I A Ivan

M Rakotondrabe

N Chaillet

High Coupling Factor Piezoelectric Materials for Bending Actuators: Analytical and Finite Elements Modeling Results

Keywords: piezoelectric actuators, PZT, PMN-PT materials

New giant piezoelectric factor materials such as PMN-PT and PZN-PT were researched during the last decade and are actually becoming commercially available. As they seem very attractive for actuator designs, we studied their potential in replacing PZT ceramics. In a first comparative approach, we tested a series of classic rectangular composite bimorph structures of different combinations of thicknesses. The goal consisted in evaluating the quantitative improvement in terms of maximum displacement range and blocking force value. The resulted ratio is situated between 3 to 5. Given the very high coupling factor of PMN-PT and PZN-PT that may reach 0.97 (the PZT ranges around 0.3), we noticed large differences between the existing analytical results and the finite elements simulations performed under COMSOL Multiphysics, which finally, for this type of materials, proved to be much more reliable.

Introduction

In the recent years there has been an increasing interest in growing and characterizing of xPb(Mg 1/3 Nb 2/3 )O 3 -(1-x)PbTiO 3 (PMN-PT) [START_REF] Zhang | Elastic, dielectric and piezoelectric coefficients of domain engineered 0.70Pb(Mg1/3Nb2/3)O3-0.30PbTiO3 single crystal[END_REF][2] [START_REF] Wang | Complete set of elastic, dielectric and piezoelectric constants of orthorhombic 0.71Pb(Mg1/3Nb2/3)O3 -0.29PbTiO3 single crystal[END_REF] and of xPb(Zn 1/3 Nb 2/3 )O 3 -(1-x)PbTiO 3 (PZN-PT) [START_REF] Zhang | Complete set of material constants of 0.93Pb(Zn1/3Nb2/3)O3-0.07PbTiO3 domain engineered single crystal[END_REF] [START_REF] Zhang | Complete set of elastic dielectric and piezoelectric coefficients of 0.93Pb(Zn1/3Nb2/3)O3-0.07PbTiO3 single crystal poled along [011[END_REF] solid solutions. Initially used as ceramic, they can be actually grown into single crystals by modified Bridgman or solidstate single crystal growth methods. These materials exhibit ultra-high piezoelectric properties for compositions near morphotropic phase boundary which is located at x=65% for PMN-PT and x=91.5% for PZN-PT. Compared to best PZT [6] ceramics, they are roughly 3 to 5 times more piezoelectric, finding early actuating applications [START_REF] Sitti | Development of PZT and PZN-PT Based Unimorph Actuators for Micromechanical Flapping Mechanisms[END_REF]. Electro-mechanical coupling coefficients are very high, usually ranging from 0.8 up to 0.97. Reported density is higher than that of PZT ceramics, of around 8060-8200 kg/m 3 . Electrostrictive properties are also important, being already subject to few applications such as [START_REF] Hall | Development and Electromechanical Properties of Multimaterial Piezoelectric and Electrostrictive PMN-PT Monomorph Actuators[END_REF]. Dielectric constants are also higher than that of PZT: for instance T 33 ε can exceed a value of 7500. As drawbacks, they are more mechanically fragile, their compliance is higher, and bipolar capability is limited. Their compliance coefficients are 2-3 times higher, suggesting that these single-crystal materials are best suited to low induced-stress, high strain and deflection applications as in the case of composite bimorph actuators. An accurate modeling of these materials is recommended to estimate the performance increase related to the optimal design.

Replacing PZT ceramics with the new generation of giant piezoelectric coefficient materials will primarily allow an increased displacement range for the same electric field or a reduction of the working voltage for the same displacement. Secondly, being monocrystalline, materials like PMN-PT and PZN-PT are more compatible with microtechnology processes.

We considered a simple composite bimorph (also called "unimorph") design for these preliminary tests, as in figure 1.a. The piezoelectric layer is considered fixed, of 100µm thick. The passive substrate is varying between 5 and 150 µm. An optimum thickness ratio exists for each pair of material/substrate (figures 2-5). First, the materials had to be defined into the program and then a series of comparative analytical and COMSOL simulations were performed. The clamped-free bending structure is evaluated in free displacement condition (figure 2.a) and in blocking force at null displacement condition (figure 2.b).

Considered piezoelectric materials

The substrate was considered a [110] silicon wafer. Several material types have been tested: 1. PZT-5H from [11] which is one of the most advanced types of soft PZT ceramics.

Analytical simulations 3.1 Background formulae

Let us consider a blocked-free beam of length L and width w. Passive material is denoted as p1 and piezoelectric material is denoted as p2 ; their related thickness is h p1 and h p2 respectively. Length is along X-axis while the Z-azis is vertical to the beam.

For the analytical formulation of the problem we will reference to the sets of constitutive equations developed by W.G. Smits and W-S. Choi [START_REF] Smits | The constituent equations of piezoelectric heterogeneous bimorphs[END_REF] and by M. S. Weinberg [START_REF] Weinberg | Working equations for piezoelectric actuators and sensors[END_REF].

The constituent equations set of Smits [START_REF] Smits | The constituent equations of piezoelectric heterogeneous bimorphs[END_REF] applies for directly for composite bimorphs at thermodynamic equilibrium. Starting from the well-known piezoelectric strain-charge coupled equations, the energy density is integrated and summed for the piezoelectric and the passive element. Then the canonical conjugates with the external quantities (voltage, momentum etc.) are determined by partial derivatives, providing elements for a constitutive matrix.

Tip displacement z δ is described as a function of the following external quantities: a mechanical moment M, an external transverse force F z at the free end of the beam, a uniform load p and an electrical voltage V z : Resulted blocking force Bz F is:

1 • © § - + + = z z z V K BL d p K L F Kw L M Kw L A
z Bz V L Bw d F 4
3 31 = M. Weinberg [START_REF] Weinberg | Working equations for piezoelectric actuators and sensors[END_REF] presented a rather different approach, more related to structural mechanics and dedicated for multilayer beams. Constitutive equations set was extended to include axial tension/compression. Neutral axis is determined, then piezoelectric axial force and torque per unit voltage. Curvature related to different parameters is derived and constitutive coefficients are provided. If we particularize the Weinberg equations for the multilayer structures to the given case, displacement z δ is:

z V M z M M z V L M C F L C M L C 2 2 2 3 2 + + = δ Blocking force Bz F becomes: z V Bz V L M F 2 3 - = where: 2 / 1 1 p h z - = , 2 / 2 2 p h z + = pi i wh A =
is the i layer cross-section area, Even though the deriving methods of z δ and Bz F are different, simulations showed identical results. The free displacement varies with the square of the length L while the blocking force depends linearly of the width w and is inversely proportional with the cubic length.

Analytical results

We consider fixed piezoelectric layer thickness h p2 = 100µm.an electric field of 8 kV/cm in the polarization direction, Geometry is L=5.0 mm by w= 0.3 mm. Passive layer thickness h p1 is considered as variable and its optimum value for the five given material from Section-2 is investigated.

As seen from figure 3, optimal Silicon layer thickness for best PZT-5H bending strain is 32 µm, for PMN-30PT [001] it is 19 µm and for PMN-29PT [011] it shows 13µm. In the case of the PZN-7PT [001] and [011] optimal silicon thickness are 15 µm and 14 µm respectively. Smaller stiffness coefficients of PMN-PT and PZT-PT than PZT favour displacement to blocking force (smaller blocking force gain, figure 4), which, however, is not a real drawback for microactuator applications. Maximum ratio of displacement gain is 3.5 for PMN-30PT [001] (actuator along X-direction) and 8.2 in the case of PMN-29PT [011] (actuator along Ydirection). show that blocking force obtained with FEM is similar to the analytical ones. However, significant differences in terms of maximum free tip displacement is noticed, which is lower especially for PMN-PT and PZN-PT. The relative difference between FEM and analytical results reach -37%, -69%, -50% and -58% for PMN-30PT[001],

PMN-29PT[011], PZN-7PT[001] and PZN-7PT[011] respectively. For PZT-5H the difference is more acceptable, of around -11%.

Several arguments may explain these differences. First, Smits and Choi [START_REF] Zhang | Elastic, dielectric and piezoelectric coefficients of domain engineered 0.70Pb(Mg1/3Nb2/3)O3-0.30PbTiO3 single crystal[END_REF] as well as Weinberg [START_REF] Weinberg | Working equations for piezoelectric actuators and sensors[END_REF] methods treat in-plane problems where electric field distribution is like that from a plane capacitor. The effect of mechanical strain and stress on electric field is not taken into account. Also, analytical methods don't take into account the electromechanical coupling factor There is a correlation between these coupling factor values and differences noticed and depicted the paragraph above. Obviously, working expressions (1) and ( 7) miss out those coupling coefficients that over-estimate the electrical field and increase the actual mechanical stiffness. Finally, for the analytical methods, the neutral fiber position is fixed. However, in fact, for composite bimorph design and especially for high coupling factor materials, the neutral fiber position is varying according to the applied voltage. Hence, the immediate consequence is the reduction of the free tip displacement.

Notice that in our case, the utilized width (0.3 mm) is quite comparable to total thickness (0.11…0.25 mm). The analysis of such a case should be done with 3D rather than actual 2D boundary conditions. This is why the differences between analytical and FEM methods are primarily explained by inherent simplifications of analytical model. While FEM method treats the coupling problem more "rigorously", in 3D, the lateral deformations are not taken into account by analytical methods. Hence, there exists a curvature in the YZ cross-section of the cantilever, due to interfacial stress between piezoelectric of the silicon layers, that analytical methods do not take into account.

Conclusions

In the present work we evaluated the actuating performances of the monocrystalline giant piezoelectric coupling factor materials known as PMN-PT and PZN-PT. The new material constants were defined under COMSOL Multiphysics. Alternate analytical simulations were performed under Matlab. The substrate was considered the Silicon, in an attempt for future PiezoMEMS devices. Compared to the regular PZT ceramics, the expected gain in terms of free displacement is between 200% and 300% and in terms of blocking force between 10% and 100%.

There was noticed a very large difference between the analytical results and the finite elements ones. In fact, analytical methods fail to accurately model the electric fields of these high coupling coefficient materials. Also, neither interface stress nor lateral deformations are taken into account by the analytical formulae. Finite elements results should be considered as most appropriate.

