Loïc Le Marrec

Philippe Lasaygues

Thierry Scotti

Chrysoula Tsogka

Efficient shape reconstruction of non-circular tubes using broadband acoustic measurements

Keywords: Acoustic, Non-linear inverse problem, Multi-frequency, Shape reconstruction

We propose an algorithm for reconstructing the boundaries of a non-circular cylindrical tube from broadband acoustic measurements. This algorithm is based on the minimization of a cost function, which is the averaging over frequency of the absolute difference between the estimated and the measured scattered field. The estimated field is computed efficiently (very fast) using ICBA, an analytic method that provides an approximated solution of the forward problem. Numerical results show that our algorithm is robust and provides an accurate reconstruction without any explicit regularization.

Introduction

In this paper we address the problem of reconstructing both the inner and outer section of almost circular cylinders (i.e., for which the maximum deviation from the mean radius is small). The applications we have in mind are non-destructive evaluation (identification of pipe lines) and medical imaging (bones). To solve the inverse scattering problem, a cost function accounting for the discrepancy between the measured and an estimated scattered field, is usually minimized [START_REF] Colton | Inverse Acoustic and Electromagnetic Scattering Theory[END_REF]. To overcome the ill posedness of such problems, a regularization term expressing some a priori information on the solution is added to the cost function [START_REF] Kokurin | Stable iteratively regularized gradient method for nonlinear irregular equations under large noise[END_REF][START_REF] Lambert | Caractérisation d'un sous-sol marin stratifié en situation de petits fonds par une méthode itérative[END_REF]. The relative confidence on the regularization is controlled by a real positive parameter whose choice is delicate and depends on the considered configuration. Our goal being to avoid such difficulties by increasing the amount of data (over-determine the problem), we consider the insonification of the unknown scatterer by a broadband ultrasonic pulse that provides multi-frequency measurements. For this problem, linear imaging techniques [START_REF] Belkebir | Modified gradient method and modified Born method for solving a two-dimensional inverse scattering problem[END_REF][START_REF] Crocco | Testing the contrast source extended Born inversion method against real data: the TM case Inverse Problems[END_REF] are generally well adapted but their validity is restricted to low-frequency insonifications or low-contrast objects. On the other hand, non-linear inversion methods [START_REF] Abubakar | Application of the multiplicative regularized contrast source inversion method on TM-and TE-polarized experimental Fresnel data[END_REF][START_REF] Marklein | Linear and nonlinear iterative scalar inversion of multi-frequency multi-bistatic experimental electromagnetic scattering data[END_REF][START_REF] Duchêne | Inversion of experimental data using linearized and binary specialized nonlinear inversion schemes[END_REF], which use an exact forward solver to compute the estimated field, are computationally more intense and increasing the number of data becomes very expensive. Thus it appears as essential for non-linear inversion methods to employ a forward solver, which is fast and efficient in a large frequency range.

To enhance the speed of computations in the proposed algorithm we use the Intercepting Canonical Body Approximation (ICBA) as forward solver during the inversion. This consists in approximating the scattered field along the measurement direction of the real almost circular cylinder by the scattering due to a centred circular cylinder, which has locally (in the measured direction) the same radius as the object [START_REF] Scotti | Shape reconstruction using diffracted waves and canonical solutions[END_REF]. Our inversion algorithm is therefore computationally efficient as it uses an analytic expression for computing the scattered field. The ICBA approximation has proved its efficiency as forward solver in and beyond the resonance region [START_REF] Wirgin | Wide-band approximation of the sound field scattered by an impenetrable body of arbitrary shape[END_REF] and has been successfully used for real time inversion of monochromatic data. The algorithm has been extended to the case of two or three frequencies where an unique solution is obtained by solving the inverse problem for each insonification and then choosing the "closest" common solution [START_REF] Ogam | Non-ambiguous boundary identification of a cylindrical object by acoustic waves[END_REF], the "closest" criterion depending on the user. The problem is here extended to broadband ultrasonic scattering and an algorithm adapted to multiple frequencies is proposed. The method is first validated for the reconstruction of the shape of a plain cylinder. In a second step the method is extended to the reconstruction of both the inner and outer sections of a tube with particular attention on the a priori information and on the inversion procedure. In comparison with the previously proposed algorithm it presents the advantage of requiring only one minimization (in contrast to two or as many as the frequencies available). Moreover, the solution does not depend on the adjustment of any parameter. Although the forward problem is solved approximately during the inversion, the proposed method is efficient, and thus suitable for systematic use in real time imaging.

Problem setup 2.1 Description of the scattering problem

Let a plane wave probing, normally to its generator, a cylindrical object defined by the domain Ω 2 embedded in an infinite medium Ω 1 (Fig. 1). The generator of the cylinder is normal to the (x, y) plane and the origin O of the polar coordinate system lies in Ω 2 . Both materials occupying domains Ω 1 and Ω 2 are assumed to be non-absorbing, homogeneous, isotropic, acoustic materials and thus the problem is reduced to a two-dimensional linear scalar acoustic scattering problem. The total acoustic fields P j=1,2 in domains Ω j=1,2 are governed by the corresponding Helmholtz equation and satisfy the continuity conditions at the boundary. The object has penetrable boundary conditions. In Ω 1 , the scattered field P = P 1 -P i satisfies the Sommerfeld radiation condition at infinity. For a given frequency f , P i = A(f )e ik 1 cos(θ i -θm) express the incident field along θ m is expressed by , where θ i is the incident angle, A(f ) the complex amplitude, k 1 = 2πf /č 1 the wave number, and č1 the velocity of the surrounding medium. Note that the e -iωt is henceforth implicit.

Measurements

The measurements are synthetic data obtained by solving numerically the 2D acoustic wave equation. The method used is based on mixed finite elements for the space discretization and on a centered 2 nd order finite difference scheme for the time discretization [START_REF] Bécache | An analysis of new mixed finite elements for the approximation of wave propagation problems[END_REF]. The incident pulse is a normalized second derivative of a Gaussian with central frequency f c . The backscattered field P is recorded on a centered measurement circle of radius r m . In order to analyze the frequency dependence of the inversion, synthetic time-domain measurements for a given θ m are converted to frequency-domain using a Fast Fourier Transform algorithm. This results to a set of backscattered data at L consecutive frequencies with a sampling frequency δf .

Approximation

During the inversion, the (true) synthetic scattered field along the measurement direction is approximated by P e , the scattered field due to a centered circular cylinder, which has the same acoustical properties as the object and in the measured direction the same radius as the object. We denote here this radius by r 1 . Remark that for this approximation to be valid it is necessary to assume that the center of the coordinate belongs to the body. At a single frequency, the

[kHz] k 1 a = 2πfc ř1 (θm) c 1 N (for f c ) δf [kHz] L r m [
P e (x m , f ) = A(f ) N n=0 B n (θ m , f )i n ǫ n H (1) n (k 1 r m ) cos(n(θ i -θ m )) , N = Ent(4.05k 1 a 1/3 +k 1 a+5) (1)
where Ent is the entire part function, H

n is the first-kind Hankel function of order n, ǫ n the Neumann factor (ǫ 0 = 1, ǫ n≥1 = 2) and the scattering coefficient B n is exactly evaluated using Rayleigh-Fourier method [START_REF] Doolittle | Sound scattering by elastic cylindrical shells[END_REF]. The final order N of the series is chosen according to the method proposed in [START_REF] Barber | Light Scattering by Particles: Computational Methods[END_REF].

Numerical example

We first consider the cylinder, called clover, whose section is presented in Fig. 2. The boundary of the object is described by the following equation, ř1 (θ) = 5.875 + 0.625 cos(3θ) ; 0 ≤ θ < 2π.

