J P Boufflet

M Dambrine

G Dupire

P Villon

On the necessity of Nitsche term. Part II: an alternative approach

The aim of this article is to explore the possibility of using a family of fixed finite elements shape functions that does not match the domain to solve a boundary value problem with Dirichlet boundary condition. The domain is embedded in a bounding box and the finite element approximation is associated to a regular structured mesh of the box. The shape of the domain is independent of the discretization mesh. In these conditions, a meshing tool is never required. This may be especially useful in the case of evolving domains, for examples shape optimization or moving interfaces. Nitsche method has been intensively applied. However, Nitsche is weighted with the mesh size h and therefore is a purely discrete point of view with no interpretation in terms of a continuous variational approach associated with a boundary value problem. In this manuscript, we introduce an alternative to Nitsche method which is associated with a continuous bilinear form. This extension has strong restrictions: it needs more regularity on the data than the usual method. We prove the well-posedness of our formulation and error estimates. We provide numerical comparisons with Nitsche method.

Introduction

The use of non matching regular grids for solving PDE has been explored since a long time. In the case of variational techniques, the main difficulty is to answer at this fundamental question: how to take account of the essential boundary conditions ? One of the first ideas is to use Lagrange multipliers [START_REF] Babuška | The finite element method with Lagrangian multipliers[END_REF][START_REF] Barbosa | Boundary Lagrange multipliers in finite element methods: error analysis in natural norms[END_REF] or penalty method [START_REF] Barrett | Finite element approximation of the Dirichlet problem using the boundary penalty method[END_REF] . This can be coupled with Nitsche method [START_REF] Nitsche | Über ein Variationsprinzip zur Lösung von Dirichlet-problemen bei Verwendung von Teilräumen, die Keinen Randbedingungen unterworfen sind[END_REF][START_REF] Hansbo | Nitsche's method for interface problems in computational mechanics[END_REF] to obtain error estimators [START_REF] Stenberg | On some techniques for approximating boundary conditions in the finite element method[END_REF]. This has been successfully applied to domain decomposition [START_REF] Stenberg | Mortaring by a method of[END_REF][START_REF] Tallec | Domain decomposition with nonmatching grids: augmented Lagrangian approach[END_REF] and to introduce the fictitious domain method [START_REF] Glowinski | A fictitious domain method for Dirichlet problem and applications[END_REF]. The drawback of this approach is the ill-conditioning of the optimality system. More recently, combining the X-FEM approximation [START_REF] Sukumar | Extended finite element method for threedimensional crack modeling[END_REF] and the level set technique [START_REF] Osher | Level set methods and dynamic implicit surfaces[END_REF][START_REF] Sethian | Level set methods and fast marching methods, volume 3 of Cambridge Monographs on Applied and Computational Mathematics[END_REF] some new approaches have been developed. There are applications in the domain of shape optimization [START_REF] Belytschko | Topology optimization with implicit functions and regularization[END_REF], fluid interfaces [START_REF] Chessa | An enriched finite element method and level sets for axisymmetric two-phase flow with surface tension[END_REF][START_REF] Chessa | An extended finite element method for two-phase fluids[END_REF][START_REF] Chessa | The extended finite element method (XFEM) for solidification problems[END_REF] and stochastic mechanics [START_REF] Nouy | X-sfem, a computational technique based on x-fem to deal with random shapes[END_REF]. What we propose in this article is an alternative way based on a modified bilinear form.

One of the question posed by Nitsche formulation is that it has no sense in a continuous setting: the regularization term depends on the mesh size. In [START_REF] Dumont | Meshfree methods and boundary conditions[END_REF], a meshfree method has been introduced that is roughly speaking a unregularized Nitsche method. Let us precise the model problem we will work on. Let Ω be a domain of R d (d ≥ 1) with a smooth boundary ∂Ω. We consider the following model boundary value problem of Dirichlet type

-∆u = f in Ω, u = g on ∂Ω, (1.1)
where f ∈ L 2 (Ω) and g ∈ H 1/2 (∂Ω). The idea introduced by Dumont and al in [START_REF] Dumont | Meshfree methods and boundary conditions[END_REF] and explored here consists in the introduction of the variational problem:

        
Find v in an appropriate space H such that B(v, ϕ) = L 2 (ϕ) for all ϕ ∈ H where

B(v, ϕ) = Ω ∇u • ∇ϕ - ∂Ω (ϕ∂ n u + u∂ n ϕ) and L 2 (ϕ) = Ω f ϕ - ∂Ω g∂ n ϕ.
(1.2)

Roughly speaking, this formulation can be seen as a continuous version of a Nitsche method without penalization. Since one should give a sense to the normal derivative, the form B is not defined on H 1 (Ω) but only on H 1 ∆ (Ω) = u ∈ H 1 (Ω) /∆u ∈ L 2 (Ω) . Endowed with the norm .

H 1 ∆ (Ω) defined by u 2 H 1 ∆ (Ω) = u 2 L 2 (Ω) + ∇u 2 L 2 (Ω) + ∆u 2 L 2 (Ω)
, it is an Hilbert space.

A notable point about B is that this form is not coercive. Elementary computations show that B(1, 1) = 0 and that for any harmonic function u, B(u, u) ≤ 0. Following Brezzi-Babuska-Ladyzhenskaya's theory ( [START_REF] Babuška | Error-bounds for finite element method[END_REF][START_REF] Babuška | The finite element method with Lagrangian multipliers[END_REF]), an inf sup argument is used in [START_REF] Dumont | Meshfree methods and boundary conditions[END_REF] to prove the well posedness on the continuous level. The former variational problem leads to a discrete method based on a Galerkin approximation built on a grid that does not have to match the domain. A discrete inf sup condition should hold uniformly in order to obtain a robust numerical method, otherwise penalization is mandatory.

In the first part of this work [START_REF] Dupire | On the necessity of nitsche term[END_REF], we have discussed this method: convergence is proved in dimension one, the limitations of the method are explored by numerical experiments. In dimension two, these experiments show that the method is unstable on complex geometries and that regularization is then needed. We considered Nitsche method in [START_REF] Dupire | On the necessity of nitsche term[END_REF]. Let us briefly recall its basics: the interested reader can find more details in [START_REF] Becker | A finite element method for domain decomposition with non-matching grids[END_REF]. On the discrete space V h , consider the bilinear form a h and the linear form l h defined as

a N h (u h , u h ) = Ω ∇u h • ∇v h - ∂Ω (u h ∂ n v h + v h ∂ n u h ) + β h ∂Ω u h v h , (1.3) b N h (v h ) = Ω f v h - ∂Ω g∂ n v h + β h ∂Ω gv h . (1.4)
The number β is a parameter called Nitsche parameter. Nitsche problem can be written as follows:

Find u h in V h such that a N h (u h , v h ) = b N h (v h ) for all v h ∈ V h (1.5)
When β is large enough, the bilinear form a N h is coercive in the weighted norm . h defined as

u h 2 h = ∇u h 2 L 2 (Ω) + 1 h u h 2 L 2 (∂Ω) + h ∇u h 2 L 2 (∂Ω) .
In the context of non matching grid, we introduced in [START_REF] Dupire | On the necessity of nitsche term[END_REF] a second regularization of geometrical type: a compatibility condition between the geometry and the computational mesh has to be satisfied to obtain good approximation properties. This condition expresses that a uniform lower bound on the size of intersection between the geometries and the regular mesh should hold. Not every domain satisfies this condition. This is nevertheless the case after a small modification of the domain.

In this work, we introduce an alternative way to deal with the essential boundary condition. Our point of view is the following: instead of looking to

u = g on ∂Ω, (1.6)
we consider it after two differentiations

∆ τ u = ∆ τ g on ∂Ω, (1.7)
where ∆ τ is the Laplace-Beltrami operator on the manifold ∂Ω . This is only possible if g is regular enough to give a sense to ∆ τ g. We shall make the assumption that g ∈ H 1 (∂Ω) such that ∆ τ g is defined in H -1 (∂Ω).

The manuscript is organized as follows. In Section 2, we first introduce the new formulation based on this idea, then we prove that any solution of this variational problem also solves the original boundary value problem (2.11). In Section 3, we adapt the study of Nitsche method to justify our approach, we obtain existence and uniqueness of the solution to the discretized version of our variational formulation and we obtain error estimates. Finally, in Section 4, we present some numerical experiments to illustrate the proposed method and we provide a numerical comparison of our method with Nitsche method.

2 A new variational formulation.

Definitions and setting of the problem.

Let us fix the notations and the variational space. Let Ω be an open, bounded and smooth subset of R d (d ≥ 1) with a smooth boundary ∂Ω. We first recall some facts of differential calculus on surfaces that will be of great use in the following, the reader can find more details in [START_REF] Henrot | Variation et optimisation de formes[END_REF]. The unit outwards normal to ∂Ω is denoted by n. For u ∈ C ∞ (Ω), the tangential gradient of u on ∂Ω is the function defined as

∇ τ u = ∇u -(∇u • n) n. (2.8)
Then, if u is defined only on ∂Ω, ∇ τ u is is defined as ∇ τ ũ, where ũ is an arbitrary extension of u since this quantity is independent of the choice of the extension. In particular, the tangential gradient ∇ τ u of a function u ∈ H 1 (∂Ω) belongs to L 2 (∂Ω). Then, the tangential divergence of a vector field v defined on ∂Ω is the L 2 (∂Ω)-adjoint operator of the tangential gradient. Therefore, it is defined by the relation: for all u ∈ C 1 (∂Ω),

∂Ω ∇ τ u • v = - ∂Ω u div τ v (2.9)
Finally, the Laplace-Beltrami operator ∆ τ is defined by ∆ τ u = div τ (∇ τ u) so that we have the following integration by parts formula

∂Ω ∆ τ u v = - ∂Ω ∇ τ u • ∇ τ v (2.10)
for a smooth function.Then, this formula extended by density, gives a sense to ∆ τ u as a element of H -1 (∂Ω) for all u ∈ H 1 (∂Ω).

A fundamental remark is that no boundary term appears in the former formula (2.10) as long as ∂Ω is a manifold with no boundary. Formula (2.10) is the keystone of our method. Therefore, the approach we will develop in the following is restricted to the case when the Dirichlet boundary condition is asked on a manifold Γ D with no border. This assumption excludes the case of mixed boundary conditions except when the Neumann and Dirichlet conditions are applied on parts Γ N and Γ D with disjoint closures Γ D ∩ Γ N = ∅ .