Legend: 1 -Figure 1 .

 11 Figure 1. Cross section of different piezoelectric bender structures: a) composite bimorph (unimorph), b) three-layer bimorph, c) two-layer bimorph.

Figure 2 .

 2 Figure 2. A composite bimorph piezo-cantilever model with Silicon substrate a) under external blocking force and b) free ended. Dimensions are 5.0x0.3x0.14mm3.

  .5⋅10 -12 m 2 /N ; 2. PMN-30PT poled along [001] which proves a larger transverse piezoelectric coefficient but also a larger compliance than the above PZT: 31 d = -921⋅10 -12 m/V; 2 11 p s = 52⋅10 -12 m 2 /N ; material constants were entered from ref. [1]. 3. PMN-29PT poled along [011] from [3]. This type of poling is quite particular for these domain engineered single crystals. It enables a -7PT poled along [001]. Although less available on the market than their homologue PMN-PT, due to growth and mechanical stability problems, we simulated from [4-7PT poled along [011] and analogue to the above PMN-29PT [011] possess also important piezoelectric properties: 32 d = -1460⋅10 -12 m/V, data from [5].

  fiber position (the origin of the Zaxis is considered at the interface).

Figure 3 .

 3 Figure 3. Analytical comparative diagram of z δ free tip displacement. Piezoelectric layer is 0.1 mm thick, length is 5 mm, width is 0.3 mm and applied voltage 80V. Silicon layer thickness h p1 is varied from 5 to 150 µm.

Figure 4 .

 4 Figure 4. Analytical comparative diagram of F Bz free tip displacement. Same conditions and legend as in figure 3.

Figures 5

 5 Figures 5 and 6 depict simulation results performed by COMSOL analysis considering the same bimorphs as in previous section. Resultsshow that blocking force obtained with FEM is similar to the analytical ones. However, significant differences in terms of maximum free tip displacement is noticed, which is lower especially for PMN-PT and PZN-PT. The relative difference between FEM and analytical results reach -37%, -69%, -50% and -58% for PMN-30PT[001],PMN-29PT[011], PZN-7PT[001] and PZN-7PT[011] respectively. For PZT-5H the difference is more acceptable, of around -11%.Several arguments may explain these differences. First, Smits and Choi[START_REF] Zhang | Elastic, dielectric and piezoelectric coefficients of domain engineered 0.70Pb(Mg1/3Nb2/3)O3-0.30PbTiO3 single crystal[END_REF] as well as Weinberg[START_REF] Weinberg | Working equations for piezoelectric actuators and sensors[END_REF] methods treat in-plane problems where electric field distribution is like that from a plane capacitor. The effect of mechanical strain and stress on electric field is not taken into account. Also, analytical methods don't take into account the electromechanical coupling factor

  PZT-5H k 31 equals 0.38 and for [001]-poled single crystals it larger: 0.49 and 0.58 for PMN-30PT and PZN-7PT respectively. For materials poled along [011], homologue k 32 factors are 0.94 and 0.86 for PMN-29PT and PZN-7PT.

Figure 5 .

 5 Figure 5. Free tip displacement w.r.t substrate thickness (FEM analysis). Same conditions and legend as in figure 3.

Figure 6 .

 6 Figure 6. Blocking force w.r.t passive Si substrate thickness (FEM analysis). Same conditions as in figure 4. Same legend as in figure 3.


Lugiez Mathieu

Dubois Sloven

M Énard

Michel El Hamidi

Abdallah L3i

F Éd

P Ôle

Spatiotemporal Extension of color Decomposition model and Dynamic Color structure-Texture extraction

A new issue in texture analysis is its extension to temporal domain, a field known as Dynamic Texture analysis. Dynamic, or temporal, texture is a spatially repetitive, time-varying visual pattern that forms an image sequence with a certain temporal stationarity. Following recent work, color image decomposition into geometrical, texture and a noise components appears as a good way to extracting meaningful information, i.e texture component, independently of noise and geometrical information.

In this way, we propose to extend spatial color decomposition model to spatiotemporal domain, and attempt to separate static texture present in video and real dynamic texture. To our best knowledge, no such time adaptation is currently available.

Introduction

Motivation

A new issue in texture analysis is its extension to temporal domain, a field known as Dynamic Texture analysis. Dynamic, or temporal, texture is a spatially repetitive, time-varying visual pattern that forms an image sequence with a certain temporal stationarity. In Dynamic Texture, the notion of self-similarity central to conventional image textures is extended to spatiotemporal aspect.

Dynamic textures are typically result from processes such as water flows, smoke, fire, a flag blowing in the wind, a moving escalator, or a walking crowd. Important tasks are thus detection, segmentation and perceptual characterization of dynamic textures. The ultimate goal is to be able to support video queries based on the recognition of the actual natural and artificial dynamic texture processes.

Following recent work, color image decomposition into geometrical, texture and a noise components appears as a good way to reach this aim in extracting meaningful information, i.e texture component, independently of noise and geometrical information. In this way, we propose to extend spatial color decomposition model to spatiotemporal domain, and attempt to separate static texture present in video and real dynamic texture. To our best knowledge, no such time adaptation is currently available.

Overview of the paper

The aim of this work is to extend a model, which decompose color image into three components, a first one containing geometrical structure: U, a second V, holding the textural information and the last one containing the noise: W, to spatiotemporal domain. So, we aim to deal with color image sequences in extending to time existing reliable model. Moreover, in the decomposition texture component, we attempt to determinate spatial texture from texture showing a real dynamicity, which will be suit for future work on dynamic texture.

In the first place of this paper we introduce the extended minimization functional problem and the associate discrete framework in which we place ourself and which is an appropriate one in image sequence processing. In a second part we present the extended time decomposition model and its grayscale implementation. Then in the third part, we will be examining the various challenges arising from the introduction of color model, our proposition to solve this problem and its numerical implementation. Finally, in the last part, we present two way to discriminate static from dynamic textures in image sequence. We will present too, choice and influence of parameters and show and discuss some significant results.

Time extension of decomposition model

Spatiotemporal structure

In order to decompose image sequences in suitable components we propose to extend the Aujol-Chambolle [START_REF] Aujol | Dual norms and image decomposition models[END_REF] decomposition model. Its rely on dual norms derived from BV 1 ,G 2 and E 3 spaces. Authors propose to minimize the following discretized functional:

in f (u,v,w)∈X 3 F(u, v, w) = J(u) Regularization TV + J * v µ Texture extraction + B * w δ
Noise extraction by shrinkage

+ 1 2λ f -u -v -w 2 X Residual part ( 1
)
where X is the Euclidian space R N×N .

To take into account the spatiotemporal structure, we consider a video as an 3-D image [START_REF] Aubert | Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations[END_REF], i.e a volume, so that we can apply 2-D image algorithms extended to the 3-D case. We assume that we have a given image sequence f ∈ L 2 (Ω), where Ω is an open and bounded domain on R 3 , with Lipschitz boundary. In order to recover u, v, w from f , we propose:

• An extended total variation definition:

t Ω ∇ xyt u dx dy dt (2)
where ∇ xyt u the spatiotemporal gradient of u.

1 BV (Ω) is the subspace functions u ∈ L 1 (Ω) such that the following quantity, called the total variation of u, is finite:

J(u) = sup Ω u(x)div(ξ (x))dx such that ξ ∈ C 1 c (Ω, R 2 ), ||ξ || L ∞ (Ω) ≤ 1 2 G
is the subspace introduced by Y.Meyer for oscillating patterns. 3 E is a dual space to model oscillating patterns:

E = . B ∞ -1,∞ dual space of . B 1 1,1
• A new definition of G extended to a third dimension:

G is the Banach space composed of the distributions f which can be written

f = ∂ 1 g 1 + ∂ 2 g 2 + ∂ 3 g 3 = div xt (g)
with g 1 , g 2 and g 3 in L ∞ (Ω). On G, the following norm is associate:

f G = in f { g L ∞ (Ω,R 3 ) ; f = div(g)} (3)

Discretization

From now on, we consider the discrete case. We take here the same notation as in [START_REF] Aujol | Dual norms and image decomposition models[END_REF] and we present the Total Variation discretization. Let ∇u the gradient vector given by:

(∇u) i, j,k = (∇u) 1 i, j,k , (∇u) 2 i, j,k , (∇u) 3 i, j,k (4)
(∇u) 1 i, j,k = u i+1, j,k -u i, j,k if i < N 0 if i = N (∇u) 2 i, j,k = u i, j+1,k -u i, j,k if j < N 0 if j = N (∇u) 3 i, j,k = u i, j,k -u i, j,k-1 if k < N 0 if k = N
The discrete TV of u is given by:

J(u) = (∇u) 1 i, j,k + (∇u) 2 i, j,k + c(∇u) 3 i, j,k (5)
we introduce the c constant to maintain homogeneity between space and time component. It's mainly for numerical implementation, to avoid discretization problem due to quantization step, which be different along space and time dimension. In practice, we often set it to one, but user can adapt it to less, more or in function of frame per second, or quickness of movement present in sequence, to ensure most reliability and homogeneity.

Spatiotemporal grayscale decomposition

The Chambolle's projection algorithm [START_REF] Chambolle | An algorithm for total variation minimization and applications[END_REF] is a smart way to numerically solve the different minimization problems induced by the functional (1), using fixed point method: p 0 = 0, and

p n+1 i, j,k = p n i, j,k + τ(∇(div(p n ) -f /λ )) i, j,k 1 + τ (∇(div(p n ) -f /λ )) i, j,k (6)
As shown in [START_REF] Chambolle | An algorithm for total variation minimization and applications[END_REF] if τ is small enough, that ensure the convergence of the algorithm.

