plugin to extend the applicability of Cellpose and Omnipose to partial annotations. From a methodological aspect, we developed a theory making it possible to use distance functions, despite having only access to partial information on the objects boundaries. From a more practical viewpoint, we developed an interactive interface within Napari, which facilitates efficient online learning with a real-time visualization of the training progress. The multi-threaded im-plementation allows users to continue annotating while the neural network trains or infers.

The new training procedure was tested in three different frames: i) training a neural network from scratch and just a few strokes, ii) improving the weights of a pre-trained network (a.k.a. transfer learning or human in the loop), iii) training with massive, but partial annotations.

For point i), frugal annotation seems to works surprisingly well despite really limited information. Just a few strokes are already enough to provide results on par -or better -than pre-trained networks.

For point ii), our experiments demonstrated the potential benefits of using transfer learning. That is, starting with a pre-trained Omnipose models, we can further refine it using our methodology.

As for point iii), we showed that training the model with just one fourth of the annotations leads to near identical performance, with a substantial reduction in annotation time. This suggests that partial, but less timeintensive annotations, might be an effective strategy to generate training sets without sacrificing the quality of the final model.

The plugin also shows a few limitations. First, the method would benefit from faster training times to make the method even more interactive. We plan to improve this aspect in the forthcoming versions. Second, it is important to mention that our formalism is currently restricted to the two dimensional setting with two labels (background/foreground). The proposed strategy could be used if the user was able to delineate a surface surrounding the objects of interest, but not just curves. In 3D, this would result in an empty valid distance set (see Theorem 1) and unadapted loss functions. This limitation of the method must be put into perspective by the fact that even the Cellpose 3D model is based on 2D predictions only, which are aggregated in post-processing.

In summary, the proposed method demonstrated numerous qualities in 2D for partial annotations, but further developments are needed to accelerate the training process and for an multi-class extension in 3D. We start with a basic observation.

Proposition 1 (Properties of the distance function).

•

A 1 ⊂ A 2 ⇒ ∀x ∈ X, dist(x, A 2 ) ≤ dist(x, A 1 ). • A 1 ⊂ A 2 and x ∈ A 1 ⇒ dist(x, 𝜕A 1 ) ≤ dist(x, 𝜕A 2 ).
Proof. The first item is direct:

dist(x, A 1 ) = inf x ′ ∈ A 1 dist(x ′ , x) ≥ inf x ′ ∈ A 2 dist(x ′ , x) = dist(x, A 2 ).
Here is one proof of the second iten by separating the two cases: either 𝑥 ∈ Å1 or 𝑥 ∈ 𝜕A 1 .

• Case 1: 𝑥 ∈ 𝜕A 1 . This case is trivial since dist(x, 𝜕A 1 ) = 0 ≤ dist(x, 𝜕A 2 ) by positivity of the distance.

• Case 2: 𝑥 ∈ Å1 . In that case, the key argument is to show that the open ball of radius dist(x, 𝜕A 1 ) centered in 𝑥 is included in Å1 . Precisely

B def = B (𝑥, dist(x, 𝜕A 1 )) ⊆ Å1 ⊆ Å2 . (17)
Indeed having Equation ( 17) established implies by contraposition that

𝜕A 2 ⊆ Å2 𝑐 ⊆ B𝑐 (18)
where the first inclusion is given by 𝜕A

2 def = Ā2 \ Å2 ⊆ Å2
𝑐 . Therefore, taking infimum with respect to these sets, implies the following inequalities and by the way the intended result.

dist(x, 𝜕A 2 ) = inf 𝑧 ∈𝜕A 2 ∥𝑧 -𝑥∥ ≥ inf 𝑧 ∈ Å2 𝑐 ∥𝑧 -𝑥∥ ≥ inf 𝑧 ∈ B𝑐 ∥𝑧 -𝑥∥ = dist(x,
= 𝑥 + (1 -1 𝑛 ) (𝑧 * -𝑥) ∈ [𝑥, 𝑧 * [⊆ Å1 converges to 𝑧 * . The contradiction comes from dist(x, 𝜕A 1 ) ≤ ∥𝑧 * -𝑥∥ ≤ ∥𝑧 -𝑥∥ < dist(x, 𝜕A 1 ).
The first inequality holds because 𝑧 * ∈ 𝜕A 1 , the second one because 𝑧 * ∈ [𝑥, 𝑧] and last one because 𝑧 ∈ B.