We consider the following model boundary value problem of Dirichlet type

-∆u = f in Ω, u = g on ∂Ω, (2.11)
where f ∈ L 2 (Ω) and g ∈ H 1 (∂Ω). We consider the space H(Ω) defined by

H(Ω) = {u ∈ H 1 (Ω) such that ∆u ∈ L 2 (Ω) and u |∂Ω ∈ H 1 (∂Ω)}, (2.12)
where u |∂Ω denotes the usual trace in H 1/2 (∂Ω) of u. Endowed with the norm

u 2 H(Ω) = u 2 L 2 (Ω) + ∇u 2 L 2 (Ω) + ∆u 2 L 2 (Ω) + u 2 L 2 (∂Ω) + ∇ τ u 2 L 2 (∂Ω) , (2.13)
the space H(Ω) is hilbertian with scalar product

u, v H = Ω (u v + ∇u • ∇v + ∆u ∆v) + ∂Ω (u v + ∇ τ u • ∇ τ v) .

Derivation of the variational formulation

Assume that u is the solution of (2.11). Then, multiplying it by a function ϕ ∈ H(Ω) and integrating by parts leads to

Ω ∇u • ∇ϕ - ∂Ω ∂ n u ϕ = Ω f ϕ. (2.14)
Now, we use the boundary condition u = g on ∂Ω to notice that

∂Ω u ϕ = ∂Ω g ϕ. (2.15)
Then, we use again the boundary condition u = g on ∂Ω under the form (1.7) to see that

∆ τ u, ϕ H -1 (∂Ω)×H 1 (∂Ω) = ∆ τ g, ϕ H -1 (∂Ω)×H 1 (∂Ω) .
Applying integration by parts formula (2.10), we get

∂Ω ∇ τ u • ∇ τ ϕ = ∂Ω ∇ τ g • ∇ τ ϕ. ( 2
               Find u in H(Ω) such thatA γ (u, ϕ) = L γ (ϕ) for all ϕ ∈ H(Ω)
where

A γ (u, ϕ) = Ω ∇u • ∇ϕ - ∂Ω (ϕ ∂ n u + u ∂ n ϕ) + γ ∂Ω ∇ τ u • ∇ τ ϕ ; L γ (ϕ) = Ω f ϕ - ∂Ω g ∂ n ϕ + γ ∂Ω ∇ τ g • ∇ τ ϕ.
(2.17)

Here, γ > 0 is a parameter to be chosen later.

Remark 2.1 The quadratic form associated to A γ is not coercive: one checks A γ (1, 1) = 0 for any value of γ. Parameter γ plays the same role than Nitsche parameter β in Nitsche formulation (1.5). Like the weighted tangential mass term

1 h ∂Ω u h v h
in Nitsche method, the tangential rigidity term

∂Ω ∇ τ u • ∇ τ ϕ
is added to compensate, after discretization, the fact that B defined in (1.2) is not coercive in H(Ω). Like Nitsche parameter β, the parameter γ is to be chosen large enough to assure some positivity of the bilinear form A γ but only once discretized. Like for Nitsche method, the key of the proof that validates this method will be an inverse inequality that provides an upper bound for ∂ n u h L 2 (∂Ω) in terms of ∇u h L 2 (Ω) . We will state it in Lemma 3.2.

Equivalence with the boundary value problem.

Now, we assume that u ∈ H(Ω) solves variational Problem (2.17). We want to characterize u as the solution of a boundary value problem.

Theorem 2.1 If γ > 0, there is equivalence between u solves the variational problem (2.17) and u solves the boundary value problem (2.11).

Proof:

By construction of the variational problem, any solution of boundary value problem (2.11) also solves (2.17). Therefore, it remains to prove that any solution of variational Problem (2.17) also solves boundary value problem (2.11).

We first take any ϕ ∈ D(Ω) as test function in (2.17). Then, by cancellation of ϕ and its derivatives in the neighborhood of ∂Ω, we have

Ω ∇u.∇ϕ = Ω f ϕ.
After an integration by parts on Ω, we check that -∆u = f in Ω. It remains to check the boundary condition. A reverse integration by parts gives that for any ϕ ∈ H(Ω)

Ω ∇u • ∇ϕ - ∂Ω ∂ n u ϕ = Ω f ϕ.
(2.18)

Now, for any ψ ∈ C ∞ (∂Ω), take an extension Ψ ∈ H(Ω) such that ∂ n Ψ = 0 on ∂Ω. Such an extension can by defined by extending ψ as a constant along the orbits of the gradient of the signed distance function to ∂Ω then multiply it by a cutoff function with respect to the distance to ∂Ω. Then,

A γ (u, Ψ) = Ω ∇u • ∇Ψ - ∂Ω ψ ∂ n u + γ ∂Ω ∇ τ u • ∇ τ ψ and L γ (Ψ) = Ω f Ψ + γ ∂Ω ∇ τ g.∇ τ ψ.
Therefore, since γ = 0, we have for any

ψ ∈ C ∞ (∂Ω) ∂Ω ∇ τ u • ∇ τ ψ = ∂Ω ∇ τ g • ∇ τ ψ. (2.19)
By integration by parts on ∂Ω, we get by density -∆ τ u = -∆ τ g in H -1 (∂Ω) and that (2.19) also holds for ψ ∈ H 1 (∂Ω). Then, ug is constant on any connected component of ∂Ω with H d-1 measure defined as

H d-1 (Γ) = Γ 1.
3 The associated discrete method.

The computational discretized domain Ω h is obtained by considering a regular grid (with a square of size h as elementary cell) overlapping Ω (see Figure 1). Then, ∂Ω h is a polygonal line connecting vertices defined as the intersection points of the boundary ∂Ω with the grid (see Figure 2). The regularity properties assumed on ∂Ω provide upper bounds on the curvature on ∂Ω (except on possible vertices of ∂Ω) so that there exists a constant C > 0 such that

|∂Ω ∩ C| ≤ Ch,
for each cell C of size h. In the simulations we will present in the sequel, the domain, Ω is represented as a level line of a function that is discretized on the grid. The vertices of ∂Ω h are then computed by linear interpolation. As a consequence, Ω h ∩ C is a polygon as shown on Figure 2. We choose

V k h = u ∈ C 0 (Ω)/u |K∩Ω ∈ P k (K ∩ Ω), for all K ∈ T h , (3.20)
where T h denotes the set of the squares of the regular grid and P k (K) denotes the polynomial's space with degrees less than k. To simplify the notations, we will omit the dependency with respect to the degree of approximation k and simply write V h instead of V k h . Let us emphasize that V h is not a subset of H(Ω): this comes from the assumption made on the laplacian of function of H(Ω). As usually in Nitsche's method, we consider the weighted norm . h defined on V h as

u h 2 h = ∇u h 2 L 2 (Ω) + 1 h u h 2 L 2 (∂Ω) + h ∇u h 2 L 2 (∂Ω) . (3.21)
As in [START_REF] Dupire | On the necessity of nitsche term[END_REF], we make an additional geometrical assumption in the spirit of the one made for fictitious domain methods or of assumption 2.2 in [START_REF] Becker | A finite element method for domain decomposition with non-matching grids[END_REF]. In these methods, the size of the surface element should be comparable with the size of the volume element in order to prove convergence. In this work, we make the following geometrical assumption. We consider a cartesian grid of mesh size h.

Definition 3.1 The domain Ω and the grid satisfy a uniform roundness condition if there exists a constant γ ∈ (0, 1) such that, for each cell grid K intersected by the domain Ω,

ρ(K ∩ Ω) ≥ γh, (3.22)
where ρ(E) denotes the diameter of the largest disk included in E.

As explained in [START_REF] Dupire | On the necessity of nitsche term[END_REF], adding this condition provides two keys ingredient: the following interpolation result Proposition 3.1 and the trace lemma 3.1 below quoted from [START_REF] Dupire | On the necessity of nitsche term[END_REF].

Proposition 3.1 Let Ω and a grid satisfying the roundness condition in the sense of Definition 3.1. Let (K, P, Σ) be a finite element and let k, m be two integers such that m < k. Assume that Π ∈ L(H k+1 (K), P ) and P ⊂ H m (K ∩ Ω). Then, there is a constant C such that

|v -Πv| m,K∩Ω ≤ Ch k+1-m |v| k+1,Ω∩K . (3.23)
The proof of Proposition 3.1 is a straightforward adaptation of the usual interpolation result in finite elements theory.

Let us recall the usual Nitsche's formulation: on the discrete space V h , consider the bilinear form a h and the linear form b h defined as

a N h (u h , u h ) = Ω ∇u h • ∇v h - ∂Ω (u h ∂ n v h + v h ∂ n u h ) + β h ∂Ω u h v h , (3.24)
b N h (v h ) = Ω f v h - ∂Ω g∂ n v h + β h ∂Ω gv h . (3.25)
The number β is a parameter called Nitsche parameter. In the approach we follow here, the added surface mass term is changed in a surface rigidity term. On the discrete space V h , we consider the bilinear form a h and the linear form b h defined as

a h (u h , v h ) = Ω ∇u h • ∇v h - ∂Ω (u h ∂ n v h + v h ∂ n u h ) + γ h ∂Ω ∇ τ u h • ∇ τ v h , (3.26) b h (v h ) = Ω f v h - ∂Ω g ∂ n v h + γ h ∂Ω ∇ τ g • ∇ τ v h . (3.27)
For reasons that will be explained with Lemma 3.3, we shall work on V h,0 , the subspace of V h of the functions with a null mean on the boundary defined as

V h,0 = u ∈ C 0 (Ω)/u |K∩Ω ∈ P k (K ∩ Ω), for all K ∈ T h such that ∂Ω u h = 0 . (3.28)
Therefore, we restrict ourselves to right hand side g of zero mean value. Let us emphasize that this is not a restriction thanks to the linearity of the Laplace operator. Let g be any function of H 1 (∂Ω), define its mean value The discrete variational problem on V h,0 can be written as follows :

[g] = 1 |∂Ω| ∂Ω g,
(P h,0 ) Find u h in V h,0 such that a h (u h , v h ) = b h (v h ) for all v h ∈ V h,0 (3.30)
To study this problem, we follow the usual steps in the proof of convergence for Nitsche method: we first prove trace and inverse inequalities, then we show that a h is coercive in the norm . h and finally we conclude by the error estimate.

Lemma 3.1 There is a non negative constant C 1 such that, for any function

u h in V h , u h 2 L 2 (∂Ω) ≤ C 1 h u h 2 L 2 (Ω) , (3.31)
We refer to [START_REF] Dupire | On the necessity of nitsche term[END_REF] for the proof: the geometrical condition is needed to justify that C 1 is not domain and mesh size dependent. Since ∂ n u h = ∇u h •n, Pythagore's equality provides the following inverse inequality result as a direct consequence of (3.31) applied to ∇u.

