I R Ène

Alain Trouv

EXISTENCE AND CONTINUITY OF MINIMIZERS FOR THE ESTIMATION OF GROWTH MAPPED EVOLUTIONS FOR CURRENT DATA TERM AND COUTEREXAMPLES FOR VARIFOLD DATA TERM

In the field of computational anatomy, the complexity of changes occurring during the evolution of a living shape while it is growing, aging or reacting to a disease, calls for more and more accurate models to allow subject comparison. Growth mapped evolutions have been introduced to tackle the loss of homology between two ages of an organism following a growth process that involves creation of new material. They model the evolution of longitudinal shape data with partial mappings. One viewpoint consists in a progressive embedding of the shape into an ambient space on which acts a group of diffeomorphisms. In practice, the shape evolves through a time-varying dynamic called the growth dynamic.

The concept of shape space has now been widely studied and successfully applied to analyze the variability of a population of related shapes. Time-varying dynamics subsequently enlarge this framework and open the door to new optimal control problems for the assimilation of longitudinal shape data. We address in this paper an interesting problem in the field of the calculus of variations to investigate the existence and continuity of solutions for the registration of growth mapped evolutions with the growth dynamic. This theoretical question highlights the unexpected role of the data term grounded either on current or varifold representations. Indeed, in this new framework, the spatial regularity of a continuous scenario estimated from a temporal sequence of shapes with the growth dynamic depends on the temporal regularity of the deformation. Current metrics have the property to be more robust to this spatial regularity than varifold metrics. We will establish the existence and continuity of global minimizers for current data term and highlight two counterexamples for varifold data term.

In the field of computational anatomy, the diffeomorphometry, as introduced by Miller, Trouvé, and Younes [START_REF] Miller | Diffeomorphometry and geodesic positioning systems for human anatomy[END_REF]33], consists in modeling and analyzing the variability of a population of embedded shapes through the action of a group of diffeomorphisms. The viewpoint of optimal control offers an elegant framework for diffeomorphic registration [START_REF] Allassonniere | Geodesic shooting and diffeomorphic matching via textured meshes[END_REF][START_REF] Trouvé | Shape Splines and Stochastic Shape Evolutions: A Second Order Point of View[END_REF]2]. Initially, the matching between two embedded shapes S and S tar in R d , like curves or surfaces, consists in the problem [START_REF] Allassonniere | Geodesic shooting and diffeomorphic matching via textured meshes[END_REF] min

φ R(φ) + A(φ)
where R is a regularization term on the deformation φ : R d → R d and A measures the discrepancy between the target shape S tar and the deformed source shape φ(S). The construction of diffeomorphism groups with Riemannian metrics allows to address this problem in the setting of high-dimensional deformation space. A group is defined by the space V of vector fields that models its tangent space at the identity. An element of the group is then generated as the end point of a flow (φ v t ) t∈[0,1] which satisfies the ordinary differential equation: (2) φt = v t • φ t , φ 0 = Id where v t ∈ V for all t. The existence of geodesics in the group allows to rewrite the variational problem

(3) min v R(v) + A(φ v 1 ) over square integrable vector fields of V and the introduction of Reproducing Kernel Hilbert Spaces (RKHS) to model V leads to efficient numerical methods like the Large deformation diffeomorphic metric mapping (LDDMM).

1.2. Growth mapped evolutions. With the progress achieved in medical imaging analysis, the interest for longitudinal data set has substantially increased in the last years and requires the processing of complex changes, which especially appear during growth or aging phenomena. Various methods derive from the concept of shape spaces as Riemannian manifolds ranging from parallel transport [37], Riemannian splines [START_REF] Trouvé | Shape Splines and Stochastic Shape Evolutions: A Second Order Point of View[END_REF], geodesic regression [START_REF] Niethammer | Geodesic Regression for Image Time-Series[END_REF][START_REF] Vialard | 3d image registration via geodesic shooting using and efficient adjoint calculation[END_REF][START_REF] Fletcher | Geodesic regression and the theory of least squares on riemannian manifolds[END_REF] including the inference from a population of a prototype scenario of evolution and its spatio-temporal variability [START_REF] Durrleman | Toward a comprehensive framework for the spatiotemporal statistical analysis of longitudinal shape data[END_REF]. In another context, the unbalanced optimal transport [START_REF] Lombardi | Eulerian models and algorithms for unbalanced optimal transport[END_REF]36] extends the optimal transport framework to tackle the problem of mass creation. Up to now, longitudinal analysis has been limited to the study of data sets with homologous observations. Yet, during the growth or the degeneration of an organism, the changes occurring over time cannot always be explained by diffeomorphic transformations, at least in a biological sense.

In [START_REF] Kaltenmark | Trouvé Partial Matchings and Growth Mapped Evolutions in Shape Spaces[END_REF], we introduced the concept of growth mapped evolutions (GME) that consists in a set of embedded shapes (S t ) t∈T indexed by a time interval T ⊂ R and equipped with a flow (φ t ) t∈T on the embedding space R d . The constraint of exhaustive homology between any two shapes S s and S t is relaxed, as illustrated in Figure 1, by an inclusion condition: for any pair s ≤ t in T , [START_REF] Arguillère | The general setting for shape analysis[END_REF] φ s,t (S s ) ⊂ S t , where φ s,t = φ t • φ -1 s . A growth mapped evolution is therefore a nested sequence of all ages of the shape through the flow. The shape S t is composed partly by the image φ s,t (S s ) of a previous state and by the new points that appeared between the time s and t calling to consider the time of birth of each point, induced by the colors in the discrete sketch of Figure 1. The viewpoint of coordinate systems with a temporal component allows to describe the history of a creation process and to define more specific populations of GME sharing a common growth pattern. A biological coordinate system consists in a space X called the coordinate space and a function τ : X → T called the birth tag whose lower sets generate a collection of nested shapes X t = {x ∈ X | τ (x) ≤ t} of X forming a canonical growth scenario. Any individual can then be parametrized with a collection of smooth mappings q t : X t → R d so that S t = q t (X t ) . The level sets X {t} = {x ∈ X | τ (x) = t} are the new coordinates of X t created at time t and q t (X {t} ) models the new points of S t . We aim to build mappings (q t ) t that are consistent with the flow, meaning that for any s < t ∈ T and any x ∈ X s , q t (x) = φ s,t (q s (x)) .

If (φ t ) t is the flow of a time-varying vector field v : T → V , it follows that qt (x) = v t (q t (x)) .

The mappings (q t ) t are extended to X to anticipate the appearance of the new points. If the point associated to a coordinate x ∈ X does not exist at time t, q t (x) returns its future place of birth. This leads to require that q t (x) = q 0 (x) for any t ≤ τ (x) and that ( 5) qt (x) = 1 1 τ (x)≤t v t (q t (x)) .

This new dynamic that depends both on the control v t and the time variable t is called the growth dynamic. See [START_REF] Kaltenmark | Trouvé Estimation of a Growth Development with Partial Diffeomorphic Mappings[END_REF]23] for more details and for a first study of the following variational problem. Throughout this paper, all scenarios evolve in the fixed time interval T = [0, 1].

1.3. Contributions. Consider a population of growth mapped evolutions modeled on a biological coordinate system (X, τ ). The estimation of a such growth mapped evolution, given an initial condition q 0 and a target shape S tar to reach at the final time of the development, can be expressed as a minimization problem on an energy of the type ( 6)

E(v) = 1 2 1 0 |v t | 2 V dt + A(v) ,
where v belongs to L 2 ([0, 1], V ), the initial mapping q 0 is fixed, q satisfies the growth dynamic [START_REF] Avants | Symmetric diffeomorphic image registration with cross-correlation: evaluating automated labeling of elderly and neurodegenerative brain[END_REF] and A : L 2 V → R measures the discrepancy between the target and the final shape q 1 (X). In the usual approach of shape spaces, the matchings of two shapes consists in searching a geodesic in a chosen space G V of diffeomorphisms with constraints on the ends. Retrieving an optimal growth scenario via the growth dynamic does not only constrain the ends of the flow of diffeomorphisms. Indeed, the final status q 1 of a solution cannot be written as an image of the initial mapping q 0 by the final state of the flow φ v 1 . It depends instead on the whole evolution of the flow over time and the variational problem (3) does not encompass this difficulty. The optimal flow to reach a final target is usually not a geodesic of G V . This paper examines the existence and the continuity of global minimizers v to the optimization problem [START_REF] Beg | Computing large deformation metric mappings via geodesic flows of diffeomorphisms[END_REF]. In Section 2, we recall that the spatial regularity of q 1 depends on the temporal regularity of v. The continuity of optimal solutions v * ∈ C([0, 1], V ) is thus a crucial point. The usual representations of shapes by currents or varifolds require to ensure that the shapes are sufficiently regular as they are a priori neither of class C 1 nor rectifiable. For this purpose, we will exploit in Section 2 the density of C([0, 1], V ) in L 2 ([0, 1], V ) as continuous vector fields actually generate mappings q 1 of C 1 (X, R d ).

Section 3 presents a rather unexpected result. We exhibit a setting with a data attachment term built on the representation by varifolds where no global minimizer is continuous. The difference between representations by currents or varifolds, regarding the models associated to the growth dynamic, is explained by the fact that oscillations in time of the vector field v generate oscillations in space for the shapes q t (X t ). The currents through their cancellation effect on these spatial oscillations can prevent this behavior.

In Section 4.1, the existence of solutions in L 2 ([0, 1], V ) for a current data term attachment is shown in a general setting. The proof is based on the linearity to the tangential component of the current representation that allows to deduce the lower semi-continuity of A. The continuity of an optimal vector field v does not follow immediately and will be established in Section 4.2.

1.4. Data representation. As many registration problems in shape analysis, the reconstitution of a growth mapped evolution requires to use a shape similarity metric. The seminal work of [START_REF] Vaillant | Surface matching via currents[END_REF][START_REF] Glaunès | Large deformation diffeomorphic metric curve mapping[END_REF] was the first to exploit the idea from geometric measure theory of representing shapes as elements of a certain space of distributions leading to distances directly invariant to parametrization. This approach is based on the representations of oriented curves or surfaces as mathematical currents. Later on, [START_REF] Charon | The Varifold Representation of Non-oriented Shapes for Diffeomorphic Rregistration[END_REF] introduced the alternative but orientation-invariant representation known as varifolds, with a straightforward extension to oriented varifolds [START_REF] Kaltenmark | Charon A general framework for curve and surface comparison and registration with oriented varifolds[END_REF], before the higher order model of normal cycles recently investigated in [START_REF] Roussillon | Kernel Metrics on Normal Cycles and Application to Curve Matching[END_REF]. Another recent approach that still stems from the representation of shapes by distributions, integrates unbalanced optimal transport to define a new similarity measure [START_REF] Feydy | Optimal Transport for Diffeomorphic Registration[END_REF].

The focus of this paper is set on currents and varifolds. In the case of currents, the metric is define by a specific kernel linear with respect to the tangential data. This property is well known to make this metric robust to the noise. See Figure 2. Note however, as a downside, that this linearity also prevents the capture of structures like sharp spines or tails. As we will see in this paper, this central difference between current and varifold metrics will have an important impact on how they drive the reconstitution of growth mapped evolutions.

Figure 2. Denote X the noisy red curve and Y the smooth one. From a current point of view, µ X ≈ µ Y . Conversely, with varifolds, the length of X is about twice the length of Y and this approximation no longer holds.

