Pierre Badin
email: pierre.badin@icp.inpg.fr

Antoine Serrurier
email: antoine.serrurier@icp.inpg.fr

Three-dimensional modeling of speech organs: Articulatory data and models

Keywords: Speech production, articulatory modeling, MRI, linear analysis, PCA

Volume images of tongue, jaw, velum, nasopharyngeal wall, etc., were acquired by MRI on one French subject uttering a corpus of sustained articulations. Supplementary images of jaw, hard palate, and nasal cavities were acquired by CT. The three-dimensional outlines of these organs are represented by the vertices of triangular surface meshes. Using linear component analysis, the tongue was found to possess five degrees of freedom, including one for jaw movement, and the velum and nasopharyngeal port two degrees of freedom. These parameters can be interpreted in phonetic / biomechanical terms control a linear articulatory model of speech production.

Introduction

For a very long time, articulatory modeling of vocal tract and speech production organs has been essentially limited to the midsagittal plane. But progress and refinements brought into this domain led to a point where three-dimensional (3D) modeling has become unavoidable. The first motivation for such an approach is that the area function needed for calculating the sounds produced by an articulation specified in the midsagittal plane has to be inferred from the sole associated midsagittal contours, problem which is obviously impossible to solve as long as no other information is available on the transverse shape and size of the vocal tract. In particular, there are articulatory situations that can lead to occlusions in the midsagittal plane, while the vocal tract is not actually occluded on each side of this plane: (i) lateral consonants that are characterized by the presence of a complete apico-alveolar closure in the midsagittal plane while lateral channels are maintained open; (ii) nasal vowels, for which the uvula -an appendix of the velum in the midsagittal region -is often in contact with both the back of the tongue and the nasopharyngeal wall, creating an occlusion in the midsagittal plane, while lateral channels remain open. The detailed 3D knowledge of the vocal tract shape is important to deal in a more realistic way with the aerodynamic models needed for speech production studies. Finally, a 3D display of articulators can be useful in the domain of speech rehabilitation.

We had conducted a first 3D modeling study [START_REF] Badin | Three-dimensional articulatory modeling of tongue, lips and face, based on MRI and video images[END_REF]) using a set of three complementary stacks of MRI images (one axial stack in the laryngo-pharyngeal region, one oblique stack in the velar region, and one coronal stack in the front region including lips). We experienced difficulties in determining tongue tip, lips or velar region with satisfactory accuracy. It was thus decided to develop similar models with much better defined tongue tip, sublingual cavity, lateral cheek cavities, velum and lips, based on sets of 25 sagittal MRI images that allow a better accuracy in a number of situations and a more reliable reconstruction of transverse images.

The present article reports our attempts to reconstruct 3D jaw, tongue, velum and nasopharyngeal wall positions and shapes from MRI and CT data for a single subject uttering a more comprehensive corpus of sustained articulations in French, and to develop corresponding 3D linear articulatory models.

A subject-oriented linear modeling approach

Our modeling approach is described in details in [START_REF] Badin | Three-dimensional articulatory modeling of tongue, lips and face, based on MRI and video images[END_REF]. In the framework of speech robotics, the speech apparatus is viewed as a plant (an articulatory model) driven by a controller so as to recruit articulators and coordinate their movements in order to generate audio-visual speech. This concept implies the notion of a relatively small number of degrees of freedom (henceforth DoF) for the articulatory plant, i.e. the specification, for each articulator, of the limited set of movements that it can execute independently of the other articulators. The 3D geometry of the various non-rigid organs is thus modeled as the weighted sum of a small number of linear components or DoFs. These components are extracted by linear analysis from a set of vocal and nasal tract shapes representative of the speech production capabilities of the subject. The analysis is based on both Principal Component Analysis (PCA) and linear regression. The weights of the sum constitute the articulatory control parameters associated with the components: a given set of values of these parameters produces a given shape of the organs.

The corpus consisted of a set of artificially sustained articulations designed as to cover the maximal range of French articulations: the oral and nasal vowels [a ϯ e i y u o ø ѐ oe Ϫ ϯ oe ѐ ], the consonants [p t k f s Ѐ m n ల l] in three symmetrical contexts [a i u], and two specific articulations, a rest and a prephonatory position. Though limited to 46 French phonemes, this corpus proved to be sufficient for developing midsagittal articulatory models with nearly the same accuracy as corpora 40 times larger [START_REF] Beautemps | Linear degrees of freedom in speech production: Analysis of cineradio-and labio-film data and articulatory-acoustic modeling[END_REF]).

[Invited Paper]

Note that our approach is subject-oriented, i.e. that the model is based on a single subject. This avoids merging the physiological characteristics and the coarticulatory or compensatory strategies that can vary fairly much between various subjects.

In the framework of our linear approach, one DoF may be defined for a given speech articulator as one variable that can control completely a specific variation of shape and position of this articulator, and that is linearly uncorrelated with the other DoFs over the set of tasks considered. We aim to exploit the correlations in the articulatory data to reduce the number of DoFs of the articulators. However, this approach must be carefully balanced by another criterion, the biomechanical likelihood, i.e. making sure that the DoFs are not related to control strategies actually used by the subject during the task but are really associated with movements that are plausible from the viewpoint of biomechanics.

Each linear DoF is iteratively determined by a mixture of PCA applied to part or whole of the organ region and of multiple regression of the data against control parameters either arbitrarily imposed such as jaw height or determined by the preceding PCA. Note that the solution of this type of linear decomposition is not unique in general: while PCA delivers optimal factors explaining the maximum of data variance with a minimum number of components, our approach allows some freedom to decide the nature and distribution of the variance explained by the components (for instance to make them more interpretable in terms of control), at the cost of a sub-optimal variance explanation and of weak correlation between components.

Determination of the organ shapes from MR and CT images

Preliminary remarks on speech organs anatomy

The final result of an articulatory model is the shape of the complete vocal and nasal tracts, needed for simulating the aerodynamic / acoustic stage of the speech production process. The seemingly most straightforward way to deal with this problem was to develop tract or duct models (cf. e.g. [START_REF] Badin | A three-dimensional linear articulatory model based on MRI data[END_REF]). However, this approach is not well suited for taking precisely into account the complex geometry of the various speech organ / cavities, e.g. tongue tip, sublingual cavity, velopharyngeal port or epiglottis. An organ-based modeling approach, where each organ and cavity are modeled separately, seems more appropriate, as it allows to reconstruct the oral and nasal tracts subsequently in more details, and to obtain more reliable area functions.

As the purpose of the study was not to develop biomechanical muscle models, but models of organs / cavities boundaries as a whole, attention was directed towards organ outlines behavior in regions where they define the vocal tract. Therefore, compromises were established: (1) different groups of muscle fibers, including sometime connective tissues, were grouped together; (2) boundaries that do not always correspond to actual organ boundaries were drawn in order to allow manipulating organs as closed volume objects. For example, the external part of tongue extrinsic muscles such as palatoglossus, styloglossus, or stylohyoid, muscles excluded; as well, only a part of the levator veli palatini muscle was included in the velum contours. Note that special care was devoted to distinguish tongue tip from mouth floor (see below). Similarly, the epiglottis was systematically excluded from the tongue contours, but will be modeled independently.

The following sections describe the methods used for obtaining the 3D surface representations of the various speech organs and cavities from MR and CT images.