Lemma 3.2 There is a non negative constant C such that, for any function

u h ∈ V h , ∂ n u h 2 L 2 (∂Ω) ≤ C h ∇u h 2 L 2 (Ω) and ∇ τ u h 2 L 2 (∂Ω) ≤ C h ∇u h 2 L 2 (Ω) . (3.32)
Finally, to deal with the last term of a h , we need a control of the L 2 (∂Ω) norm of u by the L 2 (∂Ω) norm of its tangential gradient ∇ τ u. Obviously, this is not possible for every function as shown by the example of the constant function. This is why we imposed in this first step the restriction

∂Ω u h = 0,
and work only on V h,0 . The control of u h 2 L 2 (∂Ω) is classically obtained by Wirtinger inequality (in dimension two, the boundary is a curve). Lemma 3.3 There is a non negative constant C W that depends on Ω such that, for any function

u h ∈ V h,0 , u h L 2 (∂Ω) ≤ C W ∇ τ u h L 2 (∂Ω) . (3.33)
Let us sketch the proof of this lemma. In dimension two, the boundary is a curve, u h is a periodic function and the proof is a direct consequence of Parseval formula for Fourier series. In higher dimension, things are not elementary. Following [START_REF] Taylor | Pseudodifferential operators[END_REF], the Laplace-Beltrami operator -∆ τ on ∂Ω is a positive, formally self-adjoint, elliptic pseudo-differential operator of real symbol of order two for which the spectrum (as an unbounded operator on H s (∂Ω)) is independent of s and is made of a series of real eigenvalues growing to infinity. Its kernel is the space of constant functions to which V h,0 is L 2 (∂Ω) orthogonal, hence the Courant Fischer formulae for the Rayleigh quotient of the Laplace Beltrami operator provides the result.

Let emphasize that this control is not an inverse inequality: it is based on global arguments. One could ask if a better estimate with a gain in h holds.

It is an open question: it is true on each element but the global property of null mean value plays a crucial role and cannot be handled easily at the element level. This fact is the main obstacle to a unweighted discretization of the variational formulation studied in Section 2.

We now prove that the quadratic form a h defined in (3.26) is coercive in the weighted norm . h for γ large enough. Set γ 0 = CC W where C is the constant given by the inverse inequality, Lemma 3.2 and C W the constant in Wirtinger's lemma.

Proposition 3.2 If γ > γ 0 , there is a non negative constant α independent of h but not of γ such that for all

u h ∈ V h,0 a h (u h , u h ) ≥ α u h 2 h . (3.34)
Then, the discrete problem (P h,0 ) defined in (3.30) has a unique solution.

Proof:

By Cauchy-Schwarz inequality, one has

a h (u h , u h ) = Ω |∇u h | 2 -2 ∂Ω u h ∂ n u h + γ h ∂Ω |∇ τ u h | 2 , ≥ ∇u h 2 L 2 (Ω) -2 h -1/2 u h L 2 (∂Ω) h 1/2 ∂ n u h L 2 (∂Ω) + γ h ∇ τ u h 2 L 2 (∂Ω) .
We introduce a parameter ε to be fixed later. By Cauchy inequality, we have

a h (u h , u h ) ≥ ∇u h 2 L 2 (Ω) -ε h -1/2 u h 2 L 2 (∂Ω) - 1 ε h 1/2 ∂ n u h 2 L 2 (∂Ω) + γ h ∇ τ u h 2 L 2 (∂Ω) .
Then, by the inverse inequality (3.32) and Wirtinger inequality (3.33), we get

a h (u h , u h ) ≥ ε -C ε ∇u h 2 L 2 (Ω) + γ h ∇ τ u h 2 L 2 (∂Ω) - ε h u h 2 L 2 (∂Ω) , ≥ ε -C ε ∇u h 2 L 2 (Ω) + γ -C W ε h ∇ τ u h 2 L 2 (∂Ω) , ≥ ε -C ε ∇u h 2 L 2 (Ω) + γ -C W ε C W h u h 2 L 2 (∂Ω) .
Now, we first take ε > C and then γ > C W ε, hence γ 0 can be chosen as CC W . We have proven the coercivity result (3.34) for α = min((ε -C)/ε, (γ

-C W ε)/C W )) > 0.
The continuity of the forms is automatic in the finite dimensional context: nevertheless, the constants of continuity depend on h and of γ. Here, using Lemma 3.2 to dominate the surface rigidity term or directly the definition of Nitsche's norm (3.21), we have only the poor

|a h (u h , v h )| ≤ C 1 h -2 u h h v h h . (3.35)
Next lemma, stating Galerkin orthogonality, will be used to prove the convergence.

Lemma 3.4 Consider (2.11) with the condition [g] = 0. Let u be its solution and let u h be the solution of the discrete problem (3.30) then a h ((u hu), ϕ h ) = 0 for all ϕ h ∈ V h .

Proof:

Fix ϕ h ∈ V h,0 and compute:

a h (u, ϕ h ) = Ω ∇u • ∇ϕ h - ∂Ω (ϕ h ∂ n u + u ∂ n ϕ h ) + γ h ∂Ω ∇ τ u • ∇ τ ϕ h = Ω ∇u • ∇ϕ h - ∂Ω ϕ h ∂ n u - ∂Ω g ∂ n ϕ h + γ h ∂Ω ∇ τ g • ∇ τ ϕ h .
Since V h ⊂ H 1 (Ω), we can integrate by parts and we find:

a h (u, ϕ h ) = - Ω ϕ h ∆u - ∂Ω g ∂ n ϕ h + γ h ∂Ω ∇ τ g • ∇ τ ϕ h = Ω f ϕ h - ∂Ω g ∂ n ϕ h + γ h ∂Ω ∇ τ g • ∇ τ ϕ h .
The conclusion follows by considering u h ∈ V h , the solution of the discrete problem (3.30) which satisfies:

a h (u h , ϕ h ) = Ω f ϕ h - ∂Ω g ∂ n ϕ h + γ h ∂Ω ∇ τ g • ∇ τ ϕ h .
We now assume that the uniform roundness compatibility condition is satisfied in order to have Proposition 3.1 at our disposal. We can state the error estimate.

u -u h h ≤ Ch s-3 u H s (Ω) . ( 3

.36)

Proof:

We proceed in the usual way. For any function ϕ h ∈ V h , we get by triangular inequality

u -u h h ≤ u -ϕ h h + u h -ϕ h h .
Using the continuity and coercivity of a h (3.34) and (3.35), and also Lemma 3.4, there is C 1 > 0 such that:

u h -ϕ h 2 h ≤ 1 α a h (u h -ϕ h , u h -ϕ h ) ≤ 1 α a h (u h -ϕ h , u -ϕ h ) ≤ C 1 α h -2 u h -ϕ h h u -ϕ h h .
Gathering the two inequalities, we get

u h -ϕ h h ≤ (1 + C 1 α h -2 ) inf ϕ h ∈V h u -ϕ h h . (3.37)
We introduce the trianglewise defined norm :

ϕ 2 h,H 2 (Ω) = K∈T h ϕ 2 H 2 (Ω∩K) .
to recall the following upper bound of the weighted norm proven in [START_REF] Thomée | Galerkin finite element methods for parabolic problems[END_REF] for example: there is C 2 > 0 such that, for any function ϕ ∈ H 2 (Ω):

ϕ h ≤ C 2 h -1 ϕ L 2 (Ω) + h ϕ H 1 (Ω) + h 2 ϕ h,H 2 (Ω) . (3.38)
Thanks to Proposition 3.1 on the local approximation, there is a constant C 3 > 0 such that :

u -Π(u) L 2 (Ω) + h u -Π(u) H 1 (Ω) + h 2 u -Π(u) h,H 2 (Ω) ≤ C 3 h s u H s (Ω) . (3.39)
Combining (3.38) with (3.39), we obtain inf

ϕ h ∈V h u -ϕ h h ≤ C 2 C 3 h s-1 u H s (Ω) . (3.40)
We deduce the convergence estimate (3.36) from the upper bounds (3.37) and (3.40).

The poor scaling of the surface diffusion and the choice of keeping the weighted norm of Nitsche's method explain why we have the poor convergence result: convergence is proved only if s > 3 with also k ≥ 4 at least. However, this unsatisfactory theoretical result is not observed in numerical tests as presented in next section.

4 Numerical validation and comparison with Nitsche method.

Numerical implementation.

Not every domain satisfies the compatibility condition of Definition (3.1), for example this is the case of a starfish like domain as illustrated in Figure 6. In order to force the compatibility condition, we have to modify the domain Ω by moving its boundary points that are too close from grid vertices.

Since the roundness condition (3.1) is not easily checked, we have introduced in [START_REF] Dupire | On the necessity of nitsche term[END_REF] a stronger compatibility condition that would be easy to check in implementation. We consider a regular grid G h of size h covering the domain Ω. Each cell K of the grid is then a cube of side h, its border ∂K has vertices S that are the nodes of the grid. We set ∂K * the border ∂K from which we removed the vertices. The union of the ∂K * is denoted by G * .

Definition 4.1 The domain Ω and the grid are said to be compatible if there exists a constant P c ∈ (0, 1/2) such that if S / ∈ ∂Ω is a node of the grid, then

S -Y ≥ P c h for all Y ∈ ∂Ω ∩ G * . (4.41)
The key point is (4.42) that plays the same role than the roundness of the element in the usual result. The choice of the stronger condition (4.41) in Definition 4.1 is motivated by practical consideration: it provides an easily implemented way to modify the domain in order to check the needed (4.42). Let us explain how to perform that modification.

Figure 3 illustrates the forbidden places for boundary points of ∂Ω. Once this property is satisfied, then the diameter ρ(K ∩ Ω) of the largest disk included in Ω ∩ K is uniformly in h controlled from below. In dimension two, the set K ∩ Ω contains a least an isosceles triangle of size length P c h, P c h and √ 2P c h and therefore: This is performed as illustrated in Figures 4 and5. The domain Ω is hatched. For computations, it is modified: the black area is either removed from Ω, or added to Ω and a computational domain Ωh is obtained. By an argument of continuity and derivability with respect to the domain of the solution in H 1 of (2.11), the error on the solution ũh committed by this geometrical modification is small (of order P c h 2 ). We refer to the monographs [START_REF] Allaire | Conception optimale de structures[END_REF][START_REF] Henrot | Variation et optimisation de formes[END_REF] on the subject.

ρ(K ∩ Ω) ≥ 1 2 P c h. (4.42) P c h P c h P c h P c h P c h P c h P c h P c h
According to the numerical study for the choice of this parameter performed in [START_REF] Dupire | On the necessity of nitsche term[END_REF], we have fixed the value of parameter P c = 1/10. Finally, the tangential gradients are computed thanks to (2.8): on each cell K, the boundary ∂Ω∩K is a straight line and the tangential gradient is nothing but the component of the gradient in the direction of this line. One has to be careful with the orientation. The surface rigidity term is then assembled in a loop over all cells intersected by ∂Ω.

