Irène Kaltenmark Cmla

Benjamin Charlier
email: benjamin.charlier@umontpellier.fr

Nicolas Charon
email: charon@cis.jhu.edu

A general framework for curve and surface comparison and registration with oriented varifolds

This paper introduces a general setting for the construction of data fidelity metrics between oriented or nonoriented geometric shapes like curves, curve sets or surfaces. These metrics are based on the representation of shapes as distributions of their local tangent or normal vectors and the definition of reproducing kernels on these spaces. The construction, that combines in one common setting and extends the previous frameworks of currents and varifolds, provides a very large class of kernel metrics which can be easily computed without requiring any kind of parametrization of shapes and which are smooth enough to give robustness to certain imperfections that could result e.g. from bad segmentation. We then give a sense, with synthetic examples, of the versatility and potentialities of such metrics when used in various problems like shape comparison, clustering and diffeomorphic registration.

Introduction

Context. Shape analysis has become one of the central problem in many recent applications that ranges from automatic object recognition in computer vision and robotics to the field of medical imaging where more and more advanced devices and protocols have enabled high resolution acquisitions of a variety of anatomical structures with varying morphology across individuals or along time in longitudinal studies.

The specific problems that are encountered obviously depend on the nature of the applications and data. Yet, among the issues that are most often involved in such studies, one can mention in particular the quantification of shape differences through adequate metrics, registration (i.e. esti-mation transformations between two given shapes), or the estimation of shape statistics like means, templates or generative models from an observed population of shapes. Numerous methods have been proposed to address those questions for different shape modalities. Some of these methods, [START_REF] Kendall | Shape manifolds, procrustean metrics, and complex projective spaces[END_REF]7,22,[START_REF] Holm | Soliton Dynamics in Computational Anatomy[END_REF] to mention only a few, rely on the extraction of a limited number of feature points (or landmarks). Other methods like [START_REF] Dupuis | Variational problems on flows of diffeomorphisms for image matching[END_REF][START_REF] Aronszajn | Theory of reproducing kernels[END_REF]4,3,36] instead consider shapes as entire 2D or 3D images.

In many situations however, the more natural or compact model for shapes is to work with curves or surfaces of the ambient space which may have been obtained e.g. by prior segmentation of images. This case involves particular challenges resulting from the complex nature of spaces of curves and surfaces. The general goal of this paper is to propose a fairly general construction of similarity metrics between a wide class of those objects that can be easily used in multiple problems of shape analysis.

Relation to other works. Several frameworks have been introduced in the past to define similarity metrics or measures between two given curves or surfaces. If metrics like the Hausdorff or Gromov-Hausdorff distance [27] would appear natural at first glance, they are however highly sensitive to noise or topological irregularities and not necessarily fitted for applications to inexact registration.

A class of approaches have attempted to go through parametrization functions in order to represent and compare shapes with for example square root velocity functions for closed curves [35] or q-mappings for closed surfaces [25]. In both cases, enforcing parametrization-invariance is an essential step that involves delicate and possibly costly optimization over discrete reparametrizations.

The focus of this paper is rather on shape distances that are directly invariant to parametrization. The approach we follow stems from the idea from geometric measure theory of representing shapes as elements of a certain space of distributions, which was first used for the definition of shape similarity metrics in the seminal works of [START_REF] Glaunès | Surface matching via currents[END_REF][START_REF] Glaunès | Large deformation diffeomorphic metric curve mapping[END_REF]. These were 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 registration[END_REF][START_REF] Charon | Analysis of geometric and functional shapes with extensions of currents. Application to registration and atlas estimation[END_REF] introduced the alternative but orientation-invariant representation known as varifolds before the higher order model of normal cycles was recently investigated in [31]. Since then, all these different frameworks have found numerous applications to the morphological analysis of cortical surfaces [26,[START_REF] Rekik | Multidirectional and Topography-based Dynamic-scale Varifold Representations with Application to Matching Developing Cortical Surfaces[END_REF], brain sulci [31], white matter fiber bundles [13,18] or registration of lung vessels [29]...

Contributions.

The objectives of this paper are several. We first propose to unify under one common and extended framework currents and varifolds' metrics for embedded curves and surfaces. This is based on an adaptation of the concept of oriented varifold; we show how this representation allows to construct a general class of shape similarity metrics that applies to any type of curve or surface (open or closed) or even reunions of multiple objects. Furthermore, these (Hilbert) metrics have the advantages of being independent to parametrization or to the sampling in the case of discrete shapes, efficiently computable with a closed-form expression and robust to small topological perturbations or potential segmentation issues. Besides the particular cases of currents and (unoriented) varifolds, we show that we can easily obtain from our construction new metrics with intermediate properties. We make a thorough comparison of the characteristics and behavior of a few distinct choices of those similarity measures and illustrate various possible uses. Our implementation is also freely available at [START_REF] Charlier | A short introduction to the functional shapes toolkit[END_REF].

Notation and preliminaries

In all the following, and although the framework we present can be extended to any dimension or codimension, we will restrict the exposition by considering shapes to be smooth manifolds or reunion of smooth manifolds (possibly with boundaries) of dimension 1 or 2 and embedded in the ambient space R n with n = 2 or n = 3. These include most of the usual geometrical shapes in 2D or 3D like open and closed planar curves, 3D curves, surfaces, curve bundles... Any such individual submanifold X (curve or surface) inherits the Riemannian metric and volume measure denoted vol induced from the embedding Euclidean space. At each point x ∈ X, there exists a tangent space T x X which is a linear subspace of R n of dimension 1 or 2. An orientation of X consists in giving an orientation to T x X for all x ∈ X in which case each oriented tangent space can be represented as an element of an oriented Grassmannian. With shapes of dimension or codimension 1, such oriented Grassmannian can be in both cases identified to the unit sphere S n-1 and T x X represented by the unit oriented tangent vector to the curve or unit oriented normal vector to the surface at x that we shall write t(x). Note that the choice of the opposite orientation for the tangent space will change t(x) into -t(x). A shape is said to be orientable in the usual sense if there exists a smooth orientation application x → t(x) over the entire X.