Acquisition and pre-processing of the CT and MR images

A Computer Tomography (CT) scan of the head of the subject was made, to serve as a reference. A stack of 149 axial images (512×512 pixels, 0.05 cm / pixel resolution, 0.13 cm inter slice space), was recorded for the subject at rest. These CT images, that provide a good contrast between bones, soft tissues and air, will be used to locate bony structures and to determine accurately their shapes for further use. Stacks of sagittal MR images were recorded for the subject sustaining artificially each of the 46 articulations of the corpus during about 35 sec. A set of 25 sagittal images (256×256 pixels, 0.1 cm / pixel resolution, 0.4 cm inter slice space) was obtained for each articulation. These images provide a good contrast between soft tissues and air, and also within soft tissues, but do not image clearly the bones.

Due to the complexity of the contours of the various organs, to the relatively low resolution of the images, and to the need of an accurate reconstruction of the organs, the extraction of contours has been performed manually, plane by plane. The first step is thus to edit semimanually the organ outline in each of the sagittal image where it can be seen. This is relatively straightforward for images close to the midsagittal plane, but much less so for images farther out from this midsagittal plane where the organ surface may be in tight contact with other structures or may just extend much further, such as the palatoglossus for instance. In order to improve the determination of the organ outline in these regions distant from the midsagittal plane, we created new images by reslicing the initial stack of images in planes having a more useful orientation, i.e. being more perpendicular to the organ surface. As a routine for the sagittal MRI stacks, we have used 27 transverse planes, i.e. the planes orthogonal to the sagittal ones, aligned on a semipolar grid (cf. Figure 1). The CT axial stack has been resliced into coronal and sagittal stacks.

Determination of the surface outline of the rigid bony structures

A number of structures that makes up the vocal tract can be considered as rigid: jaw, hard palate, nasal passages, nostrils and various paranasal sinuses. The contours of these structures have been manually edited from CT images in planes with appropriate orientations (e.g. the nasal passages were edited from coronal and axial images). The set of all points forming the 2D planar contours was then expanded into the common reference 3D coordinate system. These 3D points were finally processed through a 3D meshing software (Geometrica Research Group at INRIA, http://cgal.inria.fr/Reconstruction) to form a 3D surface meshing based on triangles. Figure 8 that shows tongue nomogram for jaw height illustrates also the jaw surface.

Alignment of the images on a common reference

Considering that the subject may have slightly changed position between two MR images stacks recording, it is important first to align all MRI stacks with a common reference. We use an arbitrary common 3D reference coordinate system attached to the skull of the subject. Each stack is aligned using an appropriate 3D transformation. This transformation, that corresponds to the six degrees of freedom of a solid object, 3 rotations and 3 translations, will be referred to as a (3D) rototranslation. It is obtained by aligning the rigid structures extracted from CT images (hard palate, nasal passages, paranasal sinuses) with the corresponding ones in the MR images stack, using a semi-automatic process:

(1) anchor points of the rigid structures are manually marked with care on some of the MR images of the stack, (2) the appropriate 3D rototranslation is obtained by minimizing the cumulated distance between these points recast in 3D and the corresponding nearest points on the 3D rigid structures. A similar approach was proposed by Takemoto, [START_REF] Takemoto | A method of teeth superimposition on MRI data for accurate measurement of vocal tract shape and dimensions[END_REF], the main differences being that their minimization error was the value of the volume overlap between the reference to align and the target data.

The previous procedure is also applied to the jaw for each articulation, to determine its relative position in relation to the fixed rigid structures: by combining this relative 3D rototranslation and the absolute one corresponding to the given stack, the jaw position is known in the common reference coordinate system.

Determination of the surface outline of the soft structures

The determination of the soft structures (tongue, velum, nasopharyngeal wall) is achieved in much the same way as for the rigid structures, but from the MR images of each articulation. Planar contours were edited in both sagittal and transverse images. The contours resulting from the intersection of the rigid organs surface with the plane containing the image being edited were superposed on the image in order to provide bony anchor points not visible on MRI and thus very useful for the interpretation of the image. Figure 2 and Figure 3 illustrate the edition of the velum in both MRI stacks. In addition to bony structures, previously edited soft structures are superimposed on the image to help interpretation and maintain coherence. The contour (solid lines on Figure 2 and Figure 3) is established as a 2D b-spline curve controlled by a limited number of points. The set of all 2D plane contours expanded into the 3D coordinate system forms a 3D description of the given soft organ (cf. Figure 4).

Tongue tip and sublingual boundaries

The sublingual cavity plays an important role in a number of articulations such as post-alveolar fricatives, laterals, back vowels and some consonants coarticulated with back vowels. As an illustration, note that the tongue tip is elevated enough to uncover the mouth floor -and thus to create a sublingual cavity -in 19 articulations of our corpus: [u o ѐ oe a ѐ m u Ѐ a Ѐ u p a p u k a k u l a l i l u ల a ల u ]. In these articulations, tongue tip and mouth floor contours can be easily be seen and traced (e.g. see Figure 1). For the other articulations for which the tongue tip rests on the mouth floor (see Figure 2), these contours are difficult to determine and have to be inferred indirectly.

In most of the central sagittal planes, three specific points can be identified on the tongue contours: (1) the extremity of the tip (referred to as Tip), (2) the rearmost point of the sublingual cavity, at the junction of the mouth floor outline and the inferior surface of the tongue tip (referred to as Floor), and (3) the point in the midsagittal plane where the mouth floor reaches the jaw internal surface, somewhere behind the lower incisors (referred to as Jaw).The Tip and Jaw points are easy to identify for all articulations. To simplify a little, we assume that all Jaw points in the sagittal planes are located in the same nearly axial plane orthogonal to the midsagittal plane, containing the midsagittal Jaw point and parallel to the plane tangent to the upper surface of the inferior teeth: the corresponding reference jaw arch is finally further simplified into a fourth order polynomial planar curve. The Floor point cannot be seen in articulations where the tongue tip rests on the mouth floor: it was thus decided to infer it for those cases, assuming its position being unchanging relatively to the jaw. We determined the reference mouth floor arch as the fourth order polynomial planar curve that approximates best the floor points of the [Ѐ a ] articulation. The two reference arches defined as described above are supposed attached to the jaw, and can thus be positioned accordingly for each articulation, using the jaw rototranslation.

The strategy for editing the tongue contours in sagittal planes is finally the following: (1) position reference arches in 3D, find their intersection with the sagittal image plane and display these intersection points that are supposed to reflect the position of the jaw point and the floor point, (2) edit the contours as usual whenever visible, but interpolate using the jaw point and the floor point as anchor points when the tongue tip is touching the mouth floor. In addition, the three points (Tip, Floor, Jaw) were manually identified on each sagittal contour to allow an interpolation of the three sub-contours (mouth floor, tongue tip inferior surface, tongue superior surface) with a fixed number of points that will be used in the mesh matching process (see below). Similarly, the lower edge of the velum was determined and each sagittal contour interpolated with a fixed number of point for both upper and lower faces.

Fit of generic meshes to each articulation

The plane contours defined above do not have a constant number of points and therefore cannot be used in a PCA type of analysis. In order to ensure a suitable geometric representation of all the articulations for a given soft organ, a unique generic 3D surface mesh, made of triangles, was defined for each organ and fitted by elastic deformation to each of the 46 3D shapes of the corpus. This procedure provides thus a 3D sampling of each organ surface with the same 3D vertices, which is appropriate for the development of the models. The elastic deformation of the generic mesh to fit each 3D shape is computed through a matching software developed by [START_REF] Szeliski | Matching 3-D anatomical surfaces with non-rigid deformations using octree-splines[END_REF].

