Roxane Bertrand
email: roxane.bertrand@lpl.univ-aix.fr

R Espesser
email: robert.espesser@lpl.univ-aix.fr

Voice Diversity in Conversation: a Case Study

published or not. The documents may come

Theoretical perspective

The voices to which we refer here echo the diversity of enunciative sources that a given speaker is often led to introduce in his own speech. This conception of enunciative source heterogeneity implies that the speaker is no longer conceived as the unique source of his discourse. We therefore admit that his productions are submitted to other influences as soon as the notion of self-conscious subject unicity has been challenged.

Authier [START_REF] Authier-Revuz | Hétérogénéité montrée et hétérogénéité constitutive, éléments pour une approche de l'autre dans le discours[END_REF] makes a distinction between two types of heterogeneity. ″Constitutive heterogeneity″ finds its origins in works outside linguistics (Freud, Bakhtine, Mead) and concerns an internal reality depending on the subject. ″Shown heterogeneity″, linked to language activity performed by individuals, belongs more directly to the core of linguistics insofar as it is a clear reference to the plurality of voices that subjects invoke in their speech. In this study only the second type of heterogeneity will be considered. The term ″enunciator″ [START_REF] Ducrot | Esquisse d'une théorie polyphonique de l'énonciation[END_REF] will be used to mention the different voices staged by a speaker in his own speech.

The most vivid illustration of the presence of other voices in one's speech is the direct reported speech (DRS). De Gaulmyn (1992) defines the Reported Speech as the recorded broadcast of utterances previously pronounced by identified enunciators; a case that Vincent and Dubois (1996) name ″reproduction″. These authors propose other uses of the structure of reported speech which testify the diversity of facts it encompasses, among which we find : a/ ″pseudo-reproduction″, consisting in letting believe that the utterances have been pronounced already ; b/ ″realization″ which concerns a reported speech which is the prototype of several similar events; c/ ″invention″ which consists in making say utterances that have never been pronounced [START_REF] Vincent | Le discours rapporté au quotidien[END_REF]. These three uses cover the totality of cases examined here.

DRS, far from being a mere speech-reproduction strategy, constitutes a true speech production strategy. Quoting consists in adopting utterances emanating from someone else (to achieve argumentative goals for instance) to serve one's general speech project [START_REF] Vincent | Le discours rapporté au quotidien[END_REF][START_REF] Bertrand | Etre soi avec les mots d'autrui[END_REF].

The speaker producing DRS can either quote other enunciators or himself / herself. Self-quoting supports the idea that it is rewarding to do so, since a simple direct speech might have been produced instead.

Objectives

Our objective in this study is to validate the role of prosody as a polyphonic marker through the analysis of F0 parameter. More precisely, we seek to characterize three discourse types :

-Direct Speech (DS) -Reported Speech consisting of utterances emanating from another enunciator than the speaker (DRo) -Reported Speech consisting of self quotations (DRsq).

Hypothesis

Voice changes (i.e. enunciator shift) correspond to vocal changes (i.e. shifts in pitch) [START_REF] Grosjean | Les Musiques de l'Interaction[END_REF]. These voice changes are characterised by different involvement degrees from the speaker in his DS, DRsq and DRo. We hypothesize that these involvement degrees are reflected in global variations of the F0 parameter (frequency distribution, temporal distribution, short term correlation).

Corpus and Methodology

We used excerpts from a one-hour familiar family conversation [START_REF] Traverso | La conversation familière. Analyse pragmatique des interactions[END_REF] recording between three participants. 248 DRS were produced, 188 from the main female speaker (A) studied here.

The DRS were tagged manually on the speech signal. We retained pairs constituted of a DS immediately followed by a DRS. The duration of each pair varies between 1 and 4 seconds (the mean duration is about 2 seconds) for DS and DRS respectively. The duration of each element was equalized (the longer element is clipped to the shorter duration).

The pairs were also selected on a homogeneity criterion in terms of voiced sequences. The speech signal has been segmented automatically in voiced areas delimited by their own voiced boundaries or limited to a maximal duration of 140 ms. This value, quite close to the mean value of unstressed syllables in spontaneous speech [START_REF] Astésano | Rythme et Discours : Invariance et sources de variabilité des phénomènes accentuels en français[END_REF], provided a good adjustment between the number of these voiced areas and that of real syllables in the corpus, that is close to a syllabical sequencing. We call such voiced sequences ″pseudo-syllables″ (PS).

The selected pairs consisted of a 400 ms duration of voiced speach at least and 5 PSs to avoid too big a proportion of silent pauses. Morever, overlapping pairs also were eliminated. These various constraints account for the fact that 50% of the initial DRS has been rejected. 91 pairs of DS versus DRS (among which we found 53 DRo and 38 DRsq) were part of the selection.

In this study, we considered only F0 parameter. F0 extraction combined three detection methods : comb function, AMDF and autocorrelation. F0 was computed every 10 ms.

Two types of measures were derived from the F0 files. The first type was the mean of the raw values of F0 (in Hz) of each PS. The second type is the value (in Hz) of a targetpoint (TP) obtained from a modelization routine. The F0 raw curve was modelized by a continuous and smooth curve deprived of microprosodical variations, which are not relevant for the analysis (Di Cristo [START_REF] Di Cristo | Modelling French Micromelody : Analysis and Synthesis[END_REF]. According to Di Cristo (2000), this modelization process has two main advantages: it substitutes to the raw curve a smooth curve corresponding to the continuous perception of speech melody and it ensures the emergence of the target points indicating the significant anchoring points of this curve.

Statistical analysis 5.1 F0 values distributions

Descriptive statistics

Descriptive statistics were derived from PS and TP measures. These were mean F0, standard deviation (SD) of F0, lower and upper quartiles, interquartile range (IRQ), lower and upper 10th percentiles, inter 10th percentile range (IDR). Percentile measures were computed too because they were less tied to assumptions about distribution.

Comparison of the F0 values distributions

To compare these F0 values distributions which differ from the normal distribution, we used a Kolmogorov-Smirnov test. The differences were less obvious for TPs than for PSs because there were less TPs.

Pseudo-syllabes DS vs

Proportion test

In a previous study [START_REF] Bertrand | Prosodie et discours rapporté : la mise en scène des voix[END_REF], we found a greater proportion of F0 values greater than 270 Hz in the DRS. This threshold was used here to compare the three distributions.

Global variation of F0 values

Temporal density of the TPs

TPs being located essentially at the points where the curve changes, temporal density (number of TPs per seconds) is a cue of the variability of the F0 curve. Below are compared (ttest) the temporal density of the three populations.

DS vs DRo : 3. There are no significant differences between the three discourse types. The mean temporal difference between two TPs varies from 275 to 300 ms.

Slope between two consecutive TPs

The slope between two TPs (in Hz/ms) is another cue of the variability of the F0 curve. As the mean slope is equal to 0 by construction, we compared the SD of the three populations. The three SD are significantly different with a clearcut distinction between DRsq and DRo.

To illustrate these results, we calculated the frequency value of a TP located at 300 ms from a first 250 Hz TP (mean value for the speaker in this corpus) with a slope measure of one SD (typical value of the distribution).

Table 5 : semitone ratio of TP (i+1) to estimated TP(i)

DRsq DS DRo TP(i+1)/TP(i) semitone 8.86 8.05 7.07

Temporal evolution of the F0 values

As a cue of temporal continuity, we chose the correlation coefficient (R) between two consecutive PSs F0 values and two consecutive TPs F0 values too. The table 7 shows the significance of the difference between the correlation coefficients. DRo stands out quite significantly from the other two, both by a lower variability of slopes and a higher correlation coefficient between 2 consecutive PSs (or TPs). DS and Drsq do not really differ related to these parameters.

NB : Correlations are weaker for TPs as they are more temporally scattered than PSs.

Discussion

Our results contribute to validate the role of prosody as a polyphonic marker. It is clear that the F0 parameter is relevant to differenciate the three types of discourse (DS, DRsq, Dro) that have been considered.

In our previous results two melodic effects enabled us to distinguish between DS and DRS, these effects currently belong to the Drsq and Dro types.

DRsq differs from the two other types on the F0 values distribution of the PSs and TPs. Drsq is characterised by an extension of our female speaker's pitch range as standard deviation and interpercentile ranges show (table 1a, 1b). Upper percentiles increase more than others, which confirms the significant increase of the proportion of values greater than 270 Hz (table 2, 3). Morever, the slope measures between two consecutive TPs show a more important variability for DRsq which has more abrupt slopes. The combination of these two acoustic elements contributes to make DRsq more salient in the speaker's speech. Table 5 shows that a TP estimated from a typical slope of one SD is 1.8 semitone higher for DRsq than DRo, which is perceptible.

Such results confirm the position of Vincent and Dubois, who wondering about the purpose of using reported speech with self quotations, come to the conclusion that the content of these quotations is thus highlighted. The use of DRsq implies that what is uttered is relevant for our female speaker who thus draws the attention of other interactants to some speech segments by creating a complex system.

Dro is produced with smaller slopes in a narrower pitch range than it is the case for Drsq, hence a lesser variability. Dro has a higher correlation coefficient between 2 consecutive PSs or 2 consecutive TPs, which indicates a better degree of predictability between them. We can interpret this point as reflecting a linear dependance distributed among all PSs / TPs or, more interestingly, a dependance concentrated on sequences composed of 3 or 4 consecutive PSs, which realizes a micro-serie or a reccurrent motive composed of rather smooth and continuous moves. The presence of lengthening phenomena can create this effect, and we could thus oppose the 'melodical salience' (Caélen-Haumont, 1991) of DRsq to a 'temporal salience' of DRo. On listening to the corpus, we have been able to locate the reccurrent presence of a particular ″drawling″ inflexion in DRo.

We already have established [START_REF] Bertrand | Etre soi avec les mots d'autrui[END_REF]) that DRS are relevant utterances for the speaker under study insofar as they are used to generate in her dialogue a specific self-image. It is then of prime interest to make them salient to coparticipants. She makes use of two strategies : a/ she presents herself in a high position at discourse level (she posseses the information) and subjective level (a fighting lively and dynamic person), b/ she often introduces enunciators (DRo) in low positions (discourse position of asker, or subjective position of coward). The discourse-enunciative level, in spontaneous speech notably, is not easily separated from a level belonging to the affective. Under the general heading of focalisation, Di Cristo (1999) unites aspects of discourse salience which have been dealt with separately. According to Bolinger's view, any transformation transfer is mediatised by the affective in any case, by the interest that a given speaker shows for their own message, and by the desire to impress their audience.

Our results confirm that different involvement levels can emerge by the staging of various focalisation systems (emphasis here), specific to DRsq and DRo which are ″cristallized″ (De Gaulmyn). The term 'focalisation' echoes Di Cristo's conception, who believes it plays a major part to identify and make salient speakers' involvement strategies in their speech.

DS with a small pitch range (table 1a and 1b) and a weak correlation coefficient (table 6) can be opposed to DRsq and DRo. We argue that DS are less marked because the speaker's involvement is not important as she seems more interested in focalizing DRS.