Extension of the varifold and current representations to foliated shapes

2.1. Foliation and spatial regularity of the GME's shapes. Figure 3 illustrates two scenarios generated with the growth dynamic. This time-varying dynamic allows to forecast within the initial mapping q 0 the position of the new points that will progressively be created. In these two examples, all the coordinates are initially set in a horizontal plane of R 3 . One can see in Figure 3 that at each time t, the individual shape is displaced upwards and the subset X {t} = {x ∈ X | τ (x) = t} of new coordinates is activated (their image model the new points created). When the creation process is regular, e.g. with a progressive addition of regular extensions at one boundary of the shape, the sets of new coordinates (X {t} ) t define a foliation on the coordinate space X. A foliation [38,[START_REF] Lee | Introduction to Smooth Manifolds[END_REF] looks locally like a union of parallel shapes of smaller dimension called the leaves of the foliation. In a general biological coordinate space (X, τ ), the existence of a foliation depends on the regularity of the birth tag τ . In this paper, we will consider a canonical situation where X = [0, 1] × B where B is an oriented compact manifold with corners and τ is just the projection on the first coordinate (for any (t, b)

∈ [0, 1] × B, τ (t, b) = t). Each leaf X {t} = {t} × B is diffeomorphic to B and we denote then B t .
= {t} × B. The initial condition q 0 has no reason to be an embedding of the whole coordinate space X. For the first scenario of Figure 3, all leaves of X have the same image in R d . The shape can be seen Figure 3. Two examples of individual scenarios built on a given biological coordinate system (X, τ ). The first column shows the top view of the initial position q 0 for each scenario. Each image scenario inherits the foliation of the biological coordinate system induced by the birth tag τ and enlightened by the color gradient. At each time, a close curve of new coordinates appear (we have X = [0, 1] × S 1 ).

as completely folded on itself and progressively developed in the ambient space, leaf by leaf. In order to ensure that a scenario generated by q globally corresponds to the trivial scenario induced by the coordinate system, we want each leaf of X to be embedded in R d . Some exceptions are yet allowed, typically for the outer leaves X {0} and X {1} , e.g. to model the tip of the horn. We assume thus that (H q0 ) : The restriction of q 0 to an inner leaf B t (t ∈]0, 1[) is a smooth immersion between B t and R d . This development leaf by leaf induced by the growth dynamic raises some difficulties to control the regularity of the final shape. Let us first recall that the existence and uniqueness of the GMEs generated by time-varying vector fields with the growth dynamic have been established in [23] in a general situation that we recall here. It requires a set of admissibility conditions on V

(7) (H V 1 ) i) V ⊂ C 2 (R d , R d ) .
ii) There exists c > 0 such that for any v ∈ V and any

x ∈ R d , |v(x)| R d ≤ c|v| V (|x| R d + 1) , |dv(x)| ∞ + |d 2 v(x)| ∞ ≤ c|v| V .
Consider then X a compact smooth manifold with corners and τ : X → [0, 1]. Due to the spatial discontinuity of the indicator function, the ODE of the growth dynamic

qt (x) = 1 1 τ (x)≤t v t (q t (x)) (8)
is defined in the Banach space L ∞ (X, R d ) and we have the following theorem Theorem 2.1. Under the (H V 1 ) conditions, for any initial condition q 0 ∈ L ∞ (X, R d ) and any control v ∈ L 2 ([0, 1], V ), there exists a unique solution q that satisfies (8) a.e. This solution is absolutely continuous: q ∈ AC([0, 1], L ∞ (X, R d )) and for any t ∈ [0, 1],

q t = q 0 + t 0 1 1 s≤τ v s • q s ds . Corollary 2.1. In the setting of Theorem 2.1, if q 0 ∈ C(X, R d ) and τ ∈ C(X, [0, 1]), then for any v ∈ L 2 ([0, 1], V ), q ∈ AC([0, 1], C(X, R d )).
The spatial regularity of q t at higher order requires a stronger temporal regularity of v. Even when the initial condition q 0 is of class C ∞ , if v belongs to L 2 V , we can only show that q 1 is differentiable almost everywhere. Figure 4 illustrates how the discontinuity of v impacts the generated shape (rectifiable yet on this basic example, but not C 1 ). A strong discontinuity of v at time t, meaning that t is not a Lebesgue point of v, affects the regularity of the shape along the leaf created at time t and this irregularity will hold until the end of the scenario, meaning that the final shape q 1 will be irregular along the leaf q 1 (X {t} ). However, if v is time continuous, then q 1 is of class C 1 : Proposition 2.1. In the setting of Theorem 2.1, if q 0 ∈ C 1 (X, R d ), τ ∈ C 1 (X, [0, 1]), and v ∈ C([0, 1], V ), then q 1 is of class C 1 and its differential is given for any x ∈ X by [START_REF] Durrleman | Morphometry of anatomical shape complexes with dense deformations and sparse parameters[END_REF] dq 1 (x) = dφ τ (x),1 (q 0 (x))

• dq 0 (x) -v τ (x) (q 0 (x))dτ (x) ,
where φ s,t is the flow of v on the ambient space R d .

See Proposition 4.8 in [23] for the proof. Note that it is well-known that under the (H V 1 ) conditions, this flow is of class C 1 and has a continuous differential in time and space [15,[START_REF] Younes | Shapes and Diffeomorphisms[END_REF].

Initial position, before the deformation Figure 4. The final state q 1 (X) displayed on the top left is a serrated curve with as many discontinuities as its associated vector field v given on the right as real-valued function modeling vertical translations upwards and downwards. The initial position q 0 (X) is a segment.

The lack of spatial regularity of the final shape q 1 (X) prevents the usual straightforward identification with a current or a varifold that yet applies in the restrictive setting of Proposition 2.1. We will therefore extend their definitions based on continuous trajectories v ∈ C([0, 1], V ) (setting of Proposition 2.1) to all the solutions generated by L 2 ([0, 1], V ). In the next sections, we will present for the purpose of this paper two specific situations where the shapes can be modeled by varifolds and finally we will prove that a similar extension holds for currents in a more general situation. In all these cases of interest, the key is to decompose the shape into its leaves, i.e. with the foliation induced by the birth tag, and to use the density [START_REF] Charon | The Varifold Representation of Non-oriented Shapes for Diffeomorphic Rregistration[END_REF], k-dimensional smooth shapes embedded in R d are modeled by the dual of a reproducing kernel Hilbert space (RKHS)

C([0, 1], V ) in L 2 ([0, 1], V ).

Extension for varifold representation. As presented by Charon and Trouvé

W on C 0 (R d × G k (R d ), R) where G k (R d
) is the Grassmanniann of all k-dimensional subspace of R d , e.g. lines through the origin of R d when k = 1. W represents then a space of varifolds. In all generality, the varifold µ ∈ W associated to a smooth shape Γ is given for any function ω ∈ W by

µ(ω) = R d ×G k (R d ) ω(x, V )dµ(x, V ) = Γ ω(x, T x Γ)dH k (x) , (10)
where for any Borel subset

A ⊂ R d × G k (R d ), µ(A) = H k ({y ∈ Γ | (y, T y Γ) ∈ A}).
The kernel of the RKHS W is denoted k W and given by the tensor product k E ⊗ k T of a kernel k E (x, y) on the ambient space R d and a kernel k T (U, V ) on the Grassmannian G k (R d ). Throughout this paper, we will consider a degenerate kernel k T ≡ 1 on the Grassmannian and we will write ω(x) instead of ω(x, V ) and dµ(x) instead of dµ(x, V ). 2.2.1. 2D case. We aim for a particularly simple situation to produce a counterexample. The shape is a horn modeled by a curve in R 2 . With the previous notation, we would define the coordinate space by X = [0, 1] × {-1, 1} and the birth tag τ by the projection on the first coordinate. However, to simplify the notation, we project X on the interval Ω = [-1, 1] with (t, b) → t × b, which merge the two initial coordinates that will model the tip of the horn. The curve is then parameterized by Ω and the initial condition is given by q 0 (r) = (r, 0). The horn is thus initially flattened on the horizontal segment [-1, 1] × {0} ⊂ R 2 . The birth tag is defined for any r ∈ Ω by τ (r) . = |r|. This means that points are progressively displaced starting from the points at the center and ending with the points at the boundaries of the segment. The deformations are reduced to vertical translations and the space of vector fields V is canonically identified to R. For any v ∈ L 2 ([0, 1], R), the growth dynamic is given by qt (r) = (0, v(t))1 |r|≤t and q t (r) = (r, 1 |r|≤t t |r| v(s)ds). The object of interest in this section is the shape at its final age t = 1 given by q 1 (r) = γ v (r) where

γ v : [-1, 1] → R 2 is defined by (11) γ v (r) = (r, 1 |r| v(s)ds) .
Note that the curve γ v is symmetric about the vertical axis {0} × R. Examples are displayed hereafter in Figure 6.

For any v ∈ C([0, 1], R), the varifold associated to the curve γ v is denoted µ v ∈ W and it is defined for any ω ∈ W by

µ v (ω) = Γv ω(x)dH 1 (x) = 1 -1 ω(γ v (r))| γv (r)|dr = 1 -1 ω(γ v (r)) 1 + v(|r|) 2 dr , (12)
where γv (r) is defined for any r ∈ Ω such that τ (r) / ∈ {0, 1}, i.e. r / ∈ {-1, 0, 1} (which corresponds to the boundary of X with the initial notation). We have γv

(r) = (1, -v(r)) if r ∈]0, 1[ and γv (r) = (1, v(-r)) if r ∈] -1, 0[. When v ∈ L 2 ([0, 1], R)
, γv is only defined a.e. The application [START_REF] Feydy | Optimal Transport for Diffeomorphic Registration[END_REF] Ψ W :

L 2 ([0, 1], V ), | • | L 2 V -→ W v -→ µ v : ω → 1 -1 ω(γ v (r)) 1 + v(|r|) 2 dr is yet well defined and continuous (W → C 0 (R 2 × G 1 (R 2 ), R)). Moreover, since C([0, 1], V ) is dense in L 2 ([0, 1], V ), Ψ W is the unique continuous extension of its restriction to C([0, 1], V ).
2.2.2. 3D case. The coordinate space X is now a cylinder but to simplify the notation, we instead parametrize the shapes by Ω the unit disc, equipped with the polar coordinate system. Points at their initial position are given by q 0 (θ, r) = (r cos θ, r sin θ, 0). The birth tag τ is equal to the radius τ (θ, r) = r. The growth dynamic is as before limited to vertical translations:

v ∈ L 2 ([0, 1], R), qt (θ, r) = (0, 0, v(t))1 1 r≤t .
The energy only refers to the final state of the shape. Thus, defining γ v (θ, r) = q 1 (θ, r), it follows that any time-varying vector field v ∈ L 2 ([0, 1], R) generates a surface described by the parametric function

γ v (θ, r) = (r cos θ, r sin θ, 1 r v s ds) . ( 14
) Let Jγ v be the Jacobian determinant of γ v , ∂ θ γ v (θ, r) = (-r sin θ, r cos θ, 0) , Jγ v (θ, r) = |∂ θ γ v (θ, r) ∧ ∂ r γ v (θ, r)| ∂ r γ v (θ, r) = (cos θ, sin θ, -v(r)) , = r 1 + v(r) 2 . For any v ∈ C([0, 1], R), the linear form µ v ∈ W that represents the surface is given for any ω ∈ W by µ v (ω) = 2π 0 1 0 ω(γ v (θ, r))r 1 + v(r) 2 drdθ .
Once again this expression can be extended to v ∈ L 2 ([0, 1], V ) and it defines the unique extension of v → µ v .

Extension for current representation.

Let Γ be a smooth oriented k-dimensional submanifold and denote for any x ∈ Γ, (T 1 (x), . . . , T k (x)) an orthonormal oriented basis of the tangent space

T x Γ. Then Γ is identified to the current µ ∈ C 0 (R d , ( k R d ) * ) defined for any ω ∈ C 0 (R d , ( k R d ) * ) by (15) µ(ω) = Γ ω(x)(T 1 (x) ∧ . . . ∧ T k (x))dH k (x) .
This definition can also be used for rectifiable sets which includes submanifolds with corners. We can now define the current associated to the final mapping q 1 generated by a continuous vector field.