So, to solve (1) authors propose to solve successively three different problems:

• v and w fixed:

in f u∈X 1 2λ f -u -v -w 2 X + J(u) (7)
ũ = f -u -v -w -P G λ ( f -v -w) (8)
• u and w fixed:

in f v∈G µ f -u -v -w 2 X (9) ṽ = P G µ ( f -u -w) (10)
• u and v fixed:

in f w∈δ B E f -u -v -w 2 X (11) w = P δ B E ( f -u -v) (12) = f -u -v -W ST ( f -u -v, θ ) (13)
where W ST stands for the wavelet soft-thresholding, extended to time, with non linear diffusion equations [START_REF] Welk | A four-pixel scheme for singular differential equations[END_REF], of fuv with threshold θ :

S θ (w i ) = w i 1 -13θ w 2 x + w 2 y + w 2 t + 2w 2 xy + 2w 2 xt + 2w 2 yt + 4w 2 xyt -1 if S θ (w i ) ≥ (13θ ), 0 otherwise. ( 14
)
Figure 1 present our grayscale decomposition. We can distinctly see time influence, reed's branch oscillating under water flow is clearly highlight. Moreover, waves present in basin's fountain are well regularized in U component, water dynamicity is totaly catch as texture. Reader can report to Figure 5 and6 to have comparison and difference between classic color decomposition and its spatiotemporal extension. Difference between two successive images of U, statically decomposed, presents about four times less information than our time extended decomposition.

Spatiotemporal color decomposition

In order to solve total variation minimization of color image sequence, we adapt the solution of Aujol and Ha Kang [START_REF] Aujol | Color image decomposition and restoration[END_REF], to time. In fact, Chambolle's projection is not suitable to deal with color image sequences, due to its single plan limitation. To avoid the problem of regularization, authors use classic TV minimizing functional, solving Euler-Lagrange, into Chromaticity and Brightness (CB) and don't extract noise component. To realize their numerical implementation, they use digital TV filter, based on work of Chan, Osher and Chen [START_REF] Chan | The digital TV filter and nonlinear denoising[END_REF].

Digital TV filter implementation

In order to adapt the solution of [START_REF] Aujol | Color image decomposition and restoration[END_REF] to spatiotemporal aspect we readapt their solution to temporal aspect. We reformulate the energy functional of [START_REF] Chan | The digital TV filter and nonlinear denoising[END_REF], using spatio-temporal gradient and extending neighborhood graph to time neighbors (as seen in Figure 2).

Given a noisy image's sequence u 0 , we redefine energy fonctional (presented in [START_REF] Chan | The digital TV filter and nonlinear denoising[END_REF]), adapted to spatiotemporal gradient formulation as (with λ , the Lagrange multiplier):

J(u) = t Ω ∇ xyt u dxdydt + λ 2 t Ω (u -u 0 ) 2 dxdydt (15)
The data that need to be regularized are assumed to be living on a graph. A general digital domain is modeled by a graph [Ω, E], with a finite set Ω of nodes and an edge dictionary E. If α and β are linked by an edge, whether spatially or temporally, we write α ∼ β st . A digital scalar signal u is a function on Ω, u : Ω → R. The value at node α is denoted by u α and local variation at any node is defined as |∇ α u| := ∑ β st ∼α (u β stu α ) 2 , and the regularized local variation, in its conditioned form (to avoid singularity for |∇u| in denominator of associated Euler-Lagrange equation), for any positive number ε is:

|∇ α u| ε = |∇ α u| 2 + ε 2 (16)
So, for a given noisy spatiotemporal signal u 0 , the digital TV filter, F ε,λ α , is defined as:

F ε,λ α u, u 0 = ∑ β st ∼α h αβ st (u)u β st + h αα (u)u 0 α ( 17
)
where the low-pass coefficients filters are given by:

h αβ (u) = w αβ (u) λ + ∑ γ∼α w αγ (u) h αα (u) = λ λ + ∑ γ∼α w αγ (u) with w αβ (u) = 1 |∇ α u| 2 + ε 2 + 1 |∇ β u| 2 + ε 2

Color decomposition algorithm

We present the algorithm to decompose color image sequences in two components, u and v.

(1) Initialization of f , u, v where f 0 is the original sequence

f = f 0 , u 0 = f 0 et v 0 = 0 (2) Iterate m times (a) Separate f , u and v to Brightness ( f b , u b , v b ) and Chromaticity ( f c , u c , v c ) components f b = || f || f c = f || f || u n b = ||u n || u n c = u n ||u n || v n b = ||v n || v n c = v n ||v n || (b) Iterate n times for update u c and u b u n+1 c = F ε,λ c α (u n c , f c -v n c ) u n+1 b = F ε,λ b α u n b , f b -v n b u n c = u n+1 c et u n b = u n+1 b (c) Update u and calculate the residual r u n+1 = u n+1 c * u n+1 b r n = f -u n+1 -v n (d) Iterate n times for update r r n+1 = F ε,µ α r n , f -u n+1 -v n r n = r n+1 (e) Update v v n+1 = f -u n+1 -r n+1 (f) Preparation for the next iteration u n = u n+1 v n = v n+1

Influence of parameters and numerical results

All images and results are compute from DynTex, the dynamic texture database [START_REF] Peteri | Dyntex: A comprehensive database of dynamic textures[END_REF] which provide a large and diverse database of high-quality dynamic textures. Dyntex sequences come from natural scene presenting a wide variety of moving process as flowing water, leaves blowing in wind, walking crowd... Such diversity grants user to identify and emphasize a lot of aspects in testing purpose.

Influence of parameters

The parameter which defined the wide of oscillations captured in texture is µ. It's represent, in some sense, the detail level of our decomposition in space and time. Parameters λ b and λ c in filter process control intensity of regularization process, they represent, in some sense, the scale of regularization.

To obtain week regularization, we use parameters as: µ between 0.5 and 1, λ b and λ c near to 1.

For classic parameters, which work well for most of dynamic sequence aspect, we set µ at 0.01, λ b at 0.04 and λ c at 0.01, we compute three total iterations of our algorithm and ten loops for each call of filtering.

For strong regularization and to catch lot of space and time oscillation we set λ b and λ c smaller: 0, 001 or less, µ at 0.001 or less and iterated our algorithm twenty times or more computed on sixteen or thirty-two images bloc.

Separation of dynamic from static texture

In this part we present two methods to separate real dynamicity presents in dynamic texture, from static component. In fact we consider the time part in our computation of texture component, only if enough dynamicity is present.

The first method rely on optical flow computation, and on a thresh on the norm of movement vectors.

The second one is determinate by the proximity between time and spatial gradient, thanks to ratios computing into grayscale projection algorithm.

We obtain visually good results (better with second method) and separate well dynamic from static component of the texture part. We can clearly see movement of flowing water, extract in Figure 4. So in our process we only take out moving (or non moving) objects in V component and regularized the corresponding part in U component. Such method present interest for segmentation or characterization on dynamic texture tasks.

Numerical results

We present, in Figure 3, a part of a decomposed sequence of flowing water under wood bridge. We can see the static aspect of U component, regularized in space and in time, seems to be freezed, although texture component, V, present a real dynamic, strengthened by time influence. Only moving things or objects presenting dynamicity are enhanced into V component. In this way we obtain the dynamicity present in video through oscillations along time dimension. Geometrical structures are well regularized and time varying details are strengthen and well captured with our method.

In order to prove that our dynamic decomposition method show more significant result than static decomposition, we present a comparison between two methods (static and dynamic decomposition are both computed with same classic parameters). We can easily see that time impact in result, water in Figure 3 and Figure 6 is well regularized and fluid aspect is well represent in the V component. Moreover, if user tunes parameters to obtain stronger regularization, our algorithm is able to catch wider waves in spatiotemporal texture component: see the circumference of foutain in Figure 6, more regularized (in U component) than wider waves. It's a matter of deep in spatiotemporal texture extraction, wich our algorithm is able to deal with.

In Figure 5 we can clearly see the reenforcement of moving cars texture without that static part and objects present in sequence are taken into account. For example, the simple difference between V component in dynamic and successive classic decomposition, as presented in Figure 5 and Figure 6, show a factor between two and four more details in our model (for a sequence presenting a real dynamic). Moreover reconstruction U + V is faithful to original at about nighty-six percent against about nighty percent in static model.

For more details, demonstration sequence, wider range of results and for a prsentation of similar method, rellying on real different approache [START_REF] Lugiez | Dynamic color texture modeling and color video decomposition using bounded variation and oscillatory functions[END_REF], please consult this URL: http://perso.univ-lr.fr/mlugiez.

Figure 1 .

 1 Figure 1. Spatiotemporal grayscale decomposition: Top left: f the original image from sequence, top right: U geometrical component, bottom left: W noise component, bottom right: V + 127 texture component.

Figure 2 .

 2 Figure 2. Digital TV filter at node α. The β , δ , τ and γ are α's space neighbors and t+ and t-are α's time neighbors. Each arrow means that the u value at the tail node is multiplied by the filter coefficient beside and added to α. The exception is the loop arrow at α, for which one uses the original un-regularized data u 0 , instead of the u value.

Figure 3 .Figure 4 .

 34 Figure 3. Top left: f the original image from sequence, top right: U geometrical component, bottom left: Reconstruction U + V, bottom right: V + 127 texture component

Figure 5 .

 5 Figure 5. In left, static decomposition, top: the geometrical component U, bottom: the texture component V. At center, top: image from original sequence f, bottom: simple difference between texture component from classic decomposition and from spatiotemporal decomposition. In right, top: the geometrical component U, bottom: the spatiotemporal texture component V (computed with same parameters than classic decomposition). We can clearly see that only objects in movement are reenforced in our dynamic texture component.

Figure 6 .

 6 Figure 6. In left, static decomposition, top: the geometrical component U, bottom: the texture component V. At center, top: image from original sequence f, bottom: simple difference between texture component from classic decomposition and from spatiotemporal decomposition. In right, top: the geometrical component U, bottom: the spatia-temporal texture component V (computed with same parameters than classic decomposition). We can clearly see that water seems to be freezed at the circumference of the fountain on the geometrical component of our decomposition. Moreover lot of movement details appears in undulations of water.


Guillaume Tomasini

A generalization of the category O of

L'étude des représentations irréductibles d'une algèbre de Lie simple définie sur le corps des nombres complexes a conduit Bernstein, Gelfand et Gelfand a introduire une catégorie qui fournit un cadre naturel pour les modules de plus haut poids. Le but de cette note est de présenter une construction d'une famille de catégories généralisant celle de Bernstein-Gelfand-Gelfand. Nous décrivons les modules simples de certaines de ces catégories. Cette classification permet de montrer que ces catégories sont semi-simples.

