Arianna Piccialli
email: arianna.piccialli@aeronomie.be

Julie A Rathbun

Ann Carine Vandaele

Francesca Altieri

Anni Määttänen

Anna Milillo

Alessandra Rotundi

Miriam Rengel

Pierre Drossart

Participation of women scientists in ESA solar system missions: an historical trend

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

A recent study by [START_REF] Rathbun | Participation of women in spacecraft science teams[END_REF] analyzed the participation of women in US planetary science missions. Their analysis show women scientists to be consistently under-represented in NASA's robotic planetary spacecraft missions. We plan to count science team members of 10 ESA solar system missions over a period of 38 years and to determine the percentage of women on each team.

Method

We will follow the same methodology described in [START_REF] Rathbun | Historical trends of participation of women in robotic spacecraft missions[END_REF][START_REF] Rathbun | Historical Trends of Participation of Women Scientists in Robotic Spacecraft Mission Science Teams: Effect of Participating Scientist Programs[END_REF], for consistency with their study. We therefore will consider only the original team scientists from European institutions: engineers, members of project management, students and postdocs will not be included. For each team, we search team web pages, published articles and when possible, we directly contact the Principal Investigators. In order to determine gender we rely on personal knowledge, first name or photographs.

One difficulty we are encountering is to find the original team members, as often new members are added over time. An additional difficulty is to determine the percentage of women in the field during the missions' selection year, as this information is not easy to obtain for different European countries.

Preliminary results

As preliminary analysis we counted the number of Principal Investigators of 10 ESA solar system missions and determined the percentage of PI women (See Table 1). We excluded the Cassini mission, since it is particularly difficult to determine the original team members, but we plan to add it in future. We will extend this study to the entire team members, including therefore also Co-Investigators of each mission.

Table 1 :

 1 Percentage of PI women involved in ESA

		solar system missions
	Launch year	Mission name	#PI women	Total #PI	% women
	1985	Giotto	1	9	11.1
	2003	SMART-1	0	8	0
	2000	Cluster	0	9	0
	2003	Double Star	2	9	22.2
	2003	Mars Express	0	10	0
	2004	Rosetta orbiter	1	13	7.69
	2005	Venus Express	0	8	0
	2016	ExoMars/TGO -Schiaparelli	3	10	30
	2018	BepiColombo	1	23	4.35
	2022	JUICE	1	10	10


M C Belhadjoudja

M Maghenem

E Witrant

On the Intelligent Proportional Controller Applied to Linear Systems

We analyze in this paper the effect of the wellknown intelligent proportional controller on the stability of linear control systems. Inspired by the literature on neutral timedelay systems and advanced-type systems, we derive sufficient conditions on the order of the control system, under which, the used controller fails to achieve exponential stability. Furthermore, we obtain conditions, relating the system's and the control parameters, such that the closed-loop system is either unstable or not exponentially stable. After that, we provide cases where the used controller achieves exponential stability. The obtained results are illustrated on an experimental benchmark that consists of an electronic throttle valve.

• When the latter coefficient is equal to 1, the closed-loop system fails to be exponentially stable.

I. INTRODUCTION

Model-free control (MFC) aims to regulate control systems with unknown dynamical equations. The MFC that we consider here has been introduced in [1], [2]; see also [START_REF] Thabet | Ultra-local model control based on an adaptive observer[END_REF], [START_REF] Safaei | Adaptive model-free control based on an ultra-local model with model-free parameter estimations for a generic SISO system[END_REF] for more recent formulations. Generally speaking, this approach consists of relating the input and the output by an equation, known as the ultra-local form, involving the output (and its time derivatives), the input, and an unknown function lumping whatever is unknown in the system [1]. As a result, the control input is composed of two parts: a first part designed to compensate for the unknown function, and a second part that consists of a classical linear controller, usually, a PID controller. The resulting controller is known as the intelligent PID controller. This class of controllers has been tested both numerically and experimentally on different classes of systems, such as automotive engines [START_REF] Choi | Model-free control of automotive engine and brake for Stop-and-Go scenarios[END_REF], automated vehicles [START_REF] Wang | Ultra-local model predictive control: A modelfree approach and its application on automated vehicle trajectory tracking[END_REF] and fault accommodation in greenhouses [START_REF] Lafont | A model-free control strategy for an experimental greenhouse with an application to fault accommodation[END_REF]. This being said other types of MFC techniques are available in the literature; see [START_REF] Han | From PID to active disturbance rejection control[END_REF].

Due to its easy implementation, as opposed to more advanced control strategies, MFC using intelligent PIDs is increasingly applied. However, despite this growing popularity, the rigorous analysis of these controllers is still at its early stage, to the best of our knowledge. Indeed, the stability guarantees for the resulting closed-loop system remain, mostly, unexplored. Some results along this direction have been obtained, for example, in [START_REF] Novel | A mathematical explanation via "intelligent" PID controllers of the strange ubiquity of PIDs[END_REF], where links between the sampled intelligent PID controller and the sampled classical PID controller in velocity form are established. In [START_REF] Li | A Revisit to Model-Free Control[END_REF], the robustness of intelligent PIDs is studied via sensitivity analysis. In [START_REF] Delaleau | A proof of stability of model-free control[END_REF], the discretized closed loop using MFC is shown to coincide with the Euler forward approximation of a certain class of systems. The stability The authors are with Université Grenoble Alpes, CNRS, Grenoble-INP, GIPSA-lab, F-38000, Grenoble, France (e-mail: mohamed.belhadjoudja@gipsa-lab.fr).