(

) 2
The acoustical properties used in the simulations are given in Tab.1. These values correspond to a high contrast body with respect to the surrounding medium (bone, or pipe-line in water). Two probing pulses having respectively f c = 250 kHz (Low Frequency Insonification : LFI) and f c = 1 M Hz (High Frequency Insonification : HFI) are simulated (Fig. 3). According to the mean radius of the body a and central frequency f c , the first insonification corresponds to an excitation in the resonance region (k 1 a ≈ 6) while the second is beyond resonance (k 1 a ≈ 25). In Fig. 4 and Fig. 5, measured and approximated backscattered fields are presented for θ m = π/4 and r m = 12 mm (near field). The Tab.2 resumes all the numerical characteristics of the problem for this configuration.

The local inversion method

Knowing that the center of the coordinate belongs to the body, the acoustical properties of each domain (density (ρ j ) j=1,2 , velocity (č j ) j=1,2 ), the incident P i and backscattered P numerically simulated fields at x m = (r m , θ m ) ∈ Ω 1 , we seek to reconstruct the local radius ř1 (θ m ) of the boundary.

According to ICBA, we seek solutions τ that minimize the cost function,

F(τ ; x m , f ) = | P (x m , f ) -P e (τ ; x m , f )| 2 . (3)
Here τ = r 1 , where r 1 is the radius of the circular cylinder used to estimate the backscattered field P e (τ ; x m , f ).

Single-frequency reconstruction

To analyze the role of each frequency in the inversion algorithm, the single frequency cost function ( 3) is computed for several frequencies individually. We consider here the frequency data corresponding to the scattered field at θ m = π/4 and the unknown is the local radius ř1 (π/4) = 5.433 mm of the clover. In Fig. 6, cost functions for three frequencies, f 1 = 250 kHz, f 2 = 500 kHz and f 3 = 1 M Hz, illustrate the non-uniqueness of the solution of the inverse problem by a lot of local minima. Remark that all the cost functions have a minimum (not always global) close to the exact solution. However, there exist other estimations (e.g., r 1 ≈ 2.5 mm) that are also close to minima for all the three cost functions. Therefore, careful post-processing is needed when using this algorithm with a small number of frequencies, in particularly when the data are noisy and/or the solution of the forward problem is not exact.

The main difficulty consists now in distinguishing the optimal solution. By plotting the local cost function for all the frequency-band of the HFI, between 100 kHz and 2 M Hz (58 frequencies, 2 ≤ k 1 a ≤ 50) (Fig. 7) we observe the following: (i) At fixed frequency the distribution of minima is periodic with the estimation. The period of this distribution is inversely proportional to the frequency and the number of admissible solutions increases with frequency.

(ii) A line of local minima appears for which the estimated parameter is close to the solution and varies weakly with frequency. This suggests that the reconstruction using the canonical approximation will not be perfect but a solution can be obtained with a good accuracy, for any frequency such that ka ≥ 1.

When these multi-frequency data are treated independently the number of admissible solutions increases. Seeking the minima of all cost functions corresponding to the different frequencies and then applying a post-processing algorithm to determine an unique solution requires as many minimizations as frequencies. The same holds for a method based on an iterative frequency algorithm [START_REF] Marklein | Linear and nonlinear iterative scalar inversion of multi-frequency multi-bistatic experimental electromagnetic scattering data[END_REF] (i.e., starting from the lower frequencies and then refining the solution by considering the higher available frequencies). Moreover the gain in precision for such algorithm is not ensured.

The broadband inversion method that we present in the following section is based on these remarks.

Multi-frequency reconstruction

The proposed method is based on the minimization of a single cost function, the "Mean Cost Function"(MCF), defined by,

F (τ ; x m ) = 1 L L l=1 F(τ ; x m , f l ). ( 4
)
Figure 8 presents the mean cost function (4) corresponding to the reconstruction of the local radius ř1 (π/4) with HFI. The sum in (4) is made on three frequency domains (D j ,j = 1, 2, 3) that cover respectively 33%, 66% and 99% of the energetic frequency-band of the probing wave (Fig. 3). We observe in Fig. 8 that as the frequency range increases, variations of the cost function decrease and the minimum corresponding to the optimal solution becomes deeper with respect to the other minima, and thus easier to estimate. Averaging over frequency has a constructive effect on the minimum, which is frequency invariant and a destructive effect on the others. In Fig. 9 the mean cost functions using the full frequency spectrum (domain D 3 ) for HFI and LFI are compared. We remark that the number of local minima increases with f c . Thus, for a given domain of estimation the number of initial guesses necessary for the reconstruction with a local inversion method, increases with the central frequency of the probing pulse.

In both cases, the deepest minimum is very close to the solution, thus we conclude that the central frequency does not affect the precision of the optimal estimation (absolute error : 0.019 mm for LFI and 0.048 mm for HFI).

Numerical example

All around the body, the backscattered field is recorded at 16 points x m , with r m = 12 mm and θ m = mπ/8, (m = 0, ...15). Each multi-frequency field P (θ m ) is used to reconstruct the local radius ř1 (θ m ) of the body. To examine the influence of the central frequency on the reconstruction we use both the LFI and HFI as probing waves.

The MCF is defined on D 3 (L = 110 for LFI and L = 92 for HFI). The minimization is realized by a quasi-Newton algorithm [15]. Because a local minimization algorithm is used, 15 initial guesses, equally distributed on the estimation domain [3; 10] mm, are used. Both LFI and HFI provide satisfactory reconstructions (see Fig. 10 and Fig. 11). Note that the deepest minimum is obtained more than once for the lowest frequency insonification but generally only one time for the higher frequency insonification. This underlines the difficulty in distinguishing the optimal solution from all the admissible ones. The good results obtained here can be explained by the low deviation of the section of the object from the mean radius, which validates the circular approximation used for the forward problem.

Application to external and internal boundary reconstruction

We consider now the problem of reconstructing both the inner and outer sections of a tube. In this case, the body used to estimate the forward problem during the inversion is a centered circular tube with inner radius r 2 . The objective of the inversion is to reconstruct both the inner and outer sections of the body at several discrete angles θ m . At each angle θ m , the cost function varies according to the radius r 1 and r 2 of the circular tube. This problem could be solved in the following two ways : (i) Reconstruction of both parameters at the same time. This needs a great amount of initial guesses in order to cover the whole estimation domain, in particular for high central frequencies. (ii) Estimation of the parameters in two independent steps (using some a priori estimation for the second parameter during the inversion of the first). In this case one needs to ensure the robustness of the first estimated parameter to the a priori on the second one.

In the example considered here, the hollow cylinder is filled by an acoustic material (ρ 3 = 1200 kg/m 3 and č3 = 1700 m/s) defining the domain Ω 3 . All material properties of the media are again exactly known. The outer and inner sections (Fig. 14) are described by the following polar functions,

Outersection : ř1 (θ) = 5.875 -0.875 cos(2θ) + 0.750 sin(3θ) ; 0 ≤ θ < 2π Innersection : ř2 (θ) = 3.0 -0.45 cos(2θ) ; 0 ≤ θ < 2π. ( 5
)
All the other characteristics are the same: mean external radius, properties of the media and measurement configurations. This new body is distinguished from the previous ones by its concavity and a greater deviation of the outer section from the mean radius.

In order to test the robustness of the reconstruction, the MCF for the estimation of each parameter is plotted for two cases: in the first case the a priori on the second parameter is exact and in the second case the a priori is wrong. In Fig. 12 and Fig. 13, the MCF obtained for the estimation of each parameter is plotted for each probing wave. The simulated measurements are the backscattered field at θ m = 0, which corresponds to ř1 (0) = 5.000 mm and ř2 (0) = 2.550 mm. The unknowns r 1 and r 2 are estimated in the domain [4; 8] mm and [2; 4] mm respectively. Numerical results are given in Tab.3. We remark that the estimation of the external radius is independent from the a priori on the 3.348 0.799 inner radius. The cost function and the admissible solution are of the same order, whatever the a priori information is. On the contrary for the inner radius, the estimation is more sensitive to the a priori on the outer radius. For the HFI the cost function obtained when the a priori is the exact solution is very irregular. The minimization gives a solution that is far from the exact solution with an absolute error that is of the order of half the wavelength (in the surrounding media). When the a priori on the outer radius is the admissible solution obtained previously, the admissible solution is satisfactory. Remark that for the LFI, the solutions obtained with the two a priori are of the same order. The above observations for θ m = 0 also hold for other measurements directions. The reconstruction provides better (or equivalent) results when the a priori information on r 1 is the radius estimated in the first step compared to the estimation obtained when the outer radius is the true one. This is particularly surprising especially because there is only a small discrepancy between the two a priori information (less than a quarter wavelength for both probing waves). This result suggests that the estimations of both parameters should be obtained with the same method: having the exact solution as a priori information seems not to be the best choice. We also observe that for both parameters the spectrum of the probing wave does not perturb the estimation. The frequency domain used affects the width of the attraction domain of the solution (and then the number of the admissible solutions) only for the external radius reconstruction.