By proof by contradiction, Equation ( 17) holds.

In conclusion, in all cases the inequality is verified. □ Theorem 1 can be proven in two steps. First, notice that the inclusion B ⊆ E (Assumption 1) and the first bullet in Proposition 1 implies that dist(x, E) ≤ dist(x, B) for any x ∈ X.

Let's establish the converse inequality. Let x denote an arbitrary point in D. Aiming for a proof by contradiction, assume that dist(x, E) < dist(x, B). We can proceed by separating two cases:

• Case 1: dist(x, E) = 0. This implies that x ∈ E since the set E is closed as a finite union of closed sets 𝜕X 𝑖, 𝑗 . Moreover, as x belongs to D, in particular x belongs to S. It is sufficient to apply [START_REF] Legland | Morpholibj: integrated library and plugins for mathematical morphology with imagej[END_REF] and obtain x ∈ E ∩ S ⊆ B which is inconsistent with dist(x, B) > 0.

• Case 2: 𝑟 Since 𝑃(0) = x ∈ S 𝑖 0 and 𝑃(1) = z ∉ S 𝑖 0 , the scalar 𝑠 is well defined. The remaining task is to show that 𝑃(𝑠) ∈ 𝜕S 𝑖 0 . The argument works by construction and with a topological argument. Indeed, by definition of the infimum, there exists a sequence 0 < 𝜂 𝑛 → 0 such that 𝑃(𝑠 + 𝜂 𝑛 ) ∉ S 𝑖 0 , thus 𝑃(𝑠) ∉ S𝑖 0 . Also by definition, for all 0 ≤ 𝜂 < 𝑠, 𝑃(𝜂) ∈ S 𝑖 0 , thus 𝑃(𝑠) ∈ S𝑖 0 .

In both cases, the assumption dist(x, E) < dist(x, B) leads to a contradiction. We deduce that dist(x, E) ≥ dist(x, B).

The second inequality in Theorem 1 is a consequence of the property [START_REF] Labour | Tgf𝛽1-induced recruitment of human bone mesenchymal stem cells is mediated by the primary cilium in a smad3dependent manner[END_REF]. Indeed, this property implies that we can separate the strokes S 𝑖 into connected components S 𝑖, 𝑗 . These are subsets of the connected components X 𝑖, 𝑗 ′ for some 𝑗 ′ depending on 𝑗. The inequality is then just a consequence of Proposition 1.

Data augmentation

Data augmentation is one of the most expensive parts of the code. In fact, it recalculates the flows and distance maps on the fly at each iteration. In addition, for learning to converge faster, it is important to ensure that a patch always contains labels, even with partial annotations. In the original implementation, this was done recursively, i.e. by randomly selecting patches until they contained data. This method is inefficient for sparsely annotated images. Instead, we choose to randomly select centroids of patches among annotated pixels, so that the randomly selected patches are never empty. This avoids unnecessary recursive computation of the data augmentation function.

Furthermore, since data augmentation and flow calculation are not commutative, it was not possible to calculate the flows once and then apply the data augmentation. Nevertheless, it would be possible to calculate a deterministic number of data augmentations on the images in the dataset in advance, calculate the flows and then simply pick up the data using a data loader. We will explore this feature in forthcoming versions.

