Guillaume Caron
email: guillaume.caron@u-picardie.fr

El Mustapha Mouaddib
email: mouaddib@u-picardie.fr

Eric Marchand
email: eric.marchand@irisa.fr

3D model based tracking for omnidirectional vision: a new spherical approach

Keywords: Omnidirectional vision, stereovision, spherical optimization, tracking

The current work addresses the problem of 3D model tracking in the context of monocular and stereo omnidirectional vision in order to estimate the camera pose. To this end, we track 3D objects modeled by line segments because the straight line feature is often used to model the environment. Indeed, we are interested in mobile robot navigation using omnidirectional vision in structured environments. In the case of omnidirectional vision, 3D straight lines are projected as conics in omnidirectional images. Under certain conditions, these conics may have singularities.

In this paper, we present two contributions. We, first, propose a new spherical formulation of the pose estimation withdrawing singularities, using an object model composed of lines. The theoretical formulation and the validation on synthetic images show thus the new formulation clearly outperforms the former image plane one. The second contribution is the extension of the spherical representation to the stereovision case. We consider in the paper a sensor which combines a camera and four mirrors. Results in various situations show the robustness to illumination changes and local mistracking. As a final result, the proposed new stereo spherical formulation allows to localize online a robot indoor and outdoor whereas the classical formulation fails.

Introduction

Pose estimation can be done using image features, i.e. points, lines, or other geometrical features. When considering the availability of a partial 3D model of an object or a scene, these features need to be extracted from images and associated to the 3D model. This association process is generally done in a first image and features are then tracked over time, while the object and/or the camera is moving.

When using a perspective camera, the spatial volume in which the object can be moved, and still be perceived by the camera, is limited. Hence, objects can appear partially in images. This is particularly the case when the camera is moving within an environment of which a 3D model, even partial, is tracked.

Omnidirectional cameras, thanks to their very wide field of view, allow to track image features during a long period of time, when moving. This is synonym of efficiency, precision and robustness for the tracking and pose estimation processes. Using such a camera allows to keep the object in the image even if it moves in a wide spatial volume surrounding the camera.

A complementary way to increase robustness in a tracking and pose estimation process is to add perception redundancy using several cameras acquiring synchronously images of the object. An interesting idea is thus to use a vision sensor merging stereovision and omnidirectional vision. Different omnidirectional stereovision sensors were designed [START_REF] Nayar | Determining Depth using Two Specular Spheres and a Single Camera[END_REF][START_REF] Gluckman | Real-time omnidirectional and panoramic stereo[END_REF][START_REF] Jang | Single camera catadioptric stereo system[END_REF][START_REF] Luo | A novel omnidirectional stereo vision system via a single camera[END_REF]. We propose to use the Four On One sensor (FOO), a catadioptric sensor made of a unique camera and four parabolic mirrors placed in a square at an equal distance from the camera [START_REF] Mouaddib | Stereo vision with a single camera and multiple mirrors[END_REF][START_REF] Dequen | Stochastic local search for omnidirectional catadioptric stereovision design[END_REF] (Fig. 1).

The pose estimation problem using 2D-3D correspondences can be tackled by linear formulations for various features: points [START_REF] Haralick | Analysis and solutions of the three point perspective pose estimation problem[END_REF][START_REF] Quan | Linear n-point camera pose determination[END_REF][START_REF] Paulino | Pose estimation for central catadioptric systems: An analytical approach[END_REF], lines [START_REF] Dhome | Determination of the attitude of 3d objects from a single perspective view[END_REF][START_REF] Ansar | Linear pose estimation from points or lines[END_REF], etc. A problem raised by these formulations is to deal with outliers. They appear in presence of partial occlusions or specular reflections. RANSAC [START_REF] Fischler | Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography[END_REF] techniques allow to reject outliers in an iterative procedure. Although efficient to reject outliers it is not really well suited to handle noise in image measurements. Recent works affect a weight to image measurements in linear methods to solve this issue [START_REF] Hmam | Optimal non-iterative pose estimation via convex relaxation[END_REF][START_REF] Lepetit | EPnP: An accurate O(n) solution to the PnP problem[END_REF]. However, these methods currently do not deal with outliers as efficiently as non-linear optimization methods to compute the pose.

The non-linear formulation of the pose computation issue leads to the usual optimization methods such as steepest descent, Gauss-Newton or Levenberg-Marquardt. These methods consider first order derivatives of a cost function formalizing the error in the image between the observation and the projection of some model for a given pose. The main idea is to align the forward projection of the model to image measurements. Some more recent works use derivatives of the non-linear function of a higher order [START_REF] Malis | Improving vision-based control using efficient second-order minimization techniques[END_REF]. Several works about non-linear pose computation has been proposed in the optimization community [START_REF] Breitenreicher | Model-based multiple rigid object detection and registration in unstructured range data[END_REF][START_REF] Pottmann | Geometry and convergence analysis of algorithms for registration of 3d shapes[END_REF][START_REF] Hartley | Global optimization through rotation space search[END_REF]. As well as in the perspective vision community, many works exist [START_REF] Lowe | Fitting parameterized three-dimensional models to images[END_REF][START_REF] Dementhon | Model-based object pose in 25 lines of code[END_REF][START_REF] Phong | Object pose from 2-D to 3-D point and line correspondences[END_REF], where the main differences rely on the rotation parameterization (Euler angles, quaternion, vector-angle). Reasoning on the velocities of camera pose elements, the virtual visual servoing (VVS) [START_REF] Sundareswaran | Visual servoing based augmented reality[END_REF][START_REF] Marchand | Virtual visual servoing: A framework for realtime augmented reality[END_REF], is a full scale non-linear optimization technique which can be used for pose computation [START_REF] Comport | Real-time markerless tracking for augmented reality: the virtual visual servoing framework[END_REF]. Some works have been done about 3D model based tracking and pose estimation using VVS, for perspective camera [START_REF] Comport | Real-time markerless tracking for augmented reality: the virtual visual servoing framework[END_REF], stereo perspective rig [START_REF] Dionnet | Robust stereo tracking for space robotic applications[END_REF] and even for monocular omnidirectional vision [START_REF] Barreto | General central projection systems: modeling, calibration and visual servoing[END_REF][START_REF] Marchand | Fitting 3D models on central catadioptric images[END_REF]. These works also deal with corrupted data, which is frequent in image feature extraction, using the widely accepted statistical techniques of robust M-estimation [START_REF] Huber | Robust statistics[END_REF], giving a weight, dynamically computed, that reflects the confidence we have in each measure [START_REF] Comport | Real-time markerless tracking for augmented reality: the virtual visual servoing framework[END_REF][START_REF] Marchand | Fitting 3D models on central catadioptric images[END_REF][START_REF] Drummond | Real-time visual tracking of complex structures[END_REF].

As we are interested by indoor and outdoor mobile robot navigation, we choose to consider straight lines as features because they are frequent (doors, corridors, ...) and an omnidirectional vision sensor to recover the robot pose. The main problem in the case of omnidirectional vision, is that 3D straight lines are projected as conics in the omnidirectional image. This property induces two issues: the conics extraction, which is not easy in noisy images, and the conics representation, which has a singularity when the 3D lines are in the same plane as the optical axis.

In this paper, we present two contributions. We, first, propose a new spherical formulation of the pose estimation by VVS, using an object model composed of lines, withdrawing singularities. The second contribution is an extension of this scheme to an omnidirectional stereovision sensor, which is made with four mirrors and a single camera.

The paper is organized as follows. First, the sensors used in experiments are described and their projection models recalled in section 2. Then the new formulation of this feature and its consequences are presented in section 3. Both formulations are fairly compared on synthetical images (section 3.7). The extension to stereoscopic systems is finally presented and results on real images show the achievement of the proposed approach in section 4. Several problematic cases are studied and they highlight the superiority of the new formulation for omnidirectional vision and stereovision, particularly in pose estimation experiment of a mobile robot. A discussion in section 5 analyzes, sums up the contributions of the paper and concludes on perspectives of the work.

Sensors and models

In this work, an orthographic camera combined with a paraboloidal mirror, in the monocular case, and with several paraboloidal mirrors, in the stereo case, are considered. This configuration has single viewpoint property and is also called a central camera [START_REF] Baker | A theory of single-viewpoint catadioptric image formation[END_REF].

Unified central projection model

We propose to use the unified projection model for central cameras [START_REF] Barreto | Issues on the geometry of central catadioptric imaging[END_REF]. According to this model, a single viewpoint projection can be modeled by a stereographic projection involving a unitary sphere. This is equivalent to ad hoc models of central cameras but has the advantage to be valid for a set of different cameras [START_REF] Geyer | A unifying theory for central panoramic systems and practical applications[END_REF]. Hence, a 3D point X = (X, Y, Z) T is first projected on this unitary sphere as X S , thanks to the spherical projection function pr S (.):

X S =   X S Y S Z S   = pr S (X) with      X S = X ρ Y S = Y ρ Z S = Z ρ (1) where ρ = √ X 2 + Y 2 + Z 2 .
Then, X S is projected in the normalized image plane as x by a perspective projection, i.e. x = pr(X S ), thanks to a second projection center for which the sphere has coordinates (0, 0, ξ) T . The direct relationship between X and x defines the projection function pr ξ :

x = pr ξ (X) with x = X Z+ξρ y = Y Z+ξρ (2)
x is a point of the normalized image plane and is transformed in pixelic coordinates u thanks to the intrinsic matrix K: u = Kpr ξ (X) = pr γ (X), where γ = {α u , α v , u 0 , v 0 , ξ} is the set of intrinsic parameters.

The projection from the sphere to the image plane is invertible, allowing to retrieve a spherical point from an image point:

X S = pr -1 ξ (x) =       ξ+ √ 1+(1-ξ 2 )(x 2 +y 2 ) x 2 +y 2 +1 x ξ+ √ 1+(1-ξ 2 )(x 2 +y 2 ) x 2 +y 2 +1 y ξ+ √ 1+(1-ξ 2 )(x 2 +y 2 ) x 2 +y 2 +1 -ξ       (3)
2.2. The stereo sensor : Four On One Our stereo sensor is composed by a unique camera equipped with a telecentric lens and four parabolic mirrors placed in a square to have their axes parallel to the camera main optical axis. They are at the same distance from the camera (Fig. 1). It is nicknamed the FOO (Four On One sensor) [START_REF] Mouaddib | Stereo vision with a single camera and multiple mirrors[END_REF].

A virtual omnidirectional camera, modeled by the stereographic model, can be associated to each mirror, rigidly combined. Each "camera" handles a set of intrinsic parameters γ j . Four pairs of α u and α v are thus used. This is not the minimal parameters set for the FOO since only one real camera is used and, at least, the ratio of the width of a pixel over its height should be unique for the four virtual cameras. However, this modeling is considered to propose a method which is generic to other kinds of omnidirectional stereo rigs.