In order to recover the shape of the tube, 16 backscattered fields, recorded in the same configuration as in the previous section, are used. The proposed algorithm is the following: (i) for each angle of measurement independently, the outer local radius is first estimated with an arbitrary a priori on the inner local radius, (ii) in a second step, this estimation is used as a priori for the outer radius when the inner radius is estimated. This algorithm is faster and more robust than when the two unknowns are estimated simultaneously.

The external boundary reconstruction is given in Fig. 14.

The presence of a hole seems not to disturb the reconstruction of the outer boundary. The concave part of the external boundary and the inner section are not as well reconstructed. However, the images obtained are satisfactory in and beyond the resonance region and the absolute error is always negligible with respect to the wavelength.

Conclusion

An efficient method for reconstructing the external boundary of non-circular cylindrical objects using multi-frequency data has been presented. The proposed algorithm, using an analytical solution for the local forward problem and an iterative process to recover the unknowns, presents the advantage of being fast and thus able to provide real time images of the shape. This method is robust with respect to the geometry. Even when the deviation of the section from the mean radius is greater than the wavelength the method gives very attractive results. The frequency band of the probing wave does not play an essential role: the reconstruction is of the same order in and beyond the resonance region.

Using ICBA as forward solver for the inversion is an efficient tool for reconstruction with broadband acoustic measurements. The MCF does not need any regularization operator when the a priori are chosen properly.

For such high contrast objects, reconstruction of the inner and outer shape is accurate when the material properties are exact. We are currently working on the extension of this method to the more general inverse problem where the acoustical or mechanical properties are also unknown. In this case, we need again to check the sensitivity of the estimation of the unknowns with respect to the a priori information.

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1: Problem setup.

Figure 4 :

 4 Figure 4: Simulated (-) and approximated (---) backscattered field by the clover. The probing wave is the LFI. The acquisitions are made at r m = 12 mm and θ m = π/4. The approximation is computed with the ICBA for r 1 = ř1 (θ m ).

Figure 5 :

 5 Figure 5: Simulated (-) and approximated (---) backscattered field by the clover. The probing wave is the HFI. The acquisition are made in r m = 12 mm and θ m = π/4. The approximation is computed with the ICBA for r 1 = ř1 (θ m ). Remark that the time window for the recording is different than with LFI.

Figure 6 :

 6 Figure 6: Reconstruction of ř1 (π/4) : normalized amplitude of the cost functions for 3 frequencies: f 1 = 250 kHz (--), f 2 = 500 kHz (---) and f 3 = 1 M Hz (• • •).

Figure 7 :

 7 Figure 7: Reconstruction of ř1 (π/4) : the single frequency cost function (3) as a function of frequency and the estimation parameter.

Figure 8 :

 8 Figure 8: Reconstruction of ř1 (π/4) with the HFI (f c = 1 M Hz). MCF (4) for 3 frequency bands: D 1 (• • •), D 2 (--), D 3 (-). There is no normalization.

Figure 9 :

 9 Figure9: Reconstruction of ř1 (π/4) using all the frequency bands for the each probing wave : LFI (--) and HFI (-). There is no normalization.

Figure 10 :

 10 Figure 10: Reconstruction with LFI : admissible * and optimal ⊛ estimation of the local radius. Actual sections : --.

Figure 11 :

 11 Figure 11: Reconstruction with HFI : admissible * and optimal ⊛ estimation of the local radius. Actual sections : --.

Figure 12 :

 12 Figure 12: Normalized MCF for the reconstruction of ř1 (0) with two a priori on r 2 : r 2 = ř2 (0) (-) and r 2 = 3 mm (--). Top : LFI, bottom : HFI.

Figure 13 :

 13 Figure13: Normalized MCF for the reconstruction of ř2 (0) with two a priori on r 1 : r 1 = ř1 (0) (-) and r 1 = the optimal estimation obtained for the reconstruction of ř1 (0) (--). Top : LFI, bottom : HFI.

1 Ω 2 Ω 3 Figure 14 :

 12314 Figure 14: Actual section of the tube (bold) and corresponding domains. Optimal estimation of each local radius obtained with the LFI (•) and the HFI (+). 16 equally distributed synthetic measurements are made all around the body. The optimal estimation of ř1 (θ) is used as a priori for the reconstruction of ř2 (θ).

Table 1 :

 1 Acoustical characteristics of the media.

	ρ1 [kg/m 3 ] č1 [m/s] ρ2 [kg/m 3 ] č2 [m/s]
	1000	1500	1800	4000

Table 2 :

 2 Characteristics of the simulations. Probing wave f c

Table 3 :

 3 Reconstruction of the inner and outer radius at θ m = 0 Reconstruction of ř1 (0) = 5.000 mm Probing wave a priori on r 2 [mm] estimation, r 1 [mm] r 1 -ř1 (0)[mm]

	LFI	3.000	5.384	0.384
		ř2 (0)	5.356	0.356
	HFI	3.000	5.390	0.390
		ř2 (0)	5.392	0.292
	Reconstruction of ř2 (0) = 2.550 mm
	Probing wave a priori on r 1 [mm] estimation, r 2 [mm] r 2 -ř2 (0) [mm]
	LFI	5.384	2.797	0.247
		ř1 (0)	2.727	0.177
	HFI	5.390	2.604	0.054
		ř1 (0)

Bibliography

References


Frode Rønning
email: frode.ronning@ntnu.no

Teaching the concept of angles to multilingual learners

Keywords: Geometric concepts, angles, multilingual learners, language responsive teaching

Teaching the concept of angles to multilingual learners

Introduction

The project EsCo1 involves researchers from seven countries and focuses on classroom teaching of specific mathematical topics in language responsive ways. It is known that linguistically disadvantaged learners continue to be educationally disadvantaged, and therefore it is important to gain more knowledge about the challenges these learners face. Four topics are chosen: angles, linear equations, proportional reasoning, and probability, to be taught to pupils aged 11-13 years. In the first phase of the project, three to four experienced teachers from each country have been interviewed to gain insight into the approaches they use to teach the specific topics and what challenges particularly multilingual pupils encounter with each of these topics.

The nature of geometric concepts

Geometric concepts may have an advantage in pupils' learning process because they often can be represented with concrete objects known from everyday life, and because the words used are also often well-known from everyday language. However, the connection to everyday language may also be a hindrance because words may have different meanings in the everyday register compared to the mathematical register. Discrepancies between the everyday and mathematical register may be language specific, as has been shown by Rønning andStrømskag (2017, 2019) in connection with naming elements of a polygon (edges and vertices). In the Norwegian language, there is no proper scientific term for the concept vertex. The word that is used, hjørne, is common to both the scientific and the everyday register, and corresponds to the English word corner. Polygons are named based on the number of edges (e.g., firkant = 'four edge'), but it was observed that when classifying and naming polygons, the Grade 2 pupils studied by [START_REF] Rønning | Entering the mathematical register through evolution of the material milieu for classification of polygons[END_REF] counted the vertices but claimed that they counted the edges. [START_REF] Frege | Über Sinn und Bedeutung [On sense and reference[END_REF] distinguished between the sense (Sinn) and the reference (Bedeutung) of a word. In semiotics, a word is seen as a sign. The reference is the object that the sign refers to and the sense comprises all thoughts and ideas connected to the sign. Two different signs may have the same reference but different sense depending on the situation, and also one sign corresponding to one reference may have different sense depending on the situation.