Figure 1 :

 1 Figure 1: Example of unstability of the distance map

Figure 2 :

 2 Figure 2: A sketch of the architecture and training procedure in Omnipose

  (a) Gold-standard & corresponding definitions (b) An admissible ground-truth annotation

Figure 3 :Figure 4 :

 34 Figure 3: Ground-truth and an associated admissible annotation set

Figure 5 :

 5 Figure 5: Illustration of the valid distance set theorem. All the pixels in D are closer to B than to the boundaries of S. The colormap used to represent the distance map exhibits a progressive shift in colors, transitioning from blue to red

  (a) The sparse labels (b) Omnipose cyto2 model (c) Sketchpose result, trained from scratch (≈ 2 minutes) Evaluation of the segmentation quality. Omnipose: DICE=0.99, Jaccard index=0.91. Sketchpose: DICE=0.95, Jaccard index=0.90

Figure 7 :

 7 Figure 7: (a,b,c) A training from scratch with sparse labels. Image Credit: Eduard Muzhevskyi. The colormap has been changed to improve visualization. (d) Evaluation of segmentation quality.

Figure 8 :

 8 Figure 8: Progressive training in Sketchpose.In this example, we show that it is possible to improve the segmentation performance of Sketchpose by progressively annotating at places where the network failed. Here, a quite minimal annotation set, is enough to near perfectly separate the eggs on the leaf.

  (a) The sparse labels (b) Omnipose bact_phase model (c) Sketchpose, transfer learning (≈ 1 minute)

Figure 9 :

 9 Figure 9: Improvement of bacteria segmentation starting from the Omnipose bact_phase model. The zoomed-in view shows an under-segmentation resolved by Sketchpose transfer learning. Image from[START_REF] Zhu | Effects of fixation on bacterial cellular dimensions and integrity[END_REF]

Figure 10 :

 10 Figure 10: (a,b,c) Improvement of adipocytes segmentation starting from the Omnipose cyto2 model. (d) Evaluation of segmentation quality.

  (a) The sparse labels (b) Omnipose cyto2 segmentation (c) Sketchpose, transfer learning (≈ 5 minutes) Evaluation of segmentation quality. Omnipose: DICE=0.81, Jaccard index=0.53. Sketchpose: DICE=0.90, Jaccard index=0.72

Figure 11 :

 11 Figure 11: (a,b,c) A training with a few labels on osteoclasts. (d) Evaluation of segmentation quality.

Figure 15 :

 15 Figure 15: An example of segmentation precision loss on a Cellpose test dataset image when percentage of annotated pixels in the training dataset decreases from 100 to 10%. One can see segmentation quality highly decreases with 10% of labeled pixels

def=-

  dist(x, E) > 0. The point x verifies dist(x, B) ≤ dist(x, CB) as x ∈ D. Let us define𝑟 def = dist(x, E) > 0 and 𝜀 def = dist(x, CB)dist(x, E) ≥ dist(x, B)dist(x, E) > 0 by assumption. Since x belongs to S, there exists 𝑖 0 ∈ {0, 1} such that x ∈ S 𝑖 0 . Because dist(x, E) = 𝑟, there exists a point z ∈ E such that 𝑟 ≤ ∥x -z∥ 2 = 𝑟 + 𝜀/2.-Case 2.a: z ∈ S 𝑖 0 . By assumption[START_REF] Legland | Morpholibj: integrated library and plugins for mathematical morphology with imagej[END_REF], the contradiction comes quickly since nowz ∈ S 𝑖 0 ∩ E ⊆ S ∩ E ⊆ B ⊆ CB,and this implies the contradictive inequality𝑟 + 𝜀 = dist(x, CB) ≤ ∥x -z∥ 2 ≤ 𝑟 + 𝜀/2. Case 2.b: z ∉ S 𝑖 0 .In that case, we may define the point y on the line [x, z] which is the nearest from the point x and also in 𝜕S 𝑖 0 . Since y ∈ 𝜕S 𝑖 0 ⊆ CB, it implies a contradiction as intended:𝑟 + 𝜀 = 𝑑 (x, CB) ≤ ∥x, 𝑃(𝑠) ∥ 2 = 𝑠∥x -z∥ 2 ≤ ∥x -z∥ 2 = 𝑟 + 𝜀/2.The point y is defined as 𝑃(𝑠) where the map 𝑃 : 𝑡 ↦ → 𝑡z + (1 -𝑡)x assigns to each scalar 𝑡 ∈ [0, 1] a point 𝑃(𝑡) of the line and set𝑠 def = inf 𝑃 (𝑡 ) ∈ S 𝑖 0 𝑡.