Weight modules and Generalized Verma Modules

Let g denote a simple Lie algebra over C and U(g) denote its universal enveloping algebra. Let h be a fixed Cartan subalgebra and denote by R the corresponding set of roots. For α ∈ R we denote by H α ∈ h the corresponding co-root and by g α the rootspace for the root α. We will denote by ∆ a set of simple roots. We write R + for the corresponding set of positive roots. For a subset θ ⊂ ∆, we denote by θ the set of all roots which are linear combination of elements of θ and set θ ± = θ ∩ R ± . We consider the following subalgebras of g :

l θ = h ⊕ α∈ θ g α , n ± θ = α∈R ± -θ ± g α .
The subalgebra p θ = l θ ⊕ n + θ is called the standard parabolic subalgebra associated to θ and l θ is the standard Levi subalgebra. The later is a reductive algebra. Its semisimple part is denoted l ′ θ . If θ = ∅, then l ∅ = h and we simply write n + instead of n + ∅ . We denote by M od(g) the category of all g-modules. We will investigate some full subcategories of M od(g) for which we will describe the simple modules. A module M is a weight module if it is finitely generated, and h-diagonalizable in the sense that

M = ⊕ λ∈h * M λ , M λ = {m ∈ M : H • m = λ(H)},
with weight spaces M λ of finite dimension. We will denote by M(g, h) the full subcategory of M od(g) consisting of all weight modules. This category has been studied by several authors (e.g. [START_REF] Fernando | Lie algebra modules with finite-dimensional weight spaces[END_REF], [START_REF] Futorny | The weight representations of semisimple finite dimensional Lie algebras[END_REF]). This category also appears as a particular case of generalised Harisch-Chandra modules (see [START_REF] Penkov | Generalized Harish-Chandra modules[END_REF] for a definition).

The Bernstein-Gelfand-Gelfand category O is the full subcategory of M(g, h) whose objects are n +finite (a module M is n + -finite if for all m ∈ M the set U(n + )m is a finite dimensional vector space). The cateory O was introduced by Bernstein, Gelfand and Gelfand (see [START_REF] Bernšteȋn | Gel ′ fand, Differential operators on the base affine space and a study of g-modules, in Lie groups and their representations[END_REF]). There they desbribe the simple modules and the structure of the category itself, namely they study the projective modules in O and give a certain correspondence for multiplicities (the so called BGG-correspondence). For a review of these results we refer the reader to [START_REF] Humphreys | Representations of semisimple Lie algebras in the BGG category O[END_REF]. In order to discuss the simple modules in M(g, h) we need to recall some well known facts about generalised Verma modules.

A l θ -module can be made into a p θ -module by letting n + θ act trivially. For such a module N we define the generalised Verma module (GVM)

V (θ, N ) by V (θ, N ) = U(g) ⊗ U (p θ ) N. For any g-module M we define the l θ -module M n + θ := {m ∈ M | n + θ m = 0}. Let us recall the following classical facts about GVMs: Proposition 1.1 (i) If N is a simple l θ -module, the module V (θ, N ) admits a unique simple quotient, denoted by L(θ, N ). The module L(θ, N ) is called the simple g-module induced from (l θ , N ). (ii) If M is a simple g-module such that M n + θ = {0}, then M ∼ = L(θ, M n + θ
). We refer to [7, proposition 3.8] for a proof. We refer to [START_REF] Mazorchuk | Generalized Verma modules[END_REF] for a more detailed discussion of GVM's.

To give the classification of simple weight modules, we need one more ingredient: the so-called cuspidal modules.

Cuspidal modules

Let M be a weight module. A root α ∈ R is said to be locally nilpotent (with respect to M ) if X α ∈ g α acts by a locally nilpotent operator on the whole module M . It is said to be cuspidal if X α acts injectively on the whole module M . We denote by R N (M ) the set of locally nilpotent roots and by R I (M ) the set of cuspidal roots. We shall simply denote R N and R I when the module M is clear from the context.

It is known that for a simple weight module

R = R N ⊔ R I (see [7, lemma 2.3]). A weight module is called cuspidal if R = R I . Set R N s = {α ∈ R N : -α ∈ R N }, R N a = R N -R N s .
We define R I s and R I a the same way. Recall the following theorem: Theorem 2.1 (Fernando [7,theorem 4.18], Futorny [START_REF] Futorny | The weight representations of semisimple finite dimensional Lie algebras[END_REF]) Let M be a simple weight module. Then M is induced from a cuspidal simple module of some Levi subalgebra of g.

More precisely there is a set a simple roots ∆ and a subset θ ⊂ ∆ such that θ = R I s and R +θ + = R N . Then M n + θ is a simple cuspidal module for l θ and M ∼ = L(θ, M n + θ ). The theorem of Fernando reduces the classification of simple weight g-modules to the classification of simple cuspidal weight modules for reductive Lie algebras. By standard arguments this reduces to the classification of simple cuspidal modules for simple Lie algebras. A first step towards this classification is given by the following theorem: Theorem 2.2 (Fernando [7, theorem 5.2]) Let g be a simple Lie algebra. If M is a simple cuspidal gmodule, then g is of type A or C

The classification of simple cuspidal modules was then completed in two steps. In the first step Britten and Lemire classified all simple cuspidal modules of degree 1 (see [START_REF] Britten | A classification of simple Lie modules having a 1-dimensional weight space[END_REF]) where deg(M ) = sup λ∈h * {dim(M λ )}.

Britten and Lemire, and later Benkart, Britten and Lemire, have classified all simple modules of degree 1 when g is of type A or C (see [START_REF] Benkart | Modules with bounded weight multiplicities for simple Lie algebras[END_REF]). These modules will play a key role in our theorem 3.2 below.

Later Mathieu gave the full classification of simple cuspidal modules of finite degree greater than 1 by introducing the notion of a coherent family (see [START_REF] Mathieu | Classification of irreducible weight modules[END_REF]).

The category O S,θ

Now we define a family of full subcategories of M(g, h). Fix a set of simple roots ∆. Fix two subsets S and θ of ∆ such that θ ⊂ S. The category O S,θ is the full subcategory of M(g, h) whose objects are the modules satisfying the following conditions : (O1) As a l θ -module, M is a direct sum of simple highest weight modules. (O2) For α ∈ Sθ , the element X α ∈ g α is cuspidal for M . (O3) The module M is n + S -finite. Notice than in the case S = θ = ∅, we recover the definition of the category O. Other generalisations of category O can be recovered for particular choices of (θ, S) (see [START_REF] Rocha-Caridi | Splitting criteria for g-modules induced from a parabolic and the Berňsteȋn-Gel ′ fand-Gel ′ fand resolution of a finite-dimensional, irreducible g-module[END_REF], [START_REF] Mazorchuk | Generalized Verma modules[END_REF]). We sometimes write O S,θ (g) to emphasize the Lie algebra. Now we list some easy properties of O S,θ which are generalisations of analogous properties of O. Being artinian and noetherian, every module M in O S,θ admits a finite Jordan-Hölder series whose quotients are of the form L(S, N ) (see [START_REF] Jacobson | Basic Algebra II[END_REF]). This allows us to define the multipicity of L(S, N ) in any Jordan-Hölder series of M . From proposition 3.1(iv), the description of the simple modules in O S,θ (g) first requires to consider the case where S = ∆.

Proposition 3.1 Let M, N ∈ O S,θ . (i) Then M ⊕ N is in O S,
In the sequel we assume S = ∆. We shall only consider a simple Lie algebra g and restrict ourselves to the case θ = ∅ and θ = ∆. We label the simple roots as in [START_REF] Bourbaki | Groupes et Algèbres de Lie[END_REF]. First, from theorem 2.1, every simple module in O ∆,θ is of the form L(∆θ, C) for some simple cuspidal l ∆-θ -module C. Such a module is defined by a simple cuspidal l ′ ∆-θ -module C ′ and a scalar action of the center z of l ∆-θ . We have to find those C for which L(∆θ, C) satisfies condition (O1) of the definition of the category O ∆,θ . Theorem 3.2 Assume (g, ∆-θ) is not one of the following: (B n , {e 1 }), (D n , {e 1 }), (D n , {e n-1 }), (D n , {e n }), (E 6 , {e 1 }), (E 6 , {e 6 }), (E 7 , {e 7 }). Then we have

(i) If L(∆ -θ, C) is a simple module in O ∆,θ (
g), then g is of type A n or is of type C n , ∆θ is a connected subset of ∆ and C is a simple cuspidal module of degree 1. (ii) If g = sl n+1 and ∆θ is any connected subset of ∆ other than {e 1 } and {e n }, then the simple modules in O ∆,θ (g) are modules of degree 1. Conversely, any infinite dimensional simple sl n -module of degree 1 is a simple object in some category O ∆,θ (sl n ). (iii) If g = sp n , then there are simple modules in O ∆,θ (g) if and only if ∆θ ⊃ {e n }. In this case, all the simple modules in O ∆,θ (g) are modules of degree 1. Conversely, any infinite dimensional simple sp n -module of degree 1 is a simple object in some category O ∆,θ (sp n ).

  θ . Every submodule and every quotient of M is again in O S,θ . (ii) The category O S,θ is abelian, and every module in O S,θ is noetherian and artinian. (iii) The category O S,θ is Z(g)-finite (where Z(g) is the center of U(g)). (iv) The simple modules in O S,θ are modules of the form L(S, N ) where N is a simple module in O S,θ (l S ).

Email address: tomasini@math.u-strasbg.fr (Guillaume Tomasini).Preprint submitted to the Académie des sciences December 16, 2009

From now on, we assume that g is of type A or C and that θ is neither empty nor equal to ∆. When g = A n we also assume that ∆θ is different from {e 1 } and {e n }. When the category O ∆,θ is non empty, we want to describe its structure. First we note the following fact: if F is a finite dimensional g-module of dimension greater than 1 and M ∈ O ∆,θ is simple, then F ⊗ M does not belong to the category O ∆,θ in general. The proof of this fact requires some results of Britten and Lemire Remark that the result holds trivially when θ = ∆ (this is in fact a part of the definition of the category). Note that when θ = ∅ and g is of type C, Britten, Khomenko, Lemire, Mazorchuk have proved the semisimplicity of O ∆,θ in [START_REF] Britten | Complete reducibility of torsion free Cn-modules of finite degree[END_REF]. The result does not hold when θ = ∅ and g is of type A. In [START_REF] Grantcharov | Cuspidal representations of sl(n + 1)[END_REF], Grantcharov and Serganova have described the indecomposable modules in this case. The proof of theorem 3.3 uses the example 3.3 of [START_REF] Grantcharov | Cuspidal representations of sl(n + 1)[END_REF]. Detailed proofs will be published elsewhere.