To model the FOO stereo rig, the first camera/mirror is fixed as the stereo frame origin. The three other cameras are placed relatively to the first one and their relative poses to the first camera are noted as homogeneous matrices c 2 M c 1 , c 3 M c 1 and c 4 M c 1 . These poses and the four intrinsic parameters sets model the full stereo system. One can note this model is expendable to more cameras/mirrors, knowing each γ j and c j M c 1 .

Model based tracking on the equivalent sphere

The pose estimation method optimizes the pose of the camera relatively to an object, minimizing the error between the forward projection of the object 3D model and image measurements (Fig. 2).

In this work, we consider that indoor and urban outdoor environments are made of 3D segments. Figure 3 details the model of a 3D line, its projection on the equivalence sphere as a portion of great circle (intersection between P 1 and S) and then its projection on image place, as a conic. In our previous work [START_REF] Caron | 3D model based pose estimation for omnidirectional stereovision[END_REF], we minimized the point-to-conic distance in the omnidirectional image plane. The error is computed between the latter conic and sample points like in [START_REF] Marchand | Fitting 3D models on central catadioptric images[END_REF]. The main drawback of this approach is the degeneracy of the conic when its 3D line forms a plane with the optical axis of the camera [START_REF] Barreto | Issues on the geometry of central catadioptric imaging[END_REF]. These lines project as radial straight lines in omnidirectional images. A solution could be to combine two representations, the conic and the line in the image, but the problem is still present when a 3D straight line is near the singular case. Indeed, in that case, the conic is very bad conditioned, leading to perturbations in the computation of the distance (algebraic measure), between a point and the conic. Finally, computing a geometric distance between a point and the projection of a 3D straight line, rather than an algebraic one, is more efficient when the conic is badly conditioned and is more discriminant since it is a true distance [START_REF] Sturm | Conic fitting using the geometric distance[END_REF].

To solve these issues, we propose to reformulate the point-to-line feature on the sphere of the unified projection model. This solves the first singularity problem of the projection of a 3D line as a conic. Indeed, even if a 3D straight line forms a plane with the camera axis, its projection on the sphere is not singular. Furthermore, minimizing the point-to-line distance on the equivalent sphere, which is now a point-to-great circle distance, rather than in the image plane allows to easily use a geometric distance as feature, contrary to the conic case.

Algorithm overview

The goal of the proposed algorithm is to track a known moving object in a sequence of images. As in almost all tracking algorithm, we assume that the motion of the object between two images is small. Image measurements are first of all taken and are the input of the following tracking and pose estimation procedure:

1. Consider an initial pose c M o of the object, usually the optimal pose of the previous image (manual selection for the first image of the sequence, in experiments of this paper). 2. Project the 3D model on the sphere and sample it. A set of points is thus generated for each segment of the 3D model. 3. Project these points in the current image for pose c M o . They may not fit to the actual object (Fig. 2) since it moved between the two images. 4. Apply the moving edge algorithm [START_REF] Bouthemy | A maximum likelihood framework for determining moving edges[END_REF] to each conic of the 3D model in the image to find the set of sample points on the real corresponding contour. 5. Back-project the obtained points on the sphere. 6. Compute the point-to-great circle distances and interaction matrices. 7. Apply the VVS control law to compute the update vector of the pose. Both latter items are repeated until convergence. Then, the next image of the sequence is waited and the process restart at the first item of the list.

Following sections are organized as follows. First, the spherical virtual visual servoing is presented (section 3.2) as it introduces what need to be computed to optimize the pose. Then the new point-to-great circle feature is defined in section 3.3 as well as a way to process the image to extract measurements (section 3.4), its Jacobian (sections 3.5 and 3.6), needed for the VVS. Figure 4 graphically sums up the algorithm before tackling its validation on synthetic data in section 3.7.

Virtual visual servoing 3.2.1. General formulation

The virtual spherical camera is defined by its projection function pr S (.) and the 3D object by several features o S defined in its own frame. The VVS method estimates the real pose c M o of the object in the camera frame by minimizing the error ∆ (eq. ( 4)). ∆ is computed between the set of detected spherical features s * and the position s(r) of the same features computed by forward projection. r = [t X , t Y , t Z , θu X , θu Y , θu Z ] is a vector representation of the pose c M o (3 translation and 3 rotation parameters). Considering k features, ∆ is defined as:

∆ = k i=1 (s i (r) -s * i ) 2 , with s(r) = pr S ( c M o , o S i ). (4)
i indexes the i-th feature of the set. With this formulation, a virtual spherical camera, with initial pose c M 0 o , is moved using a visual servoing control law to minimize the error ∆. At convergence, the virtual sphere reaches the pose c M * o which minimizes ∆. Assuming this non-linear estimation process converges, this pose is the real pose. To sum up, this positioning task is expressed as the regulation of the error e = s(r) -s * .

(

) 5
Imposing an exponential decoupled decrease of the error, ė = -λe leads to the optimization of the pose so that the error evolution curve has an exponential decreasing profile. ė depends on ṡ. The image features motion is linked to the virtual camera velocity v [START_REF] Espiau | A new approach to visual servoing in robotics[END_REF]:

ṡ = L s v. (6)
v = (υ, ω) T is the velocity vector of the pose with υ = (υ X , υ Y , υ Z ), the translation velocity and ω = (ω X , ω Y , ω Z ), the rotation velocity. L s is the interaction matrix (or image Jacobian) linking the features motion to the camera velocity.

Robust formulation

To make the pose estimation more robust, it has to deal with outliers and the use of M-estimator [START_REF] Huber | Robust statistics[END_REF] is an efficient solution in this way. Some functions exist in the literature and apply well to our problem [START_REF] Stewart | Robust parameter estimation in computer vision[END_REF]. They allow uncertain measures to be less likely considered and in some cases completely rejected. To add robust estimation to our objective function ∆ (eq. ( 4)), it is modified as:

∆ R = k i=1 P(s i (r) -s * i ), (7)
where P(a) is a robust function that grows subquadratically and is monotonically non-decreasing with increasing |a|. For P, we considered the Tukey's function [START_REF] Huber | Robust statistics[END_REF] because it completely rejects outliers and give them a zero weight. This is of interest in virtual visual servoing so that detected outliers have no effect on the virtual camera motion [START_REF] Comport | Statistically robust 2d visual servoing[END_REF].

Iterative Re-Weighted Least Squares is a common method of applying the M-estimator. Thus, the error to be regulated to 0 is redefined, taking the robust function into account:

e = D(s(r) -s * ), (8)
where D is a diagonal weighting matrix given by D = diag(w 1 , ..., w k ). Each w i is a weight given to specify the confidence in each feature location and is computed using statistics on the error vector [START_REF] Comport | Real-time markerless tracking for augmented reality: the virtual visual servoing framework[END_REF].

A simple control law that allows to move the virtual camera can be designed to try to ensure an exponential decoupled decrease of e:

v = -λ (DL s ) + D (s(r) -s * ) . ( 9
)
where λ is a gain that tunes the convergence rate and the matricial operator ( ) + denotes the left pseudo inverse of the matrix inside the parentheses.

The pose c M o is then updated using the exponential map e [.] of SE(3) [START_REF] Ma | An invitation to 3D vision[END_REF]:

c M t+1 o = c M t o e [v] , (10)
The feature type choice and hence the interaction matrix expression are a key point of this algorithm. As previously mentioned, we chose the pointo-great circle feature and next sections describe the spherical virtual visual servoing for it.

Definition of the point-to-great circle feature

As we can see in figure (3), the 3D line can be represented by the intersection of two planes. These two planes P 1 and P 2 are defined, in the sphere frame, by: P

1 : A 1 X + B 1 Y + C 1 Z = 0 P 2 : A 2 X + B 2 Y + C 2 Z + D 2 = 0 (11)
with the following constraints on the 3D parameters:

     A 2 1 + B 2 1 + C 2 1 = 1 A 2 2 + B 2 2 + C 2 2 = 1 A 1 A 2 + B 1 B 2 + C 1 C 2 = 0 (12)
so that the two planes are orthogonal with unit normals

N 1 = (A 1 , B 1 , C 1 ) T and N 2 = (A 2 , B 2 , C 2 ) T . D 2 is
the orthogonal distance of the 3D line to the unified model sphere origin.

The spherical projection of a 3D line is then defined as the intersection of the unitary sphere S and plane P 1 :

X 2 + Y 2 + Z 2 = 1 A 1 X + B 1 Y + C 1 Z = 0 . ( 13
)
Since features are lying on the sphere surface, the normal vector N 1 is sufficient to parameterize the great circle.

Considering a point X S on the sphere, i.e. an image contour point projected on the sphere, its signed distance to the great circle is expressed by the dot product d = N 1 .X S :

d = A 1 X S + B 1 Y S + C 1 Z S ( 14
)
This signed distance is interesting since d = 0 when the point is on the great circle and tends to ±1 when going away the great circle. The interaction matrix linked to this distance is expressed considering the time variation of d, with X S constant during the pose optimization of one image :

ḋ = Ȧ1 X S + Ḃ1 Y S + Ċ1 Z S . (15)
This clearly depends on the interaction matrix of N 1 , i.e. its variation with respect to the pose. It is described in the following section.

Measuring edge image points

We mention that edge image points are measured since they are obtained thanks to initial points computed from the 3D model projection at an initial pose. This is the low level tracking method already used in [START_REF] Bouthemy | A maximum likelihood framework for determining moving edges[END_REF]. Let us note that it is different to a point detection method.

Points used to compute the point-to-great circle distance are obtained thanks to an adaptation of the moving edge method to omnidirectional images [START_REF] Caron | 3D model based pose estimation for omnidirectional stereovision[END_REF]. In the latter work, a 3D segment of the 3D model is projected as a conic in the omnidirectional image. The method needs a regular sampling of the conic in order to use the obtained sample points as initial positions for the edge searching, along normal directions to the conic. However, a conic to circle transformation and its inverse are necessary for the regular sampling. Indeed, the direct regular sampling of the conic is not trivial, but for a circle, it is.

In the current work, as introduced in the algorithm overview (section 3.1), the sampling of the 3D line projection is done on its corresponding portion of great circle, i.e. on the sphere. It allows a direct regular sampling, contrary to the conic case. Samples are then projected in the image to find the corresponding contour thanks to the moving edge method.