  𝜕A 1 ).

	So let's prove Equation (17): by contradiction, assume
	that there exists 𝑧 ∈ B ∩ Å1	𝑐 . Notice that [𝑥, 𝑧] ∩ Å1	𝑐
	is a compact set (here [𝑥, 𝑧] denotes the closed segment
	between the points 𝑥 and 𝑧). Thus
	𝑧 * def = arg min

𝑥 ′ ∈ [ 𝑥,𝑧 ]∩ Å1 𝑐 ∥𝑥 -𝑥 ′ ∥

is well defined and the semi-open segment [𝑥, 𝑧 * [ is included in Å1 . This implies that 𝑧 * ∈ 𝜕A 1 since the sequence 𝑧 𝑛 def

Acknowledgments

We are grateful for the information provided by Kevin John Cutler about the original Omnipose implementation.

Funding Statement

C.Cazorla was a recipient of ANRT (Agence Nationale pour la Recherche et la Technologie) in the context of the CIFRE Ph.D. program (N°2020/0843) with Imactiv-3D and Institut de Mathématiques de Toulouse (IMT). P. Weiss acknowledges a support from ANR-3IA Artificial and Natural Intelligence Toulouse Institute ANR-19-PI3A-0004 and from the ANR Micro-Blind ANR-21-CE48-0008. This work was performed using HPC resources from GENCI-IDRIS (Grant 2021-AD011012210R1).

The authors would like to thank the authors of Cellpose and Omnipose for making their datasets available. The authors acknowledge Atlantic Bone Screen for providing the osteoclasts image and DIVA Expertise for providing the adipocytes image.

Source code and documentation

The code is available there: https://bitbucket. org/koopa31/napari-sketchpose/src/ master/.

The documentation is available there: https://sketchpose-doc.readthedocs.io/ en/latest/.


M Viprey

Y Xu

A Rousseau

F Abbas

A Dima

A M Schott

Adherence with brand versus generic bisphosphonates among newly treated osteoporosis patients: a retrospective cohort study in the French claims database

Results

 During the 2009-2015 period, 3,903 patients initiated an oral BP treatment for primary OP: 1,710 (43.8%) with generic BP and 2,193 (56.2%) with brand BP. Results of switches and implementation rates according to BP group are presented in Table 1.  Univariate analyses were conducted to identify factors associated with good implementation (defined by CMA7 ≥ 0.9). Factors tested were: exposition of interest (BP group) , year of initiation, frequency of BP administration, oral BP molecule, prescriber specialty, age, sex, presence of a major chronic disease, socioeconomic frailty, polypharmacy, history of osteoporotic fracture, Charlson comorbidity index.  Binary logistic regression modelling was then used to predict good implementation, by stepwise addition of exposition of interest and variables associated with good implementation at a probability level of 0.2 or less in bivariate analyses. Results of multivariate analysis are presented in Table 2.

Background and aims

Oral bisphosphonate (BP) are recommended among the first-line treatments for osteoporosis (OP), and a low adherence to oral BP has been associated with an increased risk of further fractures. Several studies documented declines in treatment adherence with generic forms of BP compared to branded forms, possibly related to increased occurrence of gastrointestinal adverse effects, while other studies did not support this relation. As this topic remains controversial, we conducted an analysis of the French national database to test the hypothesis of a reduced medication adherence with brand versus generic forms of oral BPs.

Methods

Conclusion

In multivariate analysis, once adjusted for confounding factors, there were no differences in implementation with brand or generic BP in this population.

Adherence rate was higher in our study than the mean medication possession ratio of 66.9% presented in the meta-analysis of Imaz et al. (Osteoporos Int, 2010). This difference could be due to the fact that we chose to evaluate the dimensions of persistence and implementation separately, and that our methodology of adherence calculation included hospitalizations.