Definition 2.1. For any v ∈ C([0, 1], V ), the current associated to the mapping

q 1 : X → R d is defined for any ω ∈ C 0 (R d , ( k R d ) * ) by (16) µ v (ω) = X q * 1 ω = X ω(q 1 (x)) ∂q 1 ∂x 1 (x) ∧ . . . ∧ ∂q 1 ∂x k (x) dx 1 . . . dx k .
For this purpose we rewrite equation ( 16) with the foliation of X given by its tagging function τ . Consider X = [0, 1] × B where B is an oriented compact manifold with corners so that τ is just the projection on the first coordinate (for any (t, b)

∈ [0, 1] × B, τ (t, b) = t). Denote (Y t ) 0<t<1 the set of submanifolds of R d that are the images of B t .
= {t} × B by q 0 . By definition of the growth dynamic, for any x ∈ X and any t ≥ τ (x), [START_REF] Glaunès | Large deformation diffeomorphic metric curve mapping[END_REF] q t (x) = q 0 (x)

+ t 0 1 1 s≤τ (x) v s (q s (x))ds = φ τ (x),t (q 0 (x)) .
The image of q 1 can then be rewritten

(18) q 1 (X) = t∈[0,1] φ t,1 (q 0 (B t )) = t∈[0,1] φ t,1 (Y t ) .
We recall that for almost every t ∈ [0, 1] the restriction

q 0 : B t → Y t is a C 1 diffeomorphism ((H q0 ) condition).
We can now extend for any L 2 V -scenario the definition of the current associated to its final age. The proof lies on the fact that almost all the restrictions of q 1 to the leaves X {t} are of class

C 1 . Proposition 2.2. The function v → µ v defined for v ∈ C([0, 1], V ) has a unique continuous extension (19) L 2 ([0, 1], V ), | • | L 2 V -→ C 0 (R d , (Λ k R d ) * ), | • | ∞ * v -→ µ v : ω → 1 0 Yt ι (ht-vt) φ * t,1 ω dt ,
where (φ s,t ) s≤t is the flow of v, φ * t,1 ω is the pullback of ω by φ t,1 , ι is the interior product and h t is the unique vector field on Y t defined for almost any t ∈ [0, 1] and any x ∈ B t by h t (q 0 (x)) = ∂q0 ∂t (x).

Proof. Let us call here ϕ the application v → µ v given by Definition 2.1 when v ∈ C([0, 1], V ) and ϕ the application defined here by equation [START_REF] Grenander | -Special issue on current and future challenges in the applications of mathematics[END_REF]. We will first show that ϕ and ϕ coincides on v ∈ C([0, 1], V ). Then, we will show that ϕ is indeed a continuous linear application.

We decompose X = [0, 1]×B with a partition of unity of B. Hence, we just have to consider the case of a support [0, 1]×U where (U, ψ) is a coordinate chart around a point b ∈ B (consistent with the orientation). We can thus define a local coordinate system

x = (t, b 1 , • • • , b k-1 ) on [0, 1] × U and we have ∂q1 ∂b i (x) = dφ t,1 (q 0 (x)) • ∂q0 ∂b i (x) and ∂q1 ∂t (x) = dφ t,1 (q 0 (x)) • ∂q0 ∂t (x) -v t (q 0 (x)) = dφ t,1 (q 0 (x)) • (h t -v t )(q 0 (x)). Therefore, [0,1]×U q * 1 ω = [0,1]×U ω(q 1 (t, b)) ∂q 1 ∂t (t, b) k-1 i=1 ∂q 1 ∂b i (t, b) db 1 • • • db k-1 dt = [0,1]×U ω(φ t,1 (q 0 (t, b))) dφ t,1 (q 0 (t, b)) • (h t -v t )(q 0 (t, b)) k-1 i=1 dφ t,1 (q 0 (t, b)) • ∂q 0 ∂b i (t, b) db 1 • • • db k-1 dt = 1 0 B (φ * t,1 ω)(q 0 (t, b)) (h t -v t )(q 0 (t, b)) k-1 i=1 ∂q 0 ∂b i (t, b) db 1 • • • db k-1 dt = 1 0 Yt ι (ht-vt) φ * t,1 ω dt . Now, we have sup t∈[0,1] |dφ t,1 | = g 1 (|v| 2 L 2 V
), dq 0 is also bounded on X, so that h t and v t are bounded on q 0 (X) and therefore (20)

1 0 Yt ι (ht-vt) φ * t,1 ω dt ≤ |ω| ∞ g 2 (|v| 2 L 2 V ) ,
where g 1 and g 2 are increasing functions independent of v and ω. Consequently, for any

v ∈ L 2 V , ϕ(v) = µ v belongs to C 0 (R d , (Λ k R d ) *
) * and ϕ is continuous due to the regularity of the flow and of the interior product. Hence, since

C([0, 1], V ) is dense in L 2 V , ϕ is the unique continuous extension of ϕ.
Remark 2.1. Note that µ v is not exactly the current associated to the image q 1 (X). Indeed, even if q 1 is differentiable, it might not be an embedding. Two counter-examples are presented in Figure 5. In the first case, the direction of the development is suddenly reversed twice so that the curve is folding on itself. Hence, if we refer to number of preimages of each point of q 1 (X) as a thickness of the shape, then the thickness here is equal to 1 or 3. On the second case, the curve completely overwrites itself, so that the thickness is equal to 2 on each point.

This phenomenon depends on q 0 and v and cannot be anticipated. The current associated to our shapes therefore counts these repetitions. However, in the first scenario, since the orientation is reversed twice and by linearity of the currents with respect to the tangential data, the repetition is canceled and we have µ v (ω) = q1(X) ω. On the second example, the orientation is the same on each layer so that µ v (ω) = 2 q1(X) ω. At last, note that in practice, these situations should not happen with optimal vector fields. The penalization of v should prevent these artifacts. Generating a cancel effect via an overlapping should induce an additional cost on v with yet no reduction of the data attachment term since the current would be the same without this overlapping. Likewise, the gain of thickness as in the second example is necessarily taken from spatial correspondences with the target shape and should therefore not be profitable (at least for a metric with a reasonable scale so that the position of the points are enough discriminated).

Discontinuity for varifold data term

This section highlights a counterexample to the existence of continuous minimizers of the energy when the attachment term is build on a space of varifolds. We present a first counterexample in a 2D case that will then be adapted to a 3D case.

3.1. Counterexample with curves. The growth model is here given by the setting of Section 2.2.1 where we recall that the final shape γ v : [-1, 1] → R 2 of a growth scenario is defined by [START_REF] Ito | Lagrange Multiplier Approach to Variational Problems and Applications[END_REF] γ v (r) = (r,

1 |r| v(s)ds) .
Figure 5. Two examples of scenarios. In both situations, X is a segment but q 0 (X) is reduced to a point. On top, v is given by piecewise constant vertical translations upwards and downwards, modeled by real-valued function. The final image is a segment but when v changed its sign, the curve folded on itself. The scenario is displayed again on the left but we slightly separated the multiple fibers of the curve. One can think to a magic trick where colored attached strings are pulling out from the initial position point. On the bottom, the scenario is generated by a constant rotation anticlockwise. The ambient space is exactly rotated twice during the time interval [0, 1]. We display the development of the curves with three colors depending of the thickness : dark for 1, blue for 2 and red for 3. The green star on the bottom is just displayed to highlight the evolution of one specific point.

and it is represented by a varifold denoted µ v . We also consider a target horn

γ v tar : [-1, 1] → R 2
given by the same equation for a time-varying vector field v tar ∈ L 2 ([0, 1], R). The discrepancy between the two curves γ v and γ v tar is then estimated by the distance between µ v and µ v tar in W , this is to say with the norm |µ v -µ v tar | W . Finally, the matching problem consists in minimizing the energy given by the sum of a penalization term on v and this data attachment term

(22) E λ W (v) . = 1 2 1 0 v(r) 2 dr + λ 2 |µ v -µ v tar | 2 W .
Our aim is to examine whether the regularization L 2 on v and the data attachment term on varifolds ensure the continuity of global minimizers of E λ W . We will prove the following theorem:

Theorem 3.1. There exist v tar ∈ L 2 ([0, 1], R), λ > 0 and W such that no global minizer v * of E λ W given by (22) is a continuous function on [0, 1]. Moreover, one can assume that v tar ∈ C ∞ ([0, 1], R).
We will consider a perturbation parameterized by ≥ 0 of a degenerate constant kernel k W ≡ 1. The solutions of the optimization problem associated to this kernel will be especially easy to explicit.

Definition 3.1. Define k T ≡ 1 and a set of kernels k (x, y) = ρ( |x -y| 2 R 2 )
where ρ is a positive function such that ρ(0) = 1, ρ is bounded on R and ρ(0) < 0. They generate a set of kernels k ⊗ k T on R 2 × G 1 (R 2 ) that do not see the tangential directions. Each kernel k ⊗ k T for ≥ 0 produces a RKHS denoted W . Since W depends on , the energy will be denoted

E λ ( , v) to refer to E λ W (v).
This construction could probably be extended to a symmetric situation with a perturbation k of k T . Note yet that it would require to investigate the spatial regularity of the curve γ v . Hence, we will only consider k T ≡ 1.

3.1.1. Solutions for the degenerate kernel. The first step is to study the minimizers of E λ (0, •). When = 0, the kernel k W0 = k 0 ⊗ k T is constant and W 0 is a 1-dimensional space whose elements ω are all constant. In this case, the expression of the data attachment term is particularly simple:

|µ v -µ v tar | 2 W 0 = R 2 ×R 2 k 0 (x, y)d(µ v -µ v tar )(x)d(µ v -µ v tar )(y) = R 2 1d(µ v -µ v tar )(x) 2 = ( (v) -(v tar )) 2 ,
where (v) measures the length of the curve generated by v

(23) (v) = 2 1 0 1 + v(t) 2 dt .
Finally, the energy in this case is given by ( 24)

E λ (0, v) = 1 2 1 0 v(t) 2 dt + λ 2 ( (v) -(v tar )) 2 .
The global minimizers have then an explicit expression given by the following proposition.

Proposition 3.1. Assume that 0 = 4λ (v tar ) 4λ+1 > 2. Then v * ∈ L 2 ([0, 1], R) is a global minimizer of E λ (0, •) if and only if we have at almost all time v * (t) 2 = 2 0 /4-1. In particular, if v * ∈ C([0, 1], R) then v * is constant.
Proof. We have the following elementary lemma:

Lemma 3.1. If 0 = 4λ (v tar ) 4λ+1 > 2, then 0 minimizes P ( ) . = 2 /4 -1 2 + λ 2 ( (v tar ) -) 2 on R. Moreover, if v ∈ L 2 ([0, 1], R) satisfies v(t) 2 = 2 /4 -1 a.e. where ∈ R, then E λ (0, v) = P ( ).
Define for ≥ 2, the function ρ : R → R by

ρ (z) . = z 2 2 - 2 z 2 + 1.
This function is even, tends to +∞ when |z| tends to +∞ and ρ (z

) = 0 ⇔ z -2 z √ z 2 +1 = 0 ⇔ (z = 0 or z 2 = 2 4 -1)
. Therefore, ρ admits two minimizers that satisfy

z 2 = 2 /4 -1 > 0 ρ (z) = ( 2 /4 -1)/2 -( /2) 2 /4 -1 + 1 = -( 2 /4 + 1)/2 .
It results that the minimum of

R (v) . = 1 0 ρ (v(t))dt is reached at v * ∈ L 2 ([0, 1], R) if and only if (25) v * (t) 2 = 2 /4 -1 a.e.
By construction, these minimizers are exactly the solutions of the constrained optimization problem min L 2 v(t) 2 dt with (v) = .

Indeed, if v * satisfies equation [START_REF] Kaltenmark | Trouvé Partial Matchings and Growth Mapped Evolutions in Shape Spaces[END_REF], then (v * ) = 2 Remark 3.1. Note that with the degenerate kernel k W ≡ 1, the energy has continuous global minimizers. However, they are only two of an infinite number of solutions. Figure 6 illustrates on its left the two curves generated by v + and v -for four given lengths 0 . Figure 6 illustrates on its right few examples where 0 is fixed. The condition to be a minimizer leads to a large set of different type of curves as we only control the length of the final curve. Assume now that the target is some kind of sinusoidal curve. One can then easily see that from a spatial point of view, the two curves γ v + and γ v -are probably the less optimal solutions among the complete set of solutions γ v * . Hence, as soon as the kernel k W is perturbed and allowed to capture some spatial position of the target, one can expect that the new energies associated to v + and v -are higher than the energy of at least one other solution v * . Hypothesis: There exists v * , such that for any > 0 small enough,

1 0 v * (t) 2 + 1dt = and if there exists another v ∈ L 2 ([0, 1], R) such that (v) = and 1 0 v(t) 2 dt < 1 0 v * (t) 2 dt then R (v) < R (v * ) which is absurd. Consequently, any minimizer v * ∈ L 2 ([0, 1], R) of E λ (0, •) satisfies v * (t) 2 = 2 0 /4 -1 a.e. where 0 = (v * ) is defined on [2, +∞[ and must minimize → E λ (0, v * ) = 2 /4-1 2 + λ 2 ( (v tar ) -) 2 , i.e. 0 = 4λ (v tar ) 1+4λ > 2. Moreover, there exist exactly two continuous minimizers in L 2 ([0, 1], R) ∩ C([0, 1], R) given by v + ≡ 2 0 /4 -1 and v -= -v + .
E λ ( , v + ) > E λ ( , v * ) and E λ ( , v -) > E λ ( , v * ) .
3.1.2. Perturbation of the degenerate kernel. The next step to prove the theorem is to investigate the minimizers of v → E λ ( , v) where > 0 and

(26) E λ ( , v) = 1 2 1 0 v(r) 2 dr + λ 2 |µ v -µ v tar | 2 W .
The following proposition will establish that if some of these minimizers are continuous, they necessarily lie in a neighborhood of v + or v -(the two continuous global minimizers of E λ (0, v)). Analyzing the variations of → E λ ( , v) will then indicate that in some situations these minimizers cannot be global minimizers.