Shape representation as oriented varifolds

Representation of smooth submanifolds

With the notation above, let X be a smooth submanifold of R n (curve or surface) with finite total volume vol(X) < ∞. In similar fashion as [START_REF] Glaunès | Diffeomorphic matching of distributions: A new approach for unlabelled pointsets and sub-manifolds matching[END_REF][START_REF] Glaunès | Surface matching via currents[END_REF][START_REF] Charon | The varifold representation of non-oriented shapes for diffeomorphic registration[END_REF], we associate to X an oriented varifold µ X , i.e. a distribution on the space R n × S n-1 of (position × tangent space orientation) and defined as follows:

µ X (ω) = X ω(x, t(x))dvol(x) (1)
for any smooth test function ω : R n × S n-1 → R. In the distribution sense, we may write µ X = X δ (x, t(x)) dvol(x) with the Dirac delta δ (x, t(x)) (ω) .

= ω(x, t(x)) and the identification X → µ X gives indeed an injection from the set of smooth submanifolds to the dual W * of a certain space W of test functions on R n × S n-1 provided this latter space is large enough. Specific choices of spaces W will be discussed with more details below.

Remark 1. The distribution µ X only depends on the shape and not on any particular parametrization of X, but does depend a priori on the choice of orientation given to each tangent space through the vector t(x). In addition, the oriented varifold representation is additive in the sense that reunion of two distinct submanifolds X and Y satisfy

µ X∪Y = µ X + µ Y .
Remark 2. This general shape representation includes as particular cases both the previous frameworks of currents [START_REF] Glaunès | Diffeomorphic matching of distributions: A new approach for unlabelled pointsets and sub-manifolds matching[END_REF] and (unoriented) varifolds [START_REF] Charon | The varifold representation of non-oriented shapes for diffeomorphic registration[END_REF]. Each simply consists in restricting to spaces of test functions with particular forms as we will see hereafter.

Polyhedral shapes' finite approximations

Oriented varifolds can be then further used to embed within the same distribution space W * discrete curves or surfaces or reunions of those. Indeed, such discrete shapes are typically polyhedral objects which we can write in general as X = F i=1 X i where the cells X i are either line segments in the curve case or triangles in the case of 3D triangulated surfaces and are distinct to each other (modulo their boundaries).

Following Remark 1, we associate to such X = the oriented varifold associated to the flat cell X i according to eq. ( 1). As illustrated by Figure 1, we can make the approximation:

µ Xi (ω) ≈ Xi ω(x i , t i )dvol(x) = r i ω(x i , t i ) (2)
where x i is the barycenter of all vertices defining X i , t i ∈ S n-1 the orientation vector directing the linear subspace spanned by X i and r i . = vol(X i ) the length (resp area) of the segment (resp triangle). This leads to replace each µ Xi by a single weighted Dirac r i .δ (xi, ti) . Then, writing μX = F i=1 r i .δ (xi, ti) one has µ X ≈ μX with the following approximation bound: Proposition 1. If X is a shape contained inside the compact K of R n then for any C 1 test function ω:

|µ X (ω) -μX (ω)| ≤ ∂ x ω ∞,K .vol(X). max 1≤i≤F diam(X i ) Proof. For any cell X i , since for all x ∈ X i , t(x) = t i , we have |(µ Xi -r i δ (xi, ti) )(ω)| ≤ Xi |ω(x, t(x)) -ω(x i , t i )|dvol ≤ vol(X i ) ∂ x ω K,∞ diam(X i )
with ∂ x ω ∞,K denoting the sup-norm over K of the derivatives of ω with respect to the first variable and diam(X i ) = max x,y∈Xi |x -y| the diameter of the cell. The result follows by just summing over all cells.

Therefore, the finite sum of Diracs μX provides an acceptable approximation of the polyhedral shape X in terms of oriented varifolds as long as the size of cells remains small enough.

Another interesting property of this oriented varifold setting is that it provides consistency between discrete and continuous shape representations in the sense that the sequence of discrete distributions obtained from more and more refined polyhedral approximations of a given continuous curve or surface will converge, under certain technical assumptions, to the oriented varifold of the continuous shape. Precise conditions and proof could be generalized from similar recent results on currents and varifolds that have been derived in [28] and [START_REF] Arguillère | Registration of Multiple Shapes using Constrained Optimal Control[END_REF] but this would go beyond the scope of this paper. Yet an important consequence is that such representations are robust to discrete resampling.

Kernel spaces and metric properties

We will proceed in defining a class of metric structures on spaces of oriented varifolds that will in turn induce metrics between shapes. These dual metrics on W * are obtained by introducing more specific Hilbert spaces for the set of test functions W . For such spaces, Reproducing Kernel Hilbert Spaces (RKHS) turn out to be particularly well suited for practical computations on discrete shapes.

RKHS of oriented varifolds

We remind that a positive definite kernel on any set M is a function k : M × M → R such that for any family (x 1 , . . . , x p ) ∈ M p of distinct points in M the matrix (k(x i , x j )) i,j=1,...,p is symmetric positive semidefinite. To any such kernel is associated, through Moore-Aronszajn theorem [2], a Hilbert space of functions on M called the RKHS of k. Reproducing kernels have been vastly used for instance in machine learning [20].

In the present case, we wish to construct real kernels over the product space R n × S n-1 and take the associated RKHS as the test functions' space W in the oriented varifold representation. We consider, in this paper, a particular class of separable kernels defined as follows:

Proposition 2. If k pos and k or are two positive definite kernels of class C 1 on R n and S n-1 then the tensor product k = k pos ⊗k or is a C 1 positive definite kernel on R n ×S n-1 . In addition, the RKHS of k is continuously embedded in

C 1 (R n × S n-1 , R).
This is a consequence of classical results on the tensor product of reproducing kernels (c.f [6] 1.4.6). The reproducing kernel property implies in addition that all Diracs δ (x, t) belong to W * and that the dual metric satisfies

δ (x1, t1) , δ (x2, t2) ) W * = k((x 1 , t 1 ), (x 2 , t 2 )) = k pos (x 1 , x 2 )k or ( t 1 , t 2 ) . (3)
Although this separable structure does not cover the entire possible set of kernels on R n × S n-1 , the tensor product construction has the advantage of still providing a large class of metrics through the various possible choices of k pos and k or while being easy to interpret in terms of the combination between spatial and orientation characteristics. Indeed, k pos can be thought as the measure of proximity between point positions in the embedding space R n while k or , on the other hand, quantifies the proximity between the associated tangent spaces represented by vectors on the unit sphere.