Numerical results and comparison with Nitsche method

Our theoretical results are valid only for boundary values with zero mean value, so we solve (2.11). The presented computations have been performed on two domains: a matching square [-0.4, 0.4] × [-0.4, 0.4], and on a more complex domain: the interior of the curve given in polar coordinates by ρ(θ) = 0.385 + 0.09 • cos(7 • θ + (7/π)). This domain looks like a starfish as it is shown in Figure 6. Software was developed using MATLAB (The MathWorks). Let us emphasize that the square, with just a Lipschitz boundary, does not fit the assumptions of this work. The sample steps h = 1/400, 1/220, 1/120, 1/70, 1/40, 1/20, 1/10 have been chosen so that the square exactly matches the regular grid composed of Q8-finite elements. The order two is not justified by Theorem 3.1 that suggests to use higher order elements. Nevertheless, we will see that convergence will be observed with an order much better than expected. This suggests that our theoretical analysis can be improved.

The exact solution we seek to recover is u(x, y) = e y sin x. Intensive tests have been performed ranging γ from small values (i.e. γ ∈ {1, 2, 3, 4, 5}) to very large value (up to 200). Experiments for γ = 5 to 200 have been conduced using a step of 5. The geometrical parameter P c has been set to 10% as it seems to be a good choice referring to the previous experiments presented in [START_REF] Dupire | On the necessity of nitsche term[END_REF].

From this field of investigation, we comment on the observed results. It can be observed that, for the same value of γ, convergence graphs for L 2 errors have the same behavior as the convergence graphs for H 1 errors. This has been noted on all the experiments. Figures 15 to 16 depict convergence graphs for H 1 error only using γ ∈ [30, 50] stepped by 5 for the matching square and for the starfish. As it can be observed, the curves are more smoothed as larger values for γ are used.

As it can be observed in Figures 7-10 and 15 a quasi straight line for convergence graphs is obtained when γ exceeds 2 for the matching square. If now consider Figures 11-14 and 16, we observe that a good behavior is obtained when γ exceeds 5 for the starfish.

As it can be shown on all Figures, small values of γ suffice to ensure a good enough convergence. So, we do not present the results obtained for large values of γ even though experiments have been conduced up to 200. At this stage, we are not able to provide a safe recipe for choosing γ since only two domains have been used for our experiments.

From the graphs presented in Figures 78910111213141516, we see that the value of parameter γ has to be chosen relatively small. This is remarkable in comparison with the usual Nitsche method we used in [START_REF] Dupire | On the necessity of nitsche term[END_REF] where the value of the parameter β is usually larger. Let us exhibits this observation on the starfish domain. Figure 17 shows H 1 error for the same value of the parameter γ = β = 10 while Figure 18 depicts results for γ = 10 and β = 100. At this stage, we have not an explanation about the fact that γ can be affected a smaller value than the Nitsche's parameter β. We have just observed this fact on the performed experiments.

We are now giving some details about the practical implementation. We use MATLAB and standard double-precision having a floating-point relative accuracy of eps = 2.2204e -16 around 1.0. We use the UMFPACK direct method [START_REF] Davis | Algorithm 832: Umfpack v4.3-an unsymmetric-pattern multifrontal method[END_REF] of MATLAB and the embedded default ordering.

Tables 1 and2 show the characteristics of the matrices associated with each problem to be solved. As expected, these matrices become as large as h decreases. One can also observe on Figure 15 that curves diverge when h is below 0.01. This phenomenon has been seen over the other experiments. At this stage, we have not a fully satisfactory explanation. The MATLAB relative floating-point accuracy is eps = 2.2204e -16, we probably have cumulative round-off errors since interface terms are much smaller than internal ones. The direct solver has a pivoting strategy that permits to obtain good quality solutions even on matrices having a poor condition number. But we are probably near the limits when using such a resolution strategy on large matrices issued from the proposed formulation.

When conducing the numerical tests our main aim was to validate the proposed alternative approach on few enough significative tests. On the scope of practical use, it is obvious that more intensive experiments should be necessary in order to answer on questions relative to the choice of the parameter γ and on the adequation between the intrinsic method and numerical solvers.

h

Conclusion

We have presented a new method to solve a model boundary value problem with a structured mesh on a bounding box. This method is a variant of the usual Nitsche method with an interpretation in the continuous level, it is compatible with a level-set representation of the boundary but imposes restrictive conditions. The empirical results are by far better than the theoretical one. To obtain robustness and convergence, the procedure requires to adjust two parameters: a geometrical parameter P c and a Nitsche like coefficient γ. It seems that this coefficient can be chosen significantly smaller than Nitsche parameter.

Figure 1 :

 1 Figure 1: The domain Ω and the grid.Figure 2: Computational domain Ω h and original domain Ω.

Figure 2 :

 2 Figure 1: The domain Ω and the grid.Figure 2: Computational domain Ω h and original domain Ω.

  and make the change of unknown w = u -[g]. Then, w is the unique solution of the boundary value problem -∆w = f in Ω, w = g -[g] on ∂Ω, (3.29) with a right hand side g -[g] of zero mean value. Hence, to solve the original boundary value problem (1.1), it suffices to compute the mean value [g] of g then to solve the modified boundary value problem (3.29).

Theorem 3 . 1

 31 Consider (2.11) with the condition [g] = 0. Assume γ > γ 0 . Let u be its solution and let u h be the solution of problem (3.30). If u ∈ H s (Ω) with 2 ≤ s ≤ k, and assume that the uniform roundness condition (3.22) is satisfied. Then, there is C > 0 such that:

Figure 3 :

 3 Figure 3: illustration of Definition 4.1 on a cell K of size h of the grid.

Figure 4 :Figure 5 :

 45 Figure 4: case where S -Y < P c h/2. The dark area is removed.

Figure 6 :

 6 Figure 6: A domain as a starfish plotted on a grid (h = 1/20).

Figure 7 :

 7 Figure 7: H 1 error for the matching square: γ ∈ {1, 2, 3, 4, 5}

Figure 8 :

 8 Figure 8: L 2 error for the matching square: γ ∈ {1, 2, 3, 4, 5}

Figure 9 :Figure 10 :Figure 11 :Figure 12 :

 9101112 Figure 9: H 1 error for the matching square: γ ∈ [5, 25]

Figure 13 :Figure 14 :Figure 15 :Figure 16 :

 13141516 Figure13: H 1 error for the starfish: γ ∈[START_REF] Barrett | Finite element approximation of the Dirichlet problem using the boundary penalty method[END_REF][START_REF] Taylor | Pseudodifferential operators[END_REF]

Figures 19 and 20 Figure 17 :Figure 18 : 1

 2017181 Figures 19 and 20 depict the condition number estimate for matrices for the matching square while Figures 21 and 22 show the condition number estimate for the starfish. For Figures 19 and 21 we have γ ∈ {1, 2, 3, 4, 5}. The two other Figures are plotted using γ ∈ [5, 25] stepped by 5. The condest function of MATLAB has been used to obtain the values. These Figures clearly show that matrices are as much as ill-conditioned as h decreases since the size of the linear system becomes

Figure 19 :

 19 Figure 19: Condition number estimate for the matching square: γ ∈ {1, 2, 3, 4, 5}

Figure 20 :Figure 21 :Figure 22 :

 202122 Figure 20: Condition number estimate for the matching square: γ ∈ [5, 25]

Table 1 :

 1 Size, number of non zero entries, and sparsity rate as h for the matching square domain larger and larger. Moreover, one can observe that the condition number increases as the parameter γ increases. So, pro and cons have to be weighted, oscillations of the solutions are smoothed adding the stabilization term, but as a counterpart the condition number increases.

				1/40	1/70	1/120	1/220	1/400
	size	225	833	3201	9633	28033	93633	308481
	nnz	3133 12285 48633 148289 434689 1458689 4817919
	sparsity 6.1% 1.7% 0.47% 0.15% 0.055%	0.016% 0.0050%

Table 2 :

 2 Size, number of non zero entries, and sparsity rate as h for the starfish domain

		1/10	1/20	1/40	1/70	1/120	1/220	1/400
	size	253	784	2732	7791	21988	71961	234202
	nnz	3459 11218 40640 118241 338024 1115451 3647590
	sparsity 5.4% 1.8% 0.54% 0.19% 0.069% 0.021 % 0.0066%

Fix Γ a connected component of ∂Ω. To determine the constant C Γ such that ug = C Γ on Γ, we localize around Γ: set v Γ = d Γ χ with d Γ the signed distance to Γ (seen from Ω) and χ a cutoff function with χ(x) = 1 in the neighborhood of Γ and χ(x) = 0 in the neighborhood of the complement of Γ in ∂Ω. The function v Γ lays in H(Ω) and satisfiesv Γ = ∂ n v Γ = 0 on ∂Ω \ Γ. Then, A γ (u, v Γ ) = L γ (v Γ ) reduces by cancellation on the other components of ∂Ω to Ω ∇u • ∇v Γ -Γ (v Γ ∂ n u + u ∂ n v Γ ) + γ Γ ∇ τ u • ∇ τ v Γ = Ω f v Γ -Γ g ∂ n v Γ + γ Γ ∇ τ g • ∇ τ v Γ .Taking into account (2.18) and(2.19), this leads to

Since H d-1 (Γ) = 0, C Γ = 0 and u = g on Γ. This can be done connected component by connected component to check u = g on ∂Ω.


Dr Nathalie Lewi-Dumont
email: nathalielewi@gmail.com

Blind teen-age students' expectations Focus: school-age Topic: What are the changing demands for the person with visual impairment?

 stressed that the topic of inclusion at the secondary level should be one of the main concern in several countries. In France, since the promulgation of a new law (2005) and a new organisation of mainstreaming disabled pupils, inclusion increased significantly, particularly in high schools. Blind learners, who used to be mainstreamed after mastering techniques, do so much earlier. However, one can assume the hypothesis that a successful inclusion depends on a lot of persons and facts. Being involved in a new OECD (Organization for Economic Co-operation and Development) international research on factors determining the success and failure of high school and university handicapped students in a statistical and objective way, I shall focus on the conference topic and deal with visual impairment only, presenting the feelings and expectations of students about their experience in high schools. After presenting the backgrounds of the study, I will outline the major topics students introduced.

1. Backgrounds of the study 1.1. The OECD research OECD conducts a longitudinal research in 10 countries, including France, about handicapped students in general. It aims to assess the impact of material and human helps to high school and university students: the first part is to reach as many young people as possible who were, in 2007, in last year of high school and others who were in their third year of university so they fill out questionnaires. After analyse, the case studies, for which I volunteered for visually impaired, will be performed. For this lecture, I will present a few blind students' opinions, which will help building the items for the case studies.