The lateral consonant [l a ] was chosen as reference articulation for the tongue, as the tongue tip is sufficiently raised to let clearly appear the sub-lingual cavity, i.e. the lower surface of tongue tip and the mouth floor, in this articulation. Similarly, the nasal phoneme [a ] was chosen as a reference for the velum and nasopharyngeal wall, as the contact between velum and surrounding walls is minimum in this articulation. The meshing software was used to generate the generic meshes from the points sampling the various planar contours of the reference. As the reference and target shapes may be very different, it was judged more reliable to match the reference mesh progressively, using a series of ten intermediary sets of sagittal contours, where each point is linearly interpolated between its reference and target positions.

This matching process has finally resulted in a set of organ surfaces described in terms of triangular meshes having the same number of vertices, in a common reference coordinate system: 5239 vertices for the velum (RMS reconstruction error, 0.06 cm), 2110 vertices of the nasopharyngeal wall (RMS, 0.04 cm), and 1640 vertices for the tongue, for each of the 46 articulations of the corpus (RMS, 0.06 cm). This forms the basis for the articulatory modeling, as illustrated in the next section.

Articulatory models of velum, nasopharyngeal wall, jaw and tongue

Head tilt

Head movements of the subject during the recording session could introduce a variation of the vocal tract shape as suggested by [START_REF] Kitamura | Difference in vocal tract shape between upright and supine postures: observation by an open-type MRI scanner[END_REF]. Head tilt changes in the sagittal orientation are the most important. They may increase the variability of the data without being related to a precise degree of freedom of the articulators and are therefore considered as a global articulatory degree of freedom of the head that is not specifically related to speech production. Therefore, it appears necessary, before modeling speech articulators independently, to remove the variability related to this degree of freedom from the data. Preliminary analysis has shown s strong correlation between spinal chord and nasopharyngeal wall. The horizontal coordinate of the back pharyngeal wall in the midsagittal plane at the height of uvula was used as tilt predictor, and referred to as Head Tilt parameter (HT).

The contribution of this HT parameter to the explanation of the total variance of the data has been computed by means of linear regression: it explains 17% of the cumulated variance of the central part of the velum, 31% of the variance of the nasopharyngeal wall, and 7 % of the tongue surface. In the following, the models are built from the 46 velum and nasopharyngeal wall meshes cleared of the head tilt influence, i.e. once the linear contribution of the influence of HT parameter has been removed from the raw data. The influence of HT on the tongue was not deemed high enough to take it into account.

Articulatory model of velum of the velum and of the nasopharyngeal wall

In order to determine the number and nature of the articulatory degrees of freedom of the velum, PCA was applied to the 2812×3 vertex coordinates that represent the surface of the central part of the velum over the 46 articulations corrected for tilt variations The first PCA parameter VL explains 83% of the cumulated variance of all the central velum points, while the associated RMS reconstruction error is 0.08 cm. This parameter can then be used to predict the entire velum surface based on a linear regression of the whole set of velum points on this parameter; in other words, the external regions of the velum can be predicted from the central region by means of this first PCA parameter. The effect of this parameter on the whole velum is illustrated on Figure 7 by the shape associated with the two extreme values of VL found in the data. The main movement associated with VL is a movement in an oblique direction. Considering its orientation and its prime importance for speech [START_REF] Bell-Berti | Understanding velic motor control: studies of segmental context[END_REF]), the levator veli palatini can be thought to be much involved in this movement; this control parameter is thus referred to as Velum Levator (VL).

The second velum PCA parameter, VS, explains 6% of the total variance of the central points of the velum. The cumulated variance explained by the two parameters VL and VS attains 89% while the cumulated RMS reconstructions error lowers to 0.06 cm. This second parameter is much related to a horizontal displacement coupled with a vertical elongation of the velum, which complements the velopharyngeal port closure by a front to back movement and may significantly modify the velopharyngeal port constriction. Despite a relatively small gain in variance explanation, VS appears as a coherent complementary degree of freedom of the velum: (1) it reduces the RMS reconstruction error by 25%, (2) it may modify the velopharyngeal constriction in a meaning full way.

The velopharyngeal port closure is known to be controlled by the combined contribution of velum pulling and superior constrictor sphincter action (cf. e.g. [START_REF] Bell-Berti | Understanding velic motor control: studies of segmental context[END_REF]). Expectedly, we found that the VL parameter explains 47% of the total nasopharyngeal wall variance, with a reconstruction with an RMS error of 0.07 cm. The effect of VL in the midsagittal plane occurs in the upper region, along the main direction of velum deformation, but in an antagonist way. This movement corresponds to a variation of thickness of the Passavant's Pad, related to the contraction of the palatopharyngeus muscle and of the pterygopharyngeal portion of the superior constrictor, and participates to a global sphincter behavior of the velopharyngeal port. The parameter VS, that explains only 5% of the nasopharyngeal wall variance, was thus not retained in the nasopharyngeal wall model. In another study [START_REF] Rossato | Velar movements in French: an articulatory and acoustical analysis of coarticulation[END_REF]), one of the coils of an Electro Magnetic Articulograph was attached to the velum of the same subject uttering an extensive corpus of VCV sequences. We have determined the vertex of the velum mesh that was the closest to the velum coil position for all phoneme targets, and found a rather strong similarity between the domains occupied by the coil and the vertex in the midsagittal plane. This is an additional evidence of the validity of our data acquisition and modeling approach.

Jaw displacement model

A strong correlation (R 2 = 0.93) was found between the lower incisor's height and the angle of rotation around a transverse axis, and smaller correlation with the incisor's horizontal position. The jaw height parameter JawHei has been used as a control parameter of the main jaw movement.

Tongue model

The first parameter, jaw height JH, is defined as the variable JawHei centered on its mean and normalized by its standard deviation. Its main effect is a tongue rotation around a point in the back of the tongue (see Figure 9). The next two parameters, tongue body TB, and tongue dorsum TD, are extracted by PCA from the coordinates of the midsagittal tongue contour, excluding the tongue tip region, from which the JH contribution has been removed. They control respectively the front-back and flattening-arching movements of the tongue (see Figure 9). The next two parameters, tongue tip height TT, and tongue tip advance TA, are extracted by PCA from the midsagittal tongue tip contour coordinates, from which the TB and TD contribution have been removed (see Figure 9).

Table I gives the variance, relative to the total variance of the full 3D coordinates, that is explained by each component, for both raw PCA and the controlled analysis described above. It appears that a amount of 82.6 % is explained by our controlled analysis, which is only 8% below the optimal result from a raw PCA with the same number of components.

Finally, the 3D tongue model is controlled by the five articulatory parameters JH, TB, TD, TT1, and TT2. The effects of these commands are demonstrated in Figure 8 which displays tongue shapes for two extreme values (-3 and +3) of one parameter, all other parameters being set to zero., and in Figure 9 which displays the midsagittal intersection of the 3D nomograms with sinusoidal variations of the control parameters.

Raw I: Evaluation of the tongue analysis: varex is the relative explained tongue data variance, varexcum its cumulated value, and RMS the RMS reconstruction error between the modeled mesh and the original planar contours.

Discussion and perspectives

The present work produced a number of valuable results. First, a database of 3D geometrical descriptions of tongue, velum and nasopharyngeal wall was established for a speaker sustaining a set of 46 French allophones covering the speech possibilities of the subject. Linear component analysis of these data revealed that five components could account for about 82 % of the total variance of the tongue shape, while 89 % of the velum variance is taken into account by two parameters, as well as 47 % of the nasopharyngeal wall (this is a smaller amount, but it should be noted that the absolute variance of the nasopharyngeal wall in about 10 times smaller than that of the velum itself).