Remark 3.2 (RKHS properties). Denote ω = K W (µ v -µ v tar ), where K W : W → W is the canonical isomophism of Hilbert spaces. By construction of a RKHS, ω is given at any (x, V ) ∈ R 2 × G 1 (R 2 ) by ω(x, V ) = R 2 ×G1(R 2 ) k W (x, V ), (y, V ) d(µ v -µ v tar )(y, V ) . (27) Let us recall then that |µ v -µ v tar | 2 W = (µ v -µ v tar )(ω) and ∂ ∂v 1 2 |µ v -µ v tar | 2 W = ∂ ∂v µ v (ω) . ( 28
)
Proposition 3.2. Assume that 0 = 4λ (v tar ) 4λ+1 > 2. If for any > 0 small enough, there exists a continuous global minimizer v of E λ ( , •), then [START_REF] Lombardi | Eulerian models and algorithms for unbalanced optimal transport[END_REF] lim

→0 min(|v -v + | ∞ , |v -v -| ∞ ) = 0 , where v + ≡ 2 0 /4 -1 and v -= -v + are the only continuous global minimizers of E λ (0, •). Proof. Denote ω v ( , •) = K W (µ v -µ v tar ). Since the kernel k W of W is reduced to k (x, y) = ρ( |x -y| 2 )
, the tangential component of the varifold µ v -µ v tar can be ignored and we have (see Remark 3.2)

ω v ( , •) = R 2 k (•, y)d(µ v -µ v tar )(y) .
With the symmetry of the curve γ v , the associated varifold µ v can be rewritten

µ v (ω) = 1 i=0 1 0 ω(S i (γ v (r))) 1 + v(r) 2 dr , (30)
where S 1 = S is the symmetry with respect to the vertical axis through origin and S 0 = Id. We then symmetrize ω v as follows

ω S v ( , x) . = 1 i=0 ω v ( , S i (x)) = 1 i=0 R 2 ρ( |S i (x) -y| 2 )d(µ v -µ v tar )(y) ,
so that equation ( 30) reads

µ v (K W (µ v -µ v tar )) = µ v (ω v ( , •)) = 1 0 ω S v ( , γ v (r)) v(r) 2 + 1dr .
We recall that γ v (r) = (r,

1 |r| v(s)ds) and that for any v ∈ L 2 lim →0 ω S v ( , •) ≡ 2( (v) -(v tar )) . ( 31
)
One can easily prove that v → E λ ( , v) is differentiable with respect to v and we have for any

δv ∈ L 2 ∂E λ ∂v ( , v) δv = 1 0 v(t)δv(t)dt + λ ∂ ∂v µ v δv (K W (µ v -µ v tar )) ∂E λ ∂v ( , v) δv = 1 0 v(t)δv(t)dt + λ 1 0 ∂ 2 ω S v ( , γ v (t)) 0, 1 t δv(s)ds v(t) 2 + 1 + ω S v ( , γ v (t)) v(t)δv v(t) 2 + 1 dt ,
where

∂ 2 ω S v ( , x) is the derivative of ω S v with respect to x. Denote α ,v (s) . = s 0 ∂ 2 ω S v ( , γ v (t)) (0, 1) v(t) 2 + 1dt , so that ∂E λ ∂v ( , v) = 1 + λ ω S v ( , γ v ) √ v 2 + 1 v + α ,v . Note then that |α ,v | ∞ = O( ). Indeed, ∂ 2 ω S v ( , x) = 1 i=0 S i 1 0 2(S i (x) -y) ρ( |S i (x) -y| 2 )d(µ v -µ v tar )(y)
and since ρ is bounded on R we deduce that for any bounded neighborhood of (0, 0) in

R + × L 2 ([0, 1], R), we have |∂ 2 ω S v ( , γ v )| ∞ = O( ) and |α ,v | ∞ = O( ) . (32)
Assume now that for any > 0, there exists a continuous solution v that minimizes E λ ( , •). It must then satisfy

1 + λ ω v ( , γ v ) v 2 + 1 v + λα ,v = 0 a.e. ( 33
)
For small enough, equations ( 33) and [START_REF] Miller | Diffeomorphometry and geodesic positioning systems for human anatomy[END_REF] imply that there exist M > 0 and β ≥ 0 such that at almost any time t ∈ [0, 1] we have either

|( v (t) 2 + 1 -β | ≤ M 1/2 v (t) 2 + 1 or |v (t)| ≤ M 1/2 . ( 34
)
To go further let us first show that the lengths of the curves (γ v ) ≥0 converge.

Lemma 3.2. (v ) tends to 0 = (v + ) = (v -).
Proof. We have

E λ (0, v ) ≤ E λ ( , v ) + o(1) ≤ E λ ( , v + ) + o(1) ≤ E λ (0, v + ) + o(1) . ( 35
)
Left and right inequalities result from the continuity of E λ (•, v). Since v minimizes E λ ( , •), the central inequality is also true. Consider now 0 = 4λ (v tar ) 4λ+1 > 2 and the polynomial P ( ) = 1 2 ( 2 /4-1)+ λ 2 ( (v tar )-) 2 . Lemma 3.1 says that (v + ) = 0 and E λ (0, v + ) = P ( (v + )) = P ( 0 ). Moreover, if for any > 0, we define δv ≡ (v ) 2 /4 -1, then (δv ) = (v ) and E λ (0, δv ) ≤ E λ (0, v ) (δv minimizes R , see proof of Proposition 3.1). It results from equation [START_REF] Niethammer | Geodesic Regression for Image Time-Series[END_REF] and Lemma 3.

1 that P ( (v )) = E λ (0, δv ) ≤ P ( 0 ) + o(1).
At last, since 0 minimizes P , we have

P ( 0 ) ≤ P ( (v )) ≤ P ( 0 ) + o(1) .
Hence, since P admits a unique minimizer (quadratic polynomial), we have (v ) = 0 + o(1) = (v + ) + o(1).

We can now prove that the first case of equation ( 34) is the only one true. Denote for any > 0,

A . = {t ∈ [0, 1] | | v (t) 2 + 1 -β | ≤ O( 1/2 ) v (t) 2 + 1} and . = 2(λ R (A )β + (1 -λ R (A )) (where λ R is the Lebesgue measure). Then (v ) = 2 1 0 v (t) 2 + 1dt = 2 A β dt + [0,1]\A 1dt + O( 1/2 ) = + O( 1/2 ) .
Lemma 3.2 implies then that = 0 + o(1). Moreover, according to equation ( 31), we have necessarily β = 2λ( (v tar )-(v ))+o(1) so that β = 2λ( (v tar )-)+o(1) = 2λ( (v tar )-0 )+o [START_REF] Allassonniere | Geodesic shooting and diffeomorphic matching via textured meshes[END_REF]. At last, from Proposition 3.1, we have (v tar ) -0 = 0 /(4λ) so that 2β = 0 + o(1) = + o(1). Finally, we deduce by definition of that λ R (A ) = 1 + o(1).

Therefore, there exists M > 0 such that for almost any t ∈ [0, 1],

|v (t) 2 -( 2 0 /4 -1)| ≤ M . Since v is continuous and 0 > 2, it follows that v satisfies either |v -v + | ∞ ≤ M or |v -v -| ∞ ≤ M .
And finally, lim

→0 min(|v -v + | ∞ , |v -v -| ∞ ) = 0 .
The final step to prove the theorem is to study the variations of E λ (•, v) with respect to at a global minimizer v = v * . The aim is to show that the energy around v + and v -increases too fast, with respect to , to allow any v in their neighborhood to be a global minimizer of E λ ( , •). As announced in Remark 3.1, the idea is to compare the geometric properties of all the minimizers of E λ (0, •). We will thus rewrite the gradient of this energy via some geometric descriptors. Definition 3.2. Denote x v the centroid of the curve γ v defined by

(36) x v . = 1 (v) R 2 xdµ v (x)
and V (v) the associated variance defined by

(37) V (v) . = 1 (v) R 2 |x -x v | 2 dµ v (x) . Lemma 3.3. The function → E λ ( , v
) is differentiable and for = 0, we have

∂E λ ∂ (0, v) = -λ ρ(0) (v tar ) -(v) (v)V (v) -(v tar )V (v tar ) + (v) (v tar )|x v tar -x v | 2 .
Proof. The proof depends neither on the dimension of the ambient space nor on the dimension of the varifolds. Let assume that the ambient space is R d and let us start to establish with varifolds the algebraic formulae for the variance (

V (X) = E[X 2 ] -E[X] 2 ). For any v ∈ L 2 , we have (v)V (v) = R d |x -x v | 2 dµ v (x) = R d |x| 2 + |x v | 2 -2 x, x v dµ v (x) = R d |x| 2 dµ v (x) + (v)|x v | 2 -2 x v , R d x dµ v (x) = R d |x| 2 dµ v (x) -(v)|x v | 2 .
Then, one can easily show that → E λ ( , v) is differentiable and that

∂E λ ∂ (0, v) = ∂ ∂ λ 2 µ v -µ v tar W =0 = ∂ ∂ λ 2 R d ×R d ρ( |x -y| 2 )d(µ v -µ v tar )(x)d(µ v -µ v tar )(y) =0 = λ 2 ρ(0) R d ×R d |x -y| 2 d(µ v -µ v tar )(x)d(µ v -µ v tar )(y) = λ 2 ρ(0) R d ×R d (|x| 2 + |y| 2 -2 x, y )d(µ v -µ v tar )(x)d(µ v -µ v tar )(y) = λ ρ(0) (v) -(v tar ) R d |x| 2 d(µ v -µ v tar )(x) a - R d xd(µ v -µ v tar )(x) 2 b .
The terms denoted by a and b can be rewritten as follows:

a = (v) -(v tar ) R d |x| 2 d(µ v -µ v tar )(x) = (v) -(v tar ) R d |x| 2 dµ v (x) - R d |x| 2 dµ v tar (x) = (v) -(v tar ) (v)V (v) -(v tar )V (v tar ) + (v) 2 |x v | 2 -(v) (v tar )|x v | 2 + (v tar ) 2 |x v tar | 2 -(v) (v tar )|x v tar | 2 and b = R d xd(µ v -µ v tar )(x) 2 = R d xdµ v (x) - R d xdµ v tar (x) 2 = (v)x v -(v tar )x v tar 2 = (v) 2 |x v | 2 + (v tar ) 2 |x v tar | 2 -2 (v) (v tar ) x v , x v tar .
Then a -b is equal to

a -b = (v) -(v tar ) (v)V (v) -(v tar )V (v tar ) -(v) (v tar ) |x v | 2 + |x v tar | 2 -2 x v , x v tar = -(v tar ) -(v) (v)V (v) -(v tar )V (v tar ) -(v) (v tar ) x v -x v tar 2 .
We retrieve the announced formula.

We now exhibit a condition to the existence of a sequence (v n ) n ⊂ L 2 such that n → 0 and for any n ≥ 0, v n is a continuous global minimizer of E λ ( n , .).