Interaction matrix for the great circle

The interaction between the time variation of the normal vector to the great circle and the camera motion is expressed from Ṅ1 = ( Ȧ1 , Ḃ1 , Ċ1 ) T [START_REF] Andreff | Visual servoing from lines[END_REF]:

Ṅ1 = - 1 D 2 N 1 N T 2 υ -N 1 × ω, (16)
with υ and ω, the translation and rotation velocities of the spherical camera (section 3.2.1). The time variation of N 1 leads to the interaction matrix linked to the projection of a 3D line on the sphere:

L N 1 =   L A 1 L B 1 L C 1   = - 1 D 2 N 2 N T 1 [N 1 ] × (17)
3.6. Interaction matrix for the point-to-great circle distance feature From equation [START_REF] Malis | Improving vision-based control using efficient second-order minimization techniques[END_REF], and knowing the elements of L N 1 (eq. ( 17)), the interaction matrix L d related to the point-to-great circle distance is expressed as:

L d =   X S Y S Z S   T   L A 1 L B 1 L C 1   . ( 18
)
The feature of the VVS is the signed distance d (eq. ( 14)). So, considering k features, as in equation ( 7), the stacking of the k point-to-line distances for current pose gives s(r). The goal of the optimization is to reach null distances between points to great circle and hence, s * = 0 (a k-vector with all elements equal to zero). Finally, the stacking of the k interaction matrices L d leads to L s and equation ( 8) allows to compute the update vector of the pose.

Validation on synthetic images

This section presents results obtained on images synthesized using the Pov-Ray software. The goal of these experiments is to study the behavior and accuracy of our proposed approach (S) comparing to the image plane based (IP) [START_REF] Marchand | Fitting 3D models on central catadioptric images[END_REF], and their impact on the estimated pose. The implementation has been done in C++ using the ViSP library [START_REF] Marchand | Visp for visual servoing: a generic software platform with a wide class of robot control skills[END_REF].

To do this evaluation, two kinds of omnidirectional images are processed. One kind presents a box where lines are not radial in the image (Fig. 5) and the other presents a majority of radials (Fig. 6). The latter case is a nearsingular situation for the image plane (IP) formulation in which it should be less efficient than in the former case, and also less efficient than the spherical formulation (S).

The experiments done with synthetical images allow to evaluate the convergence rate of the algorithms and their robustness to noise in the previously mentioned situations. The pose estimation error at convergence and the conditioning of interaction matrices are good tools to better understand the behavior of algorithms.

Three sets of experiments are led to evaluate the:

1. robustness to initialization: several different initial poses with a unique desired pose (robust estimation or not) (128 different initial poses). Random variations of the optimal pose are computed in order to be not too far in images for the low level moving edge to succeed.

robustness to low quality camera parameters: different level of noise

(1 %, 2.5 %, 5 %) applied to the camera intrinsic parameters in terms of percentage of parameter values (1000 random choices each) 3. robustness to low quality 3D model of the object (i.e. the model is not exactly the same as the real object, due to fabrication or measurement errors of the object): different level of noise (1 %, 2.5 %, 5 %) applied to the vertices of the object 3D model (1000 random choices each)

The two latter sets of experiments are led considering the robust estimation.

For experiments with noisy camera parameters or 3D model vertices, the initial pose of image plane and spherical VVS is the optimal one, for the perfect camera parameters and perfect model. The robustness is evaluated with respect to (i) the convergence rate, (ii) the orientation and position precision of the optimal pose and (iii) the interaction matrix conditioning.

Figure 7 shows convergence rates. We first observe that, when no 3D line segments of the model projects as a radial, image plane and spherical formulations lead to quasi identical convergence rates. A thin advantage for the spherical formulation is however noted when the object 3D model is importantly noised (Fig. 7(a), on the right side). Secondly, when the model is mainly projected as radials, the spherical formulation clearly outperforms the image plane one. This was expected from the theoretical developments and is confirmed by experiments as we can see in the figure 7 initial poses with or without robust estimation. In that case, the IP formulation converges 50% of the time when the M-Estimator is used whereas it never converges when it is not used. The former rate is due to the fact that since a conic close to be a radial is badly conditioned, important algebraic errors are encountered, even for a small geometric one, and the M-Estimator rejects these measures. Hence, no measures are kept on radials and only two segments of the model can be used to estimate the pose, which is subject to ambiguities. When the M-Estimator is not used, sample points on radials are kept in the optimization process and make the optimization unstable that finishes to diverge or to converge in a local minimum.

Still about the various initial poses experiment in the presence of radials, the spherical formulation allows to converge in all the tested cases when the robust estimation is used. However, it only converges at a rate of 50% to the actual pose when the M-Estimator is not used, because some sample points are attracted by shadows of the box in the image or samples points of a line are attracted by the contour of other object edges.

When applying a Gaussian noise on camera parameters or vertices of the 3D model, at 1%, 2.5% or 5% of their value, the convergence rate decreases when the level of noise is increasing, which is common. Of course, the noise on camera parameters has a major impact on the convergence rate. Indeed, the generalized focal length is noised up to ±10, for an original value of 200, the principal point too and ξ is noised up to ±0.05 for an original value of 1. For instance, figures 5(c) and 6(c) show the projection of the 3D model when camera parameters are noised, for the optimal pose (corresponding to noiseless parameters). The spherical formulation outperforms the image plane one, nearly doubling the convergence rate. The next diagrams have to be cross-interpreted with the latter convergence survey.

Working with synthetic images allows to evaluate the pose estimation precision itself without other unknown perturbations. We first study the position and orientation estimation errors in the non radial case (Fig. 8). In this case, as in the convergence survey, both formulations (IP and S) lead to similar results with a slightly better precision in position, and more particularly in orientation, of the spherical formulation. The "various initial poses" experiments of figure 8 show the estimation precision at convergence is, at least, doubled when using the robust estimation: a position error of around 2.5 cm with the robust estimation against 5 cm without.

In the presence of radial lines, the pose estimation precision is clearly better, as compared with the known groundtruth, for the spherical formulation than for the image plane one, even only considering the cases of convergence. Figure 9 shows that the spherical formulation allows not only to converge more frequently but the pose estimation is also more precise than with the image plane formulation, especially when the model is projected as radials in the image.

Another tool to study the behavior of non linear optimization algorithms is the conditioning of the Jacobian so, in the current work, the conditioning of the interaction matrix. The closer to 1 the condition number, the better conditioned it is. Figure 10(a) shows the median condition number of the interaction matrix in various experiments. The median is chosen and not the mean because, in some cases, the interaction matrix condition number for IP formulation is extremely high (about 10 6 ). The mean comparison with S formulation would then not be fair. Similar results are obtained by the two studied formulations of the 3D tracking, with a similar median condition number in all experiments where the optimization converged.

On the contrary, when looking at the median conditioning number of the robust interaction matrix for the various initial poses experiment (Fig. 10(b), left part), it helps well to explain the difference of convergence rate between both formulations. Indeed, the interaction matrix of the spherical formulation is twice better conditioned than the image plane formulation.

However, when using a noised set of camera parameters in the presence of radials, the condition number of both methods is similar, and even slightly better for the IP formulation (Fig. 10(b), center part). It helps to understand that, this is precisely the error computation as an algebraic distance in the IP formulation which leads to a low convergence rate, in this case, and a low precision when converging.

Finally, the noised model experiments in the presence of radial lines (Fig. 10 sensitive to a noised 3D model than the spherical one. And, more globally, about the conditioning of the Jacobian in all experiments, the spherical formulation shows to be very robust to noise, applied on the camera parameters or on the 3D model, with a quasi constant median conditioning, despite noise.

To conclude, this survey experimentally highlights the interest of the spherical modeling of the point-to-line distance, with the improvement of convergence rate and pose estimation precision over the image plane formulation.

Stereoscopic extension of the model based tracking in omnidirectional vision

Robust stereoscopic virtual visual servoing

The aim of this section is to present how to adapt the VVS method to an omnidirectional stereovision sensor considered as a rig of four single viewpoint omnidirectional cameras. The criterion ∆ (eq. ( 4)) has to be rewritten to take into account several cameras, assuming known stereo system calibration parameters [START_REF] Caron | Single viewpoint stereoscopic sensor calibration[END_REF]. So, for N cameras, ∆ is extended to:

∆ S = N j=1 k j i=1 (pr S ( c j M c 1 c 1 M o , o P i ) -c j s * i ) 2 (19)
with, obviously, c 1 M c 1 = I 4×4 and knowing the N -1 sets of relative camera poses with respect to the reference one. With this formulation, only six parameters have to be estimated, as for the monocular pose estimation problem. Setting N = 2, we retrieve a two cameras case ( [START_REF] Dionnet | Robust stereo tracking for space robotic applications[END_REF], in perspective vision) and for the FOO, N = 4 (Fig. 1).

To express the features motion in images of cameras 2, 3 and 4 with respect to the velocity v 1 of camera 1, i.e. the stereo rig velocity, a frame change is applied to velocity vectors v j of cameras j = 2, 3, 4. Velocity vector frame changes are computed thanks to the twist transformation matrix:

c j V c 1 = c j R c 1 [ c j t c 1 ] × 0 c j R c 1 (20)
where c j R c 1 and c j t c 1 are respectively the rotational bloc and the translation vector extracted from the homogeneous matrix c j M c 1 . Then, we can express v j w.r.t. v 1 :

v j = c j V c 1 v 1 , j = 2..4. (21)
So the feature velocity in the spherical image of the camera j is related to the motion of camera 1:

ṡj = L j v j = L j c j V c 1 v 1 . (22)
For four cameras, considering a M-Estimator is computed on each image, leading to four weighting matrices (D 1..4 ), the pose velocity vector of the rig, expressed in camera 1 frame, is computed as:

v 1 = -λ     D 1 L 1 D 2 L 2 c 2 V c 1 D 3 L 3 c 3 V c 1 D 4 L 4 c 4 V c 1     +     D 1 D 2 D 3 D 4         s 1 (r 1 ) -s * 1 s 2 (r 2 ) -s * 2 s 3 (r 3 ) -s * 3 s 4 (r 4 ) -s * 4     . (23)
Four M-Estimators are individually computed since the moving edges procedure is done individually in each of the four views of the FOO. Finally, L j is substituted by the stacking of each feature interaction matrix L d , computed for the j-th spherical camera of the rig. The same reasoning is followed to define current and desired features for each camera (see the end of section 3.6). v 1 is used to update the stereo rig pose, as in the unique camera case, and poses of the three other cameras are then updated using the stereo rig calibration parameters c j M c 1 :

c j M o = c j M c 1 c 1 M o .
These pose matrices are used in equation [START_REF] Marchand | Virtual visual servoing: A framework for realtime augmented reality[END_REF] to compute s j (r j ) for a new iteration.

Experimental results

The algorithm has been applied on several image sequences either with a static sensor and mobile objects in the scene and with mobile ones, embedded on a mobile robot.

The first handheld box tracking experiment is led to compare image plane based and spherical based VVS methods using the FOO sensor. Pose estimation stability of image plane and spherical formulations in the stereo case are then compared. Then, a localization experiment tracking doors of a scene shows the precision of estimation when using the spherical formulation. Finally, an experiment is led outdoor to track buildings in a more challenging situation than indoor.