For the pupils mentioned above, the sense of the word edge (Norwegian: kant) was that of something sharp, something that would cause pain if you hit it [START_REF] Rønning | Sense and reference of signifiers for elements of polygons[END_REF], and therefore they associated the word kant with the vertex (hjørne), in particular when approaching the vertex from the outside. It could also be observed that the same pupils could refer to a vertex as a hjørne when approaching it from the inside. This indicates, using the words of Frege (1892), a different sense for the same reference depending on the situation. The previous research was done with monolingual learners, and it is of interest to investigate whether there are particular challenges involved when teaching similar concepts, like angle, to multilingual learners.

The angle concept

I conducted a 40-minute semi-structured interview with the teacher Mary. The interview was audio recorded and later transcribed. Mary was asked what strategies she used when teaching the topic of angles and what linguistic challenges the pupils would encounter. Mary said that irrespective of the pupils, she would approach the concept of angles in a very practical manner. She tries to uncover the pupils' learning styles, and this gives an opportunity to observe carefully pupils with linguistic challenges. Mary: "These are often multilingual pupils, but also monolingual children". Challenges are strongly connected to the environment the children grow up in, and whether the adults use, what Mary called "a precise language". Mary finds working with concrete material and practical activities in various settings to be important, and that this gives the opportunity to uncover linguistic challenges.

I detected some issues connected to sense and reference [START_REF] Frege | Über Sinn und Bedeutung [On sense and reference[END_REF]) in Mary's account of working with angles. An angle v is classified as being acute (v<90 o ), right (v=90 o ) or obtuse (90 o <v<180 o ). The English words acute and obtuse are scientific words, with no obvious connection to the everyday register. The corresponding Norwegian words are spiss and stump. Spiss is common in everyday language having the sense of something sharp, e.g., the tip of a pencil. This can be confusing for the pupils, according to Mary, because an angle of e.g., 87 o is by definition spiss, but it does not look spiss (sharp). The word stump has a different character. It is an old-fashioned everyday word, possibly with no sense for young people. It sounds like stum, which means dumb, and Mary says that the pupils find this word strange, because "no angles can speak". A corresponding English word could be blunt.

In the interview, Mary pointed to several linguistic challenges when teaching the concept of angle, not necessarily connected to multilingualism. The topic of angles may therefore be challenging because the words used in the Norwegian language come from the everyday register, sometimes with a different sense, and sometimes with no sense at all for young learners.

Establishing a collaborative relationship for research on school mathematics teaching with linguistically disadvantaged learners. Project funded by John Fell Fund.


Philippe Lasaygues
email: lasaygues@lma.cnrs-mrs.fr

Assessing the cortical thickness of long bone shafts in children, using two-dimensional ultrasonic diffraction tomography

Keywords: Ultrasonic tomography, Born approximation, 2D-ring antenna, Children's bones imaging

Echography is one of the first-line techniques used in clinical practice to diagnose osteoarticular diseases in children. However, this technique involves the use of standard equipment, which is not adapted to the morphology or the acoustical properties of children's bones. In this study, we developed an ultrasonic tomography method for measuring the cortical thickness of children's long bones. Ultrasonic tomography gives cross-sectional images showing the spatial distribution of some of the physical components of an object, based on scattered ultrasound measurements. These measurements are carried out using variably dense sets of transmitter and receiver positions and various the wave frequencies. We solved this inverse scattering problem using a Born approximation, which yields an attractively simple linear relation between the object function and the scattered field, particularly in the far field. Experiments with a 2D-ring antenna show the applicability of the method and its various improvements to bone thickness imaging.

INTRODUCTION

Echography has long been the main first-line imaging technique used to diagnose various pediatric pathologies (such as abdominal injuries, urinary tract problems, congenital cardiac defects and brain anomalies and injuries). In the case of bone diseases, this technique is now proving to be an invaluable means of assessing painful traumatic and non traumatic disorders (such as hip dysplasia, knee trauma and stress fracture of the tibia). Cases where it is necessary to search for an intra articular joint effusion, an infectious periosteal apposition or a tumoral resurgence in contact with a stainless steel prosthesis or, more recently, for a traumatic ligament lesion, are just a few examples where the use of ultrasound is highly indicated [START_REF] Falcini | Comparison of quantitative calcaneal ultrasound and Dual Energy X-ray Absorptiometry in the evaluation of osteoporotic risk in children with chronic rheumatic diseases[END_REF][START_REF] Rand | Imaging of bone and soft tissue tumors. A case study approach[END_REF][START_REF] Enns | Sonographic detection of an isolated cuboid bone fracure not visualized on plain radiographs[END_REF][START_REF] Halaba | Quantitative ultrasound in the assessment of skeletal status in children and adolescents[END_REF][START_REF] Mayr | Musculoskeletal ultrasound in pediatric trauma[END_REF]May et al. 2004). X-ray computed tomography, magnetic resonance imaging and/or bone scintigraphy, the other widely used methods available [START_REF] Baert | [END_REF], have various disadvantages, depending on the type of examination for which they are used, such as their cost, the fact that they involve X-rays or require sedation and their lack of availability and accessibility. Echography is non invasive, non irradiant, painless, inexpensive and practicable on children in bed. However, there are some factors which make it impossible to extend the use of this technique on patients with other osteoarticular diseases. In practice, it is difficult to perform pediatric bone ultrasonography, because standard echographic devices are not very suitable for use with hyper echogenic tissues of this kind (because of amplification, gain, delay line problems etc.). The usual ultrasonic probes are not always adapted to the morphology of the area investigated (because of the linear array, dimensions, time and space resolution). The frequencies used are too high to be able to penetrate into the cortical bone and the underlying tissue (the medullary canal). The echographic map certainly reflects the varying impedance levels between tissues, but it does not depend on information about any particular acoustical parameter.

Many authors have dealt with the ultrasonic characterization and imaging of adults' and children's bones. Their main aim has usually been to assess the thickness of the diaphysis (as part of its geometrical conformation) and to calculate the speed of sound (an acoustical parameter) of a compressional wave crossing the structure. These aspects are both of great importance. For example, the cortical thickness provides an index to the risk of fracture and can serve to predict fracture loads in the radius, femur and lumbar vertebrae [START_REF] Wear | Autocorrelation and cepstral methods for measurement of tibial cortical thickness[END_REF][START_REF] Stegman | Cortical ultrasound velocity as an indicator of bone status[END_REF][START_REF] Foldes | Quantitative ultrasound of the tibia: A novel approach for assessment of bone status[END_REF]. To study the cortical area of long bones, measurements can be obtained in the plane parallel to their generator, using the "axial or transverse measurement" process [START_REF] Moilanen | Thickness sensitivity of ultrasound velocity in long bone phantoms[END_REF][START_REF] Bossy | Effect of bone cortical thickness on velocity measurements using ultrasonic axial transmission: A 2D simulation study[END_REF][START_REF] Bossy | Three-dimensional simulations of ultrasonic axial transmission velocity measurement on cortical bone models[END_REF]).

Measurements and images can also be obtained in the perpendicular plane, using the "radial or cross-sectional measurement and imaging" process (André et al. 1980;[START_REF] Lasaygues | Cancellous and cortical bone imaging by reflected tomography[END_REF]Ouedraogo et al. 2001).