Proposition 3.3. Assume that 0 = 4λ (v tar ) 4λ+1 > 2.
If there exists a decreasing sequence n → 0 such that v n is a continuous global minimizer of E λ ( n , 0) then for any global minimizer v * of E λ (0, •), we have

(38) min ∂E λ ∂ (0, v + ), ∂E λ ∂ (0, v -) ≤ ∂E λ ∂ (0, v * ) ,
where v + and v -are the only two continuous global minimizers of E λ (0, •) (they are constant and defined by v + ≡ 2 0 /4 -1 and v -= -v + ). Proof. Denote v n = v n . According to Proposition 3.2, either v + or v -is an accumulation point of (v n ) n . Assume that (v n ) n converges to v + (one can extract a subsequence if necessary) and consider v * a global minimizer of E λ (0, •). The continuity of ( , v) → ∂ E λ ( , v) on a neighborhood of (0, v + ) implies then that

E λ ( n , v * ) ≥ E λ ( n , v n ) = E λ (0, v n ) + n ∂ E λ (0, v n ) + o( n ) = E λ (0, v n ) + n ∂ E λ (0, v + ) + o( n ) ≥ E λ (0, v * ) + n ∂ E λ (0, v + ) + o( n ) ≥ E λ ( n , v * ) -n ∂ E λ (0, v * ) + n ∂ E λ (0, v + ) + o( n ) . It results that n ∂ E λ (0, v + ) -∂ E λ (0, v * ) + o(1) ≤ 0 and we deduce that ∂ E λ (0, v + ) ≤ ∂ E λ (0, v * ). Likewise, if (v n ) n converges to v -, we get that ∂ E λ (0, v -) ≤ ∂ E λ (0, v * ).
3.1.3. Construction of the counterexample. In conclusion, one needs to find a target, a well-chosen λ and v * a global minimizer of E λ (0, •) such that the inequality (38) is invalidated. There would consequently exist a deleted neighborhood of = 0 (meaning a neighborhood of = 0 without 0) for which there exists no continuous global minimizer of E λ ( , .). The sought-after vector fields v tar and v * must thus induce

(39) ∂E λ ∂ (0, v * ) < ∂E λ ∂ (0, v α ) ,
where v α ∈ {v + , v -}. Let us recall that we chose a decreasing function ρ (which is the case of most usual kernels used to model varifolds) so that ρ(0) < 0. Since all optimal curves have the same length, one can define 0 = (v * ) = (v α ) and according to Lemma 3.3, this inequality ( 39) is equivalent to

(v tar ) -0 V (v * ) + (v tar )|x v tar -x v * | 2 < (v tar ) -0 V (v α ) + (v tar )|x v tar -x v α | 2 .
Moreover, if we can have 4λ (v tar ) 4λ+1

> 2 then 0 = 4λ (v tar ) 4λ+1 and (v tar ) -0 = 1/(4λ + 1). In fine, the counterexample must satisfy (40)

V (v * ) 4λ + 1 + (v tar )|x v tar -x v * | 2 < V (v α ) 4λ + 1 + (v tar )|x v tar -x v α | 2 .
Let us construct it explicitly. Consider for example v tar (t) = a1 t≤1/2 with a > 0. The target curve c v tar is then given by t → (t, a(1/2 -t) + ) and we have (v tar ) = (1 + √ a 2 + 1), [START_REF] Tang | Miller Shape abnormalities of subcortical and ventricular structures in mild cognitive impairment and Alzheimer's disease: detecting, quantifying, and predicting[END_REF] x v tar = 0,

√ a 2 + 1 1 + √ a 2 + 1 a 4
and ( 42)

x v α = 0, α 2 2 4 -1 ,
where we assume that λ is large enough so that = 4λ (v tar ) 4λ+1 > 2. It results that the optimal continuous solution is v α = v + . Let us introduce a set of vector fields (v s, * ) s≥0 defined by

v s, * (t) = 2 /4 -1(1 t<1-2s + sign(t -(1 -s))1 t≥1-2s ) .
We have v + = v 0, * and for any s ≥ 0, (v s, * ) 2 + 1 ≡ 2 /4 so that v s, * is a global minimizer of E λ (0, •) that is not continuous when s > 0. In order to prove inequality (40), we just have to show that the derivative with respect to s of

s → V (v s, * ) 4λ + 1 + (v tar )|x v tar -x v s, * | 2
is strictly negative on a neighborhood of s = 0 + . Denote x s . = x v s, * . We have

x s = 0, s 2 + (1 -2s)(1 -2s)/2 2 /4 -1 = 0, 3s 2 -2s + 1 2 2 /4 -1 .
One can easily show that d ds (|x

v tar -x v s, * | 2 ) |s=0 < 0. It follows that d ds (V (v s, * )) |s=0 ≤ 0. If we denote x s = (x 1 s , x 2 s ) then s → x 1
s is constant and dx 2 s ds |s=0 < 0. At last, we need to show that there exist a and λ such that x 2 s -x 2 v tar > 0. Assume then that λ is close to +∞ so that = (v tar ) + o(1). Then since the sign of g(a) =

2 /4 -1/2 - √ a 2 +1 1+ √ a 2 +1 a/4 = x 2 s -x 2 v tar + o(1) where = (1 + √ a 2 + 1
) is strictly positive when a > 0 (see Figure 7), we deduce the final result. In conclusion, we showed that for any a > 0, if λ is large enough and > 0 small enough, the energy E λ ( , .) admits no global minimizer in C([0, 1], R)∩L 2 ([0, 1], R). Let us remark additionally that this is not a consequence of the discontinuity of v tar . Indeed, one can easily replace v tar by an approximation in C ∞ with respect to the L 2 -norm and deduce the same result.

Remark 3.3. Note that this counterexample could not be applied to the currents. Indeed, the choice of the kernel k T is not open and the canceling effect of this kernel on opposite tangent vectors would reduce the length of the set of curves generated by the v s, * (the pink curve displayed in Figure 7).

Extension to the 3D case.

As in the 2D case, we attempt now to show the following theorem for surfaces in R 3 . Theorem 3.2. There exist v tar ∈ L 2 ([0, 1], R), λ > 0 and W such that E λ W has no time-continuous global minimizer.

The main ideas of the proof remain the same. We consider as in Definition 3.1 a similar set of RKHS W whose kernels are given by k (x, y) = ρ( |x -y| 2 R 3 ) where ρ is positive scalar function such that ρ(0) = 1, ρ is bounded, and ρ(0) < 0. Proposition 3.4 will establish that E λ W0 admits again exactly two continuous global minimizers v + and v -among an infinite number of global minimizers. Proposition 3.5 will then show that the continuous solutions relative to > 0 necessarily lie in a neighborhood of v + or v -. At last, we will present a situation where the continuity is a constraint too restrictive as there exist global minimizers of E λ W0 more stable with respect to than v + and v -. In other words, if the energy increases more slowly around a discontinuous minimizer v * than around v + and v -, the existence of continuous global minimizers of E λ W for in a deleted neighborhood of 0 is excluded. As before, this will require to compare the gradients of E λ W0 with respect to at the minimizers of v → E λ W0 (v). We denote again E λ ( , •) = E λ W . The energy functions to minimize, associated to the spaces W , are unchanged

E λ ( , v) = 1 2 1 0 v(t) 2 dt + λ 2 |µ v tar -µ v | 2 W .
3.2.1. Solutions for the degenerate kernel. Consider now the case = 0. The kernel of W 0 is the constant unit kernel. By analogy with the 2D case, the area of the surface γ v is denoted (v) and we have

(v) = |µ v | 2 W 0 = 2π 0 1 0 r 1 + v(r) 2 drdθ (43) = 2π 1 0 r 1 + v(r) 2 dr . ( 44
)
Remark 3.4. Note that for any v ∈ L 2 ([0, 1], V ), we have (v) ≥ π. The growth process can only expand the initial unit disc.

The energy then reads

E λ (0, v) = 1 2 1 0 v(t) 2 dt + λ 2 (v tar ) -(v) 2 .
The next proposition will establish the minimizers of this energy. For this purpose, given any constant c ≥ 1, we will say that v ∈ L 2 ([0, 1], V ) satisfies the (P c ) property if

(P c ) for almost any time t ∈ [0, 1], v(t) 2 = 0 if t ≤ 1 c (ct) 2 -1 otherwise.
Proposition 3.4. For any λ ≥ 0, there exists a unique constant c 0 ≥ 1 such that:

v * ∈ L 2 ([0, 1], R) is a global minimizer of E λ (0, •)
if and only if it satisfies the (P c0 ) property. Additionally, c 0 = 1 if and only if (v tar ) ≤ π + 1/(2πλ). In this last case, v * ≡ 0 is the unique global minimizer of E λ (0, •).

Proof. The proof is similar as the one of Proposition 3.1. Introduce for c ≥ 1

ρ c (z, t) = z 2 2 -ct z 2 + 1 , defined on R × [0, 1]. Given t ∈ [0, 1], the function ρ c (z, t) reaches its minimum at z = 0 if t ≤ 1 c and at z c = ± (ct) 2 -1 otherwise. Thus v ∈ L 2 ([0, 1], R) minimizes 1 0 ρ c (v(t), t) dt = 1 2 1 0 v(t) 2 dt - c 2π (v)
if and only if it satisfies the (P c ) property. Now, if v c satisfies (P c ) then

(v c ) = 2π 1 0 t 1 + v c (t) 2 dt = 2π 1 c 0 t dt + 1 1 c t (ct) 2 dt = 2π 1 2c 2 + c t 3 3 1 1 c = 2π 3 c + π 3 1 c 2 . Denote ˆ : [1, +∞[→ R the function defined by ˆ (c) = 2π 3 c + π 3 1 c 2 (45)
and remark that ˆ is a bijection from Therefore, (P c ) also determines exactly the minimizers of E λ (0, •) when c minimizes

g(c) = E λ (0, v c ) = 1 2 1 1 c (ct) 2 -1 dt + λ 2 ( ˆ (c) -(v tar )) 2 = 1 2 c 2 t 3 3 1 1 c -(1 - 1 c ) + λ 2 2π 3 c + π 3 1 c 2 -(v tar ) 2 = c 2 6 + 1 3c - 1 2 + λ 2 2π 3 c + π 1 c 2 2 + (v tar ) 2 -2 (v tar ) 2π 3 c + π 3 1 c 2 2 = 1 6 + λ 2 2π 3 2 c 2 + C + 1 3 + λ 2 2π 3 2 1 c + λ 2 π 3 2 1 c 4 - λ 2 2π 3 (v tar ) 2c + 1 c 2 ,
where C is the constant λ 2 (v tar ) 2 -1 2 . Since the uniqueness of c is required, let us study the variations of this function. We have

g (c) = 1 3 + λ 2π 3 2 c - 1 3 + λ 2 2π 3 2 1 c 2 - λ 2 2π 3 2 1 c 5 . + λ 2π 3 (v tar ) 1 c 3 -1 and g (c) = 1 3 + λ 2π 3 2 + 2 1 3 + λ 2 2π 3 2 1 c 3 + 5 λ 2 2π 3 2 1 c 6 -3λ 2π 3 (v tar ) 1 c 4 .
For c ≥ 1, g = 0 is thus equivalent to h(c) = 0 where h(c) = c 4 g (c). The derivative of h is given by

h (c) = 4 1 3 + λ 2π 3 2 c 3 + 2 1 3 + λ 2 2π 3 2 -10 λ 2 2π 3 2 1 c 3 = c -3 Q(c 3 ) , where Q(X) = 4 1 3 + λ 2π 3 2 X 2 + 2 1 3 + λ 2 2π 3 2 X -10 λ 2 2π 3 2 .
Therefore, since Q is strictly increasing on [1, +∞[ and Q(1) = 2, h > 0 and h is strictly increasing on [1, +∞[. Moreover, (v tar ) ≥ π so there exists s ≥ 1 such that (v tar ) = sπ. Then h(1) = 1 + 2π 2 λ(1 -s) and h(1) < 0 is equivalent to s > 1 + 1/(2π 2 λ). Under this condition, g has only one zero and g is decreasing then increasing. Otherwise, g is strictly increasing.

Finally, since g (1) = 0, g has always only one global minimum on [1, +∞[. Additionally, if (v tar ) ≤ π + 1/(2πλ), the minimizer is c 0 = 1 and corresponds to the solution v * ≡ 0. Otherwise, c 0 > 1.

Remark 3.5. As in the 2D case, the energy associated to the degenerate kernel admits two continuous global minimizers

v + (t) . = 1 1 t> 1 c 0 (c 0 t) 2 -1 and v -. = -v + . ( 46
)
They are again surrounded by an infinite number of discontinuous global minimizers. However, these two solutions are not constant anymore. Indeed, in the 2D case, a constant vertical translation creates at all time the same amount of new matter measured by the length of the curve just created above the base between two times t and t + δt. In the 3D case, the surface created by a constant vertical translation between two times t and t + δt is similar to a cylinder whose radius increases with t. The penalization term on v tends thus to accelerate the creation over time. Figure 9. Plot of the norm of any optimal vector field. The (P c ) condition on this example is defined with c = 4.5 so that (v) ≈ 3π. The area of the surface has tripled with respect to its initial position.