Box tracking: sphere versus image plane

A box (30 cm × 25 cm × 20 cm) is manually moved in a major part of the static sensor field of view (Fig. 11), at a distance range from 35 cm to 75 cm, in this experimentation. The small size of the box, and hence its small size in images, even if it is not so fare, leads to challenging conditions. The tracking succeeds even in the image plane formulation thanks to the redundancy compensating the occurrence of singularities (Fig. 11). The spherical stereo tracking succeeds too and the tracking results are similar to the image plane formulation. It is however noticeable that the spheri-cal formulation is more stable than the image plane one, mainly due to the fact that it uses more image measurements, without discarding them due to a near-singularity. Figure 12 presents temporal variations for each position and orientation coordinates estimated from the box stereo sequence, and particularly on a subset of this sequence where the differences between IP and S are the most visible. The instability of the image plane formulation w.r.t. the spherical one is particularly clear, in this experiment, for the X and Z coordinates and Y and Z rotation angles.

(a) position variations in meters (b) orientation variations in degrees Figure 12: Pose variations between frames 200 and 250 of the sequence presented in figure 11, using the four mirrors. "IP" stands for "Image Plane" formulation and "S" for "Spherical" formulation.

Application to mobile robot localization

Indoor experimentation. In this experiment, we consider a massively radial model projection and mutual occlusion case. The FOO sensor is vertically mounted on a mobile robot with mirrors reflecting the ground and walls. This usual placement of the omnidirectional sensor allows to actually sense information 360 o all around the robot. Hence, it leads to the projection of vertical lines as radial lines in the image plane (Fig. 13(b)), the degenerate and even singular case of the normalized conic representation. Indeed, when using doors as the 3D model for VVS in the image plane, the tracking is lost from the first images. However, the spherical formulation of the model based VVS allows to naturally deal with the projection of vertical lines when the sensor axis is also vertical. using the spherical VVS and the trajectory estimated with an external tool, which is a camera placed at the ceiling, tracking and estimating the pose of an IR target placed on the robot. The overlapping of the two trajectories shows qualitatively the estimation precision, which is important, with stable pose estimations over time, without any filtering. Even if the trajectory obtained with an external vision tool cannot be actually considered as a "ground truth", it gives an idea of it. We then computed the error between these two trajectories and obtained 6.72 cm as mean position estimation error. This mean error leads to the mean error ratio over the trajectory length (around 11 m) of 0.61 %. Outdoor experimentation. This experiment shows the behavior of our method in more challenging conditions, traveling the robot outdoor. The robot embeds a small configuration of the FOO sensor, still with the four mirrors but with a smaller lens in order to have a sensor of the same size as a monocular one : 30 cm (fig. 15(b)). A sequence of 1500 images for a 21 m path of the robot is acquired. A building is with a partial and imprecise model of its edges. In the image sequence, the building moves in a large part of images leading to self-occlusion of the building (fig. 15). Despite these diffi- culties, the tracking succeeds all along the sequence showing the robustness of the method in real conditions with partial occlusion of the building bottom by some cars, self occlusion of the building, which also present other strong gradients on its surface. One can finally note that, in a mean, the distance between the robot and the building is around 12 m. A video presenting the two latter experiments is available at the following URL: http://home.mis.u-picardie.fr/∼g-caron/videos/MBTMobileFOO.mp4.

Discussion and conclusion

Experiments aimed to evaluate the new spherical formulation of the 3D model based tracking in omnidirectional vision and its extension to stereovision. We compared this new formulation with the image plane based one. Dealing with this spherical representation allows (i) to withdraw singular and nearly singular conic representations, (ii) to reduce the number of parameters to represent a line and, finally, (iii) to easily use an actual geometrical distance. All these theoretical interests are confirmed by the results.

Experimental results on synthetic images show exactly the situation in which the image plane formulation is defeated whereas the spherical formulation is always efficient. Based on this experimental demonstration, the results on real images show two things. First, using the image plane formulation, the redundancy brought by the FOO stereovision sensor adds enough robustness to the tracking, even if it is not perfectly stable. Then, experimental results on real images show that the use of the spherical formulation stabilizes estimations in the tracking of a box. This is furthermore the only formulation allowing to succeed to track an environment for the onboard localization of a mobile robot, when the 3D model is mainly projected as radials, which is really often the case indoor as well as outdoor. The redundancy of the four views is not enough constraining for the image plane formulation in the latter case, highlighting again the interest of the spherical formulation.

Another advantage of the technique is the processing time since, even if no particular code optimization has been led, processing is real time w.r.t. the acquisition frame rate. The tracking of a box in the monocular VVS case for images of 640 × 480 pixels (450 × 450 useful) takes around 25 ms and the tracking of doors in the last experiment with the FOO takes 45 ms for 800 × 800 pixels FOO images, with 250 moving edge sites in each view. Furthermore, the latter computation time can even be reduced since processes are parallelizable for the stereo case. Indeed, the moving edge stage is done on each image individually, as the weights computation of the M-Estimator, and these steps could be done in parallel for each view.

Of course, if image acquisition conditions are really hard for a tracking method (extremely small object in images or important displacement of the tracked object/scene between two consecutive frames), the proposed algorithm may diverge. But if the new spherical method diverges, the former image plane one will diverge too, whereas the opposite has never been observed in any of our experiments. A better behavior of the former image plane formulation over the new spherical one is, furthermore, never waited from the theoretical expressions.

To sum up, the new spherical formulation of model based pose estimation and the redundancy brought by the FOO sensor allow to make the tracking robust and stable in static or mobile situations, indoor and outdoor. Experiments show that the use of this sensor is adapted, efficient and robust, for real time onboard mobile robot localization applications.

Future works will be focused on the low level image processing side to fully adapt the moving edge to the spherical geometry and to study the interests brought by this adaptation. Finally, the mix of the proposed contour-based method with a textured-based one [START_REF] Caron | Tracking planes in omnidirectional stereovision[END_REF] is one of the perspectives of this work too, in order to merge their efficiency in textured and non-textured areas.

Figure 1 :

 1 Figure 1: The omnidirectional stereo sensor we used: orthographic camera, parabolic mirrors, 90 cm height

Figure 2 :

 2 Figure 2: Looking for the corresponding edge along normal vectors (blue) to the conic (green) in each sample (red) of the projected 3D model using the moving edge method [33].

Figure 3 :

 3 Figure 3: Projection of a 3D straight line, defined by the intersection of planes P 1 and P 2 , as a great circle on the sphere S.

Figure 4 :

 4 Figure 4: Synopsis of the algorithm.

Figure 5 :

 5 Figure 5: Projection of the 3D model of the box in images, in the perfect case (a), for another pose (b), for the "optimal pose" but with noisy intrinsic parameters (c) and finally, for the optimal pose, with a noise applied on the summits of the 3D model (d).

  (b), for various

Figure 6 :

 6 Figure 6: Same different cases than in figure 5 but the pose is such that the box edges are mainly projected as radials in the image.

Figure 7 :

 7 Figure7: Comparison of convergence rates, expressed as percentages of converged optimization over number of initializations, for the image plane (IP) and spherical (S) formulations with radials or not. "R" stands for "Robust" and "NR" for "No Robust". For the camera noised and model noised experiments, the robust estimator is always used.

Figure 8 :

 8 Figure 8: Position and orientation errors at convergence for the image plane (IP) and spherical (S) formulations without radials. "R" stands for "Robust" and "NR" for "No Robust". For the noised experiments, the robust estimator is always used.

Figure 9 :

 9 Figure 9: Position and orientation errors at convergence for the image plane (IP) and spherical (S) formulations with radials. "R" stands for "Robust" and "NR" for "No Robust". For the noised experiments, the robust estimator is always used.

Figure 10 :

 10 Figure 10: Comparison of the conditioning of interaction matrices of the VVS for the image plane (IP) and spherical (S) formulations with radials in the image plane or not. "R" stands for "Robust" and "NR" for "No Robust". For the camera noised and model noised experiments, the robust estimator is always used.

Figure 11 :

 11 Figure 11: Pose estimation for the handheld box. (a) an external view of the setup. (b-d) FOO images with the projected model (green), showing its pose is well computed all along the sequence.

Figure 13 Figure 13 :

 1313 Figure 13: Stereo spherical tracking and pose estimation of an environment mainly composed of radial lines in the image. The FOO camera is mounted on a mobile robot with mirrors reflecting the ground and walls. The spherical tracking of scene doors succeeds all along the sequence of 546 images where the robot makes a loop.

Figure 14 :

 14 Figure 14: Estimated trajectory using poses obtained from the stereo spherical VVS. The "ground truth" is obtained using an external localization tool. The important overlapping of these trajectories shows the precision of the estimation. The unit is the meter.

Figure 15 :

 15 Figure 15: Tracking buildings. (a) The outdoor environment. (b) A compact configuration of the FOO is mounted on the robot. (c) Estimated poses (red) using the stereo spherical VVS. The robot is driven along a trajectory of 21 m with 1500 acquired images. Tracking results are shown at (d) the beginning, (e) the middle and (f) the end of the robot path.


Anne Abeillé
email: abeille@linguist.jussieu.fr

Danièle Godard
email: dgodard@linguist.jussieu.fr

P Amsili

I Bartning

A-M Berthonneau

C Beyssade

O Bonami

P Cappeau

M Charolles

B Combettes

V Dahlet

M Dargnat

E Delais-Roussarie

A Delaveau

M Desmets

A Di Cristo

J Doetjes

B Fagard

F Gadet

B Kampers-Manhe

K Jonasson

J Jayez

J-P Koenig

L Kupferman

B Laca

B Lamiroy

K Lambrecht

F Landragin

J-M Léard

J-M Marandin

A Mari

L Mélis

P Miller

C Molinier

F Mouret

F Mourlhon-Dallies

W De Mulder

M Noailly

B Post

C Rossari

L Roussarie

F Sabio

C Schnedecker

I Simatos

H De Swart

D Van De Velde

M Van Peteghem

L Tasmowski

L Tovena

J Tseng

C Vet

C Vetters

M-T Vinet

R Vivès

A Zribi-Hertz

The Grande Grammaire du français project

We present a new reference Grammar of French (La Grande Grammaire du français), which is a collective project (gathering around fifty contributors), producing a book (about 2200 pages, to be published en 2011) and associated databases. Like the recent reference grammars of the other Romance Languages, it takes into account the important results of the linguistic research of the past thrity years, while aiming at a non specialist audience and avoiding formalization. We differ from existing French grammar by being focused on contemporary French from a purely descriptive point of view, and by taking spoken data into account. We include a description of all the syntactic phenomena, as well as lexical, semantic, pragmatic and prosodic insights, specially as they interact with syntax. The analysis concerns the data from contemporary written French, but also includes data from spoken corpora and regional or non standard French (when accessible). Throughout the grammar, a simple phrase structure grammar is used, in order to maintain a common representation. The analyses are modular with a strict division of labor between morphology, syntax and semantics. From the syntactic point of view, POS are also distinguished from grammatical relations (or functions). The databases include a terminological glossary, different lexical databases for certain POS, certain valence frames and certain semantic classes, and a bibliographical database.