An additional relevant property for the metric is the equivariance to rigid motion, namely that for any affine isometry x → Ax + b one has for any two Diracs in

W * δ (Ax1+b,A t1) , δ (Ax2+b,A t2) W * = δ (x1, t1) , δ (x2, t2) W * .
We have the following straightforward characterization: Proposition 3. Within the previous class of separable kernels, the metric W * is equivariant to the action of rigid transformations for kernels k of the form:

k((x 1 , t 1 ), (x 2 , t 2 )) = k pos (x 1 , x 2 )k or ( t 1 , t 2 ) = ρ(|x 1 -x 2 |)γ( t 1 • t 2 ) . (4)
Conversely, conditions on functions ρ and γ to obtain positive definite kernels have also been more precisely examined. The results of [START_REF] Ashburner | A fast diffeomorphic image registration algorithm[END_REF] have shown that admissible ρ can be defined as the Bessel transform of a finite positive Borel measure on R + . Similarly, a necessary and sufficient condition on γ (cf [37]) is that γ

(u) = ∞ k=0 a k P (λ) k (u) with a k ≥ 0, ∞ k=0 a k P (λ) k (1) < ∞ and P (λ) k
the ultraspherical polynomials of order λ = (n -1)/2. Note that another possible (but not exhaustive) way of constructing admissible γ is by restriction to the unit sphere of radial kernels on R n .

A particular subclass of the kernels of Proposition 3 are the ones such that γ(u) = u and that correspond to the framework of currents. Another subclass are the ones invariant to reorientation i.e. such that γ(-u) = γ(u) (or a k = 0 for odd k in the decomposition with ultraspherical polynomials) which results in metrics on W * also independent on the orientation: this precisely corresponds to the framework of unoriented varifolds developed previously in [START_REF] Charon | The varifold representation of non-oriented shapes for diffeomorphic registration[END_REF][START_REF] Beg | Computing large deformation metric mappings via geodesic flows of diffeomorphisms[END_REF], cf Section 4.3 for examples.

Metrics induced on shapes

We can now restrict the metric on W * defined in 4.1 to shapes through the identification of X as µ X ∈ W * . However, this identification may not be injective if the space W is too 'small' which may only result in a pseudo-metric between shapes. A sufficient condition is the following: Proposition 4. Let k pos and k or be two kernels as in Proposition 2 such that in addition k pos is a C 0 -universal kernel on R n and for all t ∈ S n-1 , k or ( t, t) > 0. Then d W * (X, Y ) = µ X -µ Y W * defines a distance on the set of shapes.

Proof. We only need to show that d W * (X, Y ) = 0 ⇒ X = Y . Let's write W pos and W or for the RKHS on R n and S n-1 associated to kernels k pos and k or . The C 0universality property of k pos precisely means that W pos is dense in the space of continuous functions on R n vanishing at ∞ [8]. Now, let's assume that X and Y are both smooth curves or surfaces (the case of reunions of those can be treated similarly) such that d W * (X, Y ) = 0 and X = Y . Then there exist x 0 ∈ X and r > 0 such that B(x 0 , r) ∩ Y = ∅. Consider the function g defined by g(•) = k or ( t(x 0 ), •) ∈ W or . Since g( t(x 0 )) = k or ( t(x 0 ), t(x 0 )) > 0, g is continuous and X is smooth, x → g( t(x)) is strictly positive on a certain ball B(x 0 , r ) ∩ X ⊂ B(x 0 , r). Now take h a continuous positive function on R n supported in B(x 0 , r ) and strictly positive at x 0 . By density, there exists a sequence (h n ) ∈ (W pos ) N converging uniformly to h. Then

0 = (µ X -µ Y )(h n ⊗ g) = X h n (x)g( t(x))dvol(x) - Y h n (y)g( t(y))dvol(y) n→∞ ----→ X∩B(x0,r ) h(x)g( t(x))dvol(x) .
But since h and x → g( t(x)) are both strictly positive on B(x 0 , r ), we end up with a contradiction.

Thanks to both eq. ( 1) and the reproducing kernel property of eq. (3), for any two submanifolds X and Y , we have the following simple and explicit expression for the metric:

µ X , µ Y W * = X×Y k pos (x, y)k or ( t(x), t(y))dvol X (x)dvol Y (y) . (5)
Note that the induced distance d W * inherits the equivariance to rigid motion for kernels chosen as in Proposition 3, i.e. that for any affine isometry S, one has

d W * (S(X), S(Y )) = d W * (X, Y ).
Similarly, with the discrete approximations of polyhedral shapes described in Section 3.2, we obtain a finite equivalent of (5) that writes:

μX , μY W * = F X i=1 F Y j=1 k pos (x i , y j )k or ( t X i , t Y j )r X i r Y j (6) for the two discretizations μX = F X i=1 r X i δ (xi, t X i ) and μY = F Y j=1 r Y j δ (yj , t Y j )
. Note that thanks to the C 1 regularity of both kernels, it is straightforward to deduce from Proposition 1 a similar bound on the approximation error for the W * metric, i.e. that µ X -μX W * is again controlled by the max diameter of all cells in X. A second consequence of the regularity is that the metric of ( 6) is differentiable with respect to the positions of shapes' vertices and the gradient is easily computed by chain rule differentiation.

Choosing kernels

From now on, we focus the discussion on kernels of the form given by (4). The particularly simple form of this family of metrics makes it very convenient for an implementation that preserves the flexibility in the choices of kernels.

For the radial kernel k pos , typical choices are the widely used Gaussian k pos (x, y) = e -|x-y| 2 σ 2

or Cauchy k pos (x, y) = 1 1+|x-y| 2 /σ 2 kernels in which parameter σ gives the spatial scale at which the metric is sensitive. Both satisfy the C 0 -universality assumption of Proposition 4, cf [8]. We shall be mostly experimenting with Gaussian kernels in the simulations of next section. Our implementation also includes sums of arbitrary number of Gaussians with different scales as an attempt to allow multiscale strategies.

The equivariant spherical kernel k or on the other hand has deep influence on the fundamental properties of the resulting metrics. Below, we examine several distinctive examples for γ, some of which retrieve, as special cases, previously introduced shape dissimilarity terms:

• γ(u) = 1 (distributions): gives metrics that discard the orientation vector information to only evaluate the relative positions between points in the two shapes. Those metrics can be equivalently obtained from the representation of shapes as standard distributions on R n , which was the initial approach of [START_REF] Glaunès | Diffeomorphic matching of distributions: A new approach for unlabelled pointsets and sub-manifolds matching[END_REF].