Fabien Lotte

Cuntai Guan

LEARNING FROM OTHER SUBJECTS HELPS REDUCING BRAIN-COMPUTER INTERFACE CALIBRATION TIME

Keywords: Brain-Computer Interfaces (BCI), subjectto-subject transfer, regularization

A major limitation of Brain-Computer Interfaces (BCI) is their long calibration time, as much data from the user must be collected in order to tune the BCI for this target user. In this paper, we propose a new method to reduce this calibration time by using data from other subjects. More precisely, we propose an algorithm to regularize the Common Spatial Patterns (CSP) and Linear Discriminant Analysis (LDA) algorithms based on the data from a subset of automatically selected subjects. An evaluation of our approach showed that our method significantly outperformed the standard BCI design especially when the amount of data from the target user is small. Thus, our approach helps in reducing the amount of data needed to achieve a given performance level.

INTRODUCTION

Brain-Computer Interfaces (BCI) are communication systems that enable subjects to send commands to computers by using only their brain activity [START_REF] Wolpaw | Brain-computer interfaces for communication and control[END_REF]. Most existing BCI are based on ElectroEncephaloGraphy (EEG) as the measure of brain activity [START_REF] Wolpaw | Brain-computer interfaces for communication and control[END_REF]. A major limitation of BCI is that many examples of the subject's EEG signals must be recorded in order to calibrate the BCI, which is inconvenient and time consuming. Indeed, due to large inter-subject variabilities, each BCI must be tuned specifically for the corresponding subject.

To our best knowledge, few studies have addressed the problem of calibration time reduction for BCI. Exceptions are the works in [START_REF] Krauledat | Towards zero training for brain-computer interfacing[END_REF][START_REF] Li | A self-training semisupervised svm algorithm and its application in an eeg-based brain computer interface speller system[END_REF]. In [START_REF] Krauledat | Towards zero training for brain-computer interfacing[END_REF], the calibration time is reduced by using data recorded from the same user, during previous BCI sessions. As such, this method cannot be used for a new BCI user, with no previous data available. In [START_REF] Li | A self-training semisupervised svm algorithm and its application in an eeg-based brain computer interface speller system[END_REF], a BCI is first built using very few signals from the target user. Then the BCI is adapted online using unsupervised learning. The limitation is that this BCI will have initially limited performance, becoming good only after a significant adaptation time.

In this paper, we propose to use EEG data from other subjects in the calibration process of the BCI. This aims at building an initially good BCI by using only a small amount of data from the new target user, hence reducing the calibration time. This paper is organized as follows: Section 2 describes the motivation and the principle of our approach. Then, Section 3 describes the evaluation of the proposed method and the obtained results. Finally, Section 4 concludes the paper.

METHOD

Our hypothesis is that despite large inter-subject variabilities, it is still possible to find common information in the EEG signals of some subjects. Thus, such information from additional subjects could be used to design a robust BCI even when little data from the target subject is available.

In this paper, we propose to use data from other subjects in the learning process of the Common Spatial Patterns (CSP) [START_REF] Blankertz | Optimizing spatial filters for robust eeg single-trial analysis[END_REF] and Linear Discriminant Analysis (LDA) algorithms [START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces[END_REF]. These are indeed among the most popular and efficient algorithms used for BCI design [START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces[END_REF][START_REF] Pfurtscheller | Motor imagery and direct brain-computer communication[END_REF][START_REF] Blankertz | Optimizing spatial filters for robust eeg single-trial analysis[END_REF]. However, it should be mentioned that our approach can be easily applied to other similar machine learning techniques.

The CSP and LDA algorithms

The CSP algorithm aims at learning spatial filters which can maximize the variance of the EEG signals from one class, while minimizing the variance of the EEG signals from the other class, hence achieving optimal discrimination [START_REF] Blankertz | Optimizing spatial filters for robust eeg single-trial analysis[END_REF]. More formally, when training the CSP, the obtained spatial filters w are the filters which maximize the following function:

wC 1 w T w(C 1 + C 2 )w T (1)
where C i is the spatial covariance matrix of the EEG signals from class i. This optimization problem is solved by jointly diagonalizing the two matrices C 1 and (C 1 + C 2 ) [START_REF] Blankertz | Optimizing spatial filters for robust eeg single-trial analysis[END_REF]. Once the filters w have been obtained, CSP feature extraction consists in spatially filtering the EEG signals using the w and then computing the variance of the resulting signals.

The LDA classifier uses a linear hyperplane to separate feature vectors from two classes [START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces[END_REF]. The intercept b and normal vector a of this hyperplane are computed as follow:

b = - 1 2 (µ 1 + µ 2 )C -1 (µ 1 -µ 2 ) T and a = C -1 (µ 1 -µ 2 ) T (2)
with µ 1 and µ 2 being the mean feature vector for each class and C the covariance matrix of both classes. With LDA, for an input feature vector x, the classification output is a T x + b.

Learning from other subjects

Both the CSP and LDA approaches are based on the estimation of the class covariance matrices. Much data is required to obtain proper estimates of these matrices, explaining why much training data is required to design a robust BCI. In order to train the CSP and LDA with less training data, hence reducing the BCI calibration time, we propose to incorporate information from other subjects as a regularization term in the estimation of the covariances matrices:

Ct = (1 -λ)C t + λ 1 |S t (Ω)| i∈St(Ω) C i (3)
where C t is a covariance matrix estimated from the data of the target subject, Ω is a set of subjects whose data has been recorded previously, |A| is the number of elements in set A, λ is the regularization parameter (λ ∈ [0, 1]), S t (Ω) is a subset of subjects from Ω, and C i is a covariance matrix estimated from the data of subject i. The aim of this regularization scheme is to obtain a better and more robust estimate of the covariance matrix by using covariance matrices from other subjects for which much data is available. This should lead to better classification performances, especially when little data is available for the target subject.

With our approach, incorporating information from other subjects into the training of CSP or LDA simply consists in replacing the covariance matrices from Eq. 1 and 2 by their regularized estimates from Eq. 3. To train the LDA classifier, the mean feature vectors µ are similarly regularized:

μt = (1 -λ)µ t + λ 1 |S t (Ω)| i∈St(Ω) µ i (4)
This approach raises two questions: 1) How to select a good subset of additional subjects S t (Ω) and 2) how to efficiently select the value of the regularization parameter λ? These two questions are addressed in the following sections.

Selecting a relevant subset of additional subjects

Even if data from many subjects is available, all these subjects may not be relevant to use due to the huge inter-subject variability. Rather, we should select a subset of subjects whose data can be used to classify the target subject's data. To do so, we propose the subject selection algorithm described in Algorithm 1. In this algorithm, the function accuracy = trainT henT est(trainingSet, testingSet) returns the accuracy obtained when training the CSP and LDA on the data set trainingSet and testing them on data set testingSet. The function

(best i , max f (i) ) = max i f (i)
returns best i , the value of i for which f (i) is maximum (this maximum value is denoted max f (i) ). In short, this algorithm sequentially selects the subject to add or to remove from the current subset of subjects, in order to maximize the accuracy obtained when training the BCI on the data from this subset of subjects and testing it on the training data of the target subject. This algorithm has the same structure as the Sequential Forward Floating Search algorithm [START_REF] Pudil | Floating search methods for feature selection with nonmonotonic criterion functions[END_REF], used to select a relevant subset of features. This ensures the convergence of our algorithm as well as the selection of a good subset of additional subjects.

Selecting an appropriate regularization parameter

Another problem to solve is the selection of the regularization parameter λ in Eq. 3 and 4. This parameter specifies the proportion of data from the additional subjects that will be used in the learning process. Here, selecting λ using crossvalidation (CV) is not recommended because 1) CV is a very time consuming approach, and 2) our method is aimed at being used with a small training set for the target subject. Therefore, it would not be reasonable to waste some precious training data when dividing the data set using CV. Rather, we propose the following computationally efficient heuristic to select an appropriate value for λ:

1. Compute the classification accuracy targetAcc obtained when using only the EEG signals from the target subject to train the BCI, by using Leave-One-Out Validation (LOOV) on the target subject training set. Indeed, LOOV removes only a single training example for evaluation.

2. Compute the classification accuracy selectedAcc obtained when classifying the target subject training set with a BCI trained using only the EEG signals from the selected subjects in S t (Ω) (see Algorithm 1).

3. Compute the accuracy RandAcc obtained by a classifier performing at the chance level (e.g., 50% for 2 classes).

4. Determine the value of λ as follows:

λ =    1 targetAcc ≤ RandAcc 0 targetAcc ≥ selectedAcc selectedAcc-targetAcc 100-RandAcc otherwise (5)
This heuristic considers that if the data from the target subject is not appropriate to train a satisfactory BCI (targetAcc ≤ RandAcc), then we should rely only on the data from the other subjects (λ = 1.0). If the classification performance expected when using only the data from the target user is better than that expected when using only the other subjects (targetAcc ≥ selectedAcc), then we should rely only on the data from the target subject (λ = 0.0). The remaining case corresponds to the accuracy expected when using only the other subjects being higher than that expected when using only the target subject (targetAcc < selectedAcc). In this case, the higher the difference between these two expected accuracies, the higher lambda should be as we should give more confidence to the EEG signals from the other subjects.

EVALUATION

EEG Data Set

Our approach was evaluated on data set 2a from BCI competition IV [START_REF] Blankertz | BCI competition IV website[END_REF], provided by the Graz group [START_REF] Naeem | Seperability of four-class motor imagery data using independent components analysis[END_REF]. This set comprises EEG signals from 9 subjects who performed left hand, right hand, foot and tongue Motor Imagery (MI). The EEG signals were recorded using 22 EEG channels. For the purpose of this study, only EEG signals corresponding to left and right hand MI were used. EEG signals were band-pass filtered in the 8-30 Hz frequency band using a 5 th order Butterworth filter. Indeed, this frequency band contains the main frequencies involved in MI [START_REF] Pfurtscheller | Motor imagery and direct brain-computer communication[END_REF].