1.2. 12 interviews in 2009 I started a modest qualitative study. Through Internet, I reached and recorded blind or partially sighted students (age 13 to 24): since too few used print, considering needs of partially sighted learners are different from blind people's, I will use the interviews of 12 blind young people living in various regions of France who experienced different ways of schooling. Some are still in high school, others are in university or work. They were happy to be interviewed, were friendly, spontaneous and confident. The differences in ages and experiences, although not chosen, will allow perceiving an evolution. I chose to let them express themselves but asked some complementary questions. I transcribed the interviews and will lean on their own words to tackle the main points: although each person is different and has different needs, studying the way these students express their needs, the strengths and the lacks they felt in their curriculum seems to be rich for professionals and for teachers' trainers. I will concentrate here on secondary school, what they told about higher education will be used elsewhere.

Schooling in France

Although with the 2005 law, mainstreaming is the normal way of schooling handicapped children, 3 ways remain possible: individual mainstreaming in the reference school, mainstreaming in a school with a resource room or class 1 , and special school. There are also 3 main steps in schooling: elementary (1 st to 5 th grade), secondary: junior secondary ("collège" 6 th to 9 th ) and high school ("lycée", 10 th t to 12 th ). The academic demands are more important at each level.

Each person told precisely what type of school she attended (see table below). Two excellent university students experienced only one way of schooling, Pamela and Jonas. The others experienced different ways of schooling. The youngest tend to have had more mainstreaming, except Max, who, coming from a foreign country, chose it.

General feelings and memories about high school time

All the young people appreciated, in special and resource schools, what they call "comfort" (Braille transcriptions) and the expertise of support services, particularly of special teachers. They also appreciated (with unpleasant exceptions) the commitment of teachers used to have visually impaired in their classes. However, feelings depend greatly upon their previous experiences: when used to be with sighted pairs, they generally have trouble to adapt to a more "closed" environment, in spite of "comfort". Jane and Patricia missed some freedom in their special school.

-"As far as I am concerned, special school has been more negative than positive" (Pat). Simon was in a resource high school, his family moved, he arrived in another school: -"I thought I would have more freedom, I would be less in a cocoon, and it was worse!" Chris, who had to go in a resource school after being mainstreamed in her town, never got used to it and "did not find it easier": Probably now, mainstreaming and special support would be better prepared and this "regression" would not happen. Except for Patricia, junior secondary school in a resource school has not been appreciated for different reasons, essentially human 2 . The youngest seem to feel comfortable alone in their schools. Some of the interviewed mentioned also the high school years as "tiring", when they got a lot of pedagogical and technical support and were far from home or just because studying is more demanding. It is also noticeable that most of them used the word "effort" needed to succeed, effort from themselves, effort from adults.

For a successful high school

Almost all the young people have dealt with four major points.

3.1. Before high school: skills and transition.

The young, spontaneously, said that to succeed in high school, primary was essential, particularly for the specific skills they have to master, primarily Braille grades 1 and 2 and technology. Tim and others insisted on the importance on good Braille teachers, all insisted upon academic skills: -"It is really important in primary school to be well trained (…) if you miss it, it's bust" (Ralph). What seems important is a progressive way to be mainstreamed. Most of them insist on the role of special teachers, with tenderness in their voices, and on the importance of more "closed" displays for who needs it ("it is good to keep the possibility not to be 1 « Classe d'intégration scolaire » (Clis) in elementary, « Unité pédagogique d'intégration » (UPI) in high schools.

2 UPI are generally considered as good half-way displays (cf. [START_REF] Lewi-Dumont | La scolarisation d'élèves déficients visuels en unité pédagogique d'intégration : la moitié du chemin ?[END_REF], but for that article I interviewed adults and not the pupils themselves) mainstreamed" Jane). In their opinion, all transitions must be well prepared, as well changing schools and professionals as changing materials (computers) or way of working. Pamela mentions: -"Until 4 th grade I had a great support teacher, but she did not follow pupils in high school, so, to facilitate the mainstreaming, so that changes would not happen altogether, she decided to pass me on to someone else (sad voice). Sure there have been strong links between us, but it has been a wise decision". She also mentions she « tamed » her new electronic display just before high school. Simon, mainstreamed in a progressive way [START_REF] Lewi-Dumont | Reverse mainstreaming in a first grade: a good way to learn together?[END_REF], and Jonas (never mainstreamed) agree: -"After second grade, they did not ask us, and that is great, because it seemed normal to us and we could not go back" (Simon) Jonas says it is important to be mainstreamed "since early childhood", otherwise "you get used to comfort and have to make too many efforts, and adolescence is an effort in itself!"

3.2.Material and support issues

Young people had support problems essentially when mainstreamed. They outline that in special schools, Braille books arrive on time, whereas it happens too frequently they don't get Braille documents in the same time as the others, although now they communicate more and more with their teachers through electronic devices. Lea says "teachers adapt themselves" when the transcriptions are not there. Tim considers it is the main condition for success. They also mentioned that they wait sometimes for computers. Lea and Cynthia pointed out their frequent breakdowns. In high school, special teachers are important, but act less directly with pupils at least in the classroom, but most of the pupils mention a positive play. Teacher assistants, who are suspected to reduce autonomy of VI impaired pupils and are contested, were helpful according to Pam: -"They dictated what the teacher wrote on the board (…) or if a Braille document did not arrive (…) teachers have other things to do than repeat for me (…) For me it has been essential, or at least very useful… " Lea, now that she knows her school "by heart" and is autonomous with her computer, says she does not need hers anymore, Cynthia finds it more "relaxing" to have one and Simon "said it was out of question"). Since school was the matter, the young spoke less about other specific helps, like orientation and mobility or daily living skills. Finally, I would say they did not insist a lot upon material problems. Nevertheless, studying what happened after high school for most of them, there is still work to do to help preparing their future.

Communication

Communication is one of the most spoken about themes: the young with their schoolmates, the young with the adults and the adults between themselves. Pamela, who has always been successfully mainstreamed, repeats several times "I have been lucky": with the special support teachers, with the different teachers' assistants, about the interactions between all the protagonists. As mentioned, three interviewees (from different schools) expressed human difficulties in junior high resource schools: Ralph and Marion mentioned other pupils being jealous of the handicapped. Marion "was so fed up to be made fun of that (she) had to change", preferring to go in a boarding special school far from home with a resource lycée. Chris missed too much her previous friends. On the contrary, Patricia keeps a great memory of the relations with her schoolmates:

-It is really the time I preferred, for example the sighted friends I have now, I all kept them from junior high.

Youngsters are mature enough to admit that some problems could be due to adolescence, theirs or their classmates'. Lea, the youngest, is very happy "now":

-I have boyfriends and girlfriends, I spend time with them, here I am like the others About communication with teachers, they never mentioned such terrible problems as reported in a previous article (1999). However, some had revolting anecdotes. For Patricia, it had consequences: she recalls with highly emotional voice her problem with a teacher, when individually mainstreamed in lycée with too little support. She had to quit and go to special school. Simon explains: " I have rather get along with teachers ".

Communication with specialised teams was generally good, but some pupils reproach them to categorize them and to think they know what is good for them better than themselves. Ralph and Marion asked to be in a more open environment, and had problems to get what they wanted: Mar: roughly they said that to go back to mainstreaming you got to deserve it (…). So I said OK you want good marks, you will have them! Pat and Jane suspect the special school not to have done their best to help mainstreaming.

Jane: Sincerely I think they put a brake for O & M and after they said I could not because I could not go to school alone! Special systems change to adapt to the new legal texts, but, as soon as a structure exists, professionals assume all the concerned pupils are going to « fit » in it. Some pupils suffered from a miscommunication between adults. Ralph remarks that teachers in his resource school had a tendency to lean on trained staff. When he changed school after a big struggle, he noticed teachers felt more responsible. "Lucky Pamela" explains how her itinerant teacher shared knowledge with teachers: -"Often, I had teachers who thought about directly with her on the possible adaptations (...) I had an extraordinary luck with teachers, from primary school to university!" 3.4 To succeed, everyone has his way They all insist on academic and specific skills in high school. Most of them were surprised from the fall of the academic performances in special schools. For Pat, it has been demotivating:

-"Rhythm was cooler than before (…) so instead of working, I'd rather had fun". However, Max and Jonas, who were not mainstreamed before university but are successful students, say:

Jo: the problem with mainstreaming is that you give your energy to adapt whereas you could put it in learning. That's why you must do it early! Max: they always say the level is higher there, but if one wants to learn more, one manages (...), tries to improve, because one is not going to always stay here, one has to go beyond what one learns in classes…" For all, mastering school standards is more important than other specific skills: they certainly would not think mobility as a prerequisite, but they insist on their importance for good relationship with friends and for independent life. Jonas refused mainstreaming before university: Jo: "I like my habits, I like my marks (...). Affectivity is important for me (…). They very much scared me: they told me "you'll never manage in university, you are not autonomous!" but when I arrived in college I felt like a fish in water!" Marion speaks for a "golden mean" where professionals are more "confident" to pupils: "they should ask their opinion".

In conclusion, it is important to recall I presented only the subjectivity of the young people, but subjectivity and affectivity are strong motors for learning. It seems there are improvements in the way teachers, generally, accept blind pupils in their classes. Of course, when comparing the oldest and the youngest, technology is of great help. However, reading and writing with computers is sometimes an excuse not to give documents in Braille, and it is often difficult for pupils. One must admit that in the country of Braille, Braille readers have still trouble to find what they need. The frequent changes in the ways pupils were schooled are extremely interesting. However, one gets the impression there is still a matter of chance in the way a blind pupil is schooled, unless his parents struggle. For others, a child happens to be mainstreamed or not depending on what exists in his region. And when only one display exists, it is supposed to be good for all blind pupils ("Naturally, we are all alike" Jane). "Schooling paths" (which means, following assessment of the child's needs, he can change school) were present before 2005 law, but changes obviously are proposed in absence of adequate support. If there is lack of specialists, there is lack of choice for families, except when they know their rights... All the young people expressed strongly the fact they wanted to choose. Many questions remain, I will mention three: partially sighted students did not answer, although more numerous; interviewed students were rather comfortable with school. All have been blind learners since childhood. The needs of pupils who have to learn new techniques, in the context of mainstreaming (Braille, O & M…) are worth a thorough study. I have hypothesis about those questions, and hopefully, the OECD research, which is supposed to reach all the disabled students of a given year, will help us to go further for a better support.

European

  Agency for Development in Special Needs Education http://www.europeanagency.org/ -Special Needs Education in Europe (Volume 2). Provision in post-primary education (2006) -Individual Transition Plans. Supporting the Move from School to Employment (2006) -Inclusive Education and Classroom Practice in Secondary Education (2005) -Young Voices Meeting Diversity in Education (2008).