3.2.2. Perturbation of the degenerate kernel. As before, we will now follow the continuous global minimizers of v → E λ ( , v) when tends to 0 and show that they belong to a neighborhood of v + or v -.

Proposition 3.5. Assume that λ > 0, (v tar ) > π + 1/(2πλ) and that for ≥ 0 small enough, there exists a global continuous minimum v of E λ ( , •). Then

(47) lim →0 min(|v -v + | ∞ , |v -v -| ∞ ) = 0
where v + and v -are the only continuous global minimizers of E λ (0, •).

Proof. We first show the convergence of the areas.

Lemma 3.4. Denote 0 . = (v + ) = ˆ (c 0 ), then (v ) tends to 0 and 0 > π.

Proof. Consider the function ˆ defined by equation [START_REF] Trouvé | Shape Splines and Stochastic Shape Evolutions: A Second Order Point of View[END_REF]. Recall that ˆ is a bijection from [1, +∞[ to [π, +∞[ and as we said in Remark 3.4, that for any v ∈ L 2 V , (v) ≥ π. Therefore, for any ≥ 0, there exists a unique c ≥ 1 such that (v ) = ˆ (c ). Let us show that

E λ (0, v ) ≤ E λ ( , v ) + o(1) ≤ E λ ( , v + ) + o(1) ≤ E λ (0, v + ) + o(1) .
Left and right inequalities result from the continuity of E λ (•, v). Since v minimizes E λ ( , •), the central inequality is also true. Moreover, Proposition 3.4 ensures that for any v c that satisfies (P c ), we also have E λ (0, v c ) ≤ E λ (0, v ). We introduced in the proof of Proposition 3.4 a function g that satisfies for any ≥ 0, g(c ) = E λ (0, v c ). Moreover, c 0 is the unique minimum of g. It results that

g(c 0 ) ≤ g(c ) ≤ E λ (0, v c ) ≤ E λ (0, v + ) + o(1) ≤ g(c 0 ) + o(1) .
Hence, g(c ) tends to g(c 0 ) and since g is continuous and increases around +∞, c tends to c 0 . The continuity of ˆ ensures at last that ˆ (c ) tends to ˆ (c 0 ) so that (v ) converges as announced to 0 = ˆ (c 0 ) = (v + ). The last assertion is a direct result of Proposition 3.4 under the assumption that (v tar ) > π + 1/(2πλ).

Let us compute the gradient with respect to v of the energy

E λ ( , v) = 1 2 1 0 v(t) 2 dt + λ 2 |µ v -µ v tar | 2 W . Consider ω v ( , •) = K W (µ v -µ v tar ) given for any x ∈ R 3 by ω v ( , x) = K W (µ v -µ v tar )(x) = R 3 ρ( |x -y| 2 ) d(µ v -µ v tar )(y) , so that µ v (K W (µ v -µ v tar )) = 2π 0 1 0 ω( , γ v (θ, r))r 1 + v(r) 2 drdθ .
We have then for any variation δv ∈ L 2 ([0, 1], R)

∂ v E λ ( , v) δv = 1 0 v(t) δv(t) dt + λ ∂ v µ v | δv (K W (µ v -µ v tar )) = 1 0 v(t) δv(t) dt +λ 2π 0 1 0 ∂ 2 ω v ( , γ v (θ, r)) (0, 0, 1 r δv s ds) r 1 + v(r) 2 + ω v ( , γ v (θ, r)) rv(r)δv(r) 1 + v(r) 2 drdθ ,
where

∂ 2 ω v ( , x) = 2 R 3 ρ( |x -y| 2 )(x -y) d(µ v -µ v tar )(y). Denote at last z ,v (r) . = 2π 0 rω v ( , γ v (θ, r)) dθ , α ,v (s) . = λ 2π 0 s 0 ∂ 2 ω v ( , γ v (θ, r)) (0, 0, 1) r 1 + v(r) 2 drdθ .
The gradient then reads

∇ v E λ ( , v) = 1 + λ z ,v √ 1 + v 2 v + α ,v (48)
and for any ≥ 0, since v is a zero of ∇ v E λ ( , v), we have

(49) 1 + λ z ,v 1 + v 2 v + α ,v = 0 a.e.
Now, on any bounded neighborhood of (0, 0) of R + × L 2 ([0, 1], V ), γ v is bounded, dµ v and dµ v tar are finite, so that with ρ bounded we have

|∂ 2 ω v ( , γ v )| ∞ = O( ) and thus |α ,v | ∞ = O( ) .
Hence, for > 0 small enough, there exist M > 0 and β ≥ 0 such that we have for almost any t ∈ [0, 1] either

(i) 1 + v (t) 2 -tβ ≤ M 1 2 1 + v (t) 2 or (ii) |v (t)| ≤ M 1 2 .
Denote as before for any > 0, A .

= {t ∈ [0, 1] | | v (t) 2 + 1 -tβ | ≤ M 1 2 v (t) 2 + 1}. Lemma 3.4 says that (v ) tends to 0 and 0 > π. Moreover, if v ≡ o(1), (v) = π + o(1).
Hence, for small enough, λ R (A ) = 0 implies then that (v ) < 0 which is absurd. For any t ∈ A ,

1 -M 1 2 ≤ (1 -M 1 2 ) 1 + v (t) 2 ≤ tβ ≤ β , so that β ≥ 1 -M 1 2 and A ⊂ [ 1-M 1 2 β , 1]. Moreover, since lim →0 z ,v (t) = -2πt( (v tar ) -(v)), Lemma 3.4 implies that β = β 0 + o(1) where β 0 = 2πλ( (v tar ) -0 ).
Consider a small α > 0 and denote

I α = [0, 1 β0 (1 + α)] and I + α =] 1 β0 (1 + α), 1]
. Both (i) (restricted on A ∩ I α ) and (ii) imply that there exists η > 0 such that for any < η and almost any t ∈ I α , [START_REF] Younes | Shapes and Diffeomorphisms[END_REF] |v (t)| 2 ≤ 3α (using 1 + α < √ 1 + 3α for α small enough). Since v is continuous, (v ) tends uniformly to 0 on [0, 1 β0 ] (i.e. for any sequence (v n ) n such that n → 0). Let us show now that for > 0 small enough,

I + α =] 1 β0 (1 + α), 1] ⊂ A .
There exists η > 0 such that for any < η, we have (1) for any

t ∈ I + α , tβ ≥ 1 + α/2, (2) 1 + M 1 2 ≤ (1 + α/2)(1 + α/3) -1 2 and M 1 2 ≤ (α/4) 1 2 . Using tβ ≤ (1 + M 1
2 ) 1 + v (t) 2 when t ∈ A and (ii) otherwise, we deduce that for any < η and any t ∈ I + α , we have

v (t) 2 ≥ α 3 if t ∈ A , v (t) 2 ≤ α 4 if t / ∈ A .
Since v is continuous, either I + α ∩ A or I + α ∩ A c is empty. Since (v ) tends to 0 > π, it results that for any < η, I + α =] 1 β0 (1 + α), 1] ⊂ A . Finally, v converges uniformly to 0 on [0, 1 β0 ] and 1 + v 2 converges uniformly to t → β 0 t on any interval I + α (for α > 0) and with [START_REF] Younes | Shapes and Diffeomorphisms[END_REF] we deduce that the convergence is uniform on [0, 1]. Since v is continuous, the limit of any sequence (v n ) n such that n → 0 is also continuous and satisfies the (P c ) property for c = β 0 . Additionally, equation (49) says that any minimizer of E λ (0, •) must also satisfy (P c ). The uniqueness of c when (P c ) characterizes the minimizers of E λ (0, •) (see Proposition 3.4) allows to conclude that there exists M > 0 such that for any > 0

|v -v + | ∞ ≤ M or |v -v + | ∞ ≤ M ,
where v + and v -are the two continuous global minimizers of E λ (0, •).

As before, this proximity of continuous global minimizers induces some constraints on the slopes of the energy with respect to . Proposition 3.6. If there exists a decreasing sequence n → 0 such that v n is a continuous global minimizers of E λ ( n , •) then for any global minimizers v * of E λ (0, •), we have

min ∂ E λ (0, v + ), ∂ E λ (0, v -) ≤ ∂ E λ (0, v * ) . (51)
Proof. According to Proposition 3.5, there exists a subsquence of (v n ) n which converges either to v + or v -. The proof of Proposition 3.3 can then be applied here.

3.2.3.

Construction of the counterexample. The final step is to exhibit an example for which inequality (51) does not occur. Recall the geometric expression of ∂ E λ given by Lemma 3.3 :

∂ E λ (0, v) = λ ρ(0) (v tar ) -(v) (v tar )V (v tar ) -(v)V (v) -(v) (v tar )|x v tar -x v | 2 , where x v = 1 (v) x dµ v (x) and V (v) = 1 (v) |x -x v | 2 dµ v (x) .
Since all global minimizers of E λ (0, •) have the same length, denote 0 = (v * ) = (v + ) and since ρ(0) < 0, a counterexample should thus lead to a couple (v * , v + ) satisfying:

V (v * ) (v tar ) -0 + (v tar )|x v tar -x v * | 2 < V (v + ) (v tar ) -0 + (v tar )|x v tar -x v + | 2 . (52)
We exclude the negative continuous solution v -as it is easy to show that for a target above the plane Z = 0, this solution will not be approached by any global minimizer of E λ for > 0. Moreover, we have explicitly

V (v + ) = V (v -) and if γ v tar ⊂ (Z ≥ 0), |x v tar -x v + | < |x v tar -x v -|.
Proposition 3.7. There exists a target such that for λ large enough inequality (52) occurs.

As we saw earlier, the minimization of E λ (0, •) admits either a unique solution (equal to 0) or an infinite number of solutions. In this last case, there are only two continuous solutions. One can observe that these solutions are those which, at a fixed area, produce the most widely deployed surface. We show with the following example that this property can be very restrictive. The partial derivative of E λ with respect to at (0, v), where v is a minimum, measures the stability of this minimum with respect to small variations of . Intuitively, the best candidate among this infinite number of solutions, is the one which generates the surface that is geometrically the closest to the target. The previous expression gives an explicit description of this closeness according to the attachment term we chose. It requires a small variance and a centroid close the target's one.

Here is then a possible example.

The idea is to create a compact accordion in order to create a surface with a large area that yet remains close to the horizontal plane. Let us recall that

γ v (θ, r) = (r cos θ, r sin θ, 1 r v s ds)
and consider a target generated by one of the following vector fields

v tar n (t) = nhs n (t) , with [0, 1] sn -→ {-1, 1} t → 1 1 nt =0[2] -1 1 nt =1[2] ,
with n ∈ N, h ∈]0, 1] a scale constant. Figure 10 . Therefore, no matter the choice of n, the target shape remains concentrated in D × [-h, +h] (where D is the unit disc). Yet, one can fix its area as large as necessary by increasing n.