More generally, these results are in conformity with the notion of « enunciative instability » [START_REF] Vion | De l'instabilité des positionnements énonciatifs dans le discours[END_REF] characterised by a fluctuating discourse marked by sucessive strong and weak moments dependent on the degree of a speaker's involvement in his speech. This instability or ″enunciative breathing″ would correspond to an ″enunciative rhythmization process″ based on the reccurrence of melodic patterns in some relevant discourse elements (DRS). The DS, around which the two DRS types are to be found, could be considered as planning stages that our female speaker neutralizes, knowing she is about to produce DRS so as to make the latter utterly salient.

  Dro: D = 0.0427; p = 0.5268 DS vs Drsq: D = 0.1155; p = 0.001554 Dro vs Drsq: D = 0.1357; p = 0.000355 Target-Points DS vs Dro: D = 0.045 ; p = 0.78 DS vs Drsq: D = 0.112 ; p = 0.041 Dro vs Drsq: D = 0.12 ; p = 0.03 DRsq is quite different from both DS and DRo. DS and DRo do not distinguish from one another.

  3 vs 3.6 t = 1.9 df = 115.7 p = 0.054 DS vs DRsq : 3.62 vs 3.31 t = -1.77 df = 70.1 p = 0.08 DRsq vs DRo : 3.62 vs 3.6 t = -0.09 df = 76.6 p = 0.92

Table 4 :

 4 SD of the slope of DRsq, DS and DRo DRsq vs DRo : F = 1.7441 p = 3.844 e-06

Table 1a :

 1a F0 mean and standard deviation of PS and TP.

		Item count mean	SD
		PS
	DS	896	248.69 66.08
	DRo	605	249.41 66.31
	DRsq	383	267.92 83
		TP
	DS	515	264.7 82.5
	DRo	348	261.5 79.3
	DRsq	220	285	99

Table 1b :

 1b F0 quartiles and 10th percentiles of PS and TP.

		q < q >	IQR	d < d >	IDR median
				PS
	DS	205 274	69	186 340	154	231
	DRo 204 280	76	188 344	155	232
	DRsq 216 303	87	190 385	195	244
				TP
	DS	207 304	97	186 385	199	240
	DRo 205 297	92	183 366	183	235
	DRsq 214 329	115	187 429	242	256

Table 2 :

 2 proportion of the PSs > 270 Hz

	PSs proportion Total count PSs count > 270 Hz
	DS	0.26	896	237
	DRo	0.28	605	172
	DRsq	0.36	383	140
	Proportion test
	DS vs DRo	p = 0.43 (ns)
	DS vs DRsq	p = 3.610 -9
	DRsq vs DRo	p = 0.009

Table 3 :

 3 proportion of the TP > 270 Hz

		TPs Proportion Total count TPs count > 270 Hz
	DS	0.35	515	181
	DRo	0.347	348	121
	DRsq	0.445	220	98
	Proportion test
		DS vs DRo	p = 0.967 (ns)
		DS vs DRsq	p = 0.02
		DRsq vs DRo	p = 0.0248
	NB : the proportion of the F0 values greater than 270 Hz is
	more important for TPs than for PSs: a PSs segmentation does
	not eliminate intermediate values while the TPs modelization
	retains extreme values preferentially and backgrounds
	intermediates values.
	These results show that DRsq stands out from DS and
	DRo which are indistinguishable. DRsq is characterized by an
	extension of the PSs and TPs distributions beyond 270 Hz.

Table 6

 6

	: correlation coeff. (R) of two consecutives F0 values
		(PS and TP)
	R	PS(i) PS(i+1) TP(i) TP (i+1)
	DS 0.449	0.048
	Dro 0.637	0.25
	Drsq 0.49	0.055

Table 7 :

 7 Comparison of Rs

		PS(i)PS(i+1) TP(i)TP(i+1)
	DS vs Dro	z: 5.11	z : 2.997
		p : 3.16 10 -7	p : 0.0029
	DS vs Drsq	z: 0.859	z : 0.086
		p : 0.39	p : 0.93
	Dro vs Drsq z: 3.31	z : 2.3
		p : 9.10 -4	p : 0.0207


MAITINE BERGOUNIOUX Romain Abraham
email: romain.abraham@univ-orleans.fr

Pierre Debs
email: pierre.debs@univ-orleans.fr

ISTO Olivier Rozenbaum

AUTOMATIC CHOICE OF THE THRESHOLD OF A GRAIN FILTER VIA GALTON-WATSON TREES. APPLICATION TO GRANITE CRACKS DETECTION

Keywords: Image processing, Galton-Watson, grain filters, cracks. MSC classification: 60J80, 68U10, 94A12

Automatic choice of the threshold of a grain filter via Galton-Watson trees. Application to granite cracks detection

Romain Abraham, Maïtine Bergounioux, Pierre

Introduction

A challenging issue in imaging processing is the identification of thin structures. It may be blood vessels, leaf veins or streets in a satellite or aerial image. From a mathematical point of view these structures have a zero Lebesgue measure and we have to use the Hausdorff measure instead. From a practical point of view, these structures may be not viewed once the discretization has been done. There are many methods to achieve this goal both from theoretical and numerical points of view (as the morphological skeletonization for example). However, they may be not very satisfactory since the images are very often corrupted with a high level of noise that, in addition, may not be Gaussian.

The goal of this paper is not the presentation of an image processing model that would identify the structures better than the existing ones but rather give a post-processing method allowing to remove any residual noise effect. Therefore, we do not report on thin structures recovering techniques, and do not quote the related works.

The example we have in mind is the recovery of cracks inside a granite block. More precisely, the sample has been imaged 1 via X-tomography process and we have to deal with a 3D image whose size is 1287 x 1287 x 99. For more details on the geological context one can refer to [START_REF] Maurício | Assessment of stone heritage decay by x-ray computed microtomography: I-a case study of portuguese braga granite[END_REF][START_REF] Rozenbaum | Versatile three-dimensional denoising and segmentation method of x-ray tomographic images: applications to geomaterials characterizations[END_REF]. Figure 1.1 shows a 2D slice of such a stack. The right hand side image is the result of a pre-processing method that we briefly present in Section 3 without any focus on it; it allows to get rid of contours that are not cracks. However, there is a residual impulse noise that we would like to remove while keeping cracks, especially micro-cracks. There are few methods devoted to crack identification in geological context. In biological context (angiography or X-ray imaging for example) mathematical morphology (skeletonization) is often used. On can mention the use of fuzzy measures in [START_REF] Linda | Crack detection in x-ray images using fuzzy index measure[END_REF] as well. In material context, there are many works that focus on cracks in roads or concrete. Different tools are used as statistical ones (co-occurence matrices in [START_REF] Moysan | Crack-like defect detection and sizing from image segmentation through co-occurrence matrix analysis[END_REF] and anisotropic diffusion with region linkage in [START_REF] Oliveira | Automatic crack detection on road imagery using anisotropic diffusion and region linkage[END_REF]), wavelets [START_REF] Subirats | Automation of pavement surface crack detection using the continuous wavelet transform[END_REF], geodesic contours [START_REF] Chambon | Detection of points of interest for geodesic contours: application on road images for crack detection[END_REF], multiscale approach [START_REF] Coudray | Multi-resolution approach for fine structure extraction: Application and validation on road images[END_REF] together with Markov modeling or image-based percolation models [START_REF] Yamaguchi | Image-based crack detection for real concrete surfaces[END_REF] and dealing with brightness and connectivity as in [START_REF] Sy Nguyen | Free-form anisotropy: A new method for crack detection on pavement surface images[END_REF].

To address this problem, we focus here on a grain filter. This filter, introduced in [START_REF] Vincent | Grayscale area openings and closings, their efficient implementation and applications[END_REF] in the framework of Mathematical Morphology and then generalized in [START_REF] Masnou | Image restoration involving connectedness[END_REF], consists in removing all the connected components of a binary image (or of the level sets for a grey level image) of size smaller than a fixed threshold. It is mainly used for image denoising but also for feature extraction, see [START_REF] Pritchard | Area thresholding and silhouette extraction of simple coloured objects using hue[END_REF]. We refer to [START_REF] Caselles | Grain filters[END_REF] for more details on this filter. In what follows, we suppose that we have a binary image containing features we want to extract (namely thin structures as cracks). The image is corrupted by an impulse noise where every pixel is 1 with probability p and 0 with probability 1 -p independently of the others. One feature that allows to distinguish cracks from the noise is that the cracks size is much larger than the connected components (that we define in the sequel) the noise may create. Therefore, a grain filter can be used in order to suppress the small connected components and keep the ones whose size is greater than some threshold a 0 . The aim of this paper is to give a method to compute the threshold of the grain filter via a statistical hypothesis testing. This method enters the large class of a contrario frameworks whose ideas first appeared in [START_REF] Jay | On the role of structure in vision[END_REF] and which is now widely used since the pioneer works [START_REF] Zhu | Embedding gestalt laws in markov random fields[END_REF][START_REF] Desolneux | Meaningful alignments[END_REF]. We refer to the book [START_REF] Desolneux | From gestalt theory to image analysis: a probabilistic approach[END_REF] for a complete review of these methods. The general idea of the a contrario method is to compute the probability of appearance of the looked after feature in a pure noise image. If this probability is very small (less than some fixed ε), the feature is said to be "meaningful" i.e. it cannot appear "accidentally" and thus can be viewed as an important characteristic of the studied image.

Throughout the paper, we consider that the image contains only an impulse noise with parameter p (hypothesis H 0 ). Under H 0 , we denote by C 0 the connected component of the image that contains a given point, say the origin (0, 0), and by C 0 its cardinal. We then fix a level ε > 0 for the test and find a 0 such that, under H 0 ,

P( C 0 ≥ a 0 ) ≤ ε.
Keeping the connected components of size greater than a 0 , we remove most of those due to the noise. The goal of this paper is to get an upper bound for the looked after probability, and hence to automatically compute the threshold a 0 , by comparing the size of C 0 with the total population size of a Galton-Watson process.

Let us stress that this problem has already been tackled in [START_REF] Coupier | Image denoising by statistical area thresholding[END_REF] where an asymptotic (as the size of the image tends to +∞) of the probability P( C 0 ≥ a 0 ) is obtained. This asymptotic contains a combinatorial term a k (namely the number of connected components one can make with exactly k pixels) which is only known (up to k = 47) for the 4connectivity. Our approach first gives an exact bound of the probability and not an approximation as in [START_REF] Coupier | Image denoising by statistical area thresholding[END_REF] (which is not very relevant as the size of our images are quite large and the approximations of [START_REF] Coupier | Image denoising by statistical area thresholding[END_REF] are sharp), second allows to use different features to discriminate cracks from noise. In Section 2, we focus on the size of a given connected component for the 8-connectivity and give all the details for the proofs but the ideas can be easily adapted to handle other definitions of connectivity, the same problem for 3D images or to use the diameter of the connected component instead of its size as developed as well.