22nd ESPACOMP conference -29 th November-1 st December, 2018 -Dublin, Ireland -ESPACOMP18-0087

 Retrospective, comparative cohort study carried out using the Generalist Sample of Beneficiaries (EGB), a

	permanent representative sample of the
	general population of subjects affiliated with the National French Health
	Insurance System.
	 Inclusion of

all patients aged 50 and older, new users of oral BP alone

	(alendronic acid, ibandronic acid, risedronic acid or etidronic acid, not
	associated with vitamin D)

for primary osteoporosis between 01/01/2009 and 31/12/2015.  Patients followed from the initiation date until

  : o switch (change from generic to brand drug or brand to generic, or change from one oral BP to another), o or discontinuation of BP (no new dispensation during a period greater than 2 times the duration of the previous supply), o or 12 months.

 Implementation evaluated using the continuous multiple-interval measure of medication availability version 7 (CMA7), for patients under treatment during at least 6 months: o dosages considered were the defined daily doses, o hospitalization were taking into account assuming that treatments were supplied by the hospital during the hospitalization period.

Table 1 :

 1 Implementation and switches with first-line BP initiated during the 2009-2015 period, according to BP group (N=3,903)

	Brand BP
	N = 2

,193 Generic BP N = 1,710 Still on first-line BP at 12 months, n (%)

	540 (24.6)	459 (26.8)

BP treatment discontinuation, n (%) before 6 months between 6 and 12 months

				Bivariate OR [IC95%] Multivariate OR [IC95%] 1 p-value 1
			BP group
			Generic	1	1
			Brand	0.846 [0.663;1.079]	0.974 [0.713;1.329]	0.8659
			Frequency of BP administration
			Daily	1	1
			Weekly	1.011 [0.213;4.806]	1.022 [0.212;4.934]	0.9784
			Monthly	0.716 [0.150;3.403]	0.733 [0.149;3.616]	0.7033
			Oral BP molecule
			Risedronic acid	1	1
			Alendronic acid	0.967 [0.713;1.312]	0.850 [0.593;1.220]	0.3790
			Ibandronic acid	0.735 [0.549;0.985]	0.986 [0.675;1.439]	0.9397
			Etidronic acid	0.531 [0.048;5.887]	0.653 [0.058;7.341]	0.7297
	Switch to the other BP group, n (%) before 6 months between 6 and 12 months Switch to another molecule, n (%) before 6 months between 6 and 12 months Mean CMA7 ± SD* Class of CMA7, n (%)* <90% ≥90%	1,105 (50.4) 189 (8.6) 152 (6.9) 80 (3.6) 82 (3.7) 45 (2.1) 0.932 ± 0.071 0.941 ± 0.071 860 (50.3) 176 (10.3) 118 (6.9) 38 (2.2) 53 (3.1) 6 (0.4) 202/854 (23.7) 652/854 (76.4) 141/679 (20.8) 538/679 (79.2)	Prescriber specialty* General practitioner Specialist practitioner Hospital practitioner Class of age (years) 50 to 59 60 to 69 70 to 79 ≥ 80 Sex Male Female	1 0.711 [0.431;1.171] 1.486 [0.750;2.946] 1 1.146 [0.808;1.626] 1.269 [0.890;1.809] 1.530 [1.034;2.265] 1 0.587 [0.363;0.949]	1 1.124 [0.812;1.556] 1.394 [0.952;2.042] 1 1.157 [0.813;1.647] 1.246 [0.871;1.784] 1.459 [0.982;2.166] 1 0.627 [0.382;1.030]	0.4824 0.0878 0.4187 0.2284 0.0614 0.0651
	* Analysis on the 1,533 patients under BP treatment during at least 6 months

Table 2 :

 2 Multivariate analysis of factors associated with good implementation (CMA7 ≥ 0.9) among patients under BP treatment during at least 6 months (N=1,530) * Data were missing for 3 patients, who were excluded from multivariate analysis