Our group has been focusing on the second type of process, which involves solving inverse scattering problems, but first the corresponding forward problem has to be solved. Several methods have been applied to the forward problem, including integral equation and hybrid finite-element methods and the geometrical theory of diffraction. The latter method provides asymptotic approximations for diffracted fields, which are valid in the case of high frequencies and large distances from the diffracting body. Actually, all these approaches involve the "classical" problem of minimizing the differences between the modeling data and the measured values. Several numerical strategies can be used to model the forward and inverse problems simply, efficiently and accurately. The strategy developed in this study was based on the use of ultrasonic tomography and the Born approximation. Ultrasonic tomography, which results from the linearization of the inverse acoustic scattering problem, makes it possible to detect any perturbations (even quite small ones) of the reference surrounding medium. In the case of media with weak inhomogeneities, such as soft tissues, the homogeneous mean medium is taken to be the reference medium. This has led to the development of a "constant background" method, which make it possible at the practical level to perform regular angular scanning with broadband pulses covering the whole spatial frequency spectrum of the object investigated, slice-by-slice. This leads to "reconstructionfrom-projections" algorithms, such as those used in X-ray computed tomography. In the case of media with considerable heterogeneities, such as bones, the problem is a non linear one and there is generally no single solution available. Ultrasound propagation in bone suffers from the severe mismatches encountered between the acoustic properties of bone and those of the surrounding mean medium (soft tissues). However, when the cause of the scattering appears to be quite small and localized in well-known media, a "variable background" method can be used, in which case, the reference background adopted is the water/specimen set [START_REF] Lefebvre | Progress in linear inverse scattering imaging : NDE application of Ultrasonic Reflection Tomography[END_REF].

In previous studies [START_REF] Lasaygues | Cancellous and cortical bone imaging by reflected tomography[END_REF], our group established that preliminary in vitro quantitative images of a human adult femur cross-section could be reconstructed using lowfrequency (< 1 MHz) ultrasonic reflection tomography. This method consists in assuming the cortical shaft to be a non circular cylindrical elastic cavity, which is taken to be weakly heterogeneous in comparison with the wavelength used (the local fluctuations in the acoustic characteristics of a long bone cross-section are weak), immersed in a quasi homogeneous water-like medium. The aim of this study was to assess the cortical thickness of children's thighbones using high-frequency (> 1 MHz) diffraction tomography [START_REF] Lefebvre | La tomographie ultrasonore en réflexion[END_REF] associated with an image-processing tool.

Our tomographic approach is first explained and the computer-controlled 2D-ring antenna is also described. Results obtained on experimental phantoms (artificial bone) and real fibulas are presented. Our initial results are promising and suggest that the geometrical characteristics of the bones, which are known to be one of the main factors contributing to the risk of fracture and disease, can be determined using this ultrasonic tomographic method.

ULTRASONIC DIFFRACTION TOMOGRAPHY

In this section, it is proposed to present the image reconstruction algorithm and to discuss the problem of ultrasonic wave propagation in bone.

The "reference" background

Assessing the geometrical properties of a non circular cylindrical body, such as a long bone, from an ultrasonic scattered field is a non linear, ill-posed inverse problem. There is generally no single solution to problems of this kind and a suitable modeling procedure (based on firstorder or canonical approximations, for example [START_REF] Wirgin | Some quasi-analytic and numerical methods for acoustical imaging of complex media[END_REF])) has to be used to avoid obtaining solutions that do not include relevant parameters.

When dealing with weakly inhomogeneous media such as soft biological tissues, a mean medium is chosen as the homogeneous reference background. In this case, the scattering is relatively weak and the problem can be linearized with a Born approximation. If the Green function of the unperturbed problem, i.e., the constant background, is known, the forward problem can be solved with the integral equation of Lippman-Schwinger and a far-field solution of the equation can be calculated [START_REF] Greenleaf | Computerized transmission Tomography[END_REF]). This "constant background" inverse Born approximation method, which requires regular angular slice-by-slice scanning with broadband pulses, yields the spatial frequency spectrum of the object under investigation. The inverse problem is then solved using the classical "reconstruction-fromprojections" algorithms focusing on any perturbations with respect to the reference problem [START_REF] Kak | Reconstruction from Projections: Applications in Computerized Tomography[END_REF]. The details of the complete algorithm were presented in a previous paper [START_REF] Lasaygues | High Resolution Low Frequency Ultrasonic Tomography[END_REF]).

When dealing with highly heterogeneous media such as bone (which involves strong contrasts and large objects with respect to the wavelength used), it is not possible to use the Born approximation with a "constant background" directly: hard biological tissues are heterogeneous and their acoustical characteristics differ greatly from those of the surrounding soft tissues (water in the present case: the body is immersed in a water tank). However, in cases such as non destructive testing situations, the focus is generally on flaws, which show up as strong inhomogeneities in well-known media. In this case, one can use a "variable background" inverse Born approximation method to reconstruct the perturbation [START_REF] Lasaygues | Etude de l'écho d'une fissure, traitement de signaux par analyse en ondelettes en contrôle non destructif[END_REF][START_REF] Haddadin | Ultrasound inverse scattering for tomographic imaging and self-focussing arrays[END_REF]. The reference background here is the water/specimen set and the flaw is the perturbation. We have used a similar approach here to the problem of imaging cortical bone thickness, consisting of identifying a water-like cavity in an elastic cylinder. The background, consisting of the cortical part without the hollow and the water, can be defined here as the set and the perturbation is therefore the hollow cavity.

The problem can be then reduced to studying the difference in acoustic impedance between the cortical part and the water. Some assumptions first have to be made about the properties of the bone.

Bio acoustical considerations

Bone tissue in humans over 10 y of age can be classified into two broad types, based on the macroscopic porosity. Cortical bone, which is also called compact bone, forms the outer shell of all long bones and its pore volume amounts to about 10%. Trabecular bone, which is also known as cancellous bone, is much more porous than cortical bone: it has a porosity of about 75%. The shaft of long bones gradually widens towards the ends, the compact wall of the bone becomes thinner and the medullary cavity is lined with an increasingly thick layer of spongy bone.

The acoustic impedance of bones (3 to 7 kg m -2 s -1 ) is more heterogeneous than that of the surrounding soft tissues (water ≈ 1 kg m -2 s -1 ) and this feature perturbs the ultrasonic propagation or, more specifically, the refraction, attenuation and scattering of the waves.

Conventional medical echography (in the reflection mode only) with high ultrasonic nominal frequencies (3 -13 MHz) therefore does not give accurate bone imaging results.

Human cortical bone is generally assumed to consist of a transverse isotropic medium in which the main axis of symmetry is oriented along the longitudinal axis of the diaphysis. This assumption was confirmed by several independent compression and shear wave velocity measurements on various specimens and in various directions of isotropy [START_REF] Ashman | Ultrasonic determination of the elastic properties of cortical bone: technique and limitations[END_REF][START_REF] Nicholson | Scattering of ultrasound in cancellous bone: prediction from a theoretical model[END_REF][START_REF] Pithioux | An ultrasonic methods to describe mechanical properties of compact bone[END_REF]. In our case, the cross-sections were therefore taken to be isotropic. The long bone shaft forms a long cylindrical cavity (a tube with a diameter of about 15 to 20 mm in the case of the fibula) with a forward bow. This shaft has thick walls of compact bone surrounding a hollow medullary cavity and it mediates the propagation of rather complex waves similar to elastic volume waves. Several types of wave packets can occur, each probably corresponding to a different pathway through the object (through the cortex alone or through the cortex and the medullary cavity) [START_REF] Cartney | Combined 2.25 MHz ultrasound velocity and bone mineral density measurements in the equine metacarpus and their in vivo applications[END_REF][START_REF] Cartney | Transverse path of ultrasound waves in thick-walled cylinders[END_REF] or even to a different propagating mode or type of wave (bulk compressional or shear wave, guided or surface waves) [START_REF] Anderson | The application of ray tracing towards a correction for refracting effects in computed tomography with diffracting sources[END_REF].

Studies conducted by our group have shown that, at low frequencies of [0.5 -1] MHz, adult femur tomography performed with a signal processing device gives an optimized resolution [START_REF] Lasaygues | Cancellous and cortical bone imaging by reflected tomography[END_REF]. In this case, the signal contributions used were assumed to correspond to the propagation of a compression wave (the fastest wave). The mean wave velocities were then included, ranging between 2000 and 4000 m/s (child and adult bones). In the novel approach presented here, similar assumptions are made, but the nominal frequency of the wave generated was 2.25 MHz.