Concerning the tongue, the five components correspond qualitatively to those found previously by [START_REF] Badin | Three-dimensional articulatory modeling of tongue, lips and face, based on MRI and video images[END_REF], on a more restricted corpus of 25 articulations only. The present study constitutes thus an extension of this study. It is also important to note that the variance explained here is about 10 % higher than in [START_REF] Badin | Three-dimensional articulatory modeling of tongue, lips and face, based on MRI and video images[END_REF], and that the tongue tip and sublingual cavities are much better defined. This will be useful for dealing with articulations such as post-alveolar fricatives, laterals, back vowels and some consonants coarticulated with back vowels. This is also one of the motivations for dealing with a 3D model. Another interest of this 3D model lies in the direct possibility to deal with tongue groove: Figure 8 illustrate well the strong groove variation related to the tongue dorsum parameter.

The velum / nasopharyngeal wall model constitutes an extension of the first model developed by Serrurier & Badin (2005, in press) on a restricted corpus of only 22 articulations.

The next modeling step will be the determination of the area functions for both oral and nasal tracts as a function of the shapes of tongue, velum and nasopharyngeal wall, and thus the determination of the acoustical characteristics of the complete tract. A longer-term objective is finally to extend all this process to the complete 3D vocal and nasal tracts in order to build a complete 3D articulatory model of speech.

Figure

  Figure 1: Grid for transverse reslicing superimposed on a midsagittal image Figure 2: Example of edited contours in the midsagittal plane

  of VL Figure 7: Velum shape for two extreme values of VL, and regular nomograms between these two values for the velum and the nasopharyngeal wall in the midsagittal plane (c left) and in the transverse plane (c right) indicated by the transverse solid line on the left graph.

Figure 8 :

 8 Figure 8 : Jaw and tongue positions for extreme values of JH and TD

Acknowledgements

We thank very sincerely Jean-François Lebas (Head of the Radiology Department, CHRU, Grenoble, France) for granting us access to the MRI equipment, Christophe Segebarth (Functional and Metabolic NeuroImagery Department, INSERM & UJF, Grenoble, France) and Monica Baciu (Psychology and NeuroCognition Department, UPMF, Grenoble, France) for managing the MRI recordings, Andreas Fabri (Geometrica team, INRIA, Sophia, France) for the letting us use his 3D meshing software, and Maxime Bérar (Image and Signal Laboratory, CNRS -INPG -UJF, Grenoble, France) for helping us with the elastic matching software.


David R Bickel
email: dbickel@uottawa.ca

Departing from Bayesian inference toward minimaxity to the extent that the posterior distribution is unreliable

Keywords: Bayes action, blended inference, discounting coefficient, L p norm, minimax action, relative risk aversion

A Bayesian model may be relied on to the extent of its adequacy by minimizing the posterior expected loss raised to the power of a discounting exponent. The resulting action is minimax under broad conditions when the sample size is held fixed and the discounting exponent is infinite. On the other hand, for any finite discounting exponent, the action is Bayes when the sample size is sufficiently large. Thus, the action is Bayes when there is enough reliable information in the posterior distribution, is minimax when the posterior distribution is completely unreliable, and is a continuous blend of the two extremes otherwise.

Introduction

According to the foundations of Bayesian decision making, an agent should choose the action that minimizes posterior expected loss. That prescription has the advantage of operationally defining probability but often cannot be freely carried out in applications of Bayesian statistics. That is because the prior distribution, if not other aspects of the statistical model, is to some extent unreliable, for a number of reasons. For example, the prior may be improper, at best approximating a probability measure and subject to the marginalization paradox [START_REF] Dawid | Marginalization paradoxes in Bayesian and structural inference[END_REF]. Or the prior, chosen for mathematical or computational tractability, may have highly arbitrary elements. It is rarely feasible to explicitly account for all relevant sources of uncertainty.

As a result, prior distributions are considered tentative, subject to revision in light of model checking on the basis of observed data [START_REF] Gelman | Philosophy and the practice of bayesian statistics[END_REF]. What do the resulting posterior probabilities mean, seeing that they are not necessarily reliable enough to use for taking the actions that minimize posterior loss?

To address the problem of the unreliable prior distribution, various robust Bayesian solutions weaken or discount the prior distribution to an extent determined by a discounting coefficient by replacing the prior with a set of prior distributions in some neighborhood around it, with larger neighborhoods for larger values of the coefficient (Augustin et al., 2014, §4.7). Since such solutions are complex and leave open the question of how to report an estimate or prediction or to make another decision on the basis of the resulting set of posterior distributions, Section 2 introduces a simple approach to decision making on the basis of an unreliable prior. It prescribes the Bayes action when the prior is completely reliable, the minimax action when the prior is completely unreliable, and otherwise an intermediate action that is discounted to the extent that the prior is unreliable, with the Bayes and minimax actions at the ends of a continuum (cf. [START_REF] Bickel | Blending Bayesian and frequentist methods according to the precision of prior information with applications to hypothesis testing[END_REF]. Some hypothesis-based examples illustrate the ideas and their applicability to set estimation (Section 3).

The last part of the paper focuses on asymptotics with illustrative examples. The minimin act as well as the minimax act is a limiting special case as the degree of discounting becomes infinite (Section 4). Section 5 proves that the discounted action eventually converges to the Bayes action as the sample size increases.

1

Consider a family F = {f (•|θ) : θ ∈ Θ} of sampling distributions such that f (•|θ) is a probability density function on a sample space X (n) for a sample size of n observations and conditional on a parameter value θ in some set Θ. The observed sample x ∈ X (n) is modeled as a realization of the random element X ∼ f (•|θ).

Let L : Θ × A → [0, ∞[ be a loss function on Θ and a set A of possible actions.

A Bayesian model (F , P), consisting of F and a prior distribution P of θ, is discounted to degree ∆ ∈ R, the discounting exponent, as follows. For a measure ν that dominates P , the Radon-Nikodym derivative p = d P /dν is the prior density. The corresponding posterior density is

p (θ|x) := f (x|θ) p (θ) f (x|θ) p (θ) dν (θ)
where dν (θ) = dθ if ν is the Lebesgue measure, and the posterior distribution

P (•|x) satisfies p (•|x) = d P (•|x) /dν. For any ∆ ∈ {-∞, ∞} ∪ R\ {0}, the ∆-discounted posterior expected loss and ∆-discounted posterior action are E p ∆,a (L |x) =                L ∆ (θ, a) p (θ|x) dν (θ) 1/ ∆ if -∞ < ∆ < ∞, ∆ = 0 ess sup P (•|x) θ∈Θ L (θ, a) if ∆ = ∞ ess inf P (•|x) θ∈Θ L (θ, a) if ∆ = -∞ a p ∆ = arg inf a∈A E p ∆,a (L |x) ,
where L ∆ (θ, a) = (L (θ, a)) ∆ , and ess sup

P (•|x) θ∈Θ h (θ)
is the essential supremum of a nonnegative-valued,

P (•|x)-measurable function h over Θ with respect to P (•|x), that is, ess sup P (•|x) θ∈Θ h (θ) = inf {H > 0 : P ({θ ∈ Θ : h (θ) ≤ H} |x) = 1}
with inf ∅ = ∞. Similarly, ess inf

P (•|x) θ∈Θ h (θ) is the essential infimum of h. In the case of no discounting (∆ = 1) , a p 1 = arg inf a∈A E p 1,a (L |x) is called the Bayes action since it mini- mizes E p 1,a (L |x), the posterior expected loss. Under complete discounting (∆ = ∞), a p ∞ = arg inf a∈A ess sup P (•|x)
θ∈Θ L (θ, a) is the essential minimax action. Between them (1 < ∆ < ∞), the effect of the model (F , P ) on the action is the opposite extreme, that of maximal inverse discounting (∆ = -∞), a p ∞ = arg inf a∈A ess inf

P (•|x)
θ∈Θ L (θ, a) is the essential minimin action, which is of less practical interest.