• γ(u) = u (currents): k or is the restriction of the linear kernel on S n-1 . Because of linearity, µ X can be interpreted as an element of the dual of the space of vector fields or in other words as a current and the family of metrics (5) recovers exactly the framework introduced in [START_REF] Glaunès | Surface matching via currents[END_REF][START_REF] Glaunès | Large deformation diffeomorphic metric curve mapping[END_REF]. Such metrics are only well-suited for orientable shapes. Given a surface or curve X, we denote X the same surface or curve but oppositely oriented. The linearity of the kernel implies in that case that µ X = -µ X .

• γ(u) = u 2 (unoriented varifolds): corresponds to the Binet kernel on S n-1 , invariant to orientation. This is the simplest example of unoriented varifold metrics used in [START_REF] Beg | Computing large deformation metric mappings via geodesic flows of diffeomorphisms[END_REF]. Other orientation-invariant kernels have also been proposed and studied in [START_REF] Charon | The varifold representation of non-oriented shapes for diffeomorphic registration[END_REF].

• γ(u) = e 2u σ 2
s (oriented varifolds): in that case, k or is the extrinsic Gaussian kernel on the sphere (the restriction of the Gaussian kernel of R n to S n-1 ). The shapes need to be orientable and the kernel is sensitive to the orientation but non-linear, the scale σ s enforces the angular sensitivity of the metric. Many other kernels with similar properties can be used, in particular the intrinsic heat kernels on S n-1 although its expression involves decomposition into the ultraspherical polynomial basis which is thus more numerically intensive.

Figure 2. Behavior of the different metrics. On left, a shape with thin elongated components in red. On right, a plot of the ratios between the norm of the red part with the norm of the entire shape for increasing scales σ for the Gaussian function ρ. Note the rapid decrease in the case of currents, due to the cancellation of oppositely oriented pieces 'close' to each other with respect to σ.

With both varifold representations, the non-linearity of k or prevents cancellations of thin structures when compared to currents, see Figure 2. Conversely, this linearity property enforces more regularity in some specific registration problems to model shape growth [23].

Experiments

We now give several illustrations of the respective interests and drawbacks of these different metrics, in particular the novel oriented varifolds' metric, on a few problems related to shape analysis and processing. We use the specific kernels described in Section 4.3 for currents, unoriented and oriented varifolds.

Shape clustering

In this section, we present the experimental results for object retrieval on the standard shape dataset Kimia-216. It consists in 18 classes with 12 objects in each. The contour of each shape is extracted from its smoothed image. We do not control the final sample of each contour. Curves have between 100 and 305 points. To address the question of scale invariance, we fix an oriented varifold's metric in accordance with the average scale of the data set and we apply a scaling to all the shapes such that, in fine, they all have the same norm. From the previous equivariant metrics on W * , we then obtain a rigid-invariant metrics on shapes by solving the rigid registration problem:

d inv (X, Y ) = inf S rigid d W * (X, S(Y )) . ( 7
)
The top 11 closest matches, obtained with the rigidinvariant current, oriented and unoriented varifolds' metrics, are shown in Table 5.1. The metrics are defined with a Gaussian radial kernel for k pos , which depends on the scale parameter σ and the three scalar functions γ introduced in Section 4.2, the last of which depends on an additional parameter σ s . We tested the classification for a small set of different parameters. The optimal results were ob-Method 1 st 2 nd 3 rd 4 th 5 th 6 th 7 th 8 th 9 th 10 th 11 th Total Shape context [START_REF] Belongie | Shape matching and object recognition using shape contexts[END_REF] 214 tained with σ = 1.5 and σ s = .5 for oriented varifolds, σ = 1.5 for unoriented varifolds and σ = 1 for currents. The final score of the classification is significantly reduced by few queries with a particularly low retrieval score as displayed in Figure 3.

Diffeomorphic registration

The deformation model

As a second class of applications, we turn to the use of the previous metrics as data fidelity terms for diffeomorphic registration algorithms of curves and surfaces. The framework we propose is sufficiently flexible to be embedded in a variety of inexact registration methods; in this paper, we focus on the Large Deformation Diffeomorphic Metric Mapping (LDDMM) model described in [4,39].

In this model, diffeomorphisms are constructed as flows of time-dependent velocity fields t ∈ [0, 1] → v t , each v t belonging to a predefined Hilbert space V of smooth vector fields. If X 0 denotes the template shape and X 1 the target, then the registration of X 0 to X 1 is written as the minimization problem: the flow equation ∂ t φ v t = v t • φ v t with φ v 0 = Id and λ > 0 is a weight parameter between the regularization and data fidelity terms. This problem is numerically solved for discrete shapes through a geodesic shooting procedure that involves the computation of both oriented varifold distances and their gradient with respect to vertex positions.

min v∈L 2 ([0,1],V ) 1 0 v t 2 V dt + λ µ φ v 1 (X0) -µ X1 2 W * (8) where φ v 1 (X 0 )

Registration of closed curves

We first compare registration results on (oriented) closed curves for current, unoriented and oriented varifolds. Figure 4 shows the template, target and registered shape as well as the final deformation grid in each case. With currents, notice the apparition of degenerate structures as well as the fact that the two humps are not well recovered: this is another manifestation of the cancellation effect previously mentioned. These effects are totally avoided in the two other cases. However, unoriented varifolds, by being insensitive to the orientation, tend to have more difficulties in reproducing the convoluted double hump by instead creating a single 'average' hump. Oriented varifolds achieve very accurate registration in that case.

Registration of set of curves

We consider now a simple example involving disconnected 'bags' of curves instead of the previous single connected shapes. These were obtained by extracting sketches from two images of faces using the Line Segment Detector al- gorithm of [START_REF] Grompone Von Gioi | LSD: a Line Segment Detector[END_REF]. In this situation, orienting consistently each separate segment is obviously a very cumbersome issue. Yet the unoriented varifold metric allows to compare those objects without requiring that orientation step.

Registration of noisy shapes

Data are often observed with an additive noise. In that situation, the cancellation property of currents can be used advantageously to suppress the effects of noise in the estimated deformation as shown in Figure 6 and7. This holds for kernel sizes σ chosen sufficiently large compared to noise induced oscillations. On the other hand, the oriented and unoriented varifold approaches are consistently much more affected by noise. Note, however, that if both the template and the target present similar type and level of noise, all three approaches will generally lead to good registrations.