On the one hand, the wavelengths occurring in bone (λ b ≈ 0.8 to 1.7 mm) are much larger than the macroscopic porosity and, in bone shafts, than the microstructural level (osteon size ≈ 100 µm in the case of children's fibulas). Cortical bones will therefore be assumed here to be only weakly heterogeneous (a homogenized equivalent medium) and the ultrasonic waves occurring in the cortical bone will therefore be less severely disturbed. This assumption is in agreement with the "linearized" propagating theory and with the asymptotic approximation.

On the other hand, the wavelength in water (λ 0 ≈ 0.6 mm) is smaller than the diameter of bones (about 15 mm). Ray theory can be applied here and straight lines running through the object can approximate the propagation paths [START_REF] Rossman | Comparison of speed of sound and ultrasound attenuation in the os calcis to bone density of the radius, femur, lumbar spine[END_REF][START_REF] Langton | Measurement of ultrasonic velocity broadband attenuation in cortical and cancellous bone[END_REF]).

In the first approximation, the perturbation of the ultrasonic wave propagation is taken to consist only of diffraction and refraction of the incident beam at the interface between the water and the cortical bone and the cortical bone and the inner cavity containing marrow.

The following experimental set-up including eight transducers was used. Transducer number 1 is the transmitter/receiver and the other transducers are receivers. The incident beam runs perpendicular to the tangential point of the scatterer (Fig. 1).

Fig. 1 approximately here

In the reflection and diffraction modes (at angles of less than ±90°, i.e., at transducers nos. 2, 3 and 7, 8), the first-arriving signal received was assumed to be the echo from the water/bone interface and the second-arriving signal received was assumed to be the compressional wave propagating due to the echo from the bone/marrow interface. All the other phenomena (shear waves, guided waves etc.) are insignificant or will occur too late to affect the compressional wave processes (fast waves).

In the transmission mode (at angles greater than ±90°, i.e., at transducers nos. 4, 5 and 6), the signals are abolished and are therefore not taken into account in the signal processing. [START_REF] Bossy | Effect of bone cortical thickness on velocity measurements using ultrasonic axial transmission: A 2D simulation study[END_REF] and [START_REF] Nicholson | Guided ultrasonic waves in long bone: Modelling, experiment and in vivo application[END_REF] recently performed numerical simulations and in vivo experiments showing that, in the axial transmission mode, the speed of sound (SOS) of the ultrasound waves was clearly dependent on the cortical thickness. Briefly, the propagation mode is compressional only in thick cortical bone (thickness to wavelength ratio > 0.5) and becomes "guided" in thin cortical bone (ratio < 0.25). In our case, at 2.25 MHz, the typical ratio varied from 1 to 10 in a child fibula. The above comment is therefore valid in cross-sectional configuration and, in a second approximation, only the compressional mode will be investigated.

Under these circumstances, the perturbation operator is the same as for a homogeneous background (in the particular case of a homogeneous half-space) and all the perturbed parameters will be defined with respect to the characteristics of the medium. The previous tomographic reconstruction algorithm can be used to image shapes and boundaries, within the conditions of the Born approximation.

MATERIAL AND METHOD

Experimental set-up and procedures

Since ultrasonic scattering data have to be collected experimentally over a broad range around the specimen in order to cover the full spatial domain, a specially-adapted experimental set-up was developed.

2D-ring antenna. An ultrasonic circular antenna consistent with fast ultrasonic radiofrequency (RF-) signal acquisition (400 ms) was designed and developed by our group. The crown of the antenna has an internal radius of 150 mm and supports eight fixed transducers distributed over an angle of 360° (Δθ fixed = 45°). The object to be imaged is placed in the presumed geometrical center of the antenna. The transducers are Imasonic ® piezo composite transducers with a 2.25 MHz nominal frequency and cylindrical focusing (in the plane of the slice). The Lecoeur Electronique ® multiplexer excites one of the transducers and organizes the signals received by all the transducers. One acquisition (i.e., one sinogram) corresponds to 64 signals (eight emitters by eight receivers) (Table 1). The ultrasonic RF-signals transmitted and received, which were composed of 4256 or 8192 samples in the present case, are digitized (8 bit, 40 MHz). The presumed geometrical centre of the antenna is determined by measuring the time-of-flight between two coaxially-aligned transducers, without the object.

Table 1 approximately here Fig. 2 approximately here

By means of step-by-step motors and an electronic device controlling the mechanical movements, the crown can be made to turn 45° in stepwise increments of (1/100)° (we denote the angular increment Δθ variable ) and to move vertically to perform many slice-by-slice measurements (we note the vertical increment Δl). In this way, an antenna comprising a larger number of transmitters and receivers was simulated. The signals recorded are then combined according to the angles of diffraction to form the complete sinogram. Transmission signals, corresponding to the direct wave field, were eliminated.

The complete duration of this ultrasonic imaging process (acquisition + signal processing + reconstruction) depends on the signal processing steps and on the capacity of the computer (CPU time). In the examples here, the time required was 1 min per image. The image is a square consisting of 255 x 255 pixels.

Calibration and off-set measurements. Let us consider an object to be imaged. The backscattering measured impulse response of this object at angle φ can be written r(φ, t) = h(φ, t) * g(φ, t),

(1)

where h(φ, t) is the impulse response (the mechanical offset) of the whole apparatus comprising the mechanics (the experimental bench) and electroacoustics (the transducers, which are assumed to be activated by a perfect electronic pulse δ(t)), and g(φ, t) is the impulse response of the object.

We are interested in the last quantity, g(φ, t), and the problem is how to determine it, i.e., how to correct the apparatus deviation, i.e., to "deconvolve" the RF signals r(φ, t) by the apparatus impulse response h(φ, t)

g(φ, t) = r(φ, t) * * h(φ, t), ( 2
)
where * * designates the "deconvolution " operation.

The problem is now that of evaluating the impulse response of the apparatus h. This is done by imaging a very thin copper wire (0.07 mm), the response of which is taken to be an ideal impulse (Fig. 3 and Fig. 4).

g(φ, t) = δ(t) ⇒ r (φ, t) = h(φ, t).

(3)

The circular data recordings on a thin cylinder allow estimating, for each transducer, the mechanical variations of the position and the water depth to the center of the bench.

Fig. 3 approximately here

Fig. 4 approximately here

For each projection, the "deconvolution " introduced by the mechanical offset was a gap in the signal in comparison with the reference signal obtained at the same angle on the wire sinogram.

Wall thickness assessment

Ultrasonic diffraction tomography, with some signal processing refinements, is ideally suited to reconstructing the external boundaries of high-contrast cylindrical bodies. Figure 5.a.

shows an example of an ultrasonic reflection tomographic image of a cylindrical plexiglas tube with an external diameter of 16 mm and an internal diameter of 5 mm, obtained under the experimental conditions described above. Relatively good agreement was observed between the external diameter detected by ultrasound tomography and that measured using calipers.

However, there was a discrepancy as to the position of the boundary of the inner cavity, because the reconstruction process assigns a constant speed of sound (i.e., wave velocity in water ≈ 1500 m/s) to every pixel in the cross-section, which biases the estimates of the inner cavity radius and the thickness. An image processing algorithm was therefore developed to detect the inner cavity radius on the image and to correct the position errors due to the reference speed of sound and the apparent distances.

where c 0 is the speed of sound in water.

RESULTS

The present ultrasonic diffraction tomographic method was applied to one geometricalmimicking phantom and to two real children's fibulas (12 y old). The sample thicknesses varied as a function of the radial position and were measured using calipers at eight locations around the scatterer. Density and velocity were measured using the ultrasonic procedures described in [START_REF] Pithioux | An ultrasonic methods to describe mechanical properties of compact bone[END_REF]. The experimental procedure fulfilled the criteria mentioned in the previous section.