The solutions v * that minimize E λ (0, •) are characterized by the (P c ) property with a optimal constant c to define and such that (v * ) = ˆ (c) denoted again 0 . One can easily show that if λ tends to +∞, 0 tends to (v tar ). For λ large enough, we have thus

0 = ˆ (c) = 2π 3 c + π 3c 2 ≈ 2π 1 + (nh) 2 = (v tar ) .
If hn is large enough, one can do the approximation c ≈ 3hn. Let us compare v + and v * n defined for any t ∈ [0, 1] by

v + (t) = 1 1 t> 1 c (ct) 2 -1 and v * n (t) = s n (t)v + (t) ,
where n is given by the choice of the target. They both satisfy (P c ). These two vector fields are displayed in Figure 12 and the surfaces that they generate are presented in Figure 11. When n increases, the continuous solution grows in space when the other one remains concentrated since

|z v * n (r)| ≤ 1 n √ c 2 -1 ≈ 3h.
More precisely, for any surface generated by v ∈ L 2 , the centroid belongs to the vertical axis through the origin. When x v + will move upwards when n increase, we have conversely for any n Likewise, for any v ∈ L 2

|x v tar n | ≤ max r |z v tar n (r)| ≤ h and |x v * n | ≤ max r |z v * n (r)| ≤ 3h .
V (v) = 1 (v) R 3 |x -x v | 2 dµ v (x) = 1 (v) (r cos θ) 2 + (r sin θ) 2 + |z v (r) -x v | 2 dµ v (x) = 1 (v) 2π 3 + 1 (v) |z v (r) -x v | 2 dµ v (x) .
It follows that

V (v * n ) ≤ 2π 3 0 + (6h) 2 and V (v + ) = 2π 3 0 + 1 0 |z v + (r) -x v + | 2 dµ v + (x) .
In fine, if nh is fixed, V (v + ) and |x v tar n -x v + | 2 are fixed and strictly positive. Yet in the same time, if h tends to 0, V (v * n ) can be reduced to the minimal variance over the vector fields that satisfy (P c ) and |x v tar n -x v * n | 2 tends to 0. Therefore, the inequality

V (v * ) (v tar ) -0 + (v tar )|x v tar -x v * | 2 < V (v + ) (v tar ) -0 + (v tar )|x v tar -x v + | 2 can be satisfied.
In conclusion, note that v * = s n v + might not be the best candidate to minimize ∂ E λ ( , •) on a neighborhood of = 0, but it was easy to demonstrate that it is strictly better than v + for n and λ large enough. As in the 2D case, one could generate similar surfaces with a smooth function s n . This counterexample is not built on the discontinuity of v tar . At last, as pointed in Remark 3.5, this 3D example highlights a property of the optimal vector field that did not appear in the 2D case. With the growth dynamic, the norm of the optimal vector field tends to increase over time.

4. Existence of continuous minimizers in the current case 4.1. Existence of global minimizers in L 2 ([0, 1], V ). In Section 2.3, we extended the current representation for any foliated shape generated by a control v ∈ L 2 ([0, 1], V ) with the growth dynamic in the general situation of a coordinate space X = [0, 1] × B.

(53) L 2 ([0, 1], V ), | • | L 2 V -→ C 0 (R d , (Λ k R d ) * ), | • | ∞ * v -→ µ v : ω → 1 0 Yt ι (ht-vt) φ * t,1 ω dt ,
where (φ s,t ) s≤t is the flow of v, φ * t,1 ω is the pullback of ω by φ t,1 , ι is the interior product and h t is the unique vector field on Y t = q 0 (B t ) defined for almost any t ∈ [0, 1] and any x ∈ B t by h t (q 0 (x)) = ∂q0 ∂t (x) where B t = {t} × B ⊂ X. Unlike the varifolds, the currents provides a data attachment term that ensures the existence of continuous minimizers of

E(v) . = 1 2 1 0 |v| 2 V dt + λ 2 |µ tar -µ v | 2 W ,
where µ tar and µ v are the currents associated to the target and the solution generated by v and W is now a RKHS embedded in the space of test functions C 0 (R d , ( k R d ) * ). However, this result is not immediate. In this section, we first prove the existence of a solution in L 2 ([0, 1], V ). The expression of the current µ v (53) enlightens the foliation of our generated shapes and isolates each leaf. It allows to show a central property of current attachment terms that is not verified by varifold attachment terms: the lower semi-continuity (l.s.c.) on

L 2 V . Proposition 4.1. For any ω ∈ C 0 (R d , (Λ k R d ) * ), the application v → µ v (ω) is continuous with respect to the weak topology of L 2 ([0, 1], V ). In particular, v → |µ tar -µ v | 2
W * is l.s.c. with respect to the weak topology.

Proof. We recall partially the assumptions on the space of vector fields V

(54) (H V 1 )
There exists c > 0 such that for any v ∈ V and any

x ∈ R d , |v(x)| R d ≤ c|v| V (|x| R d + 1) . Consider a weakly convergent sequence v n v ∞ in L 2 V , we have for any ω ∈ C 0 (R d , (Λ k R d ) * ) |µ v n (ω) -µ v ∞ (ω)| ≤ 1 0 Yt ι v n t -v ∞ t φ v ∞ , * t,1 ω dt + 1 0 Yt ι ht-v n t (φ v n , * t,1 ω -φ v ∞ , * t,1 ω) dt . (55)
The first term of the right-hand side is a continuous linear form on L 2 V evaluated on v n -v ∞ . This is where the linearity of the currents attachment terms on the tangential data plays its role. Indeed, we have for any

u ∈ L 2 V | (u)| = 1 0 Yt ι ut φ v ∞ , * t,1 ω dt ≤ sup t,y∈Yt |φ v ∞ , * t,1 ω(y)| ∞ 1 0 Yt |u t (y)| R d dH k-1 (y) dt ≤ sup t,y∈Yt |dφ v ∞ t,1 | k ∞ |ω(y)| ∞ 1 0 Yt c(|y| R d + 1)|u t | V dH k-1 (y) dt ≤ c sup t,y∈Yt |dφ v ∞ t,1 | k ∞ |ω(y)| ∞ sup t,y∈Yt (|y| R d + 1) sup t vol(Y t ) 1 0 |u t | V dt ≤ c |u| L 2 V ,
where vol(Y t ) is the volume of Y t . Consequently, since (v n ) n weakly converges to v ∞ , (v n -v ∞ ) tends to 0. The second term can be bounded as follows

1 0 Yt ι ht-v n t (φ v n , * t,1 ω -φ v ∞ , * t,1 ω) dt ≤ m 1 (n)m 2 ,
where m 1 (n) = sup t,y∈Yt |φ v ∞ , * t,1 ω(y) -φ v n , * t,1 ω(y)| ∞ tends to 0 and

m 2 = sup t vol(Y t ) sup X ∂q 0 ∂t (x) + c sup t,y∈Yt (|y| R d + 1) |v n | L 2 V ≤sup n |v n | L 2 V .
We already know that if a sequence (v n ) n weakly converges to v ∞ then (t, y) → φ vn t,1 (y) converges compactly to (t, y) → φ v∞ t,1 (y). Moreover, since (v n ) n is weakly convergent, (v n ) n is bounded, so that finally this upper bound tends to 0. Therefore, the function v → µ v , with values in C 0 (R d , (Λ k R d ) * ) * , is continuous with respect to the weak topology of L 2

V and the first result is proved.

Moreover, since W is continuously embedded into C 0 (R d , (Λ k R d ) * ), there exists c > 0 such that for any linear form

∈ C 0 (R d , (Λ k R d ) * ) * , | (ω)| ≤ | | ∞ |ω| ∞ ≤ c | | ∞ |ω| W so that | | W * ≤ c | | ∞ .
It follows that for any ω ∈ W , µ v n (ω) tends to µ v ∞ (ω), i.e. µ v n weakly converges to µ v ∞ in W * . Hence, µ v n (µ tar ) = µ v n , µ tar W * tends to µ v ∞ , µ tar W * and since the square norm of a Hilbert space is always lower semi-continuous with respect to the weak topology, we deduce that

|µ tar -µ v ∞ | 2 W * = |µ tar | 2 W * -2 µ v ∞ , µ tar W * + |µ v ∞ | 2 W * ≤ |µ tar | 2 W * -2 lim µ v ∞ , µ tar W * + lim |µ v n | 2 W * ≤ lim |µ tar | 2 W * -2 µ v n , µ tar W * + |µ v n | 2 W * ≤ lim |µ tar -µ v n | 2 W * .
This proposition induces a first main result: the existence of a solution in L 2 ([0, 1], V ) of the energy

E(v) = 1 2 |v| 2 L 2 V + λ 2 |µ tar -µ v | 2 W * .
Theorem 4.1. Consider X = [0, 1] × B where B is a compact oriented manifold with corners and τ the projection on the first coordinate of X. Assume that q 0 ∈ C ∞ (X, R d ). Consider the standard cost function

C(v) = 1 2 1 0 |v t | 2 V dt .
Under the (H q0 ) and (H V 1 ) conditions, the energy defined for any v in L 2 ([0, 1], V ) by

E(v) = C(v) + λ 2 |µ v -µ tar | 2 W *
admits a global minimizer.

Proof. Note that E is always positive. Let (v n ) n be a minimizing sequence of E. One can easily show that (v n ) n is bounded and we can then assume that v n weakly converges in L 2 V . Denote v ∞ this limit. Proposition 4.1 says that E is lower semi-continuous with respect to the weak topology of L 2

V . It follows that E(v n ) tends to E(v ∞ ) so that v ∞ minimizes E. Remark 4.1. One can generalize the previous theorem with a cost function C that satisfies C(v) tends to +∞ when |v| L 2 V tends to +∞, e.g. for cost functions of the type

C(v) = 1 2 1 0 α t |v t | 2 V dt ,
where α : [0, 1] → R + , as soon as α admits a strictly positive lower bound (see the so-called adapted norm setup [23]).

4.2. Continuity of the global minimizers. At this point, the continuity of a minimizer v * of E is not acquired. We will prove now that all minimizers belong to C([0, 1], V ), which is not true when the attachment term is defined on varifolds. The outline of the proof is simple. We show that E is differentiable with respect to v and study the critical points of E. We keep the assumptions of the previous theorem. We assume in this section that W is a RKHS embedded in the space of

C 1 differential forms C 1 0 (R d , ( k R d ) *
). We recall a standard result on the flow of a vector field. Proposition 4.2. Assume the (H V 1 ) conditions given by equation [START_REF] Bauer | Geodesic distance for right invariant sobolev metrics of fractional order on the diffeomorphism group[END_REF]. Let be v, δv ∈ L 2 V and introduce the variations v = v + δv of v in the direction δv for ∈ R. Consider φ s,t the flow of v , meaning that φ s,t = φ t • φ ,-1 s where φ t is the unique solution on [0, 1] of

φ t = Id + t 0 v s • φ s ds .
Then, the application → φ s,t (y), dφ s,t (y) is of class C

µ v (ω) = 1 0 Yt ι ht-vt φ * t,1 ω dt
with respect to the vector field v. Let be v, δv ∈ L 2 V and consider v = v + δv for ∈ R and φ s,t its flow. From the linearity of the interior product, we have

µ v (ω) = 1 0 Yt ι ht-v t φ , * t,1 ω dt (56) = 1 0 Yt ι ht-vt φ , * t,1 ω dt - 1 0 Yt ι δvt φ , * t,1 ω dt . (57)
We address the differentiation with respect to of these two terms separately. Denote

g( ) = 1 0 Yt ι ht-vt φ , * t,1 ω dt .
In order to rewrite g, let us introduce some notation. The variables are grouped in pairs:

ν s (y) = (v s (y), dv s (y)) ∈ R d × L(R d ) , ϕ s (y) = (φ s,1 (y), dφ s,1 (y)) ∈ R d × L(R d ) , δν s (y) = ∂ ∂ ν s (y) =0 = (δv s (y), dδv s (y)) , δϕ s (y) = ∂ ∂ ϕ s (y) =0 . (58) Given ω ∈ W , define f ω : (R d × L(R d )) → (Λ k R d ) * such that f ω (ϕ t (y)) = (φ , * t,1 ω) y . This is, for any k-vector ξ 1 ∧ • • • ∧ ξ k ∈ Λ k R d , f ω (ϕ t (y))(ξ 1 ∧ • • • ∧ ξ k ) = ω(φ t,1 (y)) dφ t,1 (y)ξ 1 ∧ • • • ∧ dφ t,1 (y)ξ k .
We can easily check that f ω is C 1 . At last, we get

g( ) = 1 0 Yt ι ht-vt (f ω • ϕ t ) dt .
Let us show now that g is derivable in 0 and let us explicit this derivative.

Lemma 4.1. g is derivable and there exists t → J a t in C([0, 1], V * ) such that (59) g (0) = 1 0 J a s (δv s )ds .

Proof. Denote K = s∈[0,1] Y s , i.e. K = q 0 (X) and since q 0 ∈ C(X, R d ) and X is compact, K is bounded. Proposition 4.2 implies that for any s ∈ [0, 1], there exists (t, y)

→ A s t (y) in C([0, 1] × R d , L(R d × L(R d ))) such that for any y ∈ R d , δϕ s (y) (58) is given by δϕ s (y) = ∂ ∂ ϕ s (y) =0 = 1 s A s t (y) • δν t (φ s,t (y))dt . ( 60
)
We apply the Leibniz rule to derive under the integral sign so that we get

g (0) = 1 0 Ys ι hs-vs (d ϕ f ω (ϕ s ) • δϕ s ) ds = 1 0 Ys ι hs-vs d ϕ f ω (ϕ s ) 1 s A s t • (δν t • φ s,t ) dt ds = 1 0 Ys 1 s ι hs-vs (A s t ) * d ϕ f ω (ϕ s ) • (δν t • φ s,t ) dt ds ,
where for any y ∈ R d , A s t (y) * denotes the adjoint operator of

A s t (y) ∈ L(R d , L(R d )). For any y ∈ Y s , the integrand ι hs-vs A s t (y) * d ϕ f ω (ϕ s (y)) • (δν t (φ s,t (y)
) belongs to (Λ k R d ) * and we want to bound its norm independently of y to guarantee its integrability. This will come from the (H V 1 ) conditions that gives a spatial control of the elements of V and their differential.