Current

Unoriented Oriented Noisy varifold varifold template Figure 6. Registration with noisy target. Note that non-linear kernels (oriented or not) are much more sensitive to noise as opposed to currents. In the last experiment, noise is also added to the template and the results are similar for the three metrics.

Registration of multi-objects

Multi-modal data analysis is a common problem in medical imaging. We present therefore in Figure 8 and a bundle of curves (50 and 53 curves respectively containing 9 points each). To fix the notation, X = (X 1 , X 2 ) denotes the source multi-shape and Y = (Y 1 , Y 2 ) denotes the target multi-shape where the first object corresponds to the surface part and the second one to the curve part.

To register two of these multi-shapes, we consider a variant of the optimization problem of (8) that writes

min v∈L 2 ([0,1],V ) 1 0 v t 2 V dt + λ 1 µ φ v 1 (X1) -µ Y1 2 W * 1 + λ 2 µ φ v 1 (X2) -µ Y2 2 W * 2 ,
where λ 1 , λ 2 > 0 are weighting parameters. It is important to properly calibrate the λ i 's as the order of magnitude of the two varifold data attachment terms may be very different. Finally note that the deformation φ v 1 is common to the two sub-shapes, whereas the data attachment term is specific to each sub-shape.

Registration onto an incomplete shapes

Kernel-based oriented varifold distances coupled with the LDDMM framework introduced Section 5.2.1 can be readily used even when the mesh of the data at hand has missing parts.

In our experiments, we assume that the source shape is complete whereas the target shape has a proportion of missing triangles. For instance, in Figure 9 the target dolphin mesh contains p = 30% of triangles of the complete mesh (originally containing 14207 triangles). The removed faces were picked uniformly at random. The source shape is an ellipsoid containing 20000 triangles. To give some perspective on the numerical cost, computing the oriented varifold distance between the source and the complete target typically takes 0.1 second with our CUDA implementation on a Nvidia Geforce GTX 780 Ti.

In order to get a coherent registration on such an incomplete target, it is, however, necessary to adjust for the deficiency in total area of the target. The orientation varifold representation allows to do so very conveniently by simply re weighting the target oriented varifold with a single constant equal to an estimation of the proportion of missing triangles. Then the registration problem is replaced by:

min v∈L 2 ([0,1],V ) 1 0 v t 2 V dt + λ µ φ v 1 (X) -µ Y p 2 W *
where (1 -p) is the proportion of missing triangles in the target shape Y . Figure 10 shows some experiments where the proportion p is known. Registration results look globally good even for small p and seem almost perfect for any p greater than 30%. Below this value, the registration start p = 100% p = 30% p = 5% to slightly deteriorate and spurious deformations may affect the mesh in area with complex geometries. If the true proportion p * is unknown, the quality of the registration is affected by the value of the chosen p as shown in Figure 11.

Conclusion

In this paper, we have proposed a general framework for 2D and 3D shape similarity measures, invariant to parametrization and equivariant to rigid transformations. In this simple and efficient framework that encompasses currents and unoriented varifolds, the choice of the metric reduces to scalar functions with only one or two scale parameters that parametrize families of kernels. A numerical toolbox is freely available at [START_REF] Charlier | A short introduction to the functional shapes toolkit[END_REF]. It includes all the functions introduced in this paper and allows to readily define new customized kernel metrics.

We illustrated through numerous examples that these shape representations and associated metrics are well-suited for large classes of data structures and highlighted the specific features for different sub-categories of those metrics. Beyond the use as fidelity terms for registration, we wanted to emphasize how such a framework could be also directly applied in shape clustering or classification problems.

FFigure 1 .

 1 Figure 1. Oriented varifold approximation of triangulated faces.

Figure 3 .

 3 Figure 3. Least accurate top 12 retrieval of a query shape.

  Figure 4. Diffeomorphic registration using the different metrics of a template horse (blue dotted curve) to the target (in red). The estimated deformed template is shown in blue together with deformation grid. Kernel kpos is in each case a Gaussian with spatial scale σ = 15.

Figure 5 .

 5 Figure 5. Registration between two face sketches using unoriented varifolds.

  Figure 7. Registration of a sphere (2560 vertices) on a noisy surface (1165 vertices) using currents. The template shape evolution along the deformation is shown as well as the comparison of the registered shape and the initial noise-free surface.

1 Figure 8 .

 18 Figure 8. Registration, with unoriented varifolds, of multi-shapes containing each a surface and a set of curves. Target shape is in red.

Figure 9 .

 9 Figure 9. Registration of an ellipsoid on a dolphin with 70% missing triangles, using oriented varifold data attachment term.

Figure 10 .Figure 11 .

 1011 Figure 10. Three registrations onto a target with various ratios of missing faces. The proportion (1 -p) of missing triangles is known. Corresponding target shapes are on the first row. Deformed ellipsoids are plotted on the second row. Comparison between the deformed ellipsoid and the complete target are shown on the third row.

Table 1 .

 1 Top 11 closest matching shapes for Kimia-216. The oriented varifold representation outperforms the current and unoriented varifold representations, with a total retrieval result of 93.43% in the nearest 11 matches.

		209 205 197 191 178 161 144 131 101	78	76.13
	Shock graph [34]	216 216 216 215 210 210 207 204 200 187	163 94.44
	Chordal axis transform [38] 216 216 216 214 215 212 213 208 204 195	168 95.83
	Current	214 211 205 202 200 203 197 197 189 160	150 89.56
	Unoriented varifold	214 216 212 210 210 207 202 201 191 171	150 91.92
	Oriented varifold	215 214 214 213 212 206 206 202 192 185	161 93.43
	Method	Retrieval accuracy
	Current	96.30
	Unoriented varifold	96.57
	Oriented varifold	97.92

Table 2 .