We present the computation of the threshold in the next section and Section 3 is devoted to the numerical realization (algorithms and numerical experimentation). We work with the infinite norm on Z 2 i.e. if x 1 = (i 1 , j 1 ), x 2 = (i 2 , j 2 ) are two points of Z 2 , we set

Computation of the threshold

x 1 -x 2 = max(|i 1 -i 2 |, |j 1 -j 2 |).
For x ∈ Z 2 , B(x) denotes the closed ball (with respect to the previous norm) centered at x with radius 1 and B(x) is the pointed ball B(x) \ {x}. The elements of B(x) are called the neighbors of x.

For any finite subset A of Z 2 , we denote by A the cardinal of A.

Definition 2.1. A set A ⊂ Z 2 is said to be connected if, for every x, y ∈ A, there exists a finite sequence (x 0 , . . . , x n ) such that

• x 0 = x, x n = y; • ∀i ∈ [[0, n]], x i ∈ A; • ∀i ∈ [[1, n]], x i ∈ B(x i-1 ).
We fix a positive integer N and consider (for simplicity) the domain Ω = [[-N, N ]] 2 . A binary image on Ω will be an application X : Ω -→ {0, 1}. In all the paper, we always make the confusion between the binary image X and the set X -1 (1) = {x ∈ Ω, X(x) = 1}, which is the set of the black pixels . In particular, we talk of the connected components of the image instead of the set X -1 (1).

Galton-Watson process.

A Galton-Watson process is a branching stochastic model introduced in 1873 by Francis Galton to describe population growth and is still the object of active research. We refer to [START_REF] Athreya | Branching processes[END_REF] for a comprehensive exposition on this topic. We first give a formal definition of this process. Definition 2.2. A Galton-Watson process is a stochastic process (Z n ) n∈N defined as follows: consider a probability measure ν on the set of non-negative integers (called the offspring distribution) and a family (ξ (n) i , n ≥ 0, i ≥ 1) of i.i.d. random variables with distribution ν. Then the process (Z n ) n∈N is defined recursively by

     Z 0 = 1, Z n+1 = Zn i=1 ξ (n) i for n ≥ 0 with the convention that Z n+1 = 0 if Z n = 0.
To give an intuitive picture of that process (Z n ) n∈N , let us describe it informally as a population evolution model. We start with a single individual at generation 0. This individual gives birth to a random number ξ (0) 1 of offspring with distribution ν which form generation 1. These offspring reproduce according to the same distribution ν independently of each other and independently of the past and so on. Hence, the variable Z n represents the population size at generation n whereas the variable ξ (n) i represents the number of offspring of the i-th individual of the n-th generation.

Of course, if the population dies out at some time n, then the population is zero for all further time: 0 is called an absorbing state for the process (Z n ) n∈N .

One of the initial questions raised by Francis Galton was the computation of the probability that the population dies out in finite time. This probability is called the extinction probability and is given by

P(∃n ∈ N, Z n = 0) = lim n→+∞ P(Z n = 0).
To avoid trivialities, we suppose that ν(0) > 0 (if not, the extinction probability is 0) and that ν(0) + ν(1) < 1 (if not, the extinction probability is clearly 1). The answer to the question is given by the following result (see Chapter 1 Section 5 of [START_REF] Athreya | Branching processes[END_REF]).

Proposition 2.1. Let ξ be a random variable with distribution ν. We denote by

µ = E[ξ] = +∞ k=0 kν(k)
the mean number of offspring, and by

F (s) = E[s ξ ] = +∞ k=0 ν(k)s k for s ∈ [-1, 1],
the generating function of ν.

Then, the extinction probability of the process (Z n ) n≥0 is given by the smallest nonnegative root q of the equation s = F (s).

In particular, if µ ≤ 1, q = 1 (extinction arises almost surely) whereas if µ > 1, then q < 1 (the population has a positive probability for living forever).

The offspring distribution ν and the Galton-Watson process (Z n ) n≥0 are called subcritical (resp. critical, super-critical) if µ < 1 (resp. µ = 1, µ > 1).

Stochastic domination.

Let m be the number of neighbors of one pixel. With our definition of B(x), we have m = 8 but the method remains valid for a general neighborhood. Let p ∈ (0, 1) and (Z n ) n≥0 a Galton-Watson process with the binomial B(m, p) distribution as offspring distribution ν, that is, for 0

≤ k ≤ m, ν(k) = m k p k (1 -p) m-k .
According to Proposition 2.1, in order to obtain a sub-critical Galton-Watson process, we assume that mp < 1. We denote by Z the total population size:

Z = +∞ n=0 Z n .
Theorem 2.3. Under H 0 , the size C 0 of the connected component C 0 (and thus of any connected component) is stochastically dominated by Z, that is:

∀k ∈ N, P( C 0 ≥ k) ≤ P( Z ≥ k). Proof.
The main ideas of the proof are taken from [START_REF] Mathew | Continuum percolation and Euclidian minimal spanning tree in high dimensions[END_REF] where the size of a connected component that arises in continuum percolation is compared to the size of a Galton-Watson process with Poisson offspring distribution.

Let (X (i,j) , (i, j) ∈ Z 2 ) be a family of i.i.d. Bernoulli random variables with parameter p. This models the noise of the image under H 0 . Remark that if the stochastic domination is true for an infinite image, it is a fortiori true for an image defined on

[[-N, N ]] 2 . If A is a subset of Z 2 ,

we denote by

X A = {(i, j) ∈ A, X (i,j) = 1}.
Then, X A is distributed according to a binomial distribution with parameters p and A. Moreover, if A and B are disjoint subsets of Z 2 , then X A and X B are independent.

In order to label the points of some subset in a unique way, we define a total order (denoted by ≤) on Z 2 . Let x 1 , x 2 ∈ Z 2 :

• if x 1 < x 2 then we set We define a sequence ( Wn ) n≥0 of subsets of Z 2 such that C 0 = n≥0 Wn in the following way:

x 1 ≤ x 2 . • if x 1 = x 2 = n,
• W0 = {(0, 0)}.

• Suppose that W0 , . . . , Wn are constructed with Wn = {x 1 , . . . , x k } labeled in increasing order. Then, we set

Wn+1 = k =1 X C
where

C = B(x ) \   -1 j=1 B(x j ) n-1 i=0 x∈ Wi B(x)   ,
with the convention

0 j=1 B(x j ) = ∅.
• If Wn = ∅, then we set Wk = ∅ for every k > n. The set C at step n is the set of neighbors of x that have not been involved yet in one of the C k -type set constructed during the previous steps. Consequently, these sets C are pairwise disjoint and the random variables X C are independent and distributed respectively according to a binomial B( C , p) law.

In a second step, we consider a branching random walk (W n ) n≥0 on Z 2 (we refer to [START_REF] Shi | Branching random walks[END_REF] for more details on branching random walks). This process takes its values in the set n≥0 (Z 2 ) n of finite sequences of points of Z 2 (with the convention (Z 2 ) 0 = {∅}) and is also defined recursively as follows:

• W 0 = (0, 0).

• Let us suppose that W n is given and is non-empty, say W n = (x n 1 , . . . , x n kn ). For every i ≤ k n , we consider the 8 neighbors of x n i (ranked as on Figure 2.3) (y 1 , . . . , y 8 ) and consider the sub-sequence W i n+1 formed by taking each y j with probability p, independently of the others (and independently of the other points x n j , i = j). Then W n+1 is the concatenation of all the sub-sequences (W i n+1 , 1 ≤ i ≤ k n ). Remark that a pixel may appear several times in the sequence W n+1 .

• If W n = ∅, then W n+1 = ∅.
The subsequence W i n+1 may be seen as the offspring of the pixel x i and W i n+1 is distributed according to a binomial B(8, p) distribution. Therefore, the process ( W n+1 ) n≥0 is distributed as a Galton-Watson process starting from a single individual and with offspring distribution B(8, p).

We now define a pruning procedure for the process (W n ) n≥0 which defines a set-valued process ( Ŵn ) n≥0 as follows:

• Ŵ0 = {(0, 0)}.
• Suppose that we have constructed Ŵ0 , Ŵ1 , . . . , Ŵn where each set Ŵi is composed of elements of W i , with Ŵn = {x n 1 , . . . , xn k } (ranked in increasing order). To every xn i there corresponds a point x n j i of W n and we set As the process ( Ŵn ) n≥0 is distributed as the process ( Wn ) n≥0 (since the recursive formulas are the same), we have for every a ≥ 0

Ŵ i n+1 = W j i n+1 \   i-1 =1 B(x n ) n-1 r=0 x∈ Ŵr B(x)   and then Ŵn+1 = k i=1 Ŵ i n+1 . • If Ŵn = ∅, then Ŵn+1 = ∅. (a) W1 = {(0, 1), (1, 1)} (b) W2 = W 1 2 ∪ W 2 2 where W 1 2 = {(2, 0), ( 1
P( C 0 ≥ a) = P Wn ≥ a = P Ŵn ≥ a .
Furthermore, it is clear by construction (pruning procedure) that Ŵn ≤ W n for every n ≥ 0, which yields

P Ŵn ≥ a ≤ P W n ≥ a = P( Z ≥ a)
which ends the proof.

Remark 2.2. Looking at Figure 2.2, it seems that we could in fact take m = 5 except for the first generation of the tree where we must keep m = 8. The corresponding tree is not stricto sensu a Galton-Watson tree (as the offspring distribution is not always the same), however the computations of the next subsection can probably be adapted to this case. For sake of simplicity, we take m = 8 in what follows to have a true Galton-Watson process.

2.4.

Computation of the threshold. Let (Z n ) n≥0 be a a sub-critical Galton-Watson process with offspring distribution a binomial B(m, p). For this offspring distribution to be sub-critical, recall that we must suppose that (2.1) mp < 1.