We aim at a complete description of all syntactic phenomena of French, while also taking into account some morphological, lexical, semantic, pragmatic and prosodic aspects.

Presentation of the project

Sizable comprehensive grammars, written by linguists but aiming at a larger audience, have been published for several European languages: for Italian, the Grande Grammatica italiana di consultazione (Renzi, Salvi, & Cardinaletti, 1989-1991), for Spanish the Gramatica descriptiva de la lengua española (Bosque & Demonte, 1999), for Catalan the Gramàtica del Català contemporani [START_REF] Solà | Gramàtica del Català contemporani[END_REF], for English the Cambridge Grammar of the English Language (Huddleston & Pullum, 2002).

Nothing of the sort exists for French. Most grammar books are normative, while our aim, as linguists, is purely descriptive. The best existing descriptive French grammars are smaller, written by 2 or 3 authors, and mainly based on written literary French: M. Grevisse, A. Goosse (1936[START_REF] Grevisse | Le Bon Usage[END_REF] Le Bon Usage, M. Riegel et al. (1994[START_REF] Riegel | Grammaire Méthodique du français[END_REF], La grammaire méthodique du français. While useful, they are based on a "rules and exceptions" approach, with no clear-cut distinctions between grammatical relations and categories, between syntax and semantics. They are not focused on the modern stage of the language, including references to 17 th or 18 th century authors such as Corneille or Marivaux. 1Our project, called the Grande Grammaire du français is an international research project, which started in 2001 and was sponsored by the French CNRS. It will produce a book of about 2200 pages, to be published in 2011, and several associated electronic databases. It is a collective project, with 3 editors (Anne Abeillé, Annie Delaveau, Danièle Godard), gathering more than 50 authors, from different countries. 2 Like other large grammars for other Romance languages, it includes the results of recent linguistic research and makes them accessible to a larger audience. It is systematic, fully explicit, but not formalized ; it uses both conventional and modern terminology, with explicit definitions and a rich array of examples.

intuitions as well as on large available corpora, both written and spoken. They also include regional variation (esp. Belgium, Switzerland, Quebec) and non standard data. To ensure maximal coherence and readability, it is based on a surfacist phrase structure grammar (with a fixed set of parts of speech and grammatical relations).

Our objectives are as follows:

-describe the variety of uses in modern French, both written and spoken, in France and abroad; -synthetize existing linguistic knowledge, complete it, and make it accessible also to non linguists; -clarify and stabilise grammatical terminology (often obscure, not modular, and variable from one French grammar to another).

As our focus is on « ordinary » French (to use Gadet's 1989 term), the data we analyse come from a variety of sources:

-excerpts from available written corpora (Frantext, newspapers such as Le Monde, le Soir…) -excerpts from spoken available corpora (Elicop, Ester, Corpaix, C-oral-rom, CRFP, Sankoff-Cedegren corpus…) -constructed examples for simple cases and ungrammatical forms. For language variation, we also rely on a team of "regional" experts coordinated by Françoise Gadet.

The book

The book will comprise 7 millions characters (about 2200 pages), with two parts : La phrase et ses constituants (the clause and its constituents), De la phrase au discours (from sentence to discourse). It is divided into 20 chapters concerned with all major problems of French syntax, and their interaction with the lexicon, morphology, semantics, discourse and prosody. Five chapters are each devoted to a part of speech (POS) : Verb, Noun, Adjective, Preposition, Adverb, while three are devoted to a certain type of phrase (Sentence, Relative clause) or construction (Complex predicates, Coordination). Some chapters are more semantically oriented : those on determination and quantification, on proforms, on negation, on tense, aspect and mode, on comparison or adverbial adjuncts (« circumstances »). We also have a chapter on clause types, one on word order, one on discourse, one on phonology and prosody, one on punctuation and the new writing codes (such as those found in SMS). We also include a glossary and a comprehensive bibliography.

The data

We describe a variety of data from modern French (after 1950), based on common usage, both written (newspapers, novels) and spoken (radio, interviews).

We define an appropriate marking to distinguish whether these data are attested or not, grammatical or not, standard or not, spoken or not, regional or not (see Table 1). We call «non standard» those existing forms which are sociologically stigmatized. We call « regional» forms found in several regions and not stigmatized there. For exemple, the comparative correlatives are introduced a comparative adverb (or adjective) in standard French (1a), but in Belgium and Provence, they may be introduced by au preceding the comparative form :

(1) a Plus il pleut plus il se plaint (the more it rains the more he complains) b R Au plus il pleut, au plus il se plaint

Similarly, causative constructions with faire where the causee precedes the infinitival verb are marked: usually considered as typical of Quebec, they are in fact also found in Britany and in the French West Indies, so we mark them as regional:

The terminology

We propose a consistent and workable terminology, which takes linguistic results and generalizations into account. We try to keep existing terms, but redefine them with explicit linguistic properties.

We define 10 morphosyntactic tags for POS ( We also define 11 grammatical relations (or functions) associated with words and phrases (see Table 3).

Our linguistic choices are based on modularity. POS are on morphological and distributional criteria and are carefully distinguished from valence frames and from grammatical relations. In most French grammars and lexicons, these notions are not kept apart. For example, in most lexicons, the form avant (before) receives at least three different POS: Preposition, when followed by an NP as in (3a), Adverb when used alone (as in 3b), and Subordinating Conjunction when used with 'que', as in (3c). When it comes to verbs, a verb like attendre (wait) allows for the same continuations and does not change its POS (3d-f). We thus propose to have one word avant with one POS Preposition and three different valence frames: an NP complement, no complement or a sentential complement.

(3) a Je viendrai avant le match (I'll come before the game) b C'était mieux avant (it was better before) c

Je viendrai avant que tu partes (I'll come before you leave) d J'attends (I wait) e J'attends le match (I wait for the game) f J'attends que tu partes (I wait for you to leave)

We also keep POS separated from grammatical relations.

Nouns for example can modify other nouns ; this does not turn them into adjectives, as shown by the fact that they keep their gender, as maison in (4a), and possibly their number (4b). Similarly, adjectives may modify ; they are not analysed as adverbs because they lack a characteristic property of adverbs, which is to occur between auxiliaries and past participles without an intonation break ; thus, the adjective net in (4d) contrasts with the adverb nettement in (4e).

The structures

Throughout the book, we use a simple phrase structure representation for all the constructions presented. We do not force binary branching structures, and use "flatter" than usually assumed, for exemple for tense auxiliaries (see Abeillé & Godard 2002 for discussion). Figure 1 shows the structures assumed for tense auxiliary constructions : a lu Proust (has read Proust) and est allé à Rome (is gone to Roma). We do not define Sentences by the usual NP-VP schema, which is too restricted to account for the data. We define sentences (the non elliptical ones) as saturated phrases with a predicative head. Accordingly, they may not have a syntactically realized subject nor a verbal head.

VP VP

Head Cplt Cplt Head Cplt Cplt V V NP V V PP a lu Proust est allé à Rome
We follow [START_REF] Miller | Clitics and constituents in Phrase Structure Grammar[END_REF] in analysing weak personal profroms as verbal affixes, that leads us to simpler structures than usual for certain sentences. An imperative clause with a proform complement (such as regarde-le 'watch him') is a sentence reduced to a verbal head. An interrogative clause with a postverbal proform subject (such as comment va-t-il ? how is he going') can be reduced to an extracted adverb and a verb (Figure 2).

S S

Head Extracted Head V Adv V Regarde-le ! Comment va-t-il ?
Figure 2: Weak proforms as affixes

We also analyse as sentences (clauses) certain verbless utterances, with a propositional meaning, and a non verbal head. Figure 3 shows the structure associated with an interrogative clause with an adverbial head Combien le café ? (How much for the coffee ?) and a clause with a prepositional head, with the pragmatics of an order Les petits en scène ! (the small ones on stage !): There are few places where this choice raises a difficulty. One is the case of antecedentless relative clauses (7). The first phrase is a wh-phrase which functions both as the antecedent of the relative clause and the extracted phrase in the relative clause. Thus, if the extracted phrase corresponds to a PP in the relative clause, it is only acceptable if the same PP is required in the matrix clause (hence, the contrast between (7b) and (7c)).

(7) a J'ai rencontré [qui m'avait invite] (I met who had invited me) b J'ai discuté [avec qui tu avais discuté] (I discussed with whom you had discussed) c *J'ai rencontré avec qui tu avais discuté (I met with whom you had discussed)

We can represent this structure as a fragment : a structure without a head (as answers often are, as in When will you be here ? -Tomorrow). Thus, the matrix PP direcly dominates a relative clause whose extracted constituent is also a PP. But, to express the fact that one phrase plays two roles, it is necessary to use either feature structures, or to allow a representation which is not strictly a tree, since an exoression can be dominated by several (two) branches (as do Huddleston and Pullum 2002).

PP

The databases

We define several lexical databases, for closed classes of grammatical words (like complementizers, determiners), for certain valence frames (subjectless verbs, verbs with a predicative complement), and for certain semantic classes (degree and associative adverbs, locative prepositions). An example is

  based on syntactic criteria rather than semantic ones. The subject relation for example is based on verb agreement, canonical preverbal position, the possibility of realisation by weak forms such as il and ce, and the impossibility of realisation by other weak forms like en. Objets, on the other hand, are defined by the possibility of realisation by different possible weak proforms (le,la,les for definite objects, en for indefinite objects) and canonical postverbal position. With this definition, NP subjects can play different semantic roles; for example: agent for Paul in (5a), location for ce tiroir (5b), goal for Paul in (5c). Postverbal NPs in (6a,c) on the other hand are analysed as objects, even though they correspond to the first semantic argument of the verb arrive in (6a) and of the infinitive in (6c), because it can be realised by en in the first case (indefinite), and by le in the second one (definite). (5) a lave Paul ? (What is Paul washing ?) b Ce tiroir contient de l'argent (This drawer contains money) c Paul reçu des livres (P has received books)

Figure 1 :

 1 Figure 1: Tense auxiliaries

  Figure 3: Verbless sentences

Figure 4 :

 4 Figure 4: Antecedentless relative clause (6b)

table 2

 2 Paul viendra (Paul will come), Quel plaisir ce cours (What a pleasure this class) ! Tous dehors ! (Everyone outside)

	).