 2 Bull Eye Score (retrieval rate among the 24 nearest neighbors, query included) of proposed methods for Kimia-216.


INTRODUCTION

In geotechnical engineering the effect of temperature has to be studied particularly for applications such as the storage of nuclear waste [START_REF] Rutqvist | A modeling approach for analysis of coupled multiphase fluid flow heat transfer, and deformation in fractured porous rock[END_REF], the burying of high-voltage cables [START_REF] De Lieto Vollaro | Thermal analysis of underground electrical power cables buried in non-homogeneous soils[END_REF]) and the thermo-active geostructures [START_REF] Pahud | Geothermal energy and heat storage[END_REF]. These thermo-active geostructures are used to provide sustainable heating and cooling by thermal exchange between buildings and the ground through liquids that flow through closedloop circulation systems integrated into the geostructures. Thermo-active geostructures may be piles, diaphragm walls, tunnel linings, basement slabs or walls [START_REF] Brandl | Energy foundations and other thermo-active ground structures[END_REF][START_REF] Fromentin | Pieux échangeurs: conception et règles de pré-dimensionnement[END_REF][START_REF] Laloui | Comportement d'un pieu bi-fonction, fondation et échangeur de chaleur[END_REF]. The classical way in which they are used results in temperature cyclic changes in the surrounding ground temperature. Consequently, there are important questions about the effect of temperature variations on hydro-mechanical soil properties which may affect long-term structure performance. In particular, the contraction of the elastic domain (i.e., the yield locus) with increasing temperature has been demonstrated for various clay materials on centimetric samples [START_REF] Tanaka | Stress-strain behaviour of reconstituted illitic clay at different temperatures[END_REF][START_REF] Graham | Modified Cam-Clay modelling of temperature effects in clays[END_REF][START_REF] Cekerevac | Experimental study of thermal effects on the mechanical behaviour of a clay[END_REF][START_REF] Marques | Viscous behaviour of St-Roch-de-l'Achigan clay, Quebec[END_REF][START_REF] Uchaipichat | Experimental investigation of thermo-hydro-mechanical behaviour of an unsaturated silt[END_REF]. In this study, tests on larger samples of contrasted materials are performed.

The pressuremeter test is currently used to calculate the bearing capacity of deep foundations [START_REF] Afnor | NF P94-110-1 -Sols : reconnaissance et essais Essai pressiométrique Ménard[END_REF]Standard ASTM, 1999). This test consists in introducing in the soil a cylindrical probe with a flexible cover membrane which can expand radially in a hole. As the soil has a pseudo-elastic reaction versus the probe pressure during a part of the test, it is possible to calculate a pressuremeter modulus E P . After that, at higher pressures, large displacements take place and the soil becomes plastic. The creep pressure, P f is defined as the yielding pressure between the pseudo-elastic and plastic behavior. This parameter is directly correlated to the yield locus. As a result it should be possible to quantify the potential contraction of the elastic domain using pressuremeter tests. Thus, we developed an experimental method to carry out minipressuremeter tests on homogeneous soils compacted at a controlled density and water content and subjected to monotonic and cyclic thermal variations [START_REF] Eslami | Comportement Thermo-hydromécanique des sols au voisinage des géo-structures énergétiques[END_REF]. ABSTRACT: In geotechnical engineering, the proper design of thermo-active geostructures (piles, foundations, etc.) and deep waste storage disposals requires a better understanding of the thermo-hydro-mechanical behaviour of natural and compacted soils. For design purposes, important mechanical parameters of the soils such as the pressuremeter modulus (E P ), limit pressure (P l ) and creep pressure (P f ) are usually obtained by pressuremeter in-situ tests. In the present study, pressuremeter tests were conducted in laboratory, using a mini-pressuremeter, to characterise compacted soils. The objective was to examine and quantify the influence of temperature changes on pressuremeter parameters of two different compacted soils: a clay and a loam. These soils were compacted at their optimal water content and 90% of their maximal dry density (standard Proctor) in a thermo-regulated metric scale container of 800 mm in height and 600 mm in diameter. The compacted soils were then subjected to a range of temperatures from 20 to 50 °C. Only six tests were performed in each container to prevent edge effects and interaction between the tests. The thermal cycles were applied to the soil massif as following: A heating-cooling cycle (20-40-20 °C) for the clay; three heating-cooling cycles (20-50 °C) for the loam. Pressuremeter tests were conducted at the end of several temperature steps. The obtained results showed a decrease in P l and P f with increasing temperature for both tested soils, while the variation of E P was less significant. Through the temperature range tested, a quasireversibility of the effect of a heating cycle is obtained.

In the following sections, the material and the developed experimental device are first described. Then, the results are detailed and the impact of thermal variations on the pressuremeter parameters is discussed.

MATERIAL AND METHOD

In this part, first the studied material is presented, then experimental device is described and finally the mini-pressuremeter tests are detailed.

Basic characterization of materials

The tested materials were an illitic soil and a loam taken in the Paris region (Plaisir). The soils were first dried, pulverized and sieved through 2mm sieve before quartering and then used for various experimentations. The illitic material contains 77% illite, 10% kaolinite, 12% calcite and traces of quartz and feldspar. The Plaisir loam contains 81% quartz, 7% dolomite, 5% calcite, 5% clay materials and 3% felspar. The particle-size distribution of the illitic soil was determined using a laser Malvern Mastersizer 2000® device (AFNOR, 2009) whereas the particle size distribution of the Plaisir loam was determined using sieving and sedimentometric method. Almost 85% of the particles of the illitic material are smaller than 0.002 mm (clay particles). The Plaisir loam is coarser as only 20% of the particles are smaller than 0.002 mm. Other basic parameters like liquid limit (LL), plastic limit (PL), plasticity index (PI) (AFNOR, 1993) and specific surfaces (AFNOR, 1999a) were compiled in Table 1. The optimum water contents (w OPN ) and maximum dry densities ( dmax ) were obtained from the standard Proctor curves performed for each material (AFNOR, 1999b) (Figure 2). For the same compaction energy, the maximum dry density reached with the Plaisir loam is significantly higher than with the illitic soil in accordance with its coarser granulometry and its mineralogical content. The illitic soil is classified as A3 in the French standard for soil classification [START_REF] Gtr | Réalisation des remblais et des couches de forme[END_REF] and as a fat clay, MH, according to the Unified Soil Classification System [START_REF] Standard | D2487 Standard practice for classification of soils for engineering purposes[END_REF]. The Plaisir loam is classified as A1 in the French standard for soil classification and as a sandy lean clay, CL, according to the Unified Soil Classification System.

Experimental device and methods

To perform mini-pressuremeter tests at controlled temperatures, an homogeneous massif with a large volume is required as well as an experimental device to impose thermal variations to the massif. In this part, the developed methodology to obtain a meterscale sample with homogeneous properties (temperature, water content and dry densities) is described. First, the required amount of water was added to the material in a large capacity fixed-speed (38 rpm) mixer (MIX120®) to reach the target water content (Table 2). The wet material was stored in 8 plastic drums for a minimum of 5 days to ensure good homogenization.