We need an upper bound for the probability P( Z ≥ k). For this purpose, we use Dwass formula that relates the total population size of any sub-critical (or critical) Galton-Watson process to the associated Lukasiewicz random walk, see [START_REF] Dwass | The total progeny in a branching process[END_REF]:

Proposition 2.3. Let (Z n ) n≥0 be a critical or sub-critical Galton-Watson process with offspring distribution ν. Let (Y n ) n≥1 be a sequence of i.i.d. random variables with distribution ν. We set for every n ≥ 1

S n = n i=1 Y i .
Then we have

P( Z = n) = 1 n P(S n = n -1).
Recall that ν is the binomial B(m, p) distribution; then for every n ≥ 1, S n is distributed according to a binomial B(mn, p) distribution (as the sum of i.i.d B(m, p) random variables). Therefore (2.2)

P( Z = n) = 1 n mn n -1 p n-1 (1 -p) mn-n+1 .
This exact formula allows us to compute the looked after threshold:

Proposition 2.4. We set

(2.3) q = m m p(1 -p) m-1 (m -1) m-1 , and C = m m -1 e(1 -p) pK 2 1 1 -q with K = e 3 2 (1-ln 3
2 ) . Then, for every ε > 0, a threshold is given by

(2.4) a 0 = ln ε C ln q
that is for every a ≥ a 0 , we have

P( Z ≥ a) ≤ ε.
Proof. By Proposition 2.3 and Equation (2.2), we get for every a > 0

P( Z ≥ a) = n≥a P( Z = n) ≤ n≥a mn n p n-1 (1 -p) mn-n+1 , since 1 n mn n -1 ≤ mn n .
We now use the following bounds for the factorial function whose proof is postponed to the end of this section:

Lemma 2.4. For every positive integer n, we have

(2.5) Kn n+ 1 2 e -n ≤ n! ≤ n n+ 1 2 e -n e
with K = exp 3 2 1 -ln 3 2 . Using these bounds in the binomial coefficient, we get

P( Z ≥ a) ≤ n≥a (mn) mn+ 1 2 e -mn e K 2 n n+ 1 2 e -n (mn -n) mn-n+ 1 2 e -mn+n p n-1 (1 -p) mn-n+1 ≤ m m -1 1 2 e(1 -p) pK 2 n≥a m m p(1 -p) m-1 (m -1) m-1 n n -1 2 ≤ m m -1 1 2 e(1 -p) pK 2 n≥a q n .
Let us remark that the function x -→ x(1 -x) m-1 reaches its maximum on the interval [0, 1] at x = 1 m . As we supposed that p < 1 m , then

q < m m 1 m 1 -1 m m-1 (m -1) m-1 = 1.
The previous sum is then finite and

P( Z ≥ a) ≤ m m -1 1 2 e(1 -p) pK 2 q a 1 -q = Cq a .
Therefore (recall that q < 1 which gives ln q < 0),

a ≥ ln ε C ln q ⇒ Cq a ≤ ε ⇒ P( Z ≥ a) ≤ ε.
Proof of Lemma 2.4. By the trapezoidal rule, as the logarithm function is concave, we have for every positive integer k

ln(k + 1) + ln k 2 ≤ k+1 k ln t dt. This implies n-1 k=1 ln(k + 1) + ln k 2 ≤ n 1 ln t dt ⇔ ln n! - ln n 2 ≤ n ln n -n + 1 ⇔ ln n! ≤ n + 1 2 ln n -n + 1 ⇔ n! ≤ n n+ 1 2 e
-n e which gives the upper bound.

Still using the concavity of the logarithm function, its graph is always below its tangent which is is given at point k by the equation

y = 1 k x -1 + ln k.
This yields

k+ 1 /2 k-1 /2 ln t dt ≤ ln k + 1 /2k + ln k -1 /2k 2 = ln k ⇒ n+ 1 /2 3 /2 ln tdt ≤ n k=2 ln k = ln n! ⇒ (n + 1 /2) ln(n + 1 /2) -n - 3 2 ln 3 2 + 1 ≤ ln n! ⇒ (n + 1 /2) ln n + (n + 1 /2) ln(1 + 1 /2n) -n - 3 2 ln 3 2 + 1 ≤ ln n!
To get the lower bound, it suffices to check that

(n + 1 /2) ln(1 + 1 /2n) ≥ 1 /2
which comes immediately from the inequality

∀x ≥ 0, ln(1 + x) ≥ x - x 2 2 •
2.5. Other cases. We gave all the details of the proofs with the 8-connectivity for 2D images but the arguments of the previous subsections can be adapted to many other situations. We present some of them without detailing the proof of the stochastic domination.

2.5.1. Other definitions of connectivity. We focused on the 8-connectivity but the same arguments work for the 4-connectivity (taking m = 4). Numerical tests are performed in the next section to compare the performances of each connectivity. We can also extend this work to epsilon-connectivity (see [START_REF] Popov | Approximate connectivity and mathematical morphology[END_REF] for a more precise introduction) where two pixels are said to be connected if their ∞ -distance is less than k. For instance, for k = 2, this allows to fill small gaps between cracks that may be created because of the noise and/or the thickness of the cracks. All the previous computations remain valid taking m = 24 since a neighborhood of a pixel is now a 5x5 square.

2.5.2. 3D images. We also may extend the results to 3D stacks. Recall that we consider a 3D stack whose size is 1287 x 1287 x 99. If we consider the nearest neighbors of a pixel to define connectivity, the neighborhood of a pixel is now a cube of size 3 and the results apply with m = 26. Some numerical experiments are presented in Section 3.4.

Diameter of the connected component.

Let us go back to 2D-images and 8-connectivity. Let C 0 be the connected component of the image that contains 0. For x, y ∈ C 0 , we set Γ x,y as the set of connected paths that go from x to y remaining inside C 0 . For γ ∈ Γ x,y , we denote by (γ) its length i.e. the number of pixels γ is composed with. Finally, we define the diameter of C 0 by diam(C 0 ) = max This diameter can be used to discriminate cracks from noise. It also discriminates cracks (that have a quite large diameter) from ball-like connected components with the same number of pixels. Moreover our method still apply since we have, for every positive integer a, P(diam(C 0 ) ≥ 2a) ≤ P(Z a > 0) where (Z n ) n≥0 is a Galton-Watson process with binomial B(m, p) offspring distribution with m = 8. Indeed, if diam(C 0 ) ≥ 2a, there exists two points x, y ∈ C 0 whose shortest path between them has length greater than 2a. Then, at least one of the shortest paths from the origin to x or y is greater than a and the Galton-Watson process needs at least a generations to attain this point.

To compute the looked after threshold, we use the following lemma.

Lemma 2.5. Let (Z n ) n≥0 be a Galton-Watson process with binomial B(m, p) offspring distribution. Then, for every positive integer n, we have

P(Z n > 0) ≤ (mp) n .
Proof. Let us denote by F (z) the generating function of the binomial B(m, p) distribution:

F (z) = (pz + 1 -p) m .
Let us also denote for every n ≥ 0 by F n (z) the generating function of Z n . Then we have the classical recursive formula:

∀n ≥ 0, F n+1 (z) = F (F n (z)) = F n (F (z)).
Moreover, we have P(Z n > 0) = 1 -F n (0) and P(Z 0 > 0) = 1. Therefore, we get

P(Z n+1 > 0) = 1 -F n+1 (0) = 1 -(1 -p(1 -F n (0)) m ≤ mp(1 -F n (0)) = mp P(Z n > 0).
The result follows from an easy induction.

In that case, the threshold satisfies

(2.6) a 0 = 2 log ε log(mp)
.

This second threshold gives a geometric information, since we only keep structures with large diameters. In our case, this favors the cracks, that have anisotropic structure, with respect to the noise that can be viewed as small isotropic elements. Numerical tests, are reported in subsection 3.1.4.

Application to crack detection

As already mentioned, the application we have in mind is the identification of the cracks that appear in granite samples. Here, we are interested in denoising as a post-processing final step in for a complete crack identification method (based on a variational formulation) that we briefly describe in the sequel. That is why we focus on binary images (though the method works with grey level images as well) and the impulse noise that has to be deleted. The grain filter is a very efficient tool to achieve this goal: that is why we decided to focus on some improvements and automatic thresholding.

The numerical experimentation2 has been performed with MATLAB. We do not report on the computation time (we used different machines, and the codes are not optimized, especially in the 3D case). However, we may emphasize that the most time consuming step is the computation of the size (or geodesic diameter) of connected components. We used a built in MATLAB function, but this step should/can be improved with ad-hoc methods.

3.1. Tests with a 2D synthetic black and white image. We first perform numerical tests on a synthetic image that has been built with the image of Figure 1.1(a). It has been manually modified to get a image close to real data. This will be considered as the ground truth and we add impulse noise whose level is p, for p ∈ [0.01, 0.35]. The image is presented in Figure 3.1. We assume that m (the number of neighbors in the chosen neighborhood) and ε (the test level) are given. A typical choice for m is m = 3 2 -1 = 8 for 2D images (though we shall test m = 4 as well) and m = 3 9 -1 = 26 for 3D images (using 3 x 3 x 3 neighborhoods). The parameter ε is chosen small enough. We will make this precise in the sequel, when we study the sensitivity with respect to ε. Let us mention that we should only consider p < 0.125 since we have chosen m = 8 and we have to fit the assumption (2.1). However, we have added noise up to level 0.35. In the case where p is known (it is the noise level) the algorithm writes Algorithme 0 Case where p is known Given data: black and white (noisy) image u, ε and m; p is known. Compute the threshold value a 0 with (2.3) and (2.4). 2. Find the connected components of u (and their size). 3. Suppress the connected components of u whose size is less than a 0 to get the significant contours u * . However, in the real data case, the probability p is unknown and its estimate is an important issue (that also determines the m value). We may first estimate it using the number of black pixels of the binary image divided by its total size M : p Size (u=1) M . This rough method gives an upper bound for p that is too large an estimate. Indeed, the number of black pixels includes both the noise and the cracks, whereas we only want to count the noisy pixels. So we adopt a multi-scale like strategy to perform successive updates of p once we have performed a more accurate selection of cracks pixels.

Algorithme 1 Algorithm with iterative update of p

Given data: black and white (noisy) image u, ε and m. Find the connected components of u (and their size). 0.

Initialization: k = 1, u 0 = 0. 1. Iteration k Compute u -u k and p k = Size (( u -u k ) = 1) M .
Compute the threshold value a k with (2.3) and (2.4). Suppress the connected components of u whose size is less than a k : we get u k+1 . 2. Stopping criterion. Stop when the sequence (p k ) k∈N becomes stationary.

We report in Table 3.1 the values of the computed probability p and the threshold with and without this strategy with respect to the noise level. This corresponds to the computed values at iteration 1 of Algorithm (1) and at the last iteration Algorithm (1) . We have set the maximal number of iterations at 20. 1) with (last iteration) and without (first iteration) the adaptative strategy-m = 8, ε = 10 -6 .

Noise Computed

We can also see in Figure 3.2 the behavior of the computed threshold with respect to the noise level for the different approaches. We clearly see the change of behavior for a noise level greater than 0.11. This is consistent with the theoretical assumption (2.1). Noisy image, solution given by Algorithm (1) and difference with the ground truth, for different noise levels -m = 8, ε = 10 -6 -The differences are presented as negative images for a better insight (white is equal to 0, black is maximum).

3.1.1. Sensibility with respect to ε. We now investigate the sensibility of the solution computation with respect to ε. A typical choice of ε is the inverse of the number of connected components as, by independence, the mean number of false positives would be 1 in this case.

Noise

Probability Threshold (# pixels) level ε = 10 -2 ε = 10 -4 ε = 10 -6 ε = 10 -2 ε = 10 -4 ε = 10 This comparison shows that the role of ε has not a big influence when the noise level belongs to [0, 0.125] (the image size is 1.653 10 6 ). In the sequel, we choose ε 1/M where M is the size of u that is, roughly speaking, ε = 10 -6 .

3.1.2. Sensibility with respect to the connectivity. We compare here the results obtained with these two different connectivities corresponding to m = 4 and m = 8. As expected, the behavior of the threshold value for m = 4 changes for a noise level equal to 0.25, which is consistent with Assumption (2.1).

Next figure presents the solutions that are obtained with Algorithm (1) for the same noisy data and m=4, 8 respectively. We can see on Figure 3.5 that the use of 4-connectivity gives worse results. Indeed, the use of a 4-neighborhood does not take the diagonal connections into account. Therefore, thin cracks are lost and thick ones may be disconnected.

3.1.3.