-PP (prepositional phrase) : chez lui, à Paris (in Paris), depuis mars (since March) -VP (verb phrase) : va à Paris (goes to Paris), tout lire (read everything), ne pas comprendre not understand) -Sentence :

Table 3 :

 3 Grammatical functions

Table 4 :

 4 Table4which gives an exhaustive list of French determiners. Note that we include some originally complex expressions which have evolved into lexemes. French determinersThe other items which can combine with a Noun to form a Noun Phrase are adjectives (cardinals, indefinites such as quelques and adverbs (degree quantifiers such as beaucoup). Adjectives may combine with determiners (8a,b) whereas determiners are defined as non combining with one another (8c)). Adverbs, contrary to Determiners, may not combine with unmarked Nouns, and must be followed by de (8d,e) (a preposition used with the marker function): Another example is Table5which gives an exhaustive list of non reflexive verbs taking tense auxiliary être in standard modern French.

	definite			Ce, interrogative), son le, lequel	(relative,
				aucun, un, un de ces (+ Nsg), un
				certain, certains, chacun, chaque,
	Indefinite	and	de, de ce, de son, du,	nul,
	quantifiers		plusieurs, quel, quelque (sg), zéro,
				je ne sais quel / lequel, on ne sait
				quel / lequel, Dieu sait quel / lequel
	Free choice items nimporte quel, tout
	a	Mes trois chats (my three cats)
	b	Ces quelques fleurs (these few flowers)
	c	*Ces mes chats (these my cats)
	d. *Beaucoup chats
	e. Beaucoup de chats (many cats)
				advenir, aller, (ré)apparaître,
				arriver,	décéder,	(re)devenir,
	Only être		intervenir,	naître,	(re)partir,
				parvenir,	provenir,	rester,
				(re)venir, survenir, mourir
	Être in intransitive use	(re)monter, (re)descendre, retourner, rentrer, tomber (re)sortir, (r)entrer,
	Free between être and alternation avoir	accourir, demeurer, (dis)convenir (de), éclore, échapper, échoir, expirer, passer, (dis)paraître, résulter (de)

Table 5 :

 5 Non reflexive verbs taking auxiliary être in standard FrenchIn French, the default tense auxiliary is avoir, but all reflexive verbs take être (9a,b). A small subset of intransitive verbs also take être, in at least one of their uses, and they can be analysed as (a subset of) non agentive intransitive verbs (Marandin 1999):(9) a Jean a lavé le chien (Jean has washed the dog) b Jean s'est lavé (Jean has washed himself) c. Jean est parti / mort (Jean has left/died) d. Un train a/est passé (a train has passed) Another type of database is devoted to certain polymorphic grammatical words, which are notoriously difficult to analyse. Table6shows our analysis for the form que, which can correspond to a Proform, to a Complementizer or to an Adverb.ComplementizerJe pense qu'il viendra (I think that he'll come) (complement sentence) Le livre que je lis (the book that I read) (relative clause) Plus grand que toi (bigger than you) (comparative clause)

	POS	Examples
	Proform (wh-interrogative word)	Que voulez-vous ? (what do you want?)
	Degree adverb
	(wh-exclamative	Que de bruit ! (what a noise)
	word)
	Associative	Paul ne voit que toi (Paul sees only
	Adverb	you)

Table 6 :

 6 French que

For a more detailed comparison with other grammars, includingDamourette & Pichon (1911-1946), seeAbeillé, Delaveau, Godard

Site of the project

The project has a web site with a closed part, for the authors, and an open part. http://grammaire.univ-paris-diderot.fr/


Anne Abeillé

Danièle Godard

Frédéric Sabio

Stefan Müller

. Accordingly, we rely for the data on spoken corpora (Corpaix, CFRP) as well as on our intuitions. Both constructions involve a partitioned semantics but this mode of composition is associated with different effects. One construction is characterized semantically: the preposed NP is the theme of a categorical proposition. The other construction is characterized pragmatically: it is associated with an independent declarative clause, a typical use of which is to signal a break in the interaction.

1 This study is part of the PROGRAM project on the interface of prosody with syntax and semantics in French prosody. Aspects of it have been presented at the Workshop on spoken corpora (Lyon, January 2008), at the CMLF (Paris, July 2008) and at the CIL conference (Seoul, July 2008). We thank for their comments José Deulofeu, Caroline Féry, Jacques Jayez, Manuel Leonetti, Jean-Marie Marandin as well as the audiences at these events. 2 Corpaix is a corpus of spoken French collected by the GARS at University of Provence (it consists mostly of interviews). CRFP is a spoken corpus, funded by DGLLF, collected at the beginning of this century in several French towns and balanced for sex, age, social status. We follow common transcription practice in not having punctuation marks nor capital letters in examples taken from spoken corpora. 3 Throughout the text, we give glosses rather than translations.

Introduction 1

Our aim is to contrast two types of complement fronting in French, that can be found in declarative clauses, and frequently occur in every day speech. Examples of construction 1 and construction 2 are given in ( 1) and (2), respectively. The preposed NP is italicized. 2,3 (1) a. Le chocolat j'adore. (Chocolate I adore) b. Paris je connais pas [CFRP] (Paris I don't know)

(2) a. huit ans je devais avoir [CRFP] (eight years I must have had) b. des moulins à légumes ça s'appelait [CRFP] (vegetable mills that was called)

The NP complement occurs as a left peripheral element, before the (pronominal) subject. Such examples clearly differ from well known cases of preposed NPs in French, which are commonly found in (clitic) left dislocation (3a), as well as wh questions (3b) or wh exclamative clauses (3c) :

(3) a. Marie, tout le monde l'adore. (Marie everyone loves her) b. Quel âge il avait ? (Which age he had ?) c. Quel chapeau tu as ! (What a hat you have !)

We systematically contrast the syntactic, semantic and pragmatic properties of the two constructions, before presenting a formal analysis in HPSG. While the syntactic difference does not correlate with an information structure distinction (see [START_REF] Prince | On the Limits of Syntax, with reference to Left-Dislocation and Topicalization[END_REF][START_REF] Prince | How not to mark topics: 'Topicalization' in English and Yiddish[END_REF][START_REF] Fanselow | Left peripheral focus: mismatches between syntax and information structure[END_REF] for a similar conclusion about complement fronting in English, and in German and Czech respectively), it clearly correlates with other distinctions, one construction being characterized semantically, and the other being characterized form an illocutionary point of view (it corrresponds to a speech act).

Although they have been largely overlooked in the literature, some of their properties have been studied by [START_REF] Pohl | Documents pour servir à l'étude des phrases du type Les fleurs, j'aime[END_REF][START_REF] Sabio | Micro-syntaxe et macro-syntaxe : l'exemple des compléments antéposés en français[END_REF][START_REF] Sabio | L'antéposition des compléments en français contemporain : l'exemple des objets directs[END_REF][START_REF] Zribi-Hertz | Relations anaphoriques en français; esquisse d'une grammaire générative raisonnée de la reflexivité et de l'ellipse structurale[END_REF][START_REF] Zribi-Hertz | L'Anaphore et les pronoms[END_REF][START_REF] Hakihiro | Contribution à l'étude de la valence verbale en français contemporain, la non réalisation du complément d'objet direct[END_REF]. A prosodic study still remains to be done.

Syntactic properties

In the two constructions (cx 1 vs cx 2), illustrated in (1) and ( 2) above, the preposed NP is associated with a grammatical function within the sentence. More precisely, these sentences contain a predicate with an unrealized syntactic argument (an object in (1), (2a), a predicative complement in (2b)), which is somehow linked to the preposed NP. This contrasts with a hanging topic as in (3).

(3) a. Le cinéma alors on se décide ? (the movie then we make a decision?) b. euh la mairie de Saintes on connaît le le candidat socialiste qui vient de se déterminer [CRFP] (hum the town council of Saintes we know the the socialist candidate who has just made his decision)

Let us first look at cx 1. The preposed constituent is a NP or a VP (4a) ; but the missing constituent always has the same grammatical function: it is an object. Moreover, the verb belongs to the class of verbs which take an optional complement (4b):

(4) a. Travailler seule, je ne supporte pas (Working alone, I can't stand) b. A : Tu aimes le chocolat ? (you like the chocolate?) B : J'adore.

(I adore)

Accordingly, a sentence identical with that of cx 1, but without the preposed NP, remains grammatical. The relation between the preposed NP and the missing object can be a long distance dependency; however, it does not obey island constraints (the missing object can be within a relative clause or an adjunct clause):

(5) a. (The F3 appartments, one must be crazy to stand)

In fact, construction 1 has the properties of left dislocation, with an anaphoric relation between the preposed constituent and some pronominal element in the sentence. Indeed, the missing object alternates here with the general pronoun ça ('that', 'it'), which is attested in our corpora:

(6) a. Le chocolat j'adore ça. (the chocolate I adore it) b. la montagne j'a-j'adore ça [CFRP] (the mountain I adore it) c. les expressions modernes j' j'aime pas trop ça quoi [Corpaix] (the modern expressions I don't like that that much you know)

It is well known that clitic left dislocations do not obey island constraints (e.g. (Marie, you must be crazy to bear with her)

As is the case with clitic-left-dislocations, we can have (another) leftdislocated NP, which is not ordered with respect to the preposed NP:

(8) a. Moi, le chocolat, j'adore (ça) (me, the chocolate, I love (that)) b. Le chocolat, moi, j'adore (ça) (the chocolate, me, I love (that))

Thus, construction 1 can be analyzed as involving a left dislocated NP, linked to an unrealized pronoun (Zribi Hertz 1986). 4 It remains to be shown that the verb has a null pronominal complement rather than being detransitivized. The faire causative construction provides a test for transitivity (e.g. [START_REF] Abeillé | Two Kinds of Composition in French complex In[END_REF], since the causee is marked with preposition à with a transitive infinitival verb and is unmarked otherwise. In (9b), manger (to eat) is detransitivized.

(9) a. Il fera manger la soupe aux enfants / *les enfants (He will make eat their soup to the children / the children ) b. Il fera manger les enfants / *aux enfants (He will make eat the children / to the children) c Les F3, rien ne pourra leur faire aimer / *les faire aimer (F3, nothing will make like to them / them)

In cx 1, the causee must be marked with à (9c) and is thus transitive. We propose that the object is a null pronoun, interpreted like 'ça' (that) [START_REF] Zribi-Hertz | Relations anaphoriques en français; esquisse d'une grammaire générative raisonnée de la reflexivité et de l'ellipse structurale[END_REF][START_REF] Zribi-Hertz | L'Anaphore et les pronoms[END_REF]. We thus call cx 1 the 'ça-dislocation-construction'.

In cx 2, on the other hand, the preposed constituent can be of various categories (NP, AP, PP, AdvP); it is linked to a missing constituent with various syntactic functions (object, predicative complement, oblique complement or specifier) . The fronted complement enters into long distance dependencies (11a), and can correspond to an obligatory complement (as with avoir, (11b)). Contrary to what we observed with cx 1, it is difficult to insert a pronoun (in the place of the missing constituent), or it has a different meaning (11c). We conclude that the preposed NP is a filler in cx 2. Example (10a) confirms the filler gap dependency, since it contains an object of the form de N, which is only licensed by negation or the extraction of a specifier (*Il avait de retard, le train, It had of delay, the train, vs Combien il avait de retard, le train ? How much did it have of delay, the train, Abeillé et al. 2005).