A training and a testing set were available for each subject. Both sets contained 72 trials for each class, the duration of a trial being 7 sec. Subjects performed MI within the time interval of t=3 to 7 sec of each trial. In this work, we only considered the discrete classification of the trials, i.e., we assigned a class to each trial and not to each trial sample. In order to do so, we extracted the CSP features from the time segment t=3.5 to 5.5 sec of each trial, as this was the segment used by the BCI competition winner [START_REF] Blankertz | BCI competition IV website[END_REF]. Concerning the CSP, we used the 3 most discriminative pairs of filters (corresponding to the highest eigenvalues) for feature extraction.

Results

We compared the performances obtained by our approach, which includes information from other subjects in the CSP and LDA training, with the standard BCI design, which trains the CSP and LDA using only the EEG data from the target subject. We also compared our approach with a regularized variant of CSP and LDA, based on diagonal loading. More precisely, we used regularized covariance matrices Ci = C i + κI in Eq. 1 and 2, with I being the identity matrix and κ the regularization parameter, here automatically selected using Ledoit and Wolf's method [START_REF] Ledoit | A well-conditioned estimator for large-dimensional covariance matrices[END_REF]. To our best knowledge, this BCI design, based on diagonal loading in CSP and LDA, has not been studied in the literature.

To assess whether our approach could reduce the number of training examples required from the target subject, we performed these comparisons with different sizes for the training set of the target subject. When evaluating the three approaches on a given subject, this subject was used as the target subject and the 8 other subjects were used as the set of additional subjects Ω. Evaluations were performed while using only the N first trials from each class from the target subject's training set. The data used for the additional subjects in Ω was their full training set. Once the BCI trained, it was evaluated on the test set of the target user, for which all data were used, i.e, 72 trials per class. The average classification accuracies obtained (over all subjects) are displayed on Figure 1.

On the data set used, our results suggested that incorporating information from other subjects in the training of CSP and LDA can improve the classification accuracy of the BCI, especially when the amount of training data for the target subject is small. When a sufficient amount of data was available for training, e.g., for N = 72, our BCI design and the standard BCI design reached similar performances. However, when the amount of available data was very limited, our method outperformed the standard design. In the most striking case, for N = 10, our method reached an average accuracy nearly 10 % higher than that obtained with the standard design. Interestingly enough, a paired t-test revealed that the average classification accuracy (over all subjects and all N values) obtained by our BCI design was statistically higher than that obtained by both the standard BCI design Results also suggested that this approach could be used to reduce the BCI calibration time for a new user, by reducing the amount of data required from this user. For example, on the data set used, in order to obtain the same classification accuracy as the one obtained by the standard design with 30 training trials per class, our approach needs only 10 trials per class (see Figure 1). Moreover, Blankertz et al suggested that approximately 40 trials per class were necessary to obtain reasonable BCI performances [START_REF] Blankertz | The berlin brain-computer interface: Accurate performance from first-session in bci-naive subjects[END_REF]. In this study, our approach obtained the same performance as the standard BCI design with 40 trials per class while using only 20 trials per class.

Finally, it should be noted that our approach is also computationally efficient. Indeed, on this data set, the average runtime was 92.9 s for N=72 data per class for the target subject, 55.3 s for N=40 and 36.4 s for N=10. The machine used was an Intel Core 2 Quad CPU @ 2.83 GHz with 4 Go RAM, Windows Vista and MATLAB 7.4.

CONCLUSION

In this paper, we proposed a method to reduce the BCI calibration time for a new user thanks to the incorporation of information from other subjects into the learning process of CSP and LDA. More precisely, we proposed to regularize the mean vectors and covariance matrices used in the CSP and LDA training with the mean vectors and covariance matrices estimated from a subset of relevant subjects. We proposed an algorithm to select automatically this relevant subset of subjects for a given target user, as well as a method to select automatically the extent of the regularization. The whole approach is easy to implement and computationally efficient. An evaluation of this approach on data set 2a from BCI competition IV showed that it could outperform both the standard BCI design and a regularized BCI based on diagonal loading, both using only the data from the target user. This improvement was generally higher when the amount of data from the user was small. This suggested that our approach can be used to train a BCI for a new user with less data from this user than the standard approach, hence reducing the calibration time.

Future work will be dedicated to study this method with a larger pool of subjects, including disabled subjects (e.g., stroke patients), in order to assess whether patients may also benefit from this approach. Indeed, it may be more difficult to find common information between stroke patients than between healthy users.

Fig. 1 .

 1 Fig. 1. Average classification accuracies according to the number of training data available for the target subject.

  Dt: training EEG data from the target subject. Input: Ω = {Ds}, s ∈ [0, Ns]: set of EEG data from the Ns other subjects available (Dt Ω). Output: St(Ω): a subset of relevant subjects whose data can be used to classify the data Dt of the target subject.

	Output: selectedAcc: the accuracy obtained when
	classifying Dt using a BCI trained on the subset of
	selected additional subjects St(Ω).
	selected0 = {};
	remaining0 = Ω;
	accuracy0 = 0; n = 1;
	while n < Ns do
	Step 1: (bestSubject, bestAccuracy) =
	maxs∈remaining n-1 trainT henT est(selectedn-1 +
	{Ds}, Dt);
	{D bestSubject };
	accuracyn-1 = bestAccuracy;
	n = n -1;
	go to Step 2;
	else
	go to Step 1;
	end
	end
	end
	(bestN, selectedAcc) = max n∈[1,Ns] accuracyn;
	St(Ω) = selected bestN ;
	Algorithm 1: Subject selection algorithm

Input: selectedn = selectedn-1 + {D bestSubject }; remainingn = remainingn-1 -{D bestSubject }; accuracyn = bestAccuracy; n = n + 1; Step 2: if n > 2 then (bestSubject, bestAccuracy) = max s∈selectedn trainT henT est(selectedn -{Ds}, Dt); if bestAccuracy > accuracyn-1 then selectedn-1 = selectedn -{D bestSubject }; remainingn-1 = remainingn +


G Naumenko

X Artru
email: x.artru@ipnl.in2p3.fr

A Potylitsyn

Yu Popov

L Sukhikh

M Shevelev

"Shadowing" of the electromagnetic field of relativistic charged particles

In radiation processes such as a transition radiation, diffraction radiation, etc. based on relativistic electrons passing through or near an opaque screen, the electron self-field is partly shadowed after the screen over a distance of the order of the formation length γ 2 λ. This effect has been investigated on coherent diffraction radiation (DR) by electron bunches. Absorbing and conductive half-plane screens were placed at various distances L before a standard DR source (inclined half-plane mirror). The radiation intensity was reduced when the screen was at small L and on the same side as the mirror. No reduction was observed when the screen was on the opposite side. It is worth noting that absorbing and conductive half-plane screens produce the same shadowing effect. The shadowing effect is responsible for a bound on the intensity of Smith-Purcell radiation.

Introduction

Macroscopic Maxwell equations may be used (in principle) for calculating the radiation emitted by relativistic electrons passing through or near material targets. Nevertheless phenomenological concepts like equivalent photons and formation zone are useful for an intuitive understanding of the main features. In transition radiation (TR), the formation zone is a region of length 2 f l γ λ ∼ where the Coulomb field of the fast electron and the forward TR from the target cannot be measured separately (in this paper we assume that the electron is ultra-relativistic : 1 γ ). The total electromagnetic field in this region is reduced by a destructive interference between the two fields. This effect can also be interpreted with the equivalent photons method [1] (the terms virtual quanta, quasi-real photons or pseudo-photons are also used). In this picture the target intercepts the beam of equivalent photons, leaving a shadow downstream. It takes a length f l ∼ for the electron to re-form its pseudo-photon cloud. This explains in a simple way the suppression of Optical TR in a Wartski interferometer when the distance between the two foils is small compared to f l .

The similar concept of half-naked electron was introduced in [2,3] in the physics of electron scattering on atoms or nuclei. These authors consider the Landau-Pomeranchuk effect [4] and the density effect of Ter-Mikaelian [START_REF] Ter-Mikaelian | High-Energy Electromagnetic Processes in Condensed Media[END_REF] as experimental evidences for the half-naked electron effect. In this paper it will be shown, both experimentally and theoretically, that a similar shadow region exists behind a Diffraction Radiation (DR) target (either conductive or absorbing). However, unlike for TR, the shadow is only on one side of the particle trajectory (the same side as the target), as expected from a naive application of optical geometry to the pseudo-photons. In this region radiation by a second DR target is inhibited.

Shadowing effect from different points of view

Here we present different points of view about the shadow effect. Let us consider a fast electron moving near two targets like in figure 1. In the first point of view the Coulomb field (C.F.) is considered as a beam of quasi-real photon. Scattering of this field by the first target gives Diffraction Radiation. The second target is in the shadow of the first one, therefore emits almost no radiation. The Coulomb field is gradually "repaired" during the formation zone of length l f ∼ γ 2 λ.

A precise space-time description of a "half-naked electron", in the context of the Landau-Pomeranchuk effect mentioned above, is given in [3], where figures 1.1 and 2.4 depict the restoration of the Coulomb field. In figure 2 we adapted this picture to the case of a particle passing through a narrow hole. )

f f t c v l λ/ - ∼ ∼ .
Figure 4. Illustration of the radiation formation length effect by B.M. BolotovskiI in [START_REF] Bolotovskii | [END_REF].

In this second point of view, the target T 2 of Fig. 1 is submitted to the usual Coulomb field of the particle plus the field of the forward diffraction radiation from T 1 , denoted FDR (1) . Each of these fields induces a current in T 2 and this current generates a new radiation. Outgoing from T 2 , we have then the superposition of -the ordinary diffraction radiation from T 2 , denoted DR (2) , -a part of FDR from T 1 rescattered by T 2 , denoted R 2 {FDR (1) }. If T 2 is at distance less than l f from T 1 , the total inducing field, the total induced current and the sum DR (2) + R 2 {FDR (1) } are suppressed. In this viewpoint, illustrated in Fig. 5, the inhibition of radiation from the second target is a combined result of rescattering and destructive interference.

The Tomsk experiment

In the Landau-Pomeranchuk effect the formation zone is not directly observed. In optical transition radiation, it has a macroscopic longitudinal size. However its transverse size γλ ∼ is usually small compared to the width of the electron beam (this is the effective transverse size of the Coulomb field at observed wavelength λ ). Therefore a direct investigation of its transverse properties needs higher values of the product γλ .