Comparison with other filters and/or other thresholds. Though, there are a lot of filters to deal with impulse noise we limit our comparison to the median one and a morphological one, namely an opening filter with two different structural elements: a disk of radius 1 (which corresponds to the 4-connectivity) and a square 3× 3 (which corresponds to the 8-connectivity). Results are reported in Figure 3.6 for a noise level of 0.05. Visually speaking the solution that we get with Algorithm (1) is the best one. If we compare with the ground truth, we get a quantitative estimate of the performance, with δ 1 which is the 1 norm of the difference between the ground truth and the solution. The use of the geodesic diameter is not appropriate if the noise is too large. Indeed, if mp is too close to 1 then the threshold a 0 = 2 log ε log mp becomes quite large and we loose most of the crack information. If the noise is small enough, then we get good results provided ε is not too large. However, the method seems to be robust with respect to small ε. 2) for m = 8, ε = 10 -4 and a noise level 0.09.

3.2.

Tests with 2D a real black and white image.

3.2.1.

Preprocessing. We present some results on a 2D slice of a granite sample stack, for the sake of simplicity (see Figure 1.1 (a)). Though, it is not our purpose here, we briefly present the preprocessing method:

(1) Original (2D or 3D) images are pre-processed with the method of [START_REF] Bergounioux | Mathematical Analysis of a Inf-Convolution Model for Image Processing[END_REF][START_REF] Bergounioux | Second order decomposition model for image processing : numerical experimentation[END_REF]. More precisely, we use a variational model that allows to split the image in three components denoted v, u and w:

• the v component is a smooth (continuous) function whose distributional derivative is of bounded variation: it represents the image dynamic;

• the u component is a bounded variation function and ideally, should be piecewise constant. It is called a cartoon component: the image contours are given by the jump set of u; • the w component is a L 2 function that represents the details and/or the noise. The main inconvenient of variational methods is their dependence to some parameters. The result of the above decomposition depends on the choice of such parameters. We refer to [START_REF] Bergounioux | Second order decomposition model for image processing : numerical experimentation[END_REF] to get tuning rules for the two parameters λ and µ to be used in this context. From a practical point of view we choose µ 1.4λ. The tests we present here use λ = 5, µ = 7.

The preprocessing method is robust enough with respect to the choice of parameters and we can get rid of most noise (involved in the w component). So we focus on the cartoon component u that involves all the contours. However, though the gaussian part of the noise has been removed, impulse noise still remains. The challenge is to distinguish the cracks that are thin structures from the noise (which is unstructured ). However, it will be still quite difficult to recover micro-cracks.

(2) Next, the u component has to be binarized. This is a delicate step since the threshold parameter is not easy to find. We decided to choose s = m(u) -σ(u) where m and σ are the mean value and the standard deviation respectively of u. This is an heuristic choice that we cannot fully justify by now. 3.3. Grey level images. We have considered binary images, which implies some thresholding process when dealing with gray-level images. However, following [START_REF] Coupier | Image denoising by statistical area thresholding[END_REF] We end this subsection with Algorithm (3) applied to the cartoon image of the granite image (before performing thresholding). We could now use an a posteriori thresholding process on this solution as well to recover the cracks, but this will not be very good since one can see that we have lost the microcracks structure.

3.4. 3D images. Though, we have presented the method with 2D images (whose domains are -N, N 2 ) we may extend the result to 3D stacks. Recall that we consider a 3D stack whose size is 1287 x 1287 x 99. If we consider the nearest neighbors of a pixel to define connectivity, the neighborhood of a pixel is now a cube of size 3 and the results apply with m = 26 and Algorithm (1) works similarly in the 3D case. The main difference was the choice of the built in MATLAB function to compute the connected components directly with the 3D stack (and m = 26).

The preprocessing was performed directly on the 3D stack (and not slice by slice) as explained in [START_REF] Bergounioux | Second order decomposition model for image processing : numerical experimentation[END_REF] and it was natural to use the 3D stack to perform binarization and denoising with Algorithm [START_REF] Addario-Berry | Sub-gaussian tail bounds for the width and height of conditioned Galton-Watson trees[END_REF]. Of course, the computation time is quite large since we have to estimate the size of a great number of connected components. Therefore, we have to develop further strategies to improve the 3D method as for example parallelization.

Conclusion

The automatic threshold we propose seems to be performing and we are able to recover a lot of information from the binary images. Though, we have presented the method on a 2D material image, it can be applied to any binary images arising from appropriate pre-processings. One may think of as angiography images for example.

However, we could say that the size of the connected components is rather a crude criterion for distinguishing cracks for noise and some other geometric characteristic (as the diameter) would better discriminate our features. The key point is that we need an upper bound for the tail probability of the looked after characteristic (we obtained such a bound for P( Z ≥ k) in the case of the total population size, and for P(Z n > 0) in the case of the diameter) which is not obvious in general (see for instance [START_REF] Addario-Berry | Sub-gaussian tail bounds for the width and height of conditioned Galton-Watson trees[END_REF] for recent works in this direction).

Another point that we did not really focus on is the estimation of the parameter p as it was not the central point of this work. Our method (especially after iterations) seems rather satisfactory but always overestimates this probability. Some more sophisticated methods have been developed in the literature and may be used to improve the results.

Figure 1 . 1 .

 11 Figure 1.1. Original and binarized 2D slice of a 3D granite sample

2. 1 .

 1 Notations. If k < are two integers, we denote by [[k, ]] the set {n ∈ Z, k ≤ n ≤ }.

we label the 8n points of norm n as on Figure 2 Figure 2 . 1 .Figure 2 . 2 .

 22122 Figure 2.1. Labeling of points of norm n

Figure 2 . 3 .

 23 Figure 2.3. Labeling of the neighbors of a point x

Figure 2 . 4 .

 24 Figure 2.4. Example for the two first prunings

  (a) Ground truth (synthetic image) (b) Impulse noise with p=0.15

Figure 3 . 1 .

 31 Figure 3.1. Ground truth and noisy image

Figure 3 . 2 .

 32 Figure 3.2. Evolution of the threshold with respect to the noise level in the range [0 0.35] -X axis is the noise level p and Y-axis is the threshold value (number of pixels) with log10-scale. The continuous line corresponds to the case where p is computed once (iteration 1) and the dotted curve to the computed threshold with adaptative strategy (last iteration of Algorithm (1)) -m = 8, ε = 10 -6 .

  Figure 3.3. Noisy image, solution given by Algorithm (1) and difference with the ground truth, for different noise levels -m = 8, ε = 10 -6 -The differences are presented as negative images for a better insight (white is equal to 0, black is maximum).

  (a) ε = 10 -2 . (b) ε = 10 -4 . (c) ε = 10 -6 .

Figure 3 . 4 .

 34 Figure 3.4. Examples of solutions given by Algorithm (1) for ε = 10 -2 , 10 -4 , 10 -6 -Noise =0.05.

  (a) Noise: 0.01, m=4. (b) Noise: 0.01, m=8. (c) Noise: 0.05, m=4. (d) Noise: 0.05, m=8. (e) Noise: 0.15 m=4. (f) Noise: 0.15, m=8.

Figure 3 . 5 .

 35 Figure 3.5. Comparison of the solutions computed with Algorithm (1) for m = 4, m = 8 and ε = 10 -6 .

  (a) Median filter 3× 3, δ1 = 6371. (b) Opening with disk of radius , δ1 = 9900. (c) Opening with square of size 3× 3, δ1 = 21344. (d) Algorithm (1), m=8, ε = 10 -6 , δ1 = 3621.

Figure 3 . 6 . 3 . 1 . 4 .Algorithme 2 2 .

 3631422 Figure 3.6. Comparison of the solutions computed with different filters, for a noise level equal to 0.05. Here δ 1 is the 1 norm of the difference between the ground truth and the solution, that is the number of pixels that are different.

Figure 3 . 7 .

 37 Figure 3.7. Solutions computed with Algorithm (2) for m = 8 and different values of ε.

Figures 3 .

 3 7 and 3.8 present the results for a noise level equal to 0.09 and m = 8 (a) Solution with Algorithm (1). (b) Solution with Algorithm (2). (c) Difference with the ground truth. (d) Difference with the ground truth.

Figure 3 . 8 .

 38 Figure 3.8. Comparison of the solutions computed with Algorithms (1) and (2) for m = 8, ε = 10 -4 and a noise level 0.09.

  (3) Last, we use the thresholded (binary) image and use Algorithm (1).

  (a) Original (2D) image. (b) u component. (c) v component. (d) w component.

Figure 3 . 9 .

 39 Figure 3.9. Image decomposition with λ = 5 and µ = 7 -The contrast and lightning have been emphasized to provide a better view of the images.

  (a) u component. (b) Thresholded u component.

Figure 3 . 10 .

 310 Figure 3.10. Thresholded component used as the binary data

Table 3 . 3 .

 33 Computed probability p k and corresponding threshold a k during the iterations -m = 8 and ε = 10 -6 . Next figure shows the solution with Algorithms (1) and (2) :(a) Noisy data. (b) Solution with Algorithm (1). (c) Solution with Algorithm (2).

Figure 3 . 11 .

 311 Figure 3.11. Solutions computed by Algorithms (1) and (2) with m = 8 and ε = 10 -6 .

Algorithme 3

 3 we may adapt the algorithm by considering the image level sets. Precisely, for each level k ∈ [0, 255], we consider the thresholded image u k = 1 u≥k (where 1 A is the indicator function of the set A) and we recover the grey level solution as u * = 255 k=0 u k . This gives the following algorithm Algorithm for grey level images Given data: grey level image u. 0. Initialization: k = 0, u * = 0. 1. Iteration k Compute u k with Algorithm (1) applied to bw k := 1 if u ≥ k, 0 else.u * = u * + u k . 2. End.Next figure shows the solution with the image of Lena, for m = 4 and 8 respectively.

  (a) Noisy data -noise level 0.05. (b) Solution, m=4. (c) Solution, m=8. (d) Noisy data -noise level 0.15. (e) Solution, m=4. (f) Solution, m=8.

Figure 3 . 12 .

 312 Figure 3.12. Lena denoising for two different noise levels -ε = 10 -6 and m = 4, 8.

Figure 3 . 13 .

 313 Figure 3.13. Granite denoising Cartoon part -ε = 10 -6 and m = 8.

  (a) Original Image. (b) Smooth component v. (c) Noise component w. (d) Cartoon component u.(e) Binarized cartoon component.

Figure 3 . 14 .

 314 Figure 3.14. Decomposition of the 3D stack and binarized cartoon component.

Figure 3 .

 3 Figure 3.15. Solution.

Table 3 .

 3

1. Computed probability and threshold with respect to the noise level, with Algorithm (

Table 3 . 2 .

 32 Computed probability and threshold with respect to the noise level, with the adaptative strategy Algorithm (1) and different values of ε -m=8.

	-6

Table 3

 3

	Iteration	1	2	3	4
	p k	4.6928 10 -2 3.3656 10 -2 3.0146 10 -2 2.9171 10 -2
	a k	7.4827 10 7	4.6183 10 7	4.0659 10 7	3.9236 10 7

.3 the evolution of p estimate. We see that the iterative process becomes stationary very quickly (5 iterations).