A pneumatic compactor was used to compact the material in a cylindrical stainless steel container of 800 mm in height and 600 mm in diameter (Figure 3). The compactor applied dynamic forces on a metallic plate of 4 mm in thick and 600 mm in diameter put on the top of the material to facilitate a homogeneous compaction of the soil. To ensure a homogeneous density of the massif, the compaction was performed in eleven 70-mm-thick layers (Figure 4). A stainless steel tube welded to the outside of the container was connected to a Vulcatherm® thermoregulator to facilitate the circulation of an ethylene glycol-water solution at the target temperature (maximum range: -20 to 90 °C) in the tube (Figure 3). Thus, the soil compacted in the container was heated or cooled by the container's outer lateral surface to a maximal range of temperature from 20 to 50°C (Table 2). Insulating sleeves were placed around the tube to reduce the heat exchange with the surrounding atmosphere. The top of the massif was insulated with a plastic film to preserve the initial water content. Finally, the entire device was placed in a box constructed of 40-mm-thick extruded polystyrene plates to reinforce the insulation.

Seven thermal sensors PT100 were positioned within the massif at various depths and various distances from the wall of the container. The sensors were plugged to a data logger to monitor the temperature variation inside the compacted soil.

Mini-pressuremeter tests

The principle of a pressuremeter test is to introduce a cylindrical probe with a flexible cover membrane which can expand radially in a hole [START_REF] Afnor | NF P94-110-1 -Sols : reconnaissance et essais Essai pressiométrique Ménard[END_REF]ASTM, 1999). The pressuremeter curve consists in 3 steps: (i) the probe inflates to obtain the contact with the wall of the hole, (ii) the volume increases linearly with the increasing pressure allowing the calculation of the pressuremeter modulus E P (the soil pseudo-elastic reaction against the probe pressure), and (iii) the large displacements take place and the soil becomes plastic (Figure 5). The creep pressure, P f is the boundary between the second and the third steps of the test. The limit pressure P l is the measured pressure when the injected volume reaches twice the original volume of the cavity. The pressuremeter tests were conducted with an APAGEO® mini-pressuremeter probe of 380 mm in height and 28 mm in diameter. Before each pressuremeter test, a core with a diameter equal to that of the mini-pressuremeter probe was extracted using a core sampler. The 630-mm-length core was divided into small pieces to measure the water content and density of the material as a function of depth. The center of the probe was positioned halfway along the height of the compacted soil (385 mm). The probe was connected to a GDS® pressure-volume controller. The pressure controlled test consisted in applying increasing pressure with equal increments of 25kPa for at least one minute per step. The equilibrium volume was measured for each increment and the volume was plotted as a function of pressure.

The test was stopped when the injected volume (i.e., the volume variation limit of the probe) reached 140,000 mm 3 . Immediately after the pressuremeter test at a given temperature, the borehole was filled with the same material, at the same water content, to avoid influencing the later tests. The effect of temperature variations on the resistance of the minipressuremeter membrane was measured by placing the probe in a climatic chamber at a given temperature during the calibration test of the membrane resistance.

RESULTS

The results of the pressuremeter tests performed on the illitic soil and the Plaisir loam are successively presented in this part.

Illitic soil

For the first test series, starting from 20 °C, the soil was heated up to 40 °C and then cooled down to 20 °C. Figure 6 shows the temperature variation as recorded by a thermal sensor placed at 150 mm from the edge of the massif and at the middle of its height.

The thermal equilibrium was reached after 65h within the limit of ±1 °C. The water content (w) was maintained unchanged all test long. To verify this point, the average water content of the soil close to the test area (from 200 to 580 mm in depth) was measured (Table 3). At 40 °C, (w) was slightly lower than at 20 °C. This might be due to the evaporation of water during the core drilling and the sample weighting as the (w) values measured at the end of the test (I20c,d) were identical to the (w) at the beginning of the test (I20b). The average dry densities were almost identical, confirming the homogeneity of the massif. Two mini-pressuremeter tests were performed at each temperature step: I20a, I20b, then I40a, I40b and finally I20c, I20d. The test I20a failed due to a membrane leakage. The results of the five other tests are presented in Figure 7.

The evaluation of the pressuremeter parameters (E p , P l and P f ), allowed the quantitative comparison of the tests at different temperatures (Table 3, Figure 8). The values of P l and P f decreased significantly when the massif was heated up to 40 °C. Thus, the soil was yielded under lower pressures at 40 °C than at 20 °C. In other words, the results showed a softening of the soil with heating.

The pressuremeter parameters for I20b and I20c-d tests were close that showed a good reversibility of the effect of the temperature variation in the range of 20-40 °C. The softening due to the heating and the reversibility of the observed effects on the creep pressure and the pressure limit were also noticed on the pressuremeter modulus (Ep) to a lower extent.

Loam tests

For the second test series, the test began 6 days after the compaction of the massif to ensure a good homogeneity of the moisture in the entire sample. The initial temperature of the loam compacted soil was 20 °C, the soil was submitted to three heating cycles up to 50 °C and then cooling down to 20 °C. Figure 9 shows the imposed temperature on the outer lateral surface of the container and the temperature variation as recorded by a sensor placed at 150 mm from the edge of the massif and at the middle of its height.

The thermal equilibrium was reached after 30h within the limits of ±1 °C. Each step lasted 30 to 60 h. An electric failure caused a detectable decreasing of the temperature during the first heating step. The I20a-b tests were performed after the achievement of the previously set temperature. The average water content of the soil close to the test area (from 200 to 580 mm in depth) was measured (Table 4). At 50 °C, as presented for the 1 st test series, (w) was slightly lower than at 20 °C. The average dry densities were identical, confirming the homogeneity of this parameter over the massif. Two mini-pressuremeter tests were performed at several temperature steps: L20a and L20b before the first cycle; L50a and L50b at the end of the first heating cycle and finally L20c and L20d at the end of the test after the application of the 3 rd cycle. The results of these six tests are presented in Figure 10. The evaluation of the pressuremeter parameters (E p , P l and P f ), allowed the quantitative comparison of the tests at different temperatures (Table 4, Figure 11). The values of the three pressuremeter parameters decreased significantly when the massif was heated up to 50 °C. As observed for the illitic compacted soil, the results showed a softening of the loam soil with heating. At the end of the cycles, at 20 °C, the pressuremeter parameters returned back to their initial values: results of L20c-d tests were close to those of L20a-b. A reversibility of the effect of the temperature variation in the range of 20-50 °C was noticed for this material as for the illitic soil. (3)