Examples involving hypotheses

Example 1. In hypothesis testing, one infers whether (a = 1) or not (a = 0) the unknown value of θ is in H, a measurable subset of Θ. A relevant loss function is

L (θ, a) =                1 if θ ∈ H, a = 0 c if θ / ∈ H, a = 1 0 otherwise , (1)
where the cost ratio c > 0, is the penalty for incorrectly inferring that θ ∈ H relative to that for incorrectly inferring that θ / ∈ H. Let χ {0} and χ {1} denote characteristic functions on {0, 1}. For any finite ∆ = 0,

E p ∆,a (L |x) = H χ ∆ {0} (a) p (θ|x) dν (θ) + Θ\H c ∆ χ ∆ {1} (a) p (θ|x) dν (θ) 1/ ∆ = P (H|x) χ {0} (a) + c ∆ (1 -P (H|x)) χ {1} (a) 1/ ∆ = P 1/ ∆ (H|x) χ {0} (a) + c (1 -P (H|x)) 1/ ∆ χ {1} (a) a p ∆ = arg inf a∈{0,1} E p ∆,a (L |x) =                0 if P (H|x) 1-P (H|x) 1/ ∆ < c 1 if P (H|x) 1-P (H|x) 1/ ∆ > c {0, 1} if P (H|x) 1-P (H|x) 1/ ∆ = c
, suggesting that ω ∆ (H|x) := (P (H|x) / (1 -P (H|x))) 1/ ∆ be reported as the ∆-discounted posterior odds of θ ∈ H. The corresponding ∆-discounted posterior probability that θ ∈ H would then be P ∆ (H|x) := 1/ (1 + 1/ω ∆ (H|x)), resulting in P 1 (H|x) = P (H|x) and lim ∆→∞ P ∆ (H|x) = 1/2. At ∆ = 2, lower posterior probabilities are discounted by more orders of magnitude than higher ones (Figure 1, left plot), as is desirable given the unreliability of extremely small probabilities. That is applicable not only to hypothesis testing but also to set estimation. A set estimate H (x) is a

(1 -α) 100% ∆-discounted credible set if P ∆ (H (x) |x) = (1 -α) 100%.
For example, a 95% 2-discounted credible set would be an approximately 80% 1-discounted credible set, a standard 80% credible set without discounting (Figure 1, right plot), whereas a 95% ∞-discounted credible set would be a 50% credible set. Example 2. For H, a measurable subset of Θ, let χ H denote a characteristic function on Θ. When estimating χ H (θ), the indicator of whether θ is in H, with an action in A = [0, 1] as opposed to the A = {0, 1} of Example 1, squared-error loss is appropriate:

L (θ, a) = (a -χ H (θ)) 2 .
For any finite ∆ = 0,

E p ∆,a (L |x) = (a -χ H (θ)) 2 ∆ p (θ|x) dν (θ) 1/ ∆ = H (1 -a) 2 ∆ p (θ|x) dν (θ) + Θ\H a 2 ∆ p (θ|x) dν (θ) 1/ ∆ = P (H|x) (1 -a) 2 ∆ + (1 -P (H|x)) a 2 ∆ 1/ ∆ a p ∆ = arg inf a∈[0,1] P (H|x) (1 -a) 2 ∆ + (1 -P (H|x)) a 2 ∆ = 1 + P (H|x) 1 -P (H|x) -1/(2 ∆ -1) -1 .
Although a p 1 might be considered as an elicited posterior probability that θ ∈ H (cf. Lad, 1996, §6.6.3), the a p ∆ of this example differs in general from the P ∆ (H|x) of Example 1. That may be illustrated using the default ∆ = 2: P 2 (H|x) 1 -P 2 (H|x) = P (H|x) 1 -P (H|x) = P (H|x) 1 -P (H|x)

1 3 = a p 2 .
Nonetheless, a p 1 = P (H|x) = P 1 (H|x) and lim ∆→∞ a p ∆ = 1/2 = lim ∆→∞ P ∆ (H|x).

Limiting cases of the discounting exponent

With n held fixed, the discounted posterior expected losses of finite ∆ approach those of ∆ = ±∞ under the conditions of the following proposition, which needs Lemma 1 for its proof.

Lemma 1. If h is a nonnegative-valued, P (•|x)-measurable function on Θ such that ess sup

P (•|x) θ∈Θ h (θ) < ∞, then lim ∂→∞ h ∂ (θ) d P (θ|x) 1/∂ = ess sup P (•|x) θ∈Θ h (θ) .
Proof. That directly follows from Aliprantis and Border (2006, Lemma 13.1).

Proposition 1. Suppose L (•, a) is nonnegative-valued and P (•|x)-measurable and that it satisfies ess sup

P (•|x) θ∈Θ L (θ, a) < ∞ ess sup P (•|x) θ∈Θ 1/ L (θ, a) < ∞ for some a ∈ A. Then lim ∆→±∞ E p ∆,a (L |x) = E p ±∞,a (L |x) . Proof. By Lemma 1, lim ∆→∞ E p ∆,a (L |x) = E p ∞,a (L |x), and lim ∆→-∞ E p ∆,a (L |x) = lim ∆→-∞ L -|∆| (θ, a) p (θ|x) dν (θ) -1/|∆| = lim ∂→∞ L -∂ (θ, a) p (θ|x) dν (θ) -1/∂ = lim ∂→∞ (1/ L (θ, a)) ∂ P (θ|x) 1/∂ -1 = ess sup P (•|x) θ∈Θ 1/ L (θ, a) -1 = ess inf P (•|x) θ∈Θ L (θ, a) = E p -∞,a (L |x) .
The next example compares that discounted-model approach to minimaxity with one based on a degree of risk aversion.

Example 3. Suppose W (θ, a) ≥ 0 would be the amount of future wealth in a portfolio if θ ∈ Θ is the future state of the world and a ∈ A is the investment strategy. The loss corresponding to the power utility function (Campbell and Viceira, 2002, §2.1.2) is

L (θ, a) =        L 0 + 1 ρ -1 1 W ρ -1 (θ,a) -1 if ρ = 1 L 0 -ln W (θ, a) if ρ = 1
, where ρ ∈ R is the relative risk aversion, and L 0 ∈ R such that L (θ, a) ≥ 0 for all θ ∈ Θ and a ∈ A.