All the images of this section, can be download here http://maitinebergounioux.net/PagePro/ Publications_files/Images_JMIV.rar


Learner corpus of French as a Foreign Language PAC (Phonologie de l'Anglais Contemporain):

Regional and sociolinguistic variation in contemporary English PFC (Phonologie du Français Contemporain):

Regional and sociolinguistic variation in contemporary French

PROSPECT

The phenomena observed in the preliminary recordings will be compared with future recordings of speakers of other languages in foreign and second language contexts.

ICE-IPAC

Tasks 1 )

 1 repetition task: word lists 2) reading tasks: word lists and text 3) conversations: formal interview and informal conversation Interphonology of Contemporary English -InterPhonologie de l'Anglais Contemporain PROTOCOL: ICE-IPAC PRELIMINARY RECORDING DATA: Norwegian-and French-speaking learners Mother projects IPFC (InterPhonologie du Français Contemporain):

-

  International database of L2 English phonology and phonetics -Learners with different L1s (French, Spanish, Norwegian, etc.) -Learners with different L1 varieties (Hexagonal French, Canadian French, etc.) -Learners with different L2 varieties as their target/model -Different tasks within the same dataset (repetition, reading, conversation) -Intra-speaker variability -Inter-speaker variability Word lists 1) a common list applied to all learners, regardless of their L1 2) an L1 specific list adapted to verify the difficulties frequently observed in the L2/L3 production of the learners of a given L1 Conversations 1) a formal interview led by a native Anglophone investigator 2) an informal conversation between 2 learners Two female Norwegian learners (NW1 and NW2) enrolled at UiT The Arctic University of Norway Learners -Reading of two word lists -Informal conversation between the two learners Tasks 1) Devoicing of fricatives /z, ʒ/ and affricate /dʒ/ Sounds not part of the Norwegian sound system -/z/ -> [s]: seal-zeal near-homophonous; graze [ɡɹeɪs] or [ɡɹaes] -/ʒ/ -> [ʃ]: leisure [ˈleɪʃɚ] -/dʒ/ -> [tʃ]: batch-badge both produced with [tʃ] 2) Partial neutralization of lenis-fortis in word-final stops Pre-fortis clipping observed, but voicing ceases well before release of the lenis coda -/b/ -> [p]: lap-lab near-homophonous: [laep]-[laep] -/d/ -> [t]: fat-fad near-homophonous: [faet]-[faet] -sack [saek h ] vs sag [saek]: observation anticipated by the Norwegian L1 system: the underlying contrast in the L1 of the learners is primarily expressed through aspiration on the edges of words, not voicing 3) Post-vocalic rhoticity (NW1 displays more consistent use) carter [ˈkaɹɾɚ] 4) Labial-velar approximant /w/ -> [v] Sound not part of the Norwegian sound system -word [vɔɹd], which [vɪtʃ], worthy [ˈvɜːɹdi] -hypercorrected [w] for /v/: vexed [wɛkst] -epenthesis of [v] or [w] in front of rounded, single onset: [ɹ]: written [ˈvɹɪtn], run [vɹʌn] 5) Interdental fricatives /θ, ð/ -> plosives Sounds not part of the Norwegian sound system /ð/ -> [d]: farther [ˈfɑɹdɚ], worthy [ˈvɜːɹdi] Special thanks to Helene N. Andreassen 1) Vowels /iː/ -/ɪ/; /ɔː/ -/ɒ/ No systematic difference in quality 2) <i> pronounced [aɪ] in sinner, simmer Overgeneralization of one of the L2 pronunciation rules for <i>? 3) h-dropping /h/ is deleted in here, hurry, etc. (FR2 displays more consistent use) 4) Influence by the L1 relationship between sound and writing <ch> pronounced [ʃ] in witch, China 5) Post-vocalic rhoticity carter [kaɹtɚ]; garter [ɡaɹtɚ] Special thanks to Audrey Bonfiglioli and Ioana Trifu NW FR Similar protocols -> comparability between corpora -ICE-IPAC / PAC: comparison between learners and native speakers of English -ICE-IPAC / IPFC: comparison between speakers of the same L1 learning English vs. French as a foreign / second language Text "Christmas interview of a television evangelist" If television evangelists are anything like the rest of us, all they really want to do in Christmas week is snap at their families, criticize their friends and make their neighbours' children cry by glaring at them over the garden fence. Yet society expects them to be as jovial and beaming as they are for the other fifty-one weeks of the year. If anything, more so. […] Two female French learners (FR1 and FR2) enrolled at the University of Lyon Detey, S., Racine, I., Kawaguchi, Y., Zay, F. in press. Variation among non-native speakers: the InterPhonology of Contemporary French. In: Detey, S., Durand, J., Laks, B., Lyche, C. (eds.), Varieties of Spoken French: a Source Book. Oxford: Oxford University Press. Durand, J., Przewozny, A. (2012). La Phonologie de l'Anglais Contemporain: usages, variétés et structure. Les approches de corpus à orientation phonologico-phonétique pour l'anglais. Revue Française de Linguistique Appliquée 17(1) , 25-37. Gut, U. (2009). Non-native Speech. A corpus-based analysis of the phonetic and phonological properties of L2 English and L2 German. Frankfurt: Peter Lang. Kondo, M. (2012). Design and analysis of Asian English speech corpus -How to elicit L1 phonology in L2 English data-. In: Tono, Y., Kawaguchi, Y., Minegishi, M. (eds.), Developmental and Crosslinguistic Perspectives in Learner Corpus Research. Amsterdam & Philadelphia: John Benjamins, 251-278.


Ahmad Mobasher Amini

David Dureisseix
email: dureisseix@lmgc.univ-montp2.fr

Patrice Cartraud
email: patrice.cartraud@ec-nantes.fr

Natacha Buannic
email: natacha.buannic@principia.fr

A micro-macro strategy

Keywords: FETI-DP Method, micro-macro, Ship Structure

  L'archive ouverte pluridisciplinaire

• structure details are not periodic, therefore classical homogenization methods are not easily applicable;

• edge effects are not take into account;

• zooming techniques are not easy to use: the gluing they require with the global scale often introduces artificial edge effects.

This paper presents a micro-macro strategy based on the domain decomposition FETI-DP method as the solver in analysis of ship structure. With this approach, the two scales (micro and macro) are coupled during the iterations of the solver and we can consider the structural details in areas of interest, area where the fine mesh is used and a sub-domain is located. Performances are discussed and results in term of convergence are presented for several examples.

INTRODUCTION

Analysis of a large system like ship structure, to obtain the fine scale results is one of the industrial challenges today. Since there is often only one prototype produced, which is the final product and because real-scale test of such structures are very expensive and difficult, the designers now often rely on finite element simulation. Due to the classification society rules and regulations, extensive use of numerical simulations started only recently in this domain of engineering. With the advent of new design for ships, the old rules and regulations were obsolete and engineers shifted to the numerical simulation and in particular to the finite element method. Due to different specifications, several discretized models are used:

• for global behavior of the structure, and regulations requirements of numerical tests, a so called coarse finite element model is used, see Figure 1;

• for local stresses, and to take into account all details (skylight, doors, . . . ) a model at a finer scale is required;

• for vibration analysis, . . . ;

• for fatigue life assessment , . . . ; To obtain a solution for the whole structure as well as a local solution near the details, a small scale discretized model is required. This would lead to a huge global finite element model with large number of unknowns that is very expensive to solve. Because of this large number of unknowns, memory and processors limits, computing this kind of problem with a classical direct technique is not feasible at the moment, even with last generation sparse solvers. Therefore, using a simplified global model is mandatory and engineers have used approximate method based on super-element or local-global analysis. Briefly, the local-global methods, which are used in engineering consist of several steps:

• Solving a local or microscopic problem to derive its global behavior for very parts of the structure;

• Construction of a global or macroscopic model of the ship with an equivalent homogenized constitutive behavior by the results of the first step;

• Once the global model is solved, recovering the solution on the small scale of the details with a post-processing step.

The use of such a strategy raises several difficulties: First, we can not use the periodic homogenization approach, since the structural details are not periodic. So we need the appropriate assumptions on boundary conditions for the local problems dealing with structural details. Second, the same boundary conditions are used for the post-processing re-localization phase, needed to recover the small scale solution. Depending on the selected boundary conditions, artificial edge effects may appear that pollutes the local stress computations. These edge effects, as well as approximations in the homogenized behavior, are the consequence of loosely coupled global and local computations.

To overcome to the above mentioned drawbacks, we will use domain decomposition methods that will allow to restore the coupling between the different scales along iterations. Numerous domain decomposition methods are discussed in the literature. They differ in the way gluing conditions are expressed between neighboring substructures. Another difficulty lies in the fact that complex structures as ships are assemblies of different structural elements like shells, joints and stiffeners. Usually the stiffeners are modeled with beam finite elements while various panels uses shell or plate finite element and joints use special transition elements. Therefore, we propose to use a domain decomposition approach suited to:

• couple local fine scale models of predefined areas (or sub-domain);

• glue these local models to areas without zooming, i.e, with a coarse finite element model;

• provide a mechanism to homogenize fine scale sub-domains to coarse finite elements;

• deal with previously mentioned heterogeneities.

In this article, we will not tackle the problem of gluing local models to coarse models. We propose to use the dual-primal version of the FETI method, which is briefly described in Section 2. The same Section recalls the technology used in such methods to deal with classical heterogeneities.

The Section 3 presents the problem of interface in the case of the 3D mesh of the ship structure and different heterogeneity in this structure. We will present the results of the two simple problems and a real 3D problem of a slice of the ship structure in Section 4. The results and conclusion will be in the Section 5.

GENERAL DESCRIPTION OF THE FETI-DP METHOD

Domain decomposition methods are both an efficient and flexible tool for structural analysis [START_REF] Tallec | Domain decomposition methods in computational mechanics[END_REF], [START_REF] Farhat | Implicit parallel processing in structural mechanics Computational Mechanics Advances[END_REF]. As an iterative approach, they often outperform direct methods when the size of the model increases, and they allow a parallel treatment of the resolution phase. In this article, we are not concerned with the parallelization of the resolution, but we focuss on the modularity, especially for coupling different areas (different sub-domains) that may have been modeled differently, or with different discretization levels. To reach numerically scalability, the domain decomposition method needs a multi-scale feature, mainly to be able to built a coarse problem that maintains the convergence rate weakly dependent on the number of substructures, among these methods, one can refer to BDD [START_REF] Mandel | Balancing domain decomposition[END_REF], CBDD [START_REF] Mandel | Convergence of a Balancing Domain Decomposition by Constraints and Energy Minimization Numerical Linear Algebra with Applications[END_REF], aggregation technique [START_REF] Vanek | Algebraic multigrid based on smoothed aggregation for second and fourth order problems[END_REF], [START_REF] Vanek | Convergence of Algebraic Multigrid Based on Smoothed Aggregation[END_REF], FETI [START_REF] Farhat | A method of finite element tearing and interconnecting and its parallel solution algorithm[END_REF], FETI-DP [START_REF] Farhat | A scalable dual-primal domain decomposition method Numerical Linear Algebra with Applications[END_REF], [START_REF] Farhat | FETI-DP: a dual-primal unified FETI method -part I: a faster alternative to the two-level FETI method[END_REF], LATIN micro-Macro [START_REF] Ladevèze | A micro/macro approach for parallel computing of heterogeneous structures CInternational[END_REF]. Moreover, when this coarse problem is related to an homogenized model of the same structure, this provides an efficient tool to deal with heterogeneous structures. The different versions of the FETI method (FETI-1 [START_REF] Farhat | A method of finite element tearing and interconnecting and its parallel solution algorithm[END_REF], FETI-2, FETI-DP [START_REF] Farhat | A scalable dual-primal domain decomposition method Numerical Linear Algebra with Applications[END_REF], [START_REF] Farhat | Implicit parallel processing in structural mechanics Computational Mechanics Advances[END_REF], . . . ) belongs to the family of the non overlapping domain decomposition methods with lagrange multiplier. These methods have been developed for solutions of the large scale systems of equations in structural analysis by finite element method. Among all the domain decomposition methods that were presented by many others author, we choose the FETI-DP method for the following reasons: Usually with a multi-scale domain decomposition method, the reference problem is split into sub-domains and the coarse space problem is numerically built from this fine scale description. The coarse problem can be discrete by nature, and is not always related to any finite element model.