Figure 12 shows the of the normalized parameters at each step of each test series. For the studied materials compacted at 90% of their maximal dry density, the creep pressure and the limit pressure decreased with increasing temperature that evidenced the thermal softening of both materials. These results are consistent with the literature results obtained with oedometric and triaxial tests [START_REF] Marques | Viscous behaviour of St-Roch-de-l'Achigan clay, Quebec[END_REF][START_REF] Cekerevac | Experimental study of thermal effects on the mechanical behaviour of a clay[END_REF][START_REF] Graham | Modified Cam-Clay modelling of temperature effects in clays[END_REF][START_REF] Tanaka | Stress-strain behaviour of reconstituted illitic clay at different temperatures[END_REF][START_REF] Hueckel | Thermoplasticity of saturated clays: experimental constitutive study[END_REF].

In this study, after heating-cooling cycles, for both soils all pressuremeter parameters recovered approximately their initial values (reversibility of thermal softening). Heating the illitic sample from 20 to 40 °C decreased P n.f by 15% and P n.l by 14%, while returning back to 20 °C increased these parameters respectively up to 12% and 17%. As a similar way, heating the compacted loam from 20 to 50 °C decreased P n.f by 22% and P n.l by 20%, while cooling down to 20 °C increased these parameters up to 15%. The level of P n.l and P n.f measured in the compacted loam at the end of the three thermal cycles was a little lower than the initial values. This trend has to be followed in a future study with more cycles to test a possible cumulative effect.

The comparison between these two materials shows that in spite of their difference in mineralogy, initial water content and dry densities, normalized parameters can be considered as linear. A simple linear regression (equations 4 and 5) may be used to determine P f and P l values according to the sample temperature (T): (5)

CONCLUSIONS

The objective of this paper was to quantify the effect of both monotonic and cyclic temperature variations on different soil pressuremeter parameters. A specific device was developed to measure the minipressuremeter parameters in laboratory conditions and at different temperatures. The evolution of the pressuremeter modulus (Ep), the creep pressure (P f ) and the limit pressure (P l ) was measured under imposed thermal conditions on a clayey compacted soil and a compacted loam. The results of the mini-pressuremeter tests on both materials showed a decrease in the creep pressure and the limit pressure with temperature increase (thermal softening), while the variation of the pressuremeter modulus was less significant, this could be due to the well-known higher variability of this parameter. The results tended to show the reversibility of the temperature effects on the measured parameters for one and three heating cycle both for the illitic soil in the tested temperature range (20-40 °C) and the compacted loam in the tested temperature range (20-50 °C).

Figure 1 .

 1 Figure 1. Granulometric curves of the studied materials.

Figure 2 :

 2 Figure 2: Compaction curves of the studied materials.

Figure 3 :

 3 Figure 3: Thermo-regulated metric scale container.

Figure 4 :

 4 Figure 4: Position of the pressuremeter cell.

Figure 5 :

 5 Figure 5 : The 3 steps of a pressuremeter curveOnly six mini-pressuremeter tests were performed in each massif to prevent edge effects and crossinfluence between the tests. The test points were positioned on a concentric circle with a diameter half than that of the container. All the test points were located at the same radial position for a correct comparison of the results. The pressuremeter tests were conducted with an APAGEO® mini-pressuremeter probe of 380 mm in height and 28 mm in diameter. Before each pressuremeter test, a core with a diameter equal to that of the mini-pressuremeter probe was extracted using a

Figure 6 :

 6 Figure 6: Temperature variation in the illitic compacted soil at 150 mm from the edge of the container and the chronology of the tests: heating-cooling cycle (20-40-20 °C), time of the pressuremeter tests (I20a, b, c, d, I40a, b).

Figure 7 :

 7 Figure 7: Pressuremeter curves for the tests on illitic material: I20b, I40a-b and I20c-d.

Figure 8 :

 8 Figure 8 : Impact of temperature variation on pressuremeter test parameters in illitic compacted soil.

Figure 9 :

 9 Figure 9: Temperature variation in the loam at 150 mm from the edge of the container and the chronology of the tests: 3 heating-cooling cycles (20-50-20 °C) and time of the pressuremeter tests.

Figure 10 :

 10 Figure 10: Pressuremeter curves for the tests on loam material: heating-cooling cycle (20-50 °C).

Figure 11

 11 Figure 11 Impact of temperature variation on pressuremeter test parameters in compacted loam.

Figure 12 :

 12 Figure 12: Evolution of the normalized parameters according to the test temperature for two materials.

Table 1 :

 1 Basic characterization of studied materials

	Material	PL LL PI % %	MBV g/100g	w OPN %	 dmax Mg/m 3
	Illitic soil	34	65 31	5.41	31	1.43
	Plaisir loam 21	27	6	1.85	16	1.81

Table 2 :

 2 Characteristics of the prepared samples.

Material w (%) ρ d (Mg/m 3 ) Tmin (°C) Tmax (°C) Nb cycles Illitic soil 30.5±0.5 1.31±0.01 20 40 1 Plaisir loam 16.1±0.2 1.72±0.02 20 50

Table 3 :

 3

	Test T	Cycle w
	(°C)	(%)

Pressuremeter parameters for the test performed on illitic samples: heating cycle (20-40-20 °C).

Table 4

 4

	: Pressuremeter parameters for the test per-
	formed on loam samples: 3 heating cycles (20-50-
	20 °C).
	Test T	Cycle w
	(°C)	(%)

Table 5

 5

	: Normalized pressuremeter parameters for
	both series.
	Material T (°C) Cycle P n.f	P n.l	E n.p
	Illite	20	0	1.00	1.00	1.00
	Illite	40	1	0.85	0.86	1.05
	Illite	20	1	0.97	1.03	1.19
	Loam	20	0	1.00	1.00	1.00
	Loam	50	1	0.78	0.80	0.98
	Loam	20	3	0.93	0.95	1.09

ACKNOWLEDGEMENTS

The authors acknowledge C. Fontaine from IC2MP laboratory (Poitiers, France) for the mineralogical analyses and the Region Lorraine for the financial support.