Slawomir Bak

Guillaume Charpiat

Etienne Corvee

Francois Bremond

Monique Thonnat

Sªawomir B¡k

Etienne Corvée

François Brémond

Learning to Match Appearances by Correlations in a Covariance Metric Space

Keywords: covariance matrix, re-identication, appearance matching

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Appearance matching of the same object registered in disjoint camera views is one of the most challenging issues in every video surveillance system. The present work addresses the problem of appearance matching, in which non-rigid objects change their pose and orientation. The changes in object appearance together with inter-camera variations in lighting conditions, dierent color responses, different camera viewpoints and dierent camera parameters make the appearance matching task extremely dicult.

Recent studies tackle this topic in the context of pedestrian recognition. Human recognition is of primary importance not only for people behavior analysis in large area networks but also in security applications for people tracking. Determining whether a given person of interest has already been observed over a network of cameras is referred to as person re-identication. Our approach is focused on, but not limited to, this application.

Appearance-based recognition is of particular interest in video surveillance, where biometrics such as iris, face or gait might not be available, e.g. due to sensors' scarce resolution or low frame-rate. Thus, appearance-based techniques rely on clothing assuming that individuals wear the same clothes between dierent sightings.

In this paper, we propose a novel method, which learns how to match appearance of a specic object class (e.g. class of humans). Our main idea is that dierent regions of the object appearance ought to be matched using dierent strategies to obtain a distinctive representation. Extracting region-dependent features allows us to characterize the appearance of a given object class in a more ecient and informative way. Dierent kinds of features characterizing various regions of an object is fundamental to our appearance matching method.

We propose to model the object appearance using covariance descriptor [START_REF] Tuzel | Region covariance: A fast descriptor for detection and classication[END_REF] yielding rotation and illumination invariance. Covariance descriptor has already been successfully used in the literature for appearance matching [START_REF] Bak | Boosted human re-identication using riemannian manifolds[END_REF][START_REF] Oncel | Covariance tracking using model update based on lie algebra[END_REF]. In contrast to these approaches, we do not dene a priori feature vector for extracting covariance, but we learn which features are the most descriptive and distinctive depending on their localization in the object appearance. Characterizing a specic class of objects (e.g. humans), we select only essential features for this class, removing irrelevant redundancy from covariance feature vectors and ensuring low computational cost. The output model is used for extracting the appearance of an object from a set of images, while producing a descriptive and distinctive representation.

In this paper, we make the following contributions:

• We formulate the appearance matching problem as the task of learning a model that selects the most descriptive features for a specic class of objects (Section 3). • By using a combination of small covariance matrices (4 × 4) between few relevant features, we oer an ecient and distinctive representation of the object appearance (Section 3.1). • We propose to learn a general model for appearance matching in a covariance metric space using correlation-based feature selection (CFS) technique (Section 3.2).

We evaluate our approach in Section 4 before discussing future work and concluding.

Related Work

Recent studies have focused on the appearance matching problem in the context of pedestrian recognition. Person re-identication approaches concentrate either on distance learning regardless of the representation choice [START_REF] Dikmen | Pedestrian recognition with a learned metric[END_REF][START_REF] Zheng | Person re-identication by probabilistic relative distance comparison[END_REF], or on feature modeling, while producing a distinctive and invariant representation for appearance matching [START_REF] Bazzani | Multiple-shot person re-identication by hpe signature[END_REF][START_REF] Farenzena | Person reidentication by symmetry-driven accumulation of local features[END_REF]. Learning approaches use training data to search for strategies that combine given features maximizing inter-class variation whilst minimizing intra-class variation. Instead, feature-oriented approaches concentrate on an invariant representation, which should handle view point and camera changes. Further classication of appearance-based techniques distinguishes the single-shot and the multiple-shot approaches. The former class extracts appearance using a single image [START_REF] Park | Vise: Visual search engine using multiple networked cameras[END_REF][START_REF] Wang | Shape and appearance context modeling[END_REF], while the latter employs multiple images of the same object to obtain a robust representation [START_REF] Bazzani | Multiple-shot person re-identication by hpe signature[END_REF][START_REF] Farenzena | Person reidentication by symmetry-driven accumulation of local features[END_REF][START_REF] Gheissari | Person reidentication using spatiotemporal appearance[END_REF].

Single-shot approaches: In [START_REF] Wang | Shape and appearance context modeling[END_REF], a model for shape and appearance of the object is presented. A pedestrian image is segmented into regions and their color spatial information is registered into a co-occurrence matrix. This method works well if the system considers only a frontal viewpoint. For more challenging cases, where viewpoint invariance is necessary, an ensemble of localized features (ELF ) [START_REF] Gray | Viewpoint invariant pedestrian recognition with an ensemble of localized features[END_REF] is selected by a boosting scheme. Instead of designing a specic feature for characterizing people appearance, a machine learning algorithm constructs a model that provides maximum discriminability by ltering a set of simple features. In [START_REF] Lin | Learning pairwise dissimilarity proles for appearance recognition in visual surveillance[END_REF], pairwise dissimilarity proles between individuals are learned and adapted for nearest neighbor classication. Similarly, in [START_REF] Schwartz | Learning discriminative appearance-based models using partial least squares[END_REF], a rich set of feature descriptors based on color, textures and edges is used to reduce the amount of ambiguity among human class. The high-dimensional signature is transformed into a low-dimensional discriminant latent space using a statistical tool called Partial Least Squares (PLS) in a one-against-all scheme. However, both methods demand an extensive learning phase based on the pedestrians to re-identify, extracting discriminative proles, which makes the approaches non-scalable. The person re-identication problem is reformulated as a ranking problem in [START_REF] Prosser | Person re-identication by support vector ranking[END_REF]. The authors present extensive evaluation of learning approaches and show that a ranking-based model can improve the reliability and accuracy. Distance learning is also the main topic of [START_REF] Zheng | Person re-identication by probabilistic relative distance comparison[END_REF]. A probabilistic model maximizes the probability of true match pair having a smaller distance than that of a wrong match pair. This approach focuses on maximizing matching accuracy regardless of the representation choice.

Multiple-shot approaches: In [START_REF] Hirzer | Person re-identication by descriptive and discriminative classication[END_REF], every individual is represented by two models: descriptive and discriminative. The discriminative model is learned using the descriptive model as an assistance. In [START_REF] Gheissari | Person reidentication using spatiotemporal appearance[END_REF], a spatiotemporal graph is generated for ten consecutive frames to group spatiotemporally similar regions. Then, a clustering method is applied to capture the local descriptions over time and to improve matching accuracy. In [START_REF] Farenzena | Person reidentication by symmetry-driven accumulation of local features[END_REF], the authors propose the feature-oriented approach which combines three features: (1) chromatic content (HSV histogram);

(2) maximally stable color regions (MSCR) and (3) recurrent highly structured patches (RHSP). The extracted features are weighted using the idea that features closer to the bodies' axes of symmetry are more robust against scene clutter. Recurrent patches are presented in [START_REF] Bazzani | Multiple-shot person re-identication by hpe signature[END_REF]. Using epitome analysis, highly informative patches are extracted from a set of images. In [START_REF] Cheng | Custom pictorial structures for re-identication[END_REF], the authors show that features are not as important as precise body parts detection, looking for part-to-part correspondences.

Learning approaches concentrate on distance metrics regardless of the representation choice. In the end, those approaches use very simple features such as color histograms to perform recognition. Instead of learning, feature-oriented approaches concentrate on the representation without taking into account discriminative analysis. In fact, learning using a sophisticated feature representa-tion is very hard or even unattainable. In [START_REF] Hirzer | Person re-identication by descriptive and discriminative classication[END_REF], the authors deteriorate covariance feature to apply learning. As covariances do not live on Euclidean space, it is dicult to perform learning on an unknown manifold without a suitable metric.

This work overcomes learning issues by considering a covariance metric space using an entropy-driven technique. We combine advantages of a strong descriptor with the ecient selection method, thus producing a robust representation for appearance matching. [START_REF] Oncel | Covariance tracking using model update based on lie algebra[END_REF] The approach Our appearance matching requires two operations. The st stage overcomes color dissimilarities caused by variations in lighting conditions. We apply the histogram equalization [START_REF] Hordley | Illuminant and device invariant colour using histogram equalisation[END_REF] technique to the color channels (RGB) to maximize the entropy in each of these channels and to obtain camera-independent color representation. The second step is responsible for appearance extraction (Section 3.3) using a general model learned for a specic class of objects (e.g. humans). The following sections describe our feature space and the learning, by which the appearance model for matching is generated.

Feature Space

Our object appearance is characterized using the covariance descriptor [START_REF] Tuzel | Region covariance: A fast descriptor for detection and classication[END_REF]. This descriptor encodes information on feature variances inside an image region, their correlations with each other and their spatial layout. The performance of the covariance features is found to be superior to other methods, as rotation and illumination changes are absorbed by the covariance matrix.

In contrast to [1, 2, 15], we do not limit our covariance descriptor to a single feature vector. Instead of dening a priori feature vector, we use a machine learning technique to select features that provide the most descriptive representation of the appearance of an object.

Let L = {R, G, B, I, ∇ I , θ I , . . . } be a set of feature layers, in which each layer is a mapping such as color, intensity, gradients and lter responses (texture lters, i.e . Gabor, Laplacian or Gaussian). Instead of using covariance between all of these layers, which would be computationally expensive, we compute covariance matrices of a few relevant feature layers. These relevant layers are selected depending on the region of an object (see Section 3.2). In addition, let layer D be a distance between the center of an object and the current location. Covariance of distance layer D and three other layers l (l ∈ L) form our descriptor, which is represented by a 4 × 4 covariance matrix. By using distance D in every covariance, we keep a spatial layout of feature variances, which is rotation invariant. State of the art techniques very often use pixel location (x, y) instead of distance D, yielding better description of an image region. Conversely, among our detail experimentation, using D rather than (x, y), we did not decrease the recognition accuracy in the general case, while decreasing the number of features in the covariance matrix. This discrepancy may be due to the fact that we hold spatial Fig. 1. A meta covariance feature space. Example of three dierent covariance features. Every covariance is extracted from a region (P ), distance layer (D) and three channel functions (e.g. bottom covariance feature is extracted from region P3 using layers: D, I-intensity, ∇I -gradient magnitude and θI -gradient orientation). information twofold, (1) by location of a rectangular sub-region from which the covariance is extracted and (2) by D in covariance matrix. We constraint our covariances to combination of 4 features, ensuring computational eciency. Also, bigger covariance matrices tend to include superuous features which can clutter the appearance matching. 4 × 4 matrices provide suciently descriptive correlations while keeping low computational time needed for calculating generalized eigenvalues during distance computation.