For any y ∈ Y s , the application

A s t (y) * d ϕ f ω (ϕ s (y)) belongs to L R d × L(R d ), (Λ k R d ) * and can be identified to an element of (Λ k R d ) * ⊗ (R d × L(R d )) * . Moreover, for any ζ ∈ (Λ k R d ) * ⊗ (R d × L(R d )) * , consider l ζ y : V → (Λ k R d ) * by l ζ y (u) = ζ u(y), du(y) . Then l ζ
y is linear and under the (H V 1 ) conditions, there exists c V ∈ V , such that for any u ∈ V , for any y ∈ K, if µ = (u, du), then

|l ζ y (u)| (Λ k R d ) * = |ζ(µ(y))| (Λ k R d ) * ≤ |ζ| (Λ k R d ) * ⊗(R d ×L(R d )) * |µ(y)| R d ×L(R d ) ≤ c V |u| V sup y∈K (1 + |y| R d )|ζ| (Λ k R d ) * ⊗(R d ×L(R d )) * . Hence, l ζ y belongs to (Λ k R d ) * ⊗ V * and |l ζ y | (Λ k R d ) * ⊗V * ≤ c V sup y∈K (1 + |y| R d )|ζ| (Λ k R d ) * ⊗(R d ×L(R d )) * .
We can therefore apply Fubini's theorem to get that for any u ∈ V (61)

J a t (u) . = t 0 Ys ι hs-vs l (A s t ) * dϕfω(ϕs) φs,t (u) ds .
Finally, since t → A s t (y) is continuous, we deduce easily that t → J a t is continuous.

To study the second term of µ in equation (57), we introduce the next lemma.

Lemma 4.2. Given ω ∈ C 1 0 (R d , (Λ k R d ) * ), define for any t ∈ [0, 1] and any u ∈ V (62) J b t (u) = Yt ι u (ω) .
Then J b belongs to C([0, 1], V * ).

Proof. For any t ∈ [0, 1], J b t is linear on V and with the (H V 1 ) conditions, there exists c > 0 such that for any u ∈ V ,

|J b t (u)| ≤ Yt |u(y)| R d |ω(y)| ∞ dH k-1 (y) ≤ c sup y∈K (1 + |y| R d ) vol(Y t )|ω| ∞ |u| V .
Thus, J b t ∈ V * . Moreover, we can show that t → J b t is differentiable (and in particular continuous). From the spatial regularity of any u ∈ V , we deduce that ω

u . = ι u (ω) ∈ C 1 (R d , (Λ k-1 R d ) *
). Now, under the (H q0 ) conditions, we can pull backward the integrand of J b :

J b t (u) = {t}×B q * 0 ω u .
Therefore, if ∂ ∂t is the vector field on X defined at any point (t, x B ) ∈ [0, 1] × B by (1, 0 Tx B B ), then ∂ ∂t generates a flow ψ t on X satisfying ψ t (s, x B ) = (s + t, x B ). Thus, α . = q * 0 ω u ∈ C 1 (X, Λ k-1 T * X) is a (k-1)-form on X and it results from Cartan's formula and Stokes' theorem (see Corollary 6 4.2.2. Continuity of the minimizers. Finally, we can conclude that all solutions are continuous and the next theorem recalls all assumptions.

Theorem 4.2. Consider X = [0, 1] × B where B is a compact oriented manifold with corners and τ the projection on the first coordinate of X. Assume that q 0 ∈ C ∞ (X, R d ). Under the (H q0 ) and (H V 1 ) conditions, if v * ∈ L 2 ([0, 1], V ) minimizes the energy defined by

E(v) = 1 2 1 0 |v| 2 V dt + λ 2 |µ v -µ tar | 2 W * ,
then v * belongs to C([0, 1], V ). More precisely, for any (v, δv) in L 2 V ×L 2 V , the application → g( ) .

= E(v+ δv) is differentiable at 0 and we have g (0) = where K V and L V = K -1 V are the isomorphisms between V and V * , ω = K W (µ v -µ tar ), J a , J b ∈ C([0, 1], V * ) are defined by equations (61) and (62) and h t is the unique vector field on Y t defined for almost any t ∈ [0, 1] and any x ∈ B t by h t (q 0 (x)) = ∂q0 ∂t (x). Proof. We have ∂ ∂

1 2 |µ v -µ tar | 2 W * =0 = ∂ ∂ µ v (ω) =0 .
The expression of µ v (ω) is given by equation (57) and its derivative with respect to is given above in Section 4.2.1. At last, if v * minimizes E then L V v * t = J b t -J a t for almost every t ∈ [0, 1]. Since J a and J b are continuous, t → v * t = K V (J b t -J a t ) is continuous at any t ∈ [0, 1]. Remark 4.2. One can easily generalize this theorem with a cost function on L 2 V of the type C(v) = 1 2 1 0 C(v t , t)dt. More precisely, assume that there exists ∈ C([0, 1], V * ) such that for any t ∈ [0, 1], ∂C ∂v (v, t) = t (v) and t is invertible. If v * ∈ L 2 ([0, 1], V ) minimizes the energy

E(v) = 1 2 1 0 C(v t , t)dt + λ 2 |µ v -µ tar | 2 W * ,
then for any t ∈ [0, 1] v * t = -1 t K V (J b t -J a t ) . It follows that v * ∈ C([0, 1], V ). Remark 4.3. Note that J a 0 is always null. Moreover, if H k-1 (Y t ) is null, then J b t is also null. In the case of the horn, Y 0 represents the tip of the horn. It is thus reduced to a point, so that v * 0 is necessarily vanishing.

Conclusion

We examined in this paper a growth process by foliation. In the large class of growth mapped evolutions, the existence of a foliation induced by the birth tag guaranties some regularity of the growth process. Each image growth mapped evolution of a given biological coordinate system inherits its foliation that is then a key element to describe and to overcome the lack of spatial regularity of the generated shapes and to define current and varifold representations.

The growth dynamic is the first time-varying dynamic introduced for the analysis of longitudinal shape data in the context of shape space. We studied the existence and continuity of global minimizers v of the optimization problem for the assimilation of time-varying shapes in the specific case of the growth dynamic. These questions lie on the choice of the data attachment term. We exhibited two counterexamples for the varifold representation. These situations highlighted the lack of spatial regularity of a shape generated by a discontinuous time-varying vector field t → v t . This issue is well addressed by the current representation that has a regularization effect on the shapes. We proved indeed, with a data attachment term built on a current representation, the existence of global minimizers as well as their continuity.

Annex: Reminder on differential geometry

A k-dimensional manifold with corners extends the definition of regular manifolds (in the usual sense) to allow the shape to locally resemble a semi-orthant of R k . At any x 0 ∈ X, there exists a chart (U, ψ) We denote H k the k-dimensional Hausdorff measure on R d . We remind that H k is defined as an outer measure on R d that basically measures the k-dimensional volume of a subset of R d . In particular, when k = d, we have H d = λ d the usual Lebesgue measure. If M is a p-dimensional submanifold of R d , then H k (M ) is the k-volume of M if p = k, vanishes if p < k and equals +∞ when k < p.

The interior product exhibits in Section 4.1 the linearity property of the currents with respect to the tangential data. Definition 6.1. The interior product is defined to be the contraction of a differential form with a vector field. Thus if v is a vector field on the manifold M , then

ι v : (Λ k M ) * → (Λ k-1 M ) *
is the map which sends a k-form ω to the (k-1)-form ι v ω defined by the property that for any m ∈ M , (k-1)-vector

ξ 1 ∧ • • • ∧ ξ k-1 , ξ i ∈ T m M , (ι v ω)(m) ξ 1 ∧ • • • ∧ ξ k-1 . = ω(m) v(m) ∧ ξ 1 ∧ • • • ∧ ξ k-1 .
Hence, ι is linear with respect to v.

Corollary 6.1, given hereafter, results from Stokes' theorem and Cartan's formula and plays a central role to exploit the linearity of the current representation with respect to the tangential data of a shape. It is used in Lemma 4.2. Proof. See [START_REF] Lee | Introduction to Smooth Manifolds[END_REF].

The Lie derivative of differential forms with respect to vector fields in the direction of a vector field v expresses how a current associated to a shape X varies when X is deformed in the direction of v. More precisely, given a flow φ t such that φ 0 = Id and φt| t=0 = v We apply Cartan's formula in a particularly simple case. The manifold M is embedded in [0, 1] × M and the deformation is the translation along the first coordinate.

L v ω = lim

Figure 1 .

 1 Figure 1. Inner partial mappings of a growth mapped evolution to model a creation process.

Figure 6 .

 6 Figure 6. On the left. Solutions generated by the continuous minimizers v + and v -. Each color is associated to a length 0 . The dot line is the image of the initial position, i.e. the base of the horn. On the right. Solutions generated by a set of discontinuous minimizers v * at fixed 0 (and v + on the top).

Figure 7 .

 7 Figure 7. The target c tar is the blue curve. The red curve is c v + where v + is the positive unique global continuous minimizer of E λ (0, v). The pink curve belongs to the set of curves generated by the v s, * . The three dots in the middle are the respective centroid of the curves. One can see on this figure that x 2s -x 2 v tar is strictly positive and it increases when s tends to 0 (the pink dot tends to the red dot when s tends to 0).

Figure 8 .

 8 Figure 8. Plot of the function g

[ 1 ,

 1 +∞[ to [π, +∞[. (P c ) characterizes the minimizers of the constrained optimization problem min L 2 v(t) 2 dt with (v) = ˆ (c) .

  displays an example for n = 21 and h = 0.1.

Figure 10 .

 10 Figure 10. From left to right: the target, the vector field which generates it from the unit disc with vertical translations, a radial cut of the surface (plot of the vertical component z v tar (r)). This example is essentially the 3D analogue of the 2D shape illustrated in Figure 4. For better visibility, colors indicate the height instead of the temporal tag.

Figure 11 .Figure 12 .

 1112 Figure11. Surfaces generated from two solutions for the matching of the surface displayed in Figure10. On the left: with the discontinuous vector field s n v + , on the right: with the continuous vector field v +

  R k-p × R p + x → (x 1 , . . . , x k-p , y 1 , . . . , y p ) , centered at x 0 , i.e. ψ(x 0 ) = (0, • • • , 0), between a open set U x 0 and a semi-orthant R k-p × R p + for an integer p = p(x 0 ) ≥ 0. For a regular manifold, p is always null. If p takes values only in {0, 1} on X, then X is called a manifold with boundary.

Theorem 6 . 1 (

 61 Stokes' theorem). Let M be an oriented compact k-dimensional differential manifold with corners. For any differential (k-1)-form ω of class C 1

  X) (ω) |t=0 = µ X (L v ω) . (66)Theorem 6.2 (Cartan's formula). Let ω be a differential form of class C 1 and v a vector field thenL v ω = dι v ω + ι v dω .Proof. See[START_REF] Labourie | Geométrie différentielle[END_REF] Lemma 7.2.1 and 10.3.2.

Corollary 6 . 1 .M

 61 Denote v = ∂ t the vector field defined at any point (t, m) ∈ [0, 1]×M by (1, 0 TmM ) and M t = {t} × M then dι v ω + ι v dω ,where φ t is the local flow resulting from v. Stokes' theorem allows then to conclude.

  1 . We have for any y ∈ R d ,

		∂ ∂	φ s,t (y) =0 =
	and
	∂ ∂	dφ s,t (y) =0 =

t s dφ u,t (φ s,u (y)).δv u (φ s,u (y)) du t s d 2 φ u,t (φ s,u (y))dφ s,u (y) δv u (φ s,u (y))

+ dφ u,t (φ s,u (y))dδv u (φ s,u (y))dφ s,u (y) du .

4.2.1. Differentiability of the current representation.

A first step consists in studying the directional derivative of the current