Another property shared by cx 2 and extraction constructions is the possible occurrence of the complementizer que after the fronted element in non standard varieties (noted with %):

(13) a. %Trente euros que ça m'a coûté ! (30 that it costed me) b. %Où que c'est que je vais le mettre ? [corpaix] (where that it is that I am going to put it ?)

To summarize, the preposed NP in cx 1 is a left dislocated phrase, while it is a filler in cx 2.

Semantic Properties

The two constructions also contrast semantically. First, the content of cx 1 involves a general proposition, while cx 2 is not so constrained; second, the proposition in cx 1 is categorical, while it is thetic in cx 2.

General vs unspecified proposition

With cx 1, the content of the sentence is general: neither the NP nor the sentence can be associated with the denotation of a particular. The NP denotes a kind, a type, or an abstract object. Hence the contrast between (14a) and (14b,c). If a proper name is preposed as in (14d) (from Grevisse and Goosse 2008), it cannot refer to an individual but to the property of having this name (in the context of parents choosing a name for their baby for example):

(14) a. La musique classique, je (ne) connais pas bien / j'apprécie beaucoup. [cx 1] (the classical music, I don' know very well / I appreciate a lot) b. ??Ton offre / Ton frère, tu sais que tout le monde apprécie.

(your offer / your brother, you know that everybody appreciates) c. ??La musique classique, j'ai apprécié dans ce concert.

(the classical music, I appreciated in the concert) d. Marine, j'aime bien. ((being called) Marine I like) Confirmation of the constraint is given by the fact that, instead of an NP, one can have a preposed infinitival VP object, denoting a property (4a). In addition, the verb is typically an individual level predicate with respect to its object [START_REF] Kratzer | Focus and presupposition in dynamic interpretation[END_REF], like apprécier (to appreciate), adorer (to adore), détester (to hate), haïr (to hate), ne pas supporter (to not stand), ignorer (to ignore). It can be a stage level predicate, if there is a quantification or an habitual or iterative aspectual operator, such that there is no unique situation associated with the sentence; rather it describes a period over time, or a generalization over a behavior.

(15) Ce genre de repas, simple mais avec de bons produits, [

categorical vs thetic proposition

The second semantic difference between the two constructions concerns the (logical) form of the proposition: it is categorical in cx 1, while it is thetic in cx 2. We take the distinction between thetic vs categorical sentences to be semantic rather than structure informational (see [START_REF] Ladusaw | Thetic and categorical, stage and individual, weak and strong[END_REF][START_REF] Kim | Information articulation and truth conditions of existential sentences[END_REF] argues that the distinction may induce different truth conditions). In a categorical proposition, a property is predicated of (the referent of) an argument, which is a (sentence) theme, whereas a thetic proposition describes a situation as a whole.

It is important to distinguish the notion of Discourse Topic, which belongs to the domain of Information Structure, from that of a sentence theme, which characterizes the semantic role of the distinguished argument in a categorical proposition. However, it is notoriously difficult to associate distinctive properties with the notion of sentence theme [START_REF] Jacobs | The Dimensions of Topic-Comment[END_REF]. 5 We rely here on four properties; the two first ones are borrowed from Jacobs' notion of semantic subject; the two others come from Marandin's distinction between categorical and thetic propositions ( 2007). (i) the theme is a semantic argument of the predicate; (ii) it occurs to its left and is the last to saturate it;

(iii) it shows an affinity with definite NPs; (iv) a categorical proposition favors individual level predicates (specially dispositions).

Cx 1 exhibits the four properties. The head sentence is turned into a predicate, because the null pronoun semantically contributes a variable; this predicate is saturated by the preposed NP. As we have seen, the verb denotes a disposition of its subject, due to its lexical semantics (like aimer, ignorer) or to its aspectual or quantificational environment, which induces a generalization. In addition, the preposed NP is definite (see the examples above, (1),( 3),( 5),( 7),(14a),( 15)), or has an affinity with a definite NP ( 17).

( Finally, dispositions are not favored (any predicate is possible, and there is no aspectual constraint). Since the proposition in cx 2 is not categorical, then, it is thetic. To conclude, the two constructions share their compositional mode: both are characterized by a partitioned semantics (e.g. Krifka 2001), where the sentence translates as a predicate, which is saturated by application to the (denotation of the) preposed NP. However, they crucially differ both regarding the type of content (general vs unspecified) and the type of logical form (categorical vs thetic proposition). While there is no correlation a priori between the generality of the proposition and the other properties, it is tempting to relate partitioned semantics and a categorical proposition. Construction 2 shows that this would be wrong: there is no correlation between a compositional mode and a specific logical form for the sentence. In other words, a thetic proposition is perfectly compatible with a partitioned semantics. While the distinguished element of cx 1 plays a special role in semantics (it is the theme in a categorical proposition), it is not the case in construction 2. As we see below, the charaterization of construction 2 is at the illocutionary level.

which is an answer to the question Elle était socialiste ? She was a socialist?; interaction noted on the fly); in such cases, the preposed NP may well correspond to a narrow focus, as it does in question-answer pairs. Alternatively, the partition of the sentence serves to make the utterance more dramatic, highlighting that part of the situation that seems particularly worth of notice (as in (16b)), and justifying a pause in the discourse, a reaction on the part of the audience etc. The role of the preposed NP then appears to be that of a figure (reminiscent of the figure-ground distinction in cognitive linguistics). 8 We call the whole construction a 'dramatic extraction'.

HPSG analysis

We use the construction-based version of HPSG [START_REF] Ginzburg | Interrogative Investigations[END_REF] to account for our constructions, which are both based on a partitioned semantics.

Partitioned semantics

We assume the following hierarchy of semantic objects, which are possible values for CONT, (abst(raction), appl(ication) are taken from Webelhuth 2007):

sem-obj message soa abstr appl scope-obj rel index prop question outcome param quant-rel

As in Ginzburg and Sag 2000, message is the type of content appropriate for clauses, proposition for declarative clauses, question for interrogative clauses 8 The content of cx 1 is a categorical proposition (section 3.2). On the other hand, the preposed NP is acceptable at the beginning of an interrogative sentence (25c), whose content is not of type question [START_REF] Ginzburg | Interrogative Investigations[END_REF]. This would be problematic if the expression with which the preposed NP combines were of type question.

However, there are reasons to think that it is the entire clause, including the preposed NP, that is interpreted as a question, rather than the segment of the clause which with an interrogative word (est-ce que in (25c)). This move is required for the combination of some initial adverbs with declarative and interrogative clauses, where the entire clause comprising the initial adverb (which occurs before the wh constituent) must be interpreted as an interrogative clause [START_REF] Bonami | Adverbes initiaux et types de phrases en français[END_REF]. Given that a proposition is an element of a question [START_REF] Ginzburg | Interrogative Investigations[END_REF], we assume that a proposition is available to combine with the preposed NP in (25c). and outcome for imperative clauses. Soa are descriptions of situations. Following Webelhuth 2007 who shows the necessity of a more articulated semantics than is usuallly done in HPSG, in particular for head-filler phrases, we propose the type of partitioned soa. We add the PARTITION (partitioned/ non-partitioned) dimension to the REALITY (realis/irrealis) and POLARITY dimensions (pos/neg) in Ginzburg and Sag 2000. A partitioned-soa is the basis of a categorical proposition (the content in cx 1) and of the content of a centered-clause (cx 2). It has a nucleus of type application, that applies a (lambda) abstraction to an argument:

(29) partitioned-soa => QUANTS list(quant -rel) NUCLEUS application application => FUNC abstr ARG sem -obj abstraction => VAR param BODY soa
The parameter in abstr corresponds to an argument with respect to which the predicate is not saturated (e.g. the subject combining with the content of a VP), or a SLASH value (in extraction constructions), or a STORE value (in dislocations). We assume a two dimensional classification of the content of propositions:

proposition GENERALITY LOGICAL-FORM particular general categorical thetic cx-1-content
We define the content of a categorical proposition, which is the type of content appropriate for cx 1, as based on a partitioned-soa, with a feature THEME whose value corresponds to the argument saturating the predicate. We give it a list value in order to account for relative thematicity when there are several themes (as in the case of multiple left-dislocations):

(30) categorical-proposition => SITUATION s SOA partitioned -soa NUCLEUS appl (abstr)([1]) THEME < [1] > + L

Left-dislocated constructions

For the syntax of cx 1, we rely on a more general analysis of left-dislocations.

Following Engdhal and Vallduví 1996[START_REF] Balari | Pronouns, variables and extraction[END_REF][START_REF] Alexopoulou | On Linkhood, Topicalization and Clitic Left Dislocation[END_REF], we analyze left-dislocations as follows: (i) there is a specific head-dislocated phrase (32); (ii) pronominal elements (including ça and the null pronoun in cx 1) optionally put a parameter in the STORE value (31); (iii) this parameter is coindexed with the left-dislocated phrase. The null pro has a non-canonical SYNSEM, with an index of type abstract-object, like that of ça in French:

(31) ça

CONT [1]parameter [INDEX abstract -obj ] STORE {([1])} pro non -canonical CONT parameter [1] [INDEX abstract -obj ] STORE {([1])} (32) head-dislocated-phrase CAT sentence CONT categorical -prop SOA partitioned -soa NUCLEUS appl ([3],[1]) THEME < [1] > + L STORE S HD -DTR [2] CAT sentence CONT SOA | NUCLEUS [3]] [ ] STORE { [INDEX i]]} S DTRS < CAT NP or VP CONT [1][INDEX i] , [2]>
We define ça-dislocation, the type of cx 1, as follows:

(33) ça-dislocation-cx head-dislocated-phrase & CONT general -proposition THEME < [abstract -obj ] >
An example is the following:

CAT sentence CONT general -categorical -proposition SOA | NUCLEUS appl ([3],[2]) THEME < [2] > STORE { } peripheral head CAT NP CONT [2][INDEX i] CAT sentence CONT [SOA | NUCLEUS [3] STORE {[1][INDEX i]}
le chocolat j'adore

Dramatic extraction construction

For the syntax of cx 2, we rely on the general head-filler schema proposed in Bouma et al. 2001.

(34) head-filler-phrase

CAT [HEAD [0] verbal] SLASH S HEAD -DTR [2] CAT | HEAD [0] SLASH {[1]} S DTRS < [LOC [1]],[2] >
We associate cx 2 with a type of clause with a 'center' (which can be a narrow focus or a figure in an all-focus utterance).9

(35) centered-clause

declarative-clause CONT proposition SOA partitioned -soa NUCLEUS appl ([3], [1]) CENTER [CONT [1]]
We define a centered-clause as a subtype of declarative clause, assuming the following (partial) hierarchy of clauses:

clause decl-clause inter-clause imper-clause … centered-clause …
The dramatic extraction, the type of cx 2, combines a head-filler phrase with a centered proposition (where IC stands for 'independent clause').