Dierent combinations of three feature layers produce dierent kinds of covariance descriptor. By using dierent covariance descriptors, assigned to dierent locations in an object, we are able to select the most discriminative covariances according to their positions. The idea is to characterize dierent regions of an object by extracting dierent kinds of features (e.g. when comparing human appearances, edges coming from shapes of arms and legs are not discriminative enough in most cases as every instance posses similar features). Taking into account this phenomenon, we minimize redundancy in an appearance representation by an entropy-driven selection method.

Let us dene index space Z = {(P, l i , l j , l k ) : P ∈ P; l i , l j , l k ∈ L}, of our meta covariance feature space C, where P is a set of rectangular sub-regions of the object; and l i , l j , l k are color/intensity or lter layers. Meta covariance feature space C is obtained by mapping Z → C : cov P (D, l i , l j , l k ), where cov P (φ) is the covariance descriptor [START_REF] Tuzel | Region covariance: A fast descriptor for detection and classication[END_REF] of features φ: cov

P (φ) = 1 |P |-1 k∈P (φ k -µ)(φ k -µ) T .
Fig. 1 shows dierent feature layers as well as examples of three dierent types of covariance descriptor. The dimension n = |Z| = |C| of our meta covariance feature space is the product of the number of possible rectangular regions by the number of dierent combinations of feature layers.

Learning in a Covariance Metric Space

Let

a c i = {a c i,1 , a c i,2 , . . . a c
i,m } be a set of relevant observations of an object i in camera c, where a c ij is a n-dimensional vector composed of all possible covariance features extracted from image j of object i in the n-dimensional meta covariance feature space C. We dene the distance vector between two samples a c i,j and a c ′ k,l as follows

δ(a c i,j , a c ′ k,l ) = ρ(a c i,j [z], a c ′ k,l [z]) T z∈Z , (1)
where ρ is the geodesic distance between covariance matrices [START_REF] Förstner | A metric for covariance matrices[END_REF], and

a c i,j [z], a c ′ k,l [z]
are the corresponding covariance matrices (the same region P and the same combination of layers). The index z is an iterator of C.

We cast the appearance matching problem into the following distance learning problem. Let δ + be distance vectors computed using pairs of relevant samples (of the same people captured in dierent cameras, i = k, c = c ′ ) and let δ -be distance vectors computed between pairs of related irrelevant samples (i = k, c = c ′ ). Pairwise elements δ + and δ -are distance vectors, which stand for positive and negative samples, respectively. These distance vectors dene a covariance metric space. Given δ + and δ -as training data, our task is to nd a general model of appearance to maximize matching accuracy by selecting relevant covariances and thus dening a distance. Riemannian geometry: Covariance descriptors as positive denite symmetric matrices lie on a manifold that is not a vector space (they do not lie on Euclidean space). Specifying the covariance manifold as Riemannian we can apply dierential geometry [START_REF] Pennec | A riemannian framework for tensor computing[END_REF] to perform usual operations such as mean or distance. However, learning on a manifold space is a dicult and unsolved challenge. Methods [START_REF] Bak | Boosted human re-identication using riemannian manifolds[END_REF][START_REF] Tuzel | Pedestrian detection via classication on riemannian manifolds[END_REF] perform classication by regression over the mappings from the training data to a suitable tangent plane. Dening tangent plane over the Karcher mean of the positive training data points, we can preserve a local structure of the points. Unfortunately, the models extracted using means of the positive training data points tend to overt. These models concentrate on tangent planes obtained from training data and do not have generalization properties. In [START_REF] Hirzer | Person re-identication by descriptive and discriminative classication[END_REF], authors avoid this problem by casting covariance matrices into Sigma Points that lie on an approximate covariance space.

Neither using tangent planes over the Karcher means extracted from training data, nor casting covariances into Sigma Points satises our matching model. As we want to take full advantage of the covariance manifold space, we propose to extract a general model for appearance matching by identifying the most salient features. Based on the hypothesis: A good feature subset is one that contains features highly correlated with (predictive of ) the class, yet uncorrelated with (not predictive of ) each other [START_REF] Hall | Correlation-based Feature Subset Selection for Machine Learning[END_REF], we build our appearance model using covariance features f z (f z ∈ C) chosen by a correlation-based feature selection technique.

Correlation-based Feature Selection (CFS): Correlation-based feature selection (CFS) [START_REF] Hall | Correlation-based Feature Subset Selection for Machine Learning[END_REF] is a lter algorithm that ranks feature subsets according to a correlation-based evaluation function. This evaluation function favors feature subsets which contain features highly correlated with the class and uncorrelated with each other. In our distance learning problem, we dene positive and negative class by δ + and δ -, as relevant and irrelevant pairs of samples (see Section 3.2). Further, feature f z ∈ C is characterized by a distribution of the zth elements in distance vectors δ + and δ -. The feature-class correlation and the feature-feature inter-correlation is measured using a symmetrical uncertainty model [START_REF] Hall | Correlation-based Feature Subset Selection for Machine Learning[END_REF]. As this model requires nominal valued features, we discretize f z using the method of Fayyad and Irani [START_REF] Fayyad | Multi-interval discretization of continuous-valued attributes for classication learning[END_REF]. Let X be a nominal valued feature obtained by discretization of f z .

We assume that a probabilistic model of X can be formed by estimating the probabilities of the values x ∈ X from the training data. The uncertainty can be measured by entropy H(X) = -x∈X p(x) log 2 p(x). A relationship between features X and Y can be given by H(X|Y ) = -y∈Y p(y) x∈X p(x|y) log 2 p(x|y). The amount by which the entropy of X decreases reects additional information on X provided by Y and is called the information gain (mutual information) dened as

Gain = H(X)-H(X|Y ) = H(Y )-H(Y |X) = H(X)+H(Y )-H(X, Y ).
Even if the information gain is a symmetrical measure, it is biased in favor of features with more discrete values. Thus, the symmetrical uncertainty r XY is used to overcome this problem r XY = 2 × Gain/ H(X) + H(Y ) .

Having the correlation measure, subset of features S is evaluated using function M(S) dened as

M(S) = k r cf k + k (k + 1) r f f , ( 2
)
where k is the number of features in subset S, r cf is the average feature-class correlation and r f f is the average feature-feature inter-correlation

r cf = 1 k fz∈S r cfz , r f f = 2 k (k -1) fi,fj ∈S i<j r fifj , (3)
where c is the class, or relevance feature, which is +1 on δ + and -1 on δ -. The numerator in Eq. 2 indicates predictive ability of subset S and the denominator stands for redundancy among the features. Equation 2 is the core of CFS, which ranks feature subsets in the search space of all possible feature subsets. Since exhaustive enumeration of all possible feature subsets is prohibitive in most cases, a heuristic search strategy has to be applied. We have investigated dierent search strategies, among which best rst search [START_REF] Rich | Articial Intelligence[END_REF] performs the best.

Best rst search is an AI search strategy that allows backtracking along the search path. Our best rst starts with no feature and progresses forward through the search space adding single features. The search terminates if T consecutive subsets show no improvement over the current best subset (we set T = 5 in experiments). By using this stopping criterion we prevent the best rst search from exploring the entire feature subset search space. Fig. 2 illustrates CFS method. Let Π be the output of CFS that is the feature subset of C. This feature subset Π forms a model that is used for appearance extraction and matching.

Appearance Extraction

Having the general model Π for a specic object class (e.g. humans), we compute the appearance representation using a set of frames (see Fig. 3). In the context of person re-identication, our method belongs to the group of multiple-shot approaches, where multiple images of the same person are used to extract a compact representation. This representation can be seen as a signature of the multiple instances.

Using our model Π, a more straightforward way to extract appearance would be to compute covariance matrices of a video volume directly for every f z ∈ Π. However, using volume covariance we loose information on real feature distribution (time feature characteristics are merged). Thus, similarly to [START_REF] Bak | Boosted human re-identication using riemannian manifolds[END_REF] we compute Karcher means using a Riemannian manifold. The mean covariance matrix as an intrinsic average blends appearance information from multiple images. For every feature f z ∈ Π we compute the mean covariance matrix. The set of mean covariance matrices stands for an appearance representation of an object (signature ).

Appearance Matching

Let A and B be the object signatures. The signature consists of mean covariance matrices extracted using set Π. The similarity between two signatures A and B is dened as where ρ is a geodesic distances [START_REF] Förstner | A metric for covariance matrices[END_REF], µ A,i and µ B,i are mean covariance matrices extracted using covariance feature i ∈ Π and ǫ = 0.1 is introduced to avoid the denominator approaching to zero. Using the average of similarities computed on feature set Π the appearance matching becomes robust to noise.

S(A, B) = 1 |Π| i∈Π 1 max(ρ(µ A,i , µ B,i ), ǫ) , (4)

Experimental Results

We carry out experiments on 3 i-LIDS datasets 1 : i-LIDS-MA [START_REF] Bak | Boosted human re-identication using riemannian manifolds[END_REF], i-LIDS-AA [START_REF] Bak | Boosted human re-identication using riemannian manifolds[END_REF] and i-LIDS-119 [START_REF] Zheng | Associating groups of people[END_REF]. These datasets have been extracted from the 2008 i-LIDS Multiple-Camera Tracking Scenario (MCTS) dataset with multiple nonoverlapping camera views. These datasets tackle the person re-identication problem, where appearances of the same person acquired by dierent cameras, has to be matched. The results are analyzed in terms of recognition rate, using the cumulative matching characteristic (CMC) [START_REF] Gray | Evaluating Appearance Models for Recognition, Reacquisition, and Tracking[END_REF] curve. The CMC curve represents the expectation of nding the correct match in the top n matches (see Fig. 4). and (0.8, 3π 2 , 8, 2), respectively; N is a gaussian and L is a laplacian lter. A learning process involving all possible combinations of three layers would not be computationally tractable (229296 covariances to consider in section 3.2). Thus instead, we experimented with dierent subsets of combinations and selected a 1 The Image Library for Intelligent Detection Systems (i-LIDS) is the UK government's benchmark for Video Analytics (VA) systems reasonably ecient one. Among all possible combinations of the three layers, we choose 10 combinations (C i=1...10 ) to ensure inexpensive computation. We set

C i to (R, G, B), (∇ R , ∇ G , ∇ B ), (θ R , θ G , θ B ), (I, ∇ I , θ I ), (I, G 3 , G 4 ), (I, G 2 , L), (I, G 2 , N ), (I, G 1 , N ), (I, G 1 , L), (I, G 1 , G 2 )
, respectively, separating color and texture features. Similar idea was already proposed in [START_REF] Gray | Viewpoint invariant pedestrian recognition with an ensemble of localized features[END_REF]. Note that we add to every combination C i layer D, thus generating our nal 4 × 4 covariance descriptors. The dimension of our meta covariance feature space is n = |C| = 10 × |P| = 2810.