The embedded ultrasonic tomographic software program automatically determined the inner boundaries. On the comparative graphs, each point gives the mean cortical thicknesses and the error bars give ± 1 standard deviation.

Geometrical-mimicking phantom

The phantom was a non circular homogeneous isotropic tube made of artificial resin (NEUKADUR ProtoCast 113 TM ). Its internal and external diameters measured 17 and 19 mm, respectively, and the inner cavity diameter measured 11 mm (Fig. 6).

Fig. 6 approximately here

The density of the resin was 1100 < ρ 1 < 1200 kg/m 3 and the mean velocity of the compressional wave tested was c 1 ≈ 2750 m/s.

One scanning slice was carried out 2 mm from the base. We acquired 32 sinograms of 64 signals, with an angular increment Δθ variable = 1.4°. Results are presented in Fig. 7 and Fig. 8.

Fig. 7 approximately here

Fig. 8 approximately here Table 2 approximately here

From the ultrasonic tomogram, our algorithm of wall-thickness assessment can automatically and precisely (Table 2) size the outer boundary of the tube with an accuracy of ±0.4 mm (see Z ext in Fig. 7). The resolution of the tomogram was sufficiently excellent so that the detection of the highest color intensity is practicable by each vertical line. The accuracy of the inner cavity diameter was also satisfactory, although the mean error was larger. Only the data obtained at location 7 were inaccurate. In Fig. 7, Z app is the apparent boundary obtained by using the "sliding window" algorithm. Z int is the actual position obtained by using eq n 4.

The algorithm of wall-thickness assessment used to determine the internal boundary was sensitive to the off-centering of the scatterer in the tank. Further studies are now required to assess the robustness of the algorithm versus the identification procedure. The importance of the location at which the caliper measurements are performed also needs to be analyzed, because here, the caliper measurements and the ultrasonic measurements differed by 2 mm.

But to conclude, the cortical thickness of artificial bone was on the whole accurately assessed using the present ultrasonic tomographic method.

Child fibulas

The two child bone samples were fresh fibulas we named "sample A" and "sample B" (12 y old). Dimensions are given in Fig. 9. The inner cavities contained natural marrow.

Fig. 9 approximately here

The a priori mean velocity was the compression wave velocity, radially measured through a detail cut since the fibula "sample A", and was 2990±5 m/s. The density of the cortical material was 1700 kg/m 3 . Only one scanning slice was performed, 1.5 cm from the base. We acquired 32 sinograms consisting of 64 signals, with an angular increment Δθ variable = 1.4°.

Results are presented in Fig. 10 to Fig. 13. With both fibula samples, the geometry of the outer and inner boundaries was in keeping with the usual proportions, and the cross-sectional thicknesses were in good agreement (Table 3 and Table 4). The differences between the mechanical and acoustical measurements were small (< 0.5 mm), except in some locations: n°6 in the case of "sample A" and n°7 in that of "sample B" (> 0.5 mm). There are two possible explanations for the latter differences.

First, the internal boundary assessment algorithm used was sensitive to the image resolution.

Since ultrasound tomography is basically a Fourier synthesis problem, the resolution depends on the angular spatial sampling rate and, thus, the number of available projections, which reduced the coverage of the frequency space of the scatterer. Furthermore, the bandwidth of the pulses is limited to almost an octave. To improve the resolution, one can increase the number of transducers placed on the 2D-ring antenna to obtain more than 64 signals and the low and high spectral frequencies can be restored using deconvolution algorithms. For example, one can chose a procedure of [START_REF] Papoulis | A new algorithm in spectral analysis and band-limited extrapolation[END_REF] that consists of an iterative extension of the bandwidth of the electroacoustic set-up in Fourier space, allowing to reach resolutions of λ/30, where λ is the wavelength [START_REF] Lasaygues | Cancellous and cortical bone imaging by reflected tomography[END_REF].

Secondly, the caliper measurements on the internal cavity were performed at eight locations at the base of the bone. However, it is very likely that, at the bone level at which the ultrasonic data recordings were carried out, these dimensions were not exactly the same.

CONCLUSION

Within the limits of the Born approximation, we have described a method of performing ultrasonic diffraction imaging of the cortical diaphysis of long bones and obtaining slice-byslice tomographic data in the space-frequency plane. The experiments carried out on real bone using mechanical and electronic steering systems show the results obtained with this method to be accurate. These preliminary results suggest that ultrasonic tomography is one of a means of determining geometrical characteristics such as the cortical thickness of bone, which is known to be an indicator of the state of the fragility of the bone. An important error on the measures would limit the extension of the method to the clinical cases. Furthermore, it would be useful also to discuss transition boundaries between endosteal cortical bone and marrow, more or less heterogeneous and clean, which is an additional challenge for the ultrasonic tomography.

It is thus necessary today to improve the quality of the tomogram to increase the precision of the thickness assessment. Various ways of improving this method will be investigated before increasing the number of antenna components. To increase the number of independent projections, a few rotations of the system could be performed and the transducers could be more regularly distributed. Other algorithms, such as algebraic algorithms, which are more suitable when there are only a few projections, could be also used. The latter algorithms are also known to be suitable for the non linear inversion procedures, which have to be used to deal with the main quantitative imaging problems arising when the contrast is more than a few percent.

Table 1. Acquisition sequences.

Table 2. Comparison between calipers and ultrasound-derived dimensions (geometricalmimicking phantom).

Table 3. Comparison between calipers and ultrasound-derived dimensions ("sample A").

Table 4. Comparison between calipers and ultrasound-derived dimensions ("sample B").

Fig. 5

 5 Fig. 5 approximately here

Fig

  Fig. 10 approximately here

Fig. 14 .

 14 Fig. 14. Theoretical diffraction tomography with eight transducers. Transducers nos. 1 to 8 are transmitters (and receivers) and the other transducers are receivers.

Fig. 17

 17 Fig. 17. 2.25 MHz-ultrasonic diffracted tomography of a thin copper wire (0.07 mm).

Fig. 18 .

 18 Fig. 18. Assessment of the thickness of a cylindrical plexiglas tube. (a) Reflection Tomography. (b) Polar map, Z ext is the external boundary, Z app and Z int are the "apparent" and actual boundaries of the inner cavity. (c) Polar representation.

Fig. 19 .

 19 Fig. 19. NEUKADUR ProtoCast 113 TM resin geometrical-mimicking phantom.

Fig. 21 .

 21 Fig. 21. Cortical thickness depending on the measurement site, using mechanical calipers and a 2.25 MHz-ultrasonic tomogram (on a geometrical-mimicking phantom). (a) Initial tomogram. (b) Polar map, Z ext is the external boundary, Z app and Z int are the "apparent" and the actual boundaries of the inner cavity. (c) Mean-thickness.

Fig. 22 .

 22 Fig. 22. Geometrical dimensions of child fibulas.

Fig. 23

 23 Fig. 23. 2.25 MHz-ultrasonic diffracted tomography of a child fibula ("sample A"). (a) Initial tomogram. (b) Mean-thickness.

Fig. 24 .

 24 Fig. 24. Cortical thickness at various measurement sites, determined using mechanical calipers and 2.25 MHz-ultrasonic tomogram ("sample A").

Fig. 25

 25 Fig. 25. 2.25 MHz-ultrasonic diffracted tomography of a child fibula ("sample B"). (a) Initial tomogram. (b) Mean-thickness.

Fig. 1

 1 Fig. 1

  Fig. 2

  Fig. 4

  Fig. 6

  Fig. 8

Fig. 11

 11 Fig. 11

Fig. 13

 13 Fig. 13

Table 3

 3

	approximately here

ACKNOWLEGMENTS

The author is grateful for medical assistance from Dr Ph. Petit and Dr J.-L. Jouve from Public Assistance Hospitals of Marseille, "Timone" Children's Hospital. This work benefited from the very fruitful help of J.-P. Lefebvre and S. Mensah. Many thanks to E. Debieu, A. Cosquer, S. Devic, J.-M. Corneloup, A. Kerjoant and V. Cardinalli for their technical assistance.