(

) dramatic-extraction-cx head-filler-phrase & centered -clause CAT [IC +] CONT proposition SOA partitioned -soa NUCLEUS appl ([4],[2]) CENTER [CONT [2]] HEAD -DTR [3] CONT | SOA | NUCLEUS [4] SLASH {[1]} DTRS < [LOC [1] [CONT [2] ], [3] > 36
An example is the following: contrast between two types of NP preposing in French leads to three conclusions. First, the need for constructions as clusters of unrelated properties. It is tempting to link the pragmatics of cx 2 with filler status of the NP. This would be wrong: an argument PP can be a filler, without acquiring the same pragmatic properties (for a comparable point, see Prince 1998); moreover, the preposed NP in cx 2 is not associated with a unique informational status: it can be a narrow focus or part of an all focus utterance. Second, the need for (at least) two types of saliency. A partitioned content can highlight a constituent because it is a (semantically salient) theme (in a categorical proposition) or because it is a (pragmatically salient) figure (in a centered proposition). Third, the lack of correlation between an all focus utterance and a thetic proposition. In this analysis, the content in construction 2 (the dramatic extraction construction) is a thetic proposition; yet, the proposed NP can function as a narrow focus. This non coincidence is expected if we are right to clearly distinguish between the two dimensions; but it is worth noticing, and promises new developments: we expect cross classification; for instance, do we have an all focus utterance with a categorical proposition (the sentence God is eternal may be a candidate)? Do we have a thetic proposition with an NP functioning as a figure, as well as a narrow focus (this may be the analysis for the clefted NP in C'est la police qui arrête le voisin, It is the police who are arresting the neighbor, as an answer to 'What is happening?).

(

  10) a. [Trois heures] NP , il avait de retard, le train [specifier] (3 hours, it had of delay, the train) b. [Extrême-gauche] NP , elle était. (extreme-left she was) [predicative] c. [Place de la Nation] NP , on est allés. (Nation square we went) [oblique] d. [A moitié anglaise] AP elle était. (half British she was) [predicative] e. [A une sorcière] PP tu ressembles. (to a witch you resemble)[oblique] f. [Gentiment] AdvP il s'est comporté. (kindly he behaved)[oblique]

(

  11) a. Huit ans, je crois qu'il avait à l'époque. (eight years I think he had at the time) b. A. Il avait quoi, huit ans, à l'époque ? (he had what, eight years, at the time?) B. *Il avait. (he had) c. Huit ans, elle les avait / ?elle avait ça. (she had them / that = 'at least that') Moreover, contrary to what we have seen with cx 1, cx 2 obeys island constraints: (12) a. *Huit ans, je ne vois personne [qui ait ici]. (eight years, I see nobody who has here) b. *18 ans on ne peut pas voter [sans avoir]. (18 years you can't vote without having)

  Le chocolat, elle a dit [qu'elle adorait]. (the chocolate, she said that she adored) b. Le chocolat, je ne connais personne [qui n'aime pas]. (the chocolate I know nobody who does not like) c. Les F3, il faut être fou [pour supporter].

  cx 2] (45mn, one can wait for the bus, on this line) b. et là, tu sais ce qui lui est arrivé -une antenne ils lui ont jeté sur la tête [Corpaix] [cx2] (and then, you know what happened to himan antenna they threw to his head) c. La musique classique, j'en ai justement écouté hier. (the classical music, I listened to some yesterday) [clitic left-dislocation]

cx 1] (this type of meal, simple but based on good products) a. je n'avais encore jamais mangé à Paris. (I had never eaten in P., yet) b. on trouve rarement aujourd'hui. (one rarely finds nowadays) c. ??j'ai justement mangé hier. (I ate yesterday actually) Neither cx 2, nor the dislocation with an explicit pronoun (other than ça), are similarly constrained; they can denote a general (16a) or a particular proposition (16b,c):

(16) a. 45mn, on peut attendre le bus, sur cette ligne.

[

  Horreur, je lui faisais, docteur. [cx 1] [R. Forlani, Ma chatte ma folie, 1992, 15] (horror she had of me, doctor) c Des clopinettes il m' a donné. (peanuts he gave me)

	In fact, the preposed NP is not constrained semantically; in particular, it can
	be non referential, being a predicative NP (20a,b) or an idiom chunk (20c): 6
	(20) a. j'ai écrit dans le journal local d'Aire-sur-la-Lys je me rappelle plus
	maintenant ah l'Echo de la Lys // ça s'appelait je crois bien
	[cx 2] [Corpaix] (I wrote in the local newspaper fo Aire-sur-la-Lys I
	cannot recall now Ah l'Echo de la Lys it was called I think)
	b.
	2006] (11 hours she stayed with the judges)
	b. tu l'as pas vu une seule fois aux informations -pas une fois tu l'as
	vu [cx 2] [Corpaix]
	Definite NPs are not impossible, but not favored.
	(19) a. Tu sais ce qui est arrivé ? Le candidat du patron, ils ont refusé !
	(You know what happened ? The boss' candidate they refused !)
	b. A: Je cherche mes lunettes.	(I'm looking for my glasses)
	B: Tes lunettes, tu cherches ? (Your glasses you're looking for ?)

17) a. ??un repas simple, avec de bons produits, on sert rarement au restaurant à Paris. [cx 1] (a simple meal, based on good products, one rarely finds in a restaurant in Paris) b. un repas de ce genre, on sert rarement au restaurant à Paris. [cx 1] (this type of meal one rarely finds in a restaurant in Paris)

Cx 2 clearly has properties (i) and (ii). The gap is semantically a variable, as has been proposed in a general way for filler head constructions (see

[START_REF] Webelhuth | Complex Topic comment Structures in HPSG[END_REF]

for an implementation in HPSG), hence turning the head sentence into a semantic predicate. This predicate is saturated by application to the filler. The preposed NP is also to the left of the predicate. But the parallelism between the two constructions stops there. The NP is preferably an indefinite; in particular, measure expressions are frequent (denoting duration, frequency, age, a sum of money):

(18) a. Onze heures elle est restée chez les juges [cx 2] [Canard Enchaîné,

See Laurens 2008 for similar examples of right dislocations with null pronouns.

There can be a hierarchy of themes, a question which we leave aside here, see[START_REF] Webelhuth | Complex Topic comment Structures in HPSG[END_REF]

Notice that the fronted NP in (20b) is light in the sense of[START_REF] Abeillé | French word Order and lexical Weight[END_REF] Only special stress enables it to be extracted.

It can also be used as an initial move in routine situations (such as shop interaction): -Des oranges, il me faut. (oranges, I need), which can be analyzed as genre-relevant utterances[START_REF] Ginzburg | Semantics and Conversation[END_REF].

We assume that in idiomatic expressions such as (20c) , the fronted NP makes a non null semantic contribution.

Discourse and Pragmatic properties

Information Structure properties

The two constructions have sometimes been contrasted in terms of information structure, cx 1 being called 'topicalization' and cx 2 being called 'focalization'. The focus vs ground distinction is not relevant here, if it is understood in terms of new / old information. Topicalization is wrong if it is understood as Discourse Topic, it is not wrong if it is meant for the sentence theme (see section 3.2). In cx 1, the left dislocated NP is not always part of the ground (as is sometimes said of left dislocated phrases): it can be a (partial) answer to a question (21a), thus being an informational focus; it can also introduce a (sub) discourse topic (21b In cx 2, the filler NP can be a narrow focus (20a). But it can also be part of an all focus utterance as in (16b), where the whole sentence answers a question of type 'what happened' ; it can also consist completely of repeated material (22), with an unclear informational status (it is possibly a reassertion, in which case it would be like an all focus utterance, in spite of the repetition).

( 22) Mon père il va m'acheter un petit mouton un petit mouton il va m'acheter. [cx 2] [Corpaix] (My father he is going to buy me a small lamb a small lamb he is going to buy me)

That a left peripheral complement in a given construction is compatible with several informational status has already (although not frequently) been noted in the literature. [START_REF] Prince | On the Limits of Syntax, with reference to Left-Dislocation and Topicalization[END_REF][START_REF] Prince | How not to mark topics: 'Topicalization' in English and Yiddish[END_REF] Their general conclusion, which indeed also applies to French, and construction 2 in particular, is that, contra most generative analyses, leftwards movement is not triggered by, or does not correlate with informational features.

Illocutionary status

The two French constructions under study crucially differ concerning their illocutionary properties. While cx 1 is not specified, cx 2 is pragmatically characterized. First, cx 2 differs form cx 1 with respect to the two following properties: it cannot be embedded ((25a) vs (25b)); it can only be a declarative clause (neither an interrogative (25c) vs (25d), nor an imperative (25e)), although it can have a questioning or injunctive value in context ( 26).

(25) a. Tu sais bien que le chocolat, j'adore depuis toujours. This shows that cx 2 is defined as an utterance type rather than a sentence type. We assume that the clause type is uniquely associated with a content type, but not with a speech act [START_REF] Beyssade | The Speech Act Assignment Problem Revisited: Disentangling Speaker's Commitment from Speaker's Call on Addressee[END_REF]. We analyze (26a,b) as a confirmation request and a confirmation order, respectively. Second, cx 2 can be an answer to a question, where the preposed NP functions as a narrow focus (Quel âge il avait ? -Huit ans il avait, What was his age ? -eight years he had). It is not easily an 'out of the blue utterance', but requires an antecedent (as in a 'reprise utterance', [START_REF] Godard | Reinforcing negation : the case of Italian[END_REF]. 7 But very often, it is not part of a smooth progression. for instance, it is not integrated in a narration: a continuation with a sentence where the NP is a theme is not appropriate: (27b), where ce refers to the whole preceding situation, is a felicitous continuation for (16a), not (27a), where elle refers back to the antenna. No such constraint exists on cx 1 (28). (28) Le chocolat, j'adore ; c'est délicieux, et en plus ça remonte le moral.

[cx 1] (the chocolate, I adore; it is delicious, and it lifts one's spirits)

Cx 2 frequently contributes additional information, or a correction, that is information which is 'relevant' to the discourse topic (or the topmost QUESTION UNDER DISCUSSION) rather than a straightforward continuation in a narrative [START_REF] Ginzburg | Semantics and Conversation[END_REF]. Lastly, cx 2 is typically associated with speaker's attitudes (surprise, admiration, disgust, justification etc.). We call the preposed NP a 'center' for the clause. It may be the locus for an additional information that was left unresolved in the preceding discourse (as in (20a)), or for a correction, the NP corresponding to the point of disagreement (that is the case with (10b),