If W (θ, a) < 1 and if W (•, a) satisfies the conditions of Lemma 1, then the Bayes action in the limit of increasing relative risk aversion is

lim ρ→∞ a p 1 = arg inf a∈A lim ρ→∞ E p 1,a (L |x) = arg inf a∈A lim ρ→∞ 1 W ρ -1 (θ, a) -1 p (θ|x) dν (θ) 1/ ρ = arg inf a∈A ess sup P (•|x) θ∈Θ 1 W (θ, a)
= arg sup a∈A ess inf

P(•|x) θ∈Θ W (θ, a) ,
which reduces to the special case of minimax reciprocal wealth or maximin wealth (cf. Arrow, 1973, pp. 256-257). On the other hand, Proposition 1's limit of maximal discounting (∆ → ∞) gives, with a fixed ρ = 1, Proof. For any finite ∆ = 0, the mapping theorem yields

a p ∞ = arg inf a∈A ess sup P(•|x) θ∈Θ 1 ρ -1 1 W ρ -1 (θ,
E p ∆,a (L |X) = L ∆ (θ, a) d P (θ|X) 1/ ∆ weak ---→ L ∆ (θ, a) d δ θ0 (θ) 1/ ∆ = L ∆ (θ 0 , a) 1/ ∆ = L (θ 0 , a) . Thus, E p ∆,a (L |X) -E p 1,a (L |X) weak ---→ L (θ 0 , a) -L (θ 0 , a).
The next example illustrates the result.

Figure 1 :

 1 Figure 1: The 2-discounted posterior probability P 2 (H|x) as a function of the posterior probability P (H|x).Left plot: probabilities small enough to be interesting as posterior probabilities of a null hypothesis. Right plot: probabilities large enough for use as the posterior probabilities of credible sets.

Figure 2 :

 2 Figure 2: The ∆-discounted posterior probability P ∆ (H|x) as a function of the model's dependability, the reciprocal of its discounting exponent ∆. Left plot: P (H|x) = 5%. Right plot: P (H|x) = 10 -6 .

Figure 2

 2 Figure 2 displays discounted probabilities at other values of ∆. There, the dependability D (∆) of a model is the reciprocal of the extent that it requires discounting: D (∆) = 1/ ∆. Thus, D (1) = 1, D (2) = 1/2, D (4) = 1/4, and D (∞) = 0.

  θ∈Θ W (θ, a) , implying that lim ρ→∞ a p 1 = a p ∞ . The same holds for ρ = 1.5 Convergence to the Bayes action as data arriveDiscounting the model leads to estimates and other actions that tend to be closer to the minimax action for smaller samples but closer to the Bayes action for larger samples, once there are enough data to overwhelm questionable aspects of the model. Proposition 2. Let δ θ0 denote the Dirac measure with its mass at a θ 0 ∈ Θ, and let weak ---→ denote weak convergence as n → ∞. If X ∼ f (•|θ 0 ), if P (•|X) weak ---→ δ θ0 , and if L (•, a) is continuous on Θ for some a ∈ A, then, for any finite ∆ = 0,

Example 4 .

 4 Example 2, continued. Let p (θ|H) and p (θ|Θ\H) denote the prior probability densities conditional on a null hypothesis that θ ∈ H and the alternative hypothesis that θ / ∈ H, respectively. The posterior odds of the null hypothesis isP (H|x) 1 -P (H|x) = P (H) 1 -P (H) f (x|θ) p (θ|H) dν (θ) f (x|θ) p (θ|Θ\H) dν (θ) ,the right-hand side of which is the product of the prior odds and the Bayes factor. Both quantities typically involve unverified assumptions that require some kind of discounting before decision making on the basis of the resulting P (H|x). As a concrete form of Proposition 2, Figure3displays the convergence of the discounted and Bayes estimates to the same value, 0 if the null hypothesis is false or 1 if it is true, for, as n → ∞, the Bayes factor tends to decrease if the null hypothesis is false or increase if it is true. The curves for a p 2 and a p 4 reflect the gradation of estimates between the extremes of the Bayes action a p 1 and the minimax action a p ∞ = 1/2.

Figure 3 :

 3 Figure 3: Each a p∆ as an estimate of χ H (θ) for ∆ = ∞ (the horizontal line at 1/2), ∆ = 4 (the lighter thin curve), ∆ = 2 (the darker thin curve), and ∆ = 1 (the thick curve). Following[START_REF] Benjamin | Redefine statistical significance[END_REF], the prior odds is 10:1.

Acknowledgments

This research was partially supported by the Natural Sciences and Engineering Research Council of Canada and by the Faculty of Medicine of the University of Ottawa.


Pierre Badin
email: pierre.badin@icp.inpg.fr

Antoine Serrurier
email: antoine.serrurier@icp.inpg.fr

Three-dimensional linear modeling of tongue: Articulatory data and models

Volume images of tongue were acquired by MRI from one subject uttering a corpus representative of French allophone articulations. Supplementary images of hard palate, jaw, and hyoid bone were acquired by CT. The three-dimensional tongue surface outline was represented, for each of the 46 articulations of the corpus, by a mesh obtained by fitting a generic mesh to the set of tongue contours traced from the MR images. Jaw and hyoid bone positions were also determined. The set of the 3D coordinates of all vertices of the tongue mesh constituted the variables on which linear component analysis was applied. Six linearly independent components were found to explain 87 % of the variance of the tongue data. The associated parameters that control the linear articulator tongue model are related to jaw and hyoid positions, and to the actions of tongue muscles such as the genioglossus, the hyoglossus or the styloglossus. In addition, it was shown that the full 3D tongue surface is predictable from its 2D midsagittal contour with a mere 13.6 % increase in the overall full 3D reconstruction RMS error, which confirms quantitatively previous results. Finally, the tongue volume was found to depart by at most ±5% from its mean over the corpus, which supports the hypothesis of tongue tissue incompressibility for speech.

Introduction

For a very long time, articulatory modeling of vocal tract and speech production organs has been essentially limited to the midsagittal plane. But progress and refinements brought into this domain led to a point where three-dimensional (3D) modeling has become unavoidable. The first motivation for such an approach is that the area function needed for calculating the sounds produced by an articulation specified in the midsagittal plane has to be inferred from the sole associated midsagittal contours, problem which is obviously impossible to solve as long as no other information is available on the transverse shape and size of the vocal tract. In particular, there are articulatory situations that can lead to occlusions in the midsagittal plane, while the vocal tract is not actually occluded on each side of this plane, e.g. lateral consonants that are characterized by the presence of a complete apico-alveolar closure in the midsagittal plane while lateral channels are maintained open. The detailed 3D knowledge of the vocal tract shape is important to deal in a more realistic way with the aerodynamic models needed for speech production studies.

We had conducted a first 3D modeling study [START_REF] Badin | Three-dimensional articulatory modeling of tongue, lips and face, based on MRI and video images[END_REF]) using a set of three complementary stacks of MRI images (one axial stack in the laryngo-pharyngeal region, one oblique stack in the velar region, and one coronal stack in the front region including lips). We experienced difficulties in determining tongue tip, lips or velar region with satisfactory accuracy. It was thus decided to develop similar models with much better defined tongue tip, sublingual cavity, lateral cheek cavities, velum and lips, based on sets of 25 sagittal MRI images that allow a better accuracy in a number of situations and a more reliable reconstruction of transverse images.

The present article reports our attempts to reconstruct 3D tongue positions and shapes from MRI and CT data for a single subject uttering a more comprehensive corpus of sustained articulations in French, and to develop a corresponding 3D linear articulatory tongue model.

A subject-oriented linear modeling approach

Our modeling approach is described in details in [START_REF] Badin | Three-dimensional articulatory modeling of tongue, lips and face, based on MRI and video images[END_REF]. In the framework of speech robotics, the speech apparatus is viewed as a plant (an articulatory model) driven by a controller so as to recruit articulators and coordinate their movements in order to generate audio-visual speech. This concept implies the notion of a relatively small number of degrees of freedom (henceforth DoF) for the articulatory plant, i.e. the specification, for each articulator, of the limited set of movements that it can execute independently of the other articulators. In the framework of a linear approach, one DoF may be defined for a given speech articulator as one variable that can control completely a specific variation of shape and position of this articulator, and that is linearly uncorrelated with the other DoFs over the set of tasks considered.