Principle of the experiment

The Tomsk microtron delivers an electron beam with a Lorenz-factor 12 γ . The radiation is observed in the microwave domain ( 1 λ ∼ cm). Thus the transverse scale γλ is about 12 cm, which is quite comfortable. The formation length 2 γ λ is about 1.4 m.

The energy of the radiation coming from the interaction of one electron with the targets, in the bandwidth ω ∆ , is typically of the order of α ω ∆ . In the microwave region, it is too small to be , therefore the bunch coherence amplifies the radiation by 8 orders of magnitude, making it easily observable. Shadowing is achieved by the reflection of the pseudo-photons by a half-plane mirror or their absorption in a half-plane screen. We can indeed assume that pseudo-photons are reflected in the mirror or absorbed in absorber almost like real photons. In particular, it does not matter whether we use an absorber or a mirror for shadowing. This is supported by the analysis of diffraction radiation from a "black body" absorber in [START_REF] Bolotvskiy | Pis'ma v Zhurnal Eksperimental[END_REF], although this reference does not analyze the electromagnetic field inside the radiation formation zone. The electron field in the shadowing region is probed with a second diffraction radiation target, from which one measures the intensity and angular distribution of backward diffraction radiation (BDR), according the scheme shown in figure 6. BDR is easier to measure than FDR, since the separation from the Coulomb field occurs very close to the target. For the case where targets are on opposite sides of the beam we keep the same T 2 target but over-turn T 1 to the opposite side (see figure 7). Figure 6. Experimental scheme for the investigation of shadowing. T 1 is an absorber or conductive target, T 2 is a conductive mirror.

Figure 7. Scheme with the targets on opposite beam sides

Experimental setup

The experiment was performed on the extracted electron beam of the microtron of the Tomsk Nuclear Physics Institute. The beam is extracted from the vacuum chamber through a 20 µ m thick beryllium foil. Its parameters are listed in table 1.

τ µ ≈ Bunch population N e =6⋅10 8
Bunches in a train n b ≈1.6⋅10 4 Bunch length σ≈1.3∼1.6mm

The window caused a beam divergence ( 0 08 . radian). For this reason the distance L between targets T 1 and T 2 was limited to L max = 220 mm. At larger distance, it would be necessary to increase the impact parameter b 2 of T 2 , to prevent direct interactions with the side of the beam. This would strongly reduce the diffraction radiation yield. L was changed by moving the upstream target T 1 , keeping the diffraction radiation mirror T 2 fixed. In this way, the influence of a transversal bunch form-factor on the DR yield from was T 2 avoided. The beryllium window with diameter of 34 mm may be also considered as a source of a transition radiation. A simple estimation showed that the intensity of this radiation is negligible downstream T 1 .

The coherent radiation intensity for λ > 9 mm is by 8 orders larger than the incoherent one and has a power level =1 Watt per steradian. It means that one can investigate coherent radiation in this wavelength range without difficulty.

For the radiation measurements we used the room temperature detector DP20M, with parameters described in [8]. Its main elements are a low-threshold diode, a broadband antenna and a preamplifier.

The detector efficiency in the wavelength region λ=3∼16 mm is estimated to be constant to a ±15% accuracy. The detector sensitivity is 0.3 V/mWatt. A wave-guide with a cutoff λ cut =17 mm was used to cut the long-wave background of the accelerator RF system. The high frequency limit of the wavelength interval is defined by the bunch form-factor. This limit (λ min =9 mm) was measured using discrete wave filters [9] and a grating spectrometer.

Figure 8. Scheme of the absorber test. Figure 9. Angular distribution of the radiation intensity, measured in the absorber test (figure 8). a -BDR from conductive target without absorber, b -radiation from the same target covered by absorber.

To exclude the prewave zone effect (see [10]) a parabolic telescope was used to investigate the angular distribution. This method was suggested and tested in [11] and gives the same angular distribution as in the far field zone (R γ 2 λ).

The absorber properties were checked both with a real photon beam from a source of wavelength λ =6 mm and with the measurement of reflected pseudo-photons of beam electrons, according to the scheme shown in figure 8. In figure 9 curve a is the BDR angular distribution from a conductive target without absorber and curve b is for the same target covered by the absorber. We can see that almost no pseudo-photon reflection is registered. The test with the real photon beam showed that, within 3% experimental accuracy, no real photons passed through the absorber and no photons were reflected by it.

Experimental results

Using the described method, the angular distributions of backward radiation from the conductive target T 2 were measured for inter-target distances L=0 to 220 mm with steps of 20 mm. Two configurations of the experiment were used :

-Configuration a (figure 10a) is intended to demonstrate the shadowing of the electron field. In order not to restrict the measured angular range of the radiation from T 2 at small values of L, the absorber was inclined at 45 as shown in figure 10. -In configuration b (figure 10b) the absorber is placed on the side opposite to T 2 with respect to the electron beam. In this geometry the shadow effect is expected not to be observed. Figure 11. Samples of the measured angular distributions for different distance to the absorber using scheme a (figure 10a).

Figure 12. Samples of the measured angular distributions for different distance to the absorber using scheme b (figure 10b).

Figures 13a and13b have the same intensity scale to emphasize the shadowing effect. These figures exhibit the qualitative properties of shadowing : -the intensity in scheme a is much mower than in scheme b, -it decreases together with L. By contrast, no systematic dependence on L is seen in scheme b. To show that the shadow effect does not depend on the type of the target, the experiment was repeated replacing the absorber target T 1 by a conducting screen. The double dependence on θ and L is shown in figure 14. Comparison with figure 13 confirms the independence of the shadow effect on the target type.

Figure 14. Measured dependence similar to the one shown in figure 13, using a conducting instead of an absorbing target T 1 .

Theoretical calculation and comparison with experiment

The calculation of the expected angular distribution of the radiation from the conductive target T 2 (see figures 6 and 7), for one electron, can be done with the method outlined at the end of section 2, i.e., express the amplitude as DR (2) + R 2 {FDR (1) }. The results presented below are derived in the ultra-relativistic approximation for the electron ( 1 γ

) and the paraxial approximation for the photon

( T z k k ).

Free field evolution in z

In the paraxial approximation of the Huyghens-Fresnel formula, the evolution of a free field from a transverse plane at 0 z to a transverse plane at z can be written as free 0 free 0

( ) ( ) ( ) z G z z z = - , E E (1)
In this symbolic formula, ( ) z E represents the field state in the transverse plane of abscissa z at fixed frequency ω . In the ( )

x
T T k G z i z ω ω       ; = - .           k (2)
In the mixed ( )

y x k , representation, 2 2 1 2 ( ) ( ) exp 2 2 y y k x G x k z iz i z i i z z λ ω ω ω -/     , ; = + - .       (3)
Here G E is an ordinary product in y k and a convolution product in x .

In the limit z = +∞ , Eq.( 1) gives 2 free 0 free 0

( ) e x p ( ) 2 2 T T T T T Z r z Z i Z z Z Z ω ω ω ω ω π = /       , , + + , , |           k r E r E k (4)
with 0 Z z z = -. This formula relates the asymptotic angular distribution to the T k representation at finite 0 z .

Coulomb field

The Coulomb field of an electron fast moving along the z -axis is 1

3 2 2 2 2 C ( ) ( ) 4
T T e t x y z z vt

γ γ π -/       - , , , + - E r r (5)
We have neglected the longitudinal component. Using partial Fourier transformation, it can also be represented in energy, transverse momentum and z space,

C 2 2 0 ( ) i z v T T T ie z e v q ω ω / , , = , + k E k k (6)
or in energy, y x k , and z space, C ( ) (s i g n( ) ) 2

x i z v y ie x k z e e i x v µ ω ω τ -| | / , , , = , , E (7)
with 0 ( )

q v ω γ = / , 2 2 0 y k q µ = + , y k τ µ = / .
The Coulomb field is made of virtual, or bound photons, of phase velocity

Diffraction Radiation from one target

The forward diffraction radiation field FDR E from a target T can be considered as the negative of the intercepted Coulomb field, propagating freely in the forward half-space:

FDR C ( ) ( ) ( ) t t z G z z A z = - - , E E (8)
where t z is the point where the z axis crosses the target plane and ( ) A x y , is the supporting function of the transverse area of the target. We put the screen in the x ≥ b half-space, where b is the impact parameter, therefore ( ) ( )

( ) A x y A x x b θ , = = - ( AE is an ordinary product in ( ) x y , space).
Forward transition radiation is obtained in the limit b = -∞ . FDR E is independent on the target material (once it is fully opaque), on the surface roughness and is unchanged when the screen is tilted. The field after the screen is the sum of FDR and the full Coulomb field C E of (5-7).

} back C ( ) ( ) t t S G z z A z = - - E E ( 9
)
where S is symmetry operator about the target plane. 1 We use relativistic units ( 1) c = and rational definitions of fields and charge, for instance The FDR amplitude at t z z = is most conveniently calculated in the ( )

ρ ∇ = E , 2 (4
y x k , representation (7) : FDR ( ) ( ) ( sign( ) ) 2 t i z v x y t ie x k z x b e e i x ω µ ω θ τ / -| | , , ,
= --, . E (10) We have approximated v by 1, except in the exponentials. Taking the Fourier transform in x k , we obtain the full T k representation which, for 0 b > (non-intercepting screen), reads

( ) FDR ( ) ( ) 2( ) x t ik b i z v T t x ie z e e i ik µ ω ω τ µ -+ / , , = - , . + E k (11)
Using ( 4) one obtains the well-known energy-angle spectrum of Diffraction Radiation (here

T ωθ = k ): 2 2 2 2 0 2 2 FDR 2 2 2 2 0 2 ( ) 4
y b T q k dW R e d d q µ αω ω ω π π µ - + = | , | = . Ω + E R k (12)

Generalization to two targets

Let us now consider two successive targets T 1 (absorbing or reflecting) and T 2 (reflecting), located at 1 z and 2 z and of supporting functions 1 A and 2 A . The field between T 1 and T 2 is

(1) betw FDR C ( ) ( ) ( ) z z z = + , E E E (13) with (1)

FDR ( ) z E

given by ( 8) with 1 t z z = . In the backward direction from the mirror T 2 one observes the reflection of betw

E by T 2 : { } back 2 2 2 2 betw 2 ( ) ( ) S G z z A z = - - E E (14)
where 2

S is symmetry operator about the T 2 plane. Leaving aside the symmetry about the T 2 plane, the energy-angle distribution of back 2

E

is determined by the T k representation of the field

(1) (2) 2 betw 2 2 2 1 FDR 1 FDR 2 ( ) ( ) ( ) ( ) A z A G z z z z = = - - , F E E E (15)
where (8) has been used for target T 2 in the last term.