Learning and testing: Let us assume that we have (p + q) individuals seen from two dierent cameras. For every individual, m images from each camera are given. We take q individuals to learn our model, while

-| = q × (q -1) × m 2 .
Acronym: We name our approach COrrelation-based Selection of covariance MATrIces (COSMATI).

Results

i-LIDS-MA [START_REF] Bak | Boosted human re-identication using riemannian manifolds[END_REF]: This dataset consists of 40 individuals extracted from two non-overlapping camera views. For each individual a set of 46 images is given. The dataset contains in total 40 × 2 × 46 = 3680 images. For each individual we randomly select m = 10 images. Then, we randomly select q = 10 individuals to learn a model. The evaluation is performed on the remaining p = 30 individuals. Every signature is used as a query to the gallery set of signatures from the other camera. This procedure has been repeated 10 times to obtain averaged CMC curves. COSMATI vs. C i : We rst evaluate the improvement in using dierent combinations of features for the appearance matching. We compare models based on a single combination of features with the model, which employs several combinations of features. From Fig. 5(a) it is apparent that using dierent kinds of covariance features we improve matching accuracy. COSMATI w.r.t. the number of shots: We carry out experiments to show the evolution of the performance with the number of given frames per individual (Fig. 5(b)). The results indicate that the larger number of frames, the better performance is achieved. It clearly shows that averaging covariance matrices on a Riemannian manifold using multiple shots leads to a much better recognition accuracy. It is worth noting that N ∼ 50 is usually aordable in practice as it corresponds to only 2 seconds of a standard 25 framerate camera. COSMATI vs. LCP: We compare our results with LCP [START_REF] Bak | Boosted human re-identication using riemannian manifolds[END_REF] method. This method employs a priori 11 × 11 covariance descriptor. Discriminative characteristics of an appearance are enhanced using one-against-all boosting scheme. Both methods are evaluated on the same subsets (p = 30) of the original data.

Our method achieves slightly better results than LCP as shown in Fig. 6(a). It shows that using specic descriptors for dierent regions of the object, we are able to obtain equally distinctive representation as LCP, which uses an exhaustive one-against-all learning scheme. i-LIDS-AA [START_REF] Bak | Boosted human re-identication using riemannian manifolds[END_REF]: This dataset contains 100 individuals automatically detected and tracked in two cameras. Cropped images are noisy, which makes the dataset more challenging (e.g. detected bounding boxes are not accurately centered around the people, only part of the people is detected due to occlusion). COSMATI vs. LCP: Using the models learned on i-LIDS-MA, we evaluate our approach on 100 individuals. Results are illustrated in Fig. 6(b). Our method again performs better than LCP. It is relevant to mention that once our model is learned oine, it does not need any additional discriminative analysis. Specifying informative regions and their characteristics (features) we obtain a distinctive representation of the object appearance. In contrast to [START_REF] Bak | Boosted human re-identication using riemannian manifolds[END_REF] our oine learning is scalable and do not require any reference dataset. i-LIDS-119 [START_REF] Zheng | Associating groups of people[END_REF]: i-LIDS-119 does not t very well for multiple-shot case, because the number of images per individual is very low (in average 4). However, this dataset was extensively used in the literature for evaluating the person reidentication approaches. Thus, we also use this dataset to compare with state of the art results. The dataset consist of 119 individuals with 476 images. This dataset is very challenging since there are many occlusions and often only the top part of the person is visible. As only few images are given, we extract our signatures using maximally N = 2 images. CPS [START_REF] Cheng | Custom pictorial structures for re-identication[END_REF], SDALF [START_REF] Farenzena | Person reidentication by symmetry-driven accumulation of local features[END_REF], HPE [START_REF] Bazzani | Multiple-shot person re-identication by hpe signature[END_REF], Group Context [START_REF] Schwartz | Learning discriminative appearance-based models using partial least squares[END_REF] COSMATI vs. PRDC [START_REF] Zheng | Person re-identication by probabilistic relative distance comparison[END_REF]: We compare our approach with PRDC method. This method also requires oine learning. PRDC focuses on distance learning that can maximize matching accuracy regardless of the representation choice. We reproduce the same experimental settings as [START_REF] Zheng | Person re-identication by probabilistic relative distance comparison[END_REF]. All images of p randomly selected individuals are used to set up the test set. The remaining images form the training data. Each test set is composed of a gallery set and a probe set. In contrast to [START_REF] Zheng | Person re-identication by probabilistic relative distance comparison[END_REF], we use multiple images (maximally N = 2) to create the gallery and the probe set. The procedure is repeated 10 times to obtain reliable statistics. Our results (Fig. 7(a,b)) show clearly that COSMATI outperforms PRDC. The results indicate that using strong descriptors, we can signicantly increase matching accuracy. COSMATI vs. LCP [START_REF] Bak | Boosted human re-identication using riemannian manifolds[END_REF], CPS [START_REF] Cheng | Custom pictorial structures for re-identication[END_REF], SDALF [START_REF] Farenzena | Person reidentication by symmetry-driven accumulation of local features[END_REF], HPE [START_REF] Bazzani | Multiple-shot person re-identication by hpe signature[END_REF] and Group Context [START_REF] Schwartz | Learning discriminative appearance-based models using partial least squares[END_REF]: We have used models learned on i-LIDS-MA to evaluate our approach on the full dataset of 119 individuals. Our CMC curve is generated by averaged CMC over 10 trials. Results are presented in Fig. 7 (c). Our approach performs the best among all considered methods. We believe that it is due to the informative appearance representation obtained by CFS technique. It clearly shows that a combination of the strong covariance descriptor with the ecient selection method produces distinctive models for the appearance matching problem. Model Analysis: The strength of our approach is the combination of many different kinds of features into one similarity function. Table 1 shows the percentage of dierent kinds of covariance features embedded in all our models, which were used during the evaluation. Unlike [START_REF] Gray | Viewpoint invariant pedestrian recognition with an ensemble of localized features[END_REF], our method concentrates more on texture lters than on color features. It appears that the higher the resolution of images, the more frequent the usage of texture lters is. Figure 8 illustrates that models extracted on higher resolution employ more texture features.

Using multiple images, a straightforward way to extract an appearance would be to use a background subtraction algorithm to obtain foreground regions. Unfortunately in many real scenarios, consecutive frames are not available due to gaps in tracking results. Having a still image, we could apply the extended graph-cut approach: GrabCut [START_REF] Rother | grabcut": interactive foreground extraction using iterated graph cuts[END_REF] to obtain a human silhouette. This approach can be driven by cues coming from detection result (a rectangle around the desired object). Surprisingly, employing GrabCut in our framework did not increase matching accuracy. The main reason for this is that GrabCut often mis-segments signicant part of foreground region. Further, by learning a model, our approach already focuses on features corresponding to foreground.

Conclusion

We proposed to formulate the appearance matching problem as the task of learning a model that selects the most descriptive features for a specic class of objects. Our strategy is based on the idea that dierent regions of the object appearance ought to be matched using various strategies. Our experiments demonstrate that: (1) by using dierent kinds of covariance features w.r.t. the region of an object, we obtain clear improvement in appearance matching performance;

(2) our method outperforms state of the art methods in the context of pedestrian recognition. In the future, we plan to integrate the notion of motion in our recognition framework. This would allow to distinguish individuals using their behavioral characteristics and to extract only the features which surely belong to foreground region.

Fig. 2 .

 2 Fig. 2. Correlation-based feature selection.

Fig. 3 .

 3 Fig. 3. Extraction of the appearance using the model (the set of features selected by CFS). Dierent colors and shapes in the model refer to dierent kinds of covariance features.

Fig. 4 .

 4 Fig. 4. Example of the person re-identication on i-LIDS-MA. The left-most image is the probe image. The remaining images are the top 20 matched gallery images. The red boxes highlight the correct matches.

  p individuals are used to set up the gallery set. We generate positive training examples by comparing m images of the same individual from one camera with m images from the second camera. Thus, we produce |δ + | = q×m 2 positive samples. Pairs of images coming from dierent individuals stand for negative training examples, thus producing |δ

Fig. 5 .Fig. 6 .

 56 Fig. 5. Performance comparison: (a) with models based on a single combination of features; (b) w.r.t. the number of given frames

Fig. 7 .

 7 Fig. 7. Performance comparison using CMC curves, p is the size of the gallery set (larger p means smaller training set): (a),(b) our vs. PRDC [5]; (c) our vs. LCP [2],

Fig. 8 .

 8 Fig. 8. Extracted models for (a) lower and (b) higher resolutions: red indicates color features which turn out to be more prominent in (a).Computation complexity: In our experiments, for q = 10 and m = 10 we generate |δ + | = 1000 and |δ -| = 9000 training samples. Learning on 10.000 samples takes around 20 minutes on Intel quad-core 2.4GHz. The model is composed of 150 covariance features in average. The calculation of generalized eigenvalues of 4 × 4 covariance matrices (distance computation) takes ∼ 2µs without applying any hardware-dependent optimization routines (e.g. LAPACK library can perform faster using block operations optimized for architecture).

  ∇ is the gradient magnitude; θ corresponds to the gradient orientation; G i are Gabor's lters with parameters γ, θ, λ, σ 2 set to (0.4, 0, 8, 2), (0.4, π 2 , 8, 2), (0.8, π

	4 , 8, 2)

4.1 Experimental setup Feature space: We scale every human image into a xed size window of 64×192 pixels. The set of rectangular sub-regions P is produced by shifting 32×8 and 16× 16 pixel regions with 8 pixels step (up and down). It gives |P| = 281 overlapping rectangular sub-regions. We set L = {(l, ∇ l , θ l ) l=I,R,G,B , G i=1...4 , N , L}, where I, R, G, B refer to intensity, red, green and blue channel, respectively;

Table 1 .

 1 30% 4.62% 4.37% 5.47% 12.70 % 14.29% 17.35% 12.12% 14.16 % A table showing the percent of dierent covariance features embedded in models

	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10
	9.61 % 5.

Acknowledgements

This work has been supported by VANAHEIM and ViCoMo European projects.


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  Self-induced laser line sweeping and self-pulsing in double-clad fiber lasers in Fabry-Perot and unidirectional ring cavities Pavel Peterka, P. Navrátil, Bernard Dussardier, R. Slavik, P. Honzatko, Václav Kubeček To cite this version: Pavel Peterka, P. Navrátil, Bernard Dussardier, R. Slavik, P. Honzatko, et al.. Self-induced laser line sweeping and self-pulsing in double-clad fiber lasers in Fabry-Perot and unidirectional ring cavities. SPIE Photonics Europe, Apr 2012, Brussels" Belgium. pp.843309, 10.1117/12.924298. hal-00731855