The corpus consisted of a set of 46 artificially sustained articulations designed as to cover the maximal range of French allophones: the oral and nasal vowels [a ϯ e i y u o ø ѐ oe Ϫ ϯ oe ѐ ], the consonants [p t k f s Ѐ m n ల l] in three symmetrical contexts [a i u], and two specific articulations, a rest and a prephonatory position. Note that [START_REF] Beautemps | Linear degrees of freedom in speech production: Analysis of cineradio-and labio-film data and articulatory-acoustic modeling[END_REF] demonstrated that selecting carefully the set of articulations allowed to develop midsagittal models with nearly the same accuracy as with corpora 40 times larger: this justifies the use of our restricted corpus.

Determination of the organ shapes from MR and CT images

Preliminary remarks on speech organs anatomy

The final result of an articulatory model is the shape of the complete vocal tract, needed for simulating the aerodynamic / acoustic stage of the speech production process. The seemingly most straightforward way to deal with this problem was to develop tract or duct models (cf. e.g. [START_REF] Badin | A three-dimensional linear articulatory model based on MRI data[END_REF]). However, this approach is not well suited for taking precisely into account the complex geometry of the various speech organ / cavities, e.g. tongue tip, sublingual cavity, or epiglottis. An organ-based modeling approach, where each organ and cavity are modeled separately, seems more appropriate, as it allows to reconstruct the various tracts subsequently in more details, and thus to obtain more reliable area functions.

As the purpose of the study was not to develop biomechanical muscle models, but models of organs / cavities boundaries as a whole, attention was directed towards organ outlines in regions where they define the vocal tract. Therefore, compromises were established: (1) different groups of muscle fibers, including sometime connective tissues, were grouped together; (2) boundaries that do not always correspond to actual organ boundaries were drawn in order to allow manipulating organs as closed volume objects. For example, the external parts of tongue extrinsic muscles such as palatoglossus, styloglossus, or stylohyoid muscles are excluded. Note that special care was devoted to distinguish tongue tip from mouth floor (see below). Similarly, the epiglottis was systematically excluded from the tongue contours, but will be modeled independently in the future.

The following sections describe the methods used for obtaining the 3D surface of the tongue, jaw and hyoid bone from MR and CT images. More can be found in [START_REF] Badin | Three-dimensional modeling of speech organs: Articulatory data and models[END_REF].

Acquisition and pre-processing of the CT and MR images

A Computer Tomography (CT) scan of the head of the subject at rest was made, to as a reference for bony structures. Stacks of 25 sagittal MR images were recorded for the subject sustaining artificially each of the 46 articulations of the corpus. Due to the complexity of the contours of the various organs and to the relatively low resolution of the images, the determination of the contours has been performed manually on each image. In order to improve the determination of the outline in the regions far from the midsagittal plane, the contours have also been edited from images reconstructed from the original stack in 27 transverse planes aligned on a semipolar grid (cf. Figure 1).

Determination of the surface outline of the rigid bony structures

A number of structures that make up the vocal tract can be considered rigid: jaw, hard palate, or sphenoidal sinus for instance. The contours of these structures have been manually edited from CT images in planes with appropriate orientations. The set of all points forming the 2D planar contours was then expanded into the common reference 3D coordinate system. These 3D points were finally processed through a 3D meshing software (Geometrica Research Group at INRIA, http://cgal.inria.fr/Reconstruction) to form a 3D surface mesh based on triangles.

Alignment of the images on a common reference

Considering that the subject may have slightly changed position between two MR images stacks recording, it was important first to align all MRI stacks with a common reference. We used an arbitrary common 3D reference coordinate system attached to the skull of the subject. Each stack was aligned using the appropriate 3D rototranslation that corresponds to the six degrees of freedom of a solid object, and that is obtained by aligning the rigid structures extracted from CT images (hard palate, nasal passages, paranasal sinuses) with the corresponding ones in the MR images stack. The same procedure was also applied to the jaw and to the hyoid bone for each articulation, in order to determine their relative position in relation to the fixed rigid structures.

Determination of the surface outline of the soft structures

The tongue surface is determined in much the same way as for the rigid structures, but from the MR images of each articulation. Planar contours were edited in both sagittal and transverse images. The contours resulting from the intersection of the rigid organs surface with the plane containing the image being edited were superposed on the image in order to provide bony anchor points not visible on MRI and thus very useful for the interpretation of the image. The tongue contours were established as 2D b-spline curves controlled by a limited number of points. The set of all 2D planar contours expanded into the 3D coordinate system forms a 3D description the tongue.

Tongue tip and sublingual boundaries

The sublingual cavity plays an important role in a number of articulations such as postalveolar fricatives, laterals, back vowels and some consonants coarticulated with back vowels. As an illustration, note that the tongue tip is elevated enough to uncover the mouth floor -and thus to create a sublingual cavity -in 19 articulations of our corpus: [u o ѐ oe a ѐ m u Ѐ a Ѐ u p a p u k a k u l a l i l u ల a ల u ]. In these articulations, tongue tip and mouth floor contours can be easily seen and traced (e.g. see Figure 1). For the other articulations in which the tongue tip rests on the mouth floor, these contours are difficult to determine and have to be inferred indirectly (cf. [START_REF] Badin | Three-dimensional modeling of speech organs: Articulatory data and models[END_REF]).

Fitting the generic tongue mesh to each articulation

Linear analysis methods such as Principal Component Analysis (PCA) or multiple linear regression analysis require each observation to bear on the same number of variables. But the plane contours defined above do not have a constant number of points. Therefore, in order to ensure a suitable geometric representation for all articulations, a unique generic 3D surface mesh was built for the tongue, and then fitted by elastic deformation to each of the 3D sets of planar contours. The lateral consonant [l a ] was chosen as reference articulation, as the tongue tip is sufficiently raised to let clearly appear the sublingual cavity, i.e. the lower surface of tongue tip and the mouth floor. This matching process has finally resulted in a tongue surface described as triangular meshes having the same number of vertices, in a common reference coordinate system, with 1640 vertices, for each of the 46 articulations of the corpus (RMS: 0.06 cm). This forms the basis for the articulatory modeling, as illustrated in the next section.

Tongue model

The procedure employed by [START_REF] Badin | Three-dimensional articulatory modeling of tongue, lips and face, based on MRI and video images[END_REF], based on a controlled use of PCA and multiple linear regression analysis, was applied to the tongue data. As the jaw is one of the major tongue carriers, the jaw height parameter JH, defined as the centered and normalized value of the measured lower incisor height, was used as the first control parameter of the tongue model (the associated model coefficients are obtained by the multiple linear regression of all the vertex coordinates against JH). Its main effect is a tongue rotation around a point in the back of the tongue (see Figure 3). The next two parameters, tongue body TB, and tongue dorsum TD, were extracted by PCA from the coordinates of the midsagittal tongue contour, excluding the tongue tip region, from which the JH contribution had been removed (the associated model coefficients were obtained by multiple linear regression, as for JH). They control respectively the frontback and flattening-bunching movements of the tongue (see Figure 3). The next two parameters, tongue tip vertical (TTV) and horizontal (TTH) parameters were extracted by PCA from the midsagittal tongue tip contour coordinates, from which the TB and TD contributions have been removed (the associated coefficients were also obtained by multiple linear regression).