Eric Verret

Sophie Duquesne

Anthony Collin

Multi-objective Bayesian optimization for intumescent coatings

Experimental protocol

Bayesian optimization (BO) is a sequential model-based approach that efficiently explores and exploits the search space to find the optimal solution of a black-box function with limited evaluations. BO can be used for noisy and small dataset.

1. The coating formulations were prepared using a high-speed disperser 2. A constant dry mass is applied on white fir 3. The sample is conditioned at 30°C, 40% RH before the fire testing Gaussian process fit Based on the trade-off plot, it's challenging to achieve a formulation that simultaneously offers a low degradation distance and THR.

Based on the trade-off plot, it's relatively easy to achieve formulation with high time to ignition and low distance of degradation.

••

  Prof Sophie DUQUESNE, Prof Anthony COLLIN Univ. Lille, CNRS, INRAE, Centrale Lille, UMR 8207 -UMET -Unité Matériaux et Transformations, F-59000 Lille, France/Université de Lorraine Pipeline Wood is more and more used in buildings = sustainable materials compared to concrete and steel Intumescent is an efficient method to protect wood against fire in a building • Designs of experiment are efficient but require high amount of data • Multi-objective Bayesian Optimisation has been chosen to find the optimum configuration of the paint (pareto front) and minimize the number of samples • The performance of the coating is evaluate by 3 fire tests giving 4 parameters Outlooks • Number evaluation until a satisfactory approximation of the Pareto front is obtained • Comparison between different sampling for initial points Tonooka, Y., Takaguchi, H., Yasui, K., & Maeda, T. (2014). Life cycle assessment of a domestic natural materials wood house. Energy Procedia, 61, 1634-1637. Puri, R. G., & Khanna, A. S. (2017). Intumescent coatings: A review on recent progress. Journal of Coatings Technology and Research, 14(1), 1-20. Weil, E. D. (2011). Fire-protective and flame-retardant coatings-A state-of-the-art review. Journal of fire sciences, 29(3), 259-296. Naik, D., & Shah, P. (2014). A review on image segmentation clustering algorithms. Int J Comput Sci Inform Technol, 5(3), 3289-93. Shields, B. J., Stevens, J., Li, J., Parasram, M., Damani, F., Alvarado, J. I. M., ... & Doyle, A. G. (2021). Bayesian reaction optimization as a tool for chemical synthesis. Nature, 590(7844), 89-96. Wang, Y., Chen, T. Y., & Vlachos, D. G. (2021). NEXTorch: A Design and Bayesian Optimization Toolkit for Chemical Sciences and Engineering. Journal of Chemical Information and Modeling, 61(11), 5312-5319. Heat flux: 50 KW/m² • Test time: 30 min Intumescent coating most commune system to protect wood against fire Aims/objectives Prediction The flame propagation is evaluated using a lab-made CFE radiant panel test (similar to the ISO5658-2) Design of Experiement (DoE): = 1 Input Intumescent coatings in case of fire => expanded structure limiting heat and mass transfer In Multi-objective optimisation there is typically no single best solution. The Pareto front is the set of solutions where no other solution can improve one objective without sacrificing another. It represents the optimal trade-off between conflicting objectives, providing a range of solutions, to guide decision-making.


Emine Shaka
email: shaka@uni-kassel.de

Natalie Fischer

Elisabeth Rathgeb-Schnierer

Patterns of interaction in support situations in mathematics classrooms

Keywords: Teacher-student interaction, adaptive support, subject-specific learning, primary school

Introduction

Teaching mathematics comes with various challenges considering that learners differ, inter alia, in their learning and performance abilities [START_REF] Demo | Learning mathematics in an inclusive and open environment: An interdisciplinary approach[END_REF][START_REF] Parsons | Teachers' instructional adaptations: A research synthesis[END_REF]. Concerning these individual differences, the teacher's task consists in supporting each student's learning to effectively promote their learning processes [START_REF] Parsons | Teachers' instructional adaptations: A research synthesis[END_REF]. Nonetheless, a number of studies indicate that adaptive support takes place in only a few instances [START_REF] Lotz | Kognitive Aktivierung im Leseunterricht der Grundschule. Eine Videostudie zur Gestaltung und Qualität von Leseübungen im ersten Schuljahr [Cognitive activation in reading lessons of primary school. A video study regarding design and quality of reading exercises in the first school year[END_REF]. For this reason, the project focuses on support situations in mathematics classrooms of primary school. These specific situations are characterised by primarily dyadic teacher-student interactions which take place while learners work on their tasks in a lesson and support provided by the teacher.

When teaching is adjusted to the learners, it is referred to as an active or proactive response to heterogeneity depending on the extent of adaptation and individual support given [START_REF] Weinert | Notwendige Methodenvielfalt. Unterschiedliche Lernfähigkeiten erfordern variable Unterrichtsmethoden [Necessary variety of methods. Different learning abilities require variable methods[END_REF]. Within an educational context, the term adaptivity may be delineated as the match between learning opportunities provided by the teacher and students' specific learning conditions [START_REF] Brühwiler | Adaptive teaching competency: Effects on quality of instruction and learning outcomes[END_REF][START_REF] Hardy | Adaptive teaching in research on learning and instruction[END_REF]. The learners' respective educational needs are met by applying adaptive strategies, and thus their learning outcomes are positively affected [START_REF] Brühwiler | Adaptive teaching competency: Effects on quality of instruction and learning outcomes[END_REF]. In order to provide adaptive support for every student, relevant information on their current learning level is required [START_REF] Hardy | Adaptive teaching in research on learning and instruction[END_REF]. Hence, diagnostic competence is particularly necessary on the part of the teacher (ibid.). For this purpose, the interaction between learner and teacher plays a significant role: through interaction, it is possible to detect the student's stage of learning and make adjustments so that adaptive situations develop [START_REF] Krammer | Individuelle Lernunterstützung in Schülerarbeitsphasen [Individual learning support in students' working phases[END_REF]. Consequently, individual learning support constitutes one form of adaptivity on the micro level of teaching (ibid.).

Aim and methods

In general, there is little knowledge on how students are supported in primary school [START_REF] Lotz | Kognitive Aktivierung im Leseunterricht der Grundschule. Eine Videostudie zur Gestaltung und Qualität von Leseübungen im ersten Schuljahr [Cognitive activation in reading lessons of primary school. A video study regarding design and quality of reading exercises in the first school year[END_REF]. However, there are empirical results suggesting that teacher support is rarely adaptive (ibid.). Furthermore, the helping behaviour of teachers during phases of students' individual work has been hardly examined [START_REF] Krammer | Individuelle Lernunterstützung in Schülerarbeitsphasen [Individual learning support in students' working phases[END_REF]. Therefore, adaptivity and its potential significance for subjectspecific learning in the form of mathematical depth in support situations constitute the object of research. It is of considerable interest to find out how primary school students are supported while they are working on tasks in mathematics lessons and how far mathematical content is thereby deepened.

Since support situations have been insufficiently explored, those specific teacher-student interactions are going to be analysed. For this, videotaped maths lessons in primary school, which are provided by the DFG Graduate School INTERFACH functioning as a project framework, will be used. Not only does working with this database enable gaining insights into given support in elementary school but also into the evolving mathematical subject, which is of great importance, as indicated above. As a first step, dyadic teacher-student interactions which take place while learners work on their tasks are going to be identified and described. By means of a designed coding system, the different types of support are going to be categorised. Characterising these situations in this way will be the subject of the poster while the main focus will lie on developing the coding system, which will also be depicted. Afterwards, the extracted sequences will be evaluated in regard to adaptive support. More precisely, it will be focused on whether the given support addresses the students' issues or concerns and if the children are encouraged as well as able to apply a deeper understanding for the learning content. To this end, these situations will be examined for professional contents, typical mathematical techniques as well as the teacher's orientation. These components form the basis for classifying mathematical depth in support situations.