In the case we are interested in, the coarse problem is already provided, and the reference problem is built from this coarse problem by selecting zooming area that are coarse finite elements (a panel or a stiffener, for instance). A selected coarse finite element is re-meshed at a fine scale with the corresponding structural details (skylight, door, . . . ) and is identified as a sub-domain. The initial coarse finite element possesses nodes that are suitable to be chosen as corner nodes of the FETI-DP method. Moreover, as detailed in the following (see Figure 2), the coarse finite element model constitutes a structural homogenization of the detailed sub-structures, and the compatibility of coarse displacement field is ensured automatically with neighboring coarse elements.

Basic FETI-DP method

Let us consider the domain Ω, decomposed that into N non-overlapping sub-domains (or substructures) Ω s . Let K s , u s and f s be the stiffness matrix, displacement and prescribed force vectors associated with sub-domain of Ω s , respectively. By splitting u s into corner and reminder points we will have:

u = u r u c =           u 1 r . . . u N s r u c           (1)
Where u c is the global or primal solution vector of the corners degrees of freedoms (d.o.f) and u s r is solution vector of the remaining sub-domains degrees of freedoms. With this kinematic description, the solution at corner point is globally continuous. Using this notation, we can split the sub-domain stiffness matrix into:

K s = K s rr K s rc K s T rc K s cc (2)
By writing the equilibrium equation for each sub-domain in the global form and considering the continuous boundary condition on the interface of the sub-domains we will have the following relations:

K s rr u s r + K s rc B s c u c + B s T r λ = f s r f orallsin1, ..., N s (3)
s=Ns s=1 B s T c K s T rc u s r + s=Ns s=1 B s T c K s cc B s c u c = s=Ns s=1 B s T c f s c = f c (4)
Where B s c are boolean matrices that map the local corner d.o.fs of a sub-domain to the global corner d.o.fs:

u s c = B s c u c (5)
The interface continuity condition can be written as follows:

s=Ns s=1 B s r u s r = 0 (6)
where B s r are signed boolean matrices. With the above equations and after some algebraic transformations, we obtain the following dual-primal problem. with Lagrange multiplier λ and primal displacement u c as unknowns.

F Irr F Irc F I T rc -K * cc λ u c = d r -f * c (7)
where:

F Irr = s=Ns s=1 B s r K s -1 rr B s T r ( 8
)
F Irc = s=Ns s=1 B s r K s -1 rr K s rc B s c ( 9
)
K * cc = K cc - s=Ns s=1 (K s rc B s r ) T K s -1 rr (K s rc B s c ) (10)
d r = s=Ns s=1 B s r K s -1 rr f s r f * c = f c - s=Ns s=1 B s T c K s T rc K s -1 rr f s r (11)
The symmetric positive definite dual interface problem is obtained by condensing the u c on the λ:

(F Irr + F Irc K * -1 cc F T Irc )λ = d r -F Irc K * -1 cc f * c ( 12
)
We use the preconditioning conjugate gradient algorithm (PGC), as a solver of the above interface problem. Indeed because the number of unknowns in the above equation can be large and because computing and assembling that for each substructure would be very expensive, (2.1) is not solved by a direct method.

The kernel of the equation (2.1) is the computation of the coarse FETI-DP matrix K * cc . The matrix K * cc is sparse and its pattern is that of a stiffness matrix obtained by considering only the super-elements defined by the corner nodes.

Preconditioning for heterogenous problems

Like all iterative methods, FETI-DP solver performances depend on spectral properties of the matrix of the linear system and demand a good conditioning of the equations. In this article, we do not develop a new preconditioner for FETI-DP method. The preconditioner [START_REF] Farhat | Implicit parallel processing in structural mechanics Computational Mechanics Advances[END_REF] are:

F D -1 Irr = s=Ns s=1 W s B s r 0 0 0 S s brbr B s T r W s (13)
where, for the Dirichlet preconditioner :

S s brbr = K s brbr -K s T ibr K s -1 ii K s ibr (14)
and for the Lumped preconditioner:

S s brbr = K s brbr (15)
W s is a scaling diagonal matrix for the heterogeneous cases and i, b and r are internal nodes, boundary nodes and total internal and boundary nodes, respectively. For more informations the reader can refer to [START_REF] Farhat | A method of finite element tearing and interconnecting and its parallel solution algorithm[END_REF], [START_REF] Farhat | A scalable dual-primal domain decomposition method Numerical Linear Algebra with Applications[END_REF], [START_REF] Farhat | FETI-DP: a dual-primal unified FETI method -part I: a faster alternative to the two-level FETI method[END_REF].

INTERFACE PROBLEM IN THE CASE OF SHIP STRUCTURES

The ship structure is assemblage of the very different structural elements such as shell, plate , stiffeners, joints, etc. We consider stiffeners in two categories, primary and secondary stiffeners. All secondary stiffeners is considered into the plate sub-domain and all primary are consider like the one sub-domain. As a rule we decompose the structure in the line of intersection of the all plate and primary stiffeners. For this reason in interface line there are three lagrange multiplier. We applied the DKQ element (with 6 d.o.fs u, v, w, θ x , θ y , θ z at each node) and 2 node beam element (with 6 d.o.fs u, v, w, θ x , θ y , θ z at each node).

NUMERICAL EXAMPLES

For illustration of the performance of the above mentioned method we present two examples. The problem is a model problem of a 2D stiffened plate and is used to test the convergence of the method. The second problem concerns a 3D analysis of a slice of a ship with a fine finite element model. We suppose that the stiffeners lie on the interfaces of the decomposition into sub-domain. We prescribe an uniform shear force on the top side of the two above problems (see Figure 3). The results of the above mentioned problems with different sub-domain sizes H and delement The rate of convergence is still higher for the problem without stiffeners, even when using the Dirichlet preconditioner. With diagonal scaling and three lagrange multipliers this behavior is still to be studied to recover the initial convergence rate.

A 3D analysis of a slice of ship

In order to tackle a more realistic problem, we consider a slice of the passenger ship as shown in Figure 2. Pre and post processing is performed with the FE code Cast3M (CEA sacly, France) and reduction is achieved in a prototype FE platform in Matlab environment. The finite element characteristic of this structural model as shown in Table 4

CONCLUSIONS

In this paper we was succeed to use domain decomposition method and FETI-DP method such as solver in the static analysis of the ship structure. We considered one coarse mesh that the fine mesh (structural details) is hidden behind that and also we consider all primaries stiffeners in one sub-domain. The obtained results is comparable of the CAST3M finite elements code. A Matlab code is developed for this reason and it will be continued to decrease the rate of the convergence and applicable of the method in the free vibrational analysis.

Figure 1 :

 1 Figure 1: Global finite element model

Figure 2 :

 2 Figure 2: Coarse finite element model and a fine scale sub-domain

4. 1

 1 2D stiffened plate We consider herein the 2D static deflection and plane stress analysis of a plate with (problem I) or without (problem II) stiffeners. Both are related to a rectangular domain, clamped on the bottom side. We use DKT elements (three node triangular plane stress elements with two d.o.fs at each node) made with steel (young's modulus E = 2.0 × 10 11 N/m 2 and Poisson's ratio ν = 0.3) For problem II, an assembling of plates and stiffeners is considered. The stiffeners are discretized with two node beam elements (three d.o.f for each node, two displacements, u and v, and rotation θ). The material characteristics of the stiffeners are Young's modulus E = 2.0 × 10 11 N/m 2 , Poisson's ratio ν = 0.3, cross section area S = 0.00295 m 2 and section moment of inertia to I = 1.4471 × 10 -6 m 4 for bending.

Figure 3 :

 3 Figure 3: Plane stress problem

Figure 4 :

 4 Figure 4: Convergence behavior (plate without stiffener)

Figure 5 :

 5 Figure 5: Convergence behavior (plate with stiffener)

Table 1 :

 1 Characteristic of the slice of the ship.

	.2:


Keywords: historical phonology, panchronic phonology, tonogenesis, labiovelars, final clusters

Haudricourt's main legacies to the field of Chinese historical phonology are his systematic account of tonogenesis, and his reconstruction of final *-s and of labiovelars. The method that he proposed under the term of Panchronic Phonology holds special promise for the study of the languages of China in their universal context.

Mauss obtained funding for him to go to Leningrad for one year to pursue studies in genetics with Nikolaj I. Vavilov (1887Vavilov ( -1943)) He investigated a variety of language groups -European, as well as Asian and Oceanic. As his fragile health did not allow him to conduct much fieldwork himself, he mostly worked on the basis of available reports, which at the time were relatively deficient. He excelled at identifying the shortcomings of these sources by taking into account the language background of the persons who collected the data: reconstructing, as it were, the investigators' analytic process. He also applied this approach to the analysis of the origin of the peculiarities of the Vietnamese alphabet (1949, translated 2010).

He is remembered as an impressive mentor, at his best in informal exchanges, going straight to the heart of scientific issues. His unpolished style often makes his articles hard to read, but reflects the precision and vividness of his insights. A translation of his most seminal papers is currently in preparation (ed. by Martine Mazaudon and Alexis Michaud, Berlin: De Gruyter Mouton, forthcoming); his more anecdotal pieces, including interviews [START_REF] Haudricourt | Les pieds sur terre[END_REF][START_REF] Haudricourt | Propos /I, II de André-Georges Haudricourt[END_REF][START_REF] Haudricourt | Les conditions d'apparition et de disparition, c'est ça, à mon avis, le scientifique': entretien avec André-Georges Haudricourt[END_REF], are also extremely instructive, reflecting his profound erudition and commitment to down-to-earth scientific method, as well as his nonconformism and acute sense of irony.