The hyoid bone is connected with major tongue muscles such as the hyoglossus. The vertical and horizontal coordinates of its central part in the midsagittal plane were thus determined for each articulation, and found to be explained up to 60 % by the five tongue parameters described above. An extra parameter, obtained as first PCA extracted from the residue, raised the explanation to 94 %. This parameter, called HY, was then used as the sixth control parameter for the tongue model (the associated coefficients were obtained by multiple linear regression).

Table I displays the variance, relative to the total variance of the full 3D coordinates, explained by each component, for both raw PCA and the controlled analysis described above. It appears that an amount of 87 % is explained by our controlled analysis, which is only 6 % below the optimal result from a raw PCA with the same number of components. Table I. Evaluation of the tongue analysis: varex is the relative explained tongue data variance, varexcum its cumulated value, and RMS the RMS reconstruction error between the modeled mesh and the original planar contours.

Altogether, the 3D tongue model is controlled by the six articulatory control parameters. The effects of these parameters are demonstrated in Figure 2 which displays tongue shapes for two extreme values (-2 and +2) of TD, all other parameters being set to zero, and in Figure 3 which displays the midsagittal intersection of the 3D nomograms with linear variations of the control parameters.

Interestingly, these DoFs can be compared with movements due to various tongue muscles as proposed by [START_REF] Perrier | Influences of tongue biomechanics on speech movements during the production of velar stop consonants: A modeling study[END_REF] who developed a biomechanical model based on the anatomy of the same subject. The forward raising movement associated with the posterior genioglossus and its antagonistic backward lowering movement associated with the hyoglossus contraction correspond to our tongue body parameter TB. The backward bunching associated with the styloglossus contraction and its antagonistic flattening movement induced by the anterior genioglossus contraction is controlled in our model by the tongue dorsum TD parameter. The backward retraction of the tongue tip induced by the inferior longitudinalis contraction is controlled by TTH, while the downward movement of the tongue tip due to the superior longitudinalis is controlled by TTV. Similar biomechanical interpretations of our model could be drawn from [START_REF] Dang | Construction and control of a physiological articulatory model[END_REF] as well. 5. Predictability of the 3D tongue surface mesh from its 2D midsagittal contour [START_REF] Badin | Three-dimensional articulatory modeling of tongue, lips and face, based on MRI and video images[END_REF] stated that « most 3D geometry of tongue, lips and face can beat least for speechpredicted from their midsagittal contours. » The present study, based on a larger corpus and a more elaborated model, attempted to assess quantitatively the possibility to determine the 3D shape of the tongue from its midsagittal shape. We have thus made a series of inversion experiments. The principle consisted in determining, using an unconstrained optimization procedure, the set of six control parameters that minimizes a given error between a given tongue shape target and the tongue shape produced by the model from these control parameters. Two different error measures were used: (1) the Root Mean Square (RMS) of the vertex-to-vertex distances for the vertices of the upper surface of the tongue (from tongue tip down to tongue root) close to the midsagittal plane (about a hundred vertices among the full 3D mesh vertices), and

(2) the RMS of the vertex-to-vertex distances for all the vertices of the full 3D tongue mesh. In all experiments, the model control parameters were given zero initial values.

In the first series of experiments, the targets for the inversion procedure were the tongue shapes of the corpus modelled with the six parameters: using either midsagittal or full 3D error led exactly to the same shapes and the same errors found at the model development stage (overall midsagittal RMS error: 0.19 cm, overall full 3D RMS error: 0.22 cm), which validates the inversion procedure. In the second series of experiments, the target shapes were the measured tongue shapes, i.e. the shapes obtained by elastic deformation. Using the midsagittal error measure led to mean midsagittal and full 3D RMS errors of respectively 0.17 cm and 0.25 cm, while using the full 3D error measure led to mean midsagittal and full 3D RMS errors of 0.22 cm and 0.19 cm. It is not surprising that, when the midsagittal error is used for the optimization, the final full 3D error for the measured data is greater that that found for the modelled tongue shape targets; indeed, the optimization procedure will favour a smaller midsagittal error to the detriment of a larger full 3D error. Conversely, when the full 3D error is used, the optimization procedure will favour a smaller full 3D error to the detriment of a larger midsagittal error. The error made in the prediction of the 3D tongue shapes from their midsagittal contours can finally be quantified by the difference between the overall full 3D RMS errors for the model (0.22 cm) and for the inversion based on the midsagittal error (0.25 cm): the mere 0.03 cm (13.6 %) increase of this error testifies to the very good predictability of the 3D tongue surface mesh from its 2D midsagittal contour.

Tongue volume conservation

In the literature on biomechanical tongue modeling, tissue incompressibility is commonly assumed (cf. e.g. [START_REF] Perrier | Influences of tongue biomechanics on speech movements during the production of velar stop consonants: A modeling study[END_REF], or [START_REF] Dang | Construction and control of a physiological articulatory model[END_REF]), though tongue volume conservation has never been really assessed, at least for speech tasks. We have thus estimated the volume enclosed in the tongue surface outline: with an average of 115 cm 3 , this volume ranges between 110 and 121 cm 3 , which amounts to a mere ± 5% maximal departure from the mean. Considering the diversity of tongue shapes in the corpus, this result supports rather well the hypothesis of tongue volume conservation in speech. Note however that this conclusion should be weighted by the fact that the tongue surface outline was arbitrarily drawn in some regions (though in a coherent way): for instance, only parts of the styloglossus and of the palatoglossus were included inside the tongue volume outline.

Conclusion and perspectives

The present work produced a number of valuable results. First, a database of 3D geometrical descriptions of tongue was established for a speaker sustaining a set of 46 French allophones covering the speech possibilities of the subject. Linear component analysis of these data revealed that six components could account for about 87 % of the total variance of the tongue shape. The first five components correspond qualitatively to those found previously by [START_REF] Badin | Three-dimensional articulatory modeling of tongue, lips and face, based on MRI and video images[END_REF] on a smaller corpus of 25 articulations. The present study constitutes thus an extension of this work. It is also worth noting that the variance explained in the present study is about 10 % higher than in [START_REF] Badin | Three-dimensional articulatory modeling of tongue, lips and face, based on MRI and video images[END_REF], and that the tongue tip and sublingual cavities are better defined. This will be useful for dealing with articulations such as post-alveolar fricatives, laterals, back vowels and some consonants coarticulated with back vowels. This is also one of the motivations for dealing with a 3D model. Another interest of this 3D model lies in the direct possibility to deal with tongue groove: Figure 2 illustrates well the strong groove variation controlled in particular by the tongue dorsum parameter.

The next modeling step will be the extension of the model to the complete 3D vocal and nasal tracts (cf. [START_REF] Badin | Three-dimensional modeling of speech organs: Articulatory data and models[END_REF] for a velum model) in order to build a complete 3D articulatory model of speech. Then the determination of the area functions for both oral and nasal tracts as a function of the shapes of tongue, velum and nasopharyngeal wall, will yield the acoustical characteristics of the complete tract.

Figure 3 .

 3 Figure 2. Jaw and tongue positions for extreme values of TD

Proceedings of the 7th ISSP

Badin & Serrurier

Acknowledgements

We thank very sincerely Jean-François Lebas (Radiology Department, CHRU, Grenoble) for granting us access to the MRI equipment, Christophe Segebarth (Functional and Metabolic Neuro Imagery Department, INSERM & UJF, Grenoble) and Monica Baciu (LPNC, Grenoble, France) for managing the MRI recordings, Andreas Fabri (Geometrica team, INRIA, Sophia) for the letting us use his 3D meshing software, and Maxime Bérar (LIS, Grenoble) for helping us with the elastic matching software.