Analytical results

The analytical calculation starting from ( 6) are most easily done in the ( )

y x k , representation.
Here we only give the final results for the same-side and opposite-side cases. The details of the derivation can be found in [START_REF] Naumenko | Shadowing of the electromagnetic field of a relativistic electron[END_REF]. We take 1 0 z = , 2 z L = .

Targets on the same side.

Their supporting functions are 1

1 ( ) ( ) A x y x b θ , = - and 2 2 ( ) ( ) A x y x b θ , = - with 1 2 0 b b , > . The amplitude in the T k representation is 2 ( ) ( ) ( ) { [ 2 T x ik b i L v i L x ie e i e e ik µ ω ω ω τ µ -+ / , = , - + F k 2 1 2 ( ) ( ) 1 2 2 1 1 exp Erfc (2 ) ( ) 2 2 x ik b b T x k e i L i L b b kL µ ω ω ω + - -/     - / -- /       1 exp Erfc (2 ) ( ) ]} 2 2 y k iL iL b b i L µ ω µ ω ω -/   -   + / -+ / .         (16) The function ( ) 2 1 2
Erfc( ) 2 1

Targets on opposite sides.

We have still 1

1 ( ) ( ) A x y x b θ , = - with 1 0 b > but 2 2 ( ) ( ) A x y b x θ , = -with 2 0 b < . We obtain 2 ( ) opp 2 ( ) ( ) ( ) exp( ) { x ik b i L T x x i i e i Lv e ie ik ik µ ω τ τ ω ω µ µ - -, , , = / - - + F k 1 2 ( ) 1 2 2 1 1 exp Erfc (2 ) ( ) 2 2 x ik b T x k e i L i L b b kL µ ω ω ω -+ -/     - / -- /       2 2 2 ( ) 1 2 1 2 1 exp Erfc (2 ) ( ) } 2 2 x y ik b k e i L i L b b i L µ µ ω µ ω ω -+ -/   -   + / -+ / .         (17)

Numerical results

Numerical calculations of the quantity W=|F/e| 2 , which is proportional to the spectral-angular distribution and given by ( 16) or (17), were performed for the electron beam parameters indicated above for impact-parameter b 1 =b 2 =10 mm and wavelength 12 λ = mm. First we consider the calculations for the case where targets are on the same side of the electron beam (see figure 6). In figure 15 the radiation angular distribution calculated with (16) for different values of the distance L between absorber and conductive target for value interval of 0 220 L = ∼ mm is shown. We can see the manifestation of the shadowing effect and the effect of the recovering of the electron field. Figure 16 shows the numerical results, calculated with (17), for the case where the targets are on opposite sides of the beam (figure 7). The intensity is much higher than in the same-side case and practically independent of L, which indicates that in this case the influence of shadowing is negligible. The theoretical plots of figures 15-16 are quite similar to the experimental ones of figures 13-14, in spite of the fact that concern only one wavelength. θ' is the angle with respect to the direction of specular reflection on T 2 . It corresponds to 85-θ (in degree) of figure 13 or 100-θ of figure 14.

Discussion

The present experiment has demonstrated the feasibility of a direct observation of the shadowing of an electron electromagnetic field in a macroscopic mode. The used experimental method allowed us to make a basic analysis of this phenomenon. Unfortunately the experimental conditions in this experiment could not allow us to measure the L-dependence of the Coulomb field recovering up to γ 2 λ (L is the distance from the absorber). Nevertheless, in the observed L range, the expected properties of shadowing are clearly seen and the dependence of the radiation intensity on L and on the observation angle is in a good agreement with the theoretical one. A complementary confirmation of the pseudophotons viewpoint is the identity of the shadow effects with absorbing and conductive screens. between the targets for targets on the same side.

Figure 16. Same quantity as in Fig. 15, but for targets on opposite sides

We have presented the shadow effect from two points of view:

A) For the phenomenology, the electron field properties at 2 1 γ are close to those of an electromagnetic field in free space, therefore pseudo-photons are reflected by the mirror or absorbed in the absorber almost like real photons.

B) For the calculation, we have considered the shadowed electron field as a sum of the electron field in free space and the forward diffraction radiation (FDR) emitted from the target (or, if the electron crosses the target, forward transition radiation FTR).

The qualification "emitted from the target" applied to FDR or FTR may be confusing. In a traditional interpretation (for example in [START_REF] Bolotovskii | [END_REF]), DR and TR from perfectly conducting targets are indeed considered as radiations emitted by surface currents. For non-perfectly conducting targets, surface currents are replaced by volume polarization currents. These surface or polarization currents are induced by the electromagnetic field of the relativistic electron. If we now take the viewpoint A and apply it to the absorber case, the electron electromagnetic field is evanescing in the target and no surface or polarization current may occur on the downstream side. One could then erroneously conclude (in this traditional interpretation) that no FDR or FTR is generated from an absorber.

The solution to this paradox is: either one uses Maxwell equations in matter (MEM), which include absorption, but do not take the polarizations currents as field sources, or one uses Maxwell equations in vacuum (MEV), which take the polarizations currents as sources but do not include absorption. It is therefore not correct to "screen" the field of the polarization currents. Let us show how these two formalisms are dual and give the same prediction in the case of forward optical transition radiation.

The Coulomb field can be decomposed as F C =F u +F d , where F u is the retarded field of the upstream electron trajectory (suddenly stopped electron) and F d the retarded field of the downstream trajectory (suddenly accelerated electron). In the MEM formalism, FTR is generated by the current of the traveling electron after traversal of the screen [START_REF] Artru | [END_REF]}. This current also re-creates the Coulomb field. The total field in the forward region is then F d = F FTR + F C . Far enough from the trajectory, F FTR ≈ F d . We can then say that FTR is generated during the transition of electron from an unstable "naked" state to the stable "dressed" state (here the term "transition radiation" seems particularly appropriate). A similar interpretation of this phenomenon can be found in [2]. Similarly, forward diffraction radiation can be viewed as a consequence of the transit from a "half-naked" state to the dressed state.

In the MEV formalism, F FTR is equal in the forward region to the field F m generated by the currents circulating on or under the upstream surface of the target. These currents are induced by F C , which is nearly equal to F u in the target, and screen it, therefore F m ~ -F u and the total forward field is F C + F m = F d , as in the MEM formalism. Nevertheless the MEM approach is more intuitive for FTR.

Another example where the shadow effect is at work is Smith-Purcell (SP) radiation. One can consider the SP effect as a reflection of the pseudo-photons on the ridges of the grating. Then one expects that each ridge makes a shadow on the next one. The grating period, on the energy of Smith-Purcell radiation per electron [START_REF] Artru | Interferences between light emitted by charged particles crossing or skimming past optical fibers[END_REF][START_REF] Naumenko | Shadowing of the electromagnetic field of a relativistic electron[END_REF]. C is a numerical constant, L the length of the electron path above the grating and b the impact parameter, i.e., the distance between the electron trajectory and the tops of the ridges. In fact, such a bound with the precise value C=1/(2 π) can be derived from unitarity and analyticity of the reflection coefficients and detailed bounds are obtained for fixed ω and fixed component k y of the photon momentum [START_REF] Artru | [END_REF].

Figure 1 .

 1 Figure 1. Shadowing of the target by another target.

Figure 2 .

 2 Figure 2. Field of a particle passing through a hole (analogous to figure 1.1 of [3]).

Figure 3 .r.

 3 Figure 3. Shadowing by a half-screen.

Figure 3

 3 Figure 3 depicts an analogous situation where the particle passes near the border of a screen. In this case only one side of the Coulomb field is removed. The goal of the experiment presented below is to observe this one-sided shadow region. The second point of view has been illustrated by B.M. Bolotovskii in figure 4 [6]. The traveling Coulomb field (represented by ellipses centered on the successive positions of the particle) induces current in the half-plane screen, which in turn emit Diffraction Radiation (DR), represented by small pieces of ellipses. The radiation field is such that close to the screen it kills part of the particle field. In this point of view, shadowing appears as a destructive interference effect between the Coulomb field, CF , and the forward diffraction radiation field, FDR . The interference gradually disappears (positions 3, 4, 5 of the figure) due to the different velocities, 2 1 2 v γ - -/ and 1 c = (in our units), of the Coulomb and radiation fields. These fields get out of phase after a time

Figure 5 .

 5 Figure 5. Shadow effect as a re-scattering effect.

Figure 10 .

 10 Figure 10. Scheme of experiment on the shadow effect observation. a -the targets are on the same side of the electron beam. b -the targets are on the opposite sides of the electron beam. Samples of the measured distributions using configuration a are shown in figure 11 and the simultaneous dependence on the observation angle θ and on the distance L is shown in figure 13a. The corresponding results for the configuration b (opposite-side screens) are presented in figures 12 and 13b. Figures 11 and 12 have the same units of radiation intensity.

Figure 13 .

 13 Figure 13. Measured dependence of the radiation intensity on the observation angle θ and the distance L to the absorber. Plots a and b correspond to the configurations a and b of figure 10.

1 ω.

 1 made of real photons, i.e., If the electron is suddenly stopped, or scattered at large angle, at 0 z z = the field (5-7) is "released" and propagates as a free field downstream the 0 z point.

For a mirror

  target, the backward diffraction radiation field BDR E is symmetrical of FDR E with respect to the target plane.

{

Figure 15 .

 15 Figure 15. Calculated dependence of the radiation intensity on the observation angle θ' and distance Lbetween the targets for targets on the same side.

  emission angle θ. The shadow effect suggests a limit of the form

Table 1 .

 1 Electron beam parameters.

	Electron energy	6.1 MeV ( 12 γ = ). Bunch period	380 psec
	Train duration	4 sec

Acknowledgment

This work was partly supported by the warrant-order 1.226.08 of the Ministry of Education and Science of the Russian Federation and by the Federal agency for science and innovation, contract 02.740.11.0245