Contribution to Chinese linguistics

The relationship between Vietnamese and Chinese

Haudricourt demonstrated (pace Maspero 1912) that Vietnamese and Chinese are not genetically related, teasing out layers of borrowings, and emphasizing that typological similarities, such as the presence of lexical tone, do not constitute evidence for phylogenetic groupings (1953). This represents an important contribution towards defining the boundaries of the Sino-Tibetan family.

Systematic account of tonogenesis in Chinese

Haudricourt's account of the origin of tones in Vietnamese (1954) addresses issues of tonogenesis in a range of Asian languages, including Chinese. The discussion takes one step further the insights of previous scholars (most prominently Maspero 1912), concluding:

"Old Chinese only had three tones up until the 9 th century. At that time, the voiced stops /g/, /j/ [IPA: ɟ], /d/ and /b/ devoiced to /k/, /c/, /t/, /p/, aspirated or not depending on dialects and on tones. From that point on, the musical height [as opposed to the contour] of the tone became a contrastive phonological property used to distinguish words. Two words which used to be distinct because one had an onset /k/ and the other had an onset /g/ came to be distinguished solely by the higher tone of the former. The three-tone system thus became a six-tone system" (translation by Marc Brunelle).

A systematic account of the two stages of tonogenesis in Chinese was provided by Haudricourt as part of his general model of tonogenesis in Asia (1961), including important observations on the behavior of the mid series of initials in tonogenesis:

"We have seen that Ancient Chinese [=Middle Chinese] had only one series of sonorants. These sonorants were not involved when the voiced and voiceless initial stops merged, and so they form a "mid" series of initials. For example, in Sino-Vietnamese, [=Sino-Annamese], a Chinese dialect of the 10 th century preserved in Vietnam as a language of high culture (Maspero 1912:91-93 We see the paradox that B and C have been confused in the low series, while they remained distinct in the "mid" series. The paradox is only apparent: the tonemes of the "mid" series had no reason to be modified, whereas those of the "low" series had to be distinguished from whose of the "high" series.

Modern Chinese has an analogous treatment [START_REF] Egerod | The Lungtu dialect[END_REF] [START_REF] Rivierre | Tonogenesis in New Caledonia[END_REF][START_REF] Rivierre | Tonogenesis and Evolution of the Tonal Systems in New Caledonia, the Example of Cèmuhî[END_REF]. This demonstrates the usefulness of a cross-language approach to address problems found in the study of one specific language. Haudricourt referred to this approach as "panchronic" [START_REF] Haudricourt | Méthode pour obtenir des lois concrètes en linguistique générale[END_REF][START_REF] Haudricourt | A reconstruction of Neo-grammarian "sound laws" and the possibility of establishing "panchronic phonology[END_REF][START_REF] Hagège | La phonologie panchronique[END_REF]. The aim is to formulate generalizations about sound change that are independent of any particular language or language group. Panchronic laws are obtained by induction from a typological survey of precise diachronic events whose analysis brings out their common conditions of appearance. In turn, these laws can be used To solve this puzzle, Karlgren reconstructed a voiced series of finals: /*-d/, /*-g/, and also /*-b/ in some cases.

"Already the fact that Ancient Chinese [=Middle Chinese] possessed -p, -t, -k, -m, -n, -ng but not -b, -d, -g suggests these latter; And this seems so much the more natural as the experience from other languages shows that the mediae [i.e. voiced stops] more easily fall than the tenues. To cite my own language again, there are many Swedish dialects where bed > bē but bet > bet. It is therefore rather likely that it is a final g and a final d we have to expect in words like 乍, 敝" [START_REF] Karlgren | Analytic dictionary of Chinese and Sino-Japanese[END_REF]:27-28, cited in Baxter 1992:332).

Some years later, Karlgren realized that the words that he reconstructed with final *-g sometimes rhyme with words whose phonetic series in Old Chinese has no trace of a final stop. At that point, he concluded that final *-g must be generalized to the entire set of syllables that rhymed with his *-g-final syllables. This resulted in a reconstructed language that had almost no open syllables. Karlgren's reconstruction was adopted by many scholars. This theory has been accepted by nearly all specialists in Chinese historical phonology, with the exception of some scholars in China. Li Fang-Kuei (1971:33-34) mentions Haudricourt's hypothesis but considers that the internal evidence from Chinese does not allow for any hard-and-fast conclusion, and provisionally chooses to retain Karlgren's reconstructions with final *-g. This cautious choice tends to be reinterpreted by some of Li Fang-Kuei's epigones as an outright rejection of Haudricourt's analysis. As for Wáng Lì (1989: vol. XVII, pp. 213-247), he points out that the reconstructions of final voiced stops by Bernhard Karlgren and Walter [START_REF] Simon | Zur Rekonstruktion der altchinesischen Endkonsonanten[END_REF][START_REF] Simon | Zur Rekonstruktion der altchinesischen Endkonsonanten, II. Teil[END_REF] are problematic, but he is apparently unaware of Haudricourt's proposal.

A second major finding, also reported in the 1954b article, consists in the hypothesis that there were labiovelars in Old Chinese:

"…it seems that scholars have overlooked the fact that some rhymes in the Analytic Dictionary only 陸志韋 1947 andPulleyblank 1962:67-68).

  , whose classes he had attended with great interest at the National Institute of Agriculture. In 1940, Haudricourt was awarded a position in the newly created Centre national de la recherche scientifique, in the Botany department; he switched to the Linguistics department in 1945. His special interest in Asia led him to volunteer to work at the library of the École française d'Extrême-Orient in Hanoi from 1948 to 1949. His sympathy for the People's Republic of China earned him a chance to visit the country briefly in 1955. Haudricourt's impressive scientific production includes phonological work on a variety of languages of China. The full bibliography provided by Dufour 1997 testifies to the breadth of his research interests. To him, science was interdisciplinary as a matter of course. Within the Centre National de la Recherche Scientifique (CNRS), Haudricourt co-founded in 1976 a research centre whose goal is to investigate little-documented languages within their cultural environment, combining ethnological and linguistic work: the LACITO research centre (Langues et Civilisations à Tradition Orale).

  to shed light on individual historical situations. (About panchronic phonology, one may also consult a detailed epistemological discussion: Mazaudon and Michailovsky 2007; and recent case studies: Jacques 2011; Michaud 2012; Michaud, Jacques and Rankin 2012.) 2.3. Other contributions to the reconstruction of Old Chinese Haudricourt's contribution to Old Chinese reconstruction is not limited to his account of tones. He clarified several rhyming patterns found in the Book of Odes. Words with final stops /-p -t -k/ rhyme with words in departing tone (去聲 qùsheng) according to their MC pronunciation: for instance, words in the zhà 乍and zuó 昨series (MC: *dzraeH and *dzak, respectively) rhyme, as do words in the bì 敝 and piē 瞥 series (MC : *bjiejH and *phiet).

  Haudricourt's theory (1954b) that the departing tone comes from *-s explains this phenomenon: the words with departing tone rhyming with words in final stop should be reconstructed with final clusters *-ks, *-ts or *-ps. Moreover, from the point of view of historical morphology, Haudricourt's theory of tonogenesis leads to the reconstruction of several *-s suffixes (in particular a nominalizing suffix) which can be shown to be cognate with those found in conservative languages of the Sino-Tibetan family, such as Tibetan.

  appear with velar initials (/k/, /kʰ/, /g/, /x/, and /ŋ/), for instance -iʷei [MC *-wej] 齊, -ʷâng [*-wang] 唐, -iʷäng [*-jweng] 清, -ʷâk [*-wak] 鐸, -iʷet [*-wet] 屑etc." (translation by Guillaume Jacques; the Middle Chinese reconstructions added in square brackets are from Baxter 1992).This idea was used later byYakhontov 1960a and[START_REF] Pulleyblank | The consonantal system of Old Chinese[END_REF] to revise the reconstruction of the Old Chinese vowel system, and is the basis for the six-vowel system that is common to the recent systems of[START_REF] Starostin | Rekonstrukcija drevnekitajskoj fonologičeskoj sistemy[END_REF][START_REF] Baxter | A handbook of Old Chinese phonology[END_REF] and Zhèngzhāng Shàngfāng 郑張尚芳 2003. On the other hand, two of the ideas proposed by Haudricourt have been disproved since the time of publication.First, in his discussion of the MC initials *dzy-(常 cháng) and *zy-(船 chuán), he argues that the contrast is only due to dialect mixture. However, we now know that these two initials have entirely distinct origins in Old Chinese, and reflect a genuine contrast, though this contrast is poorly preserved in modern dialects. The phonetic reconstruction of these initials in MC by Karlgren was erroneous: it was based on misleading evidence found in the 切韻 Qièyùn. The rhyme tables in the Qièyùn clearly suggest an interpretation as voiced palatal affricate and voiced palatal fricative for these two initials, but this agrees neither with earlier texts, nor with transcriptions of Sanskrit in Chinese characters (see Lù Zhìwéi

  It should be noted that the treatment of the old voiced stops depends on the tonal category, A, B or C.For words of category A the old voiced stops had become voiceless aspirates, while for the words of category C with which B had been confused, the old voiced stops had become voiceless non-aspirates.The confusions of tones which are produced before the tones are phonemicized into tonemes are thus produced between tones of the same horizontal or vertical series." (Translation by Christopher Court, Their proposals are based on an understanding of Asian languages other than Chinese: mainly Austroasiatic, and also, in the case of Ferlus's proposal, the Oceanic languages Cèmuhî and Paicî, where simple-vs.-geminate oppositions evolved into tonal oppositions (as noted by[START_REF] Haudricourt | La langue de gomen et la langue de touho en Nouvelle-Calédonie[END_REF] and further studied by

		:24):
		A	B	C
	p, pʰ, t, tʰ, k, kʰ	33 ˧˧˧	13 ˩˧	53 ˥˧
	m, n, ŋ, l	35 ˧˥	13 ˩˧	53 ˥˧
	(* b>) p~pʰ, (*d>) t~tʰ, (*g>) k~kʰ	35 ˧˥	53 ˥˧	53 ˥˧
	published in 1972).
	Haudricourt's model was recently expanded into a generalized model of registrogenesis and
	tonogenesis, which also ushers in a hypothesis about the origin of the four divisions (sì děng

四等) of Middle Chinese (MC)

[START_REF] Ferlus | What were the four divisions of Middle Chinese?[END_REF]

. This analysis, which is inspired by facts about phonation-type registers that are not familiar to most readers, is not yet widely discussed in sinological circles. Haudricourt did not speak Chinese, and neither does Ferlus.

Second, his hypothesis concerning the existence of spirants, which he set up to solve the issues raised by Karlgren's reconstruction of voiced stops in Archaic Chinese, is now known to be mistaken. Yakhontov 1960b and Pulleyblank 1962 put forward another interpretation, which is now accepted by most scholars: that Karlgren's *d-should instead be reconstructed as a lateral. A clear account of the necessity of reconstructing laterals in Old Chinese is provided by Baxter (1992:196-199).

This overview has attained its goal if it has shown (i) that Haudricourt's legacy to the field of Chinese historical phonology is not limited to advances in the study of tonogenesis, and (ii) that the method proposed by Haudricourt under the term of Panchronic Phonology holds special promise for the study of the languages of China in their universal context.