The project has been partially supported by CNRS, appel unique, 2022.

of the latter class of systems is then analyzed. However, these conclusions do not necessarily extend to the original closed-loop system under MFC. On the other hand, in some works, MFC and its intelligent linear controllers have been reinforced via different control techniques. For example, in [START_REF] Liu | Model-Free Predictive Current Control for Three-Phase Power Converters With LCL Filter[END_REF], MFC is combined with model predictive control. In [START_REF] Precup | Model-Free sliding mode control of nonlinear systems: Algorithms and experiments[END_REF], a controller combining MFC and sliding mode control is proposed. Despite their proven efficiency, these techniques are more complex to implement, as opposed to the intelligent PID controller.

In this paper, we prove the efficiency and show the limitations of the intelligent proportional controller (iP) when applied to linear control systems. That is, we prove that applying an intelligent proportional controller to a linear control system reduces to applying a PD controller to a neutral delay system. Hence, the intelligent proportional controller inherits some of the limitations of the classical PD controller. More precisely, we derive sufficient conditions on the order of the system, under which, the origin fails to be exponentially stable. Furthermore, we derive sufficient conditions, on the system's parameters and the control gains, under which, the origin is either unstable or fails to be exponentially stable. Then, based on existing results on neutral delay systems, we derive sufficient conditions for exponential stability, which illustrates situations where the iP controller guarantees better results than just asymptotic stability. We illustrate our theoretical results via an experimental benchmark, that consists of the angle-tracking problem for an electronic throttle valve.

The paper is organized as follows. The problem statement is in Section II. Then, some preliminary results are presented in Section III. Our main results are in Section IV. Finally, the experimental results are in Section V.

Due to space limitations, the proof of Theorem 2 and numerical examples are omitted and can be found in [START_REF] Belhadjoudja | On the Intelligent Proportional Controller Applied to Linear Systems[END_REF].

Notation. We denote by R n the set of n-uples of real numbers, by R >0 the set of positive real numbers and by R ≥0 the set of nonnegative real numbers. We let N := {0, 1, 2, ...}, Z := {0, ±1, ±2, . . .}, and C be the set of complex numbers. Given τ ∈ R >0 and a time-varying function y, we write y τ (t) := y(t -τ ). Given a ∈ N, we denote the a th derivative of y by y (a) , the first derivative by ẏ, and the second derivative by ÿ. Given a matrix A ∈ R n×n , we denote by ||A|| its 2-norm, by s(A) its spectral abscissa, by ρ(A) its spectral radius and by µ(A) its logarithmic norm with respect to ||.||. We denote by I n the identity matrix of dimension n, and by 0 n the zero matrix of dimension n.

II. PROBLEM STATEMENT

We consider linear control systems of the form

α 1 y (a) + . . . + α a+1 y = β 1 u (b) + . . . + β b+1 u, (1)
where

y ∈ R is the measured output, u ∈ R is the control input, a, b ∈ N with a > b, α i , β j ∈ R for (i, j) ∈ {1, 2, . . . a + 1} × {1, 2, . . , b + 1}, and α 1 β 1 ̸ = 0.
We recall in this section the structure of the intelligent proportional controller applied to system (1). For a detailed presentation of MFC approaches with various examples, we refer the reader to [1] and [2]. Consider the control system in (1) and let us define, along the trajectories of the system, the time-varying function:

F (t) := ẏ(t) -αu(t).
(

) 2
where α ̸ = 0 is a design parameter. The trajectories of (1) can, therefore, be described by the ultra-local form [1]:

ẏ(t) = αu(t) + F (t). (3)
The intelligent proportional control law is given by

u(t) := 1 α -Ky(t) -F (t) , (4)
where F is an estimate of F , and K ∈ R is a control gain.

The key idea behind the intelligent proportional control law ( 4) is that if F ≡ F , then the closed-loop system is governed by the equation ẏ = -Ky, which leads to exponential stability if and only if K > 0. Various estimates of F are proposed in the literature; see for e.g. [1], [2], [START_REF] Thabet | Ultra-local model control based on an adaptive observer[END_REF], [START_REF] Safaei | Adaptive model-free control based on an ultra-local model with model-free parameter estimations for a generic SISO system[END_REF]. The one we consider here is given by

F (t) = ẏ(t) -αu τ (t), u τ (t) := u(t -τ ), (5)
where τ ∈ R >0 is a time delay. Remark 1: When ẏ is not available for measurements, various methods to approximate it can be found in [START_REF] Fliess | Closed-loop parametric identification for continuous-time linear systems via new algebraic techniques[END_REF], [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF], [START_REF] Liu | Error analysis of Jacobi derivative estimators for noisy signals[END_REF], [2]. Although we assume here that ẏ is perfectly known, the analysis using its approximations is an interesting perspective to this work.

The structure of such a controller is illustrated in the block diagram of Figure 1.

Clearly, when applying ( 4)-( 5) to (1), the resulting closedloop system is not governed by ẏ = -Ky. It will most likely involve a time delay (except for specific cases; see Remark 3). The effect of this delay cannot be ignored in general, even if it is sufficiently small; see [START_REF] Hale | Effects of small delays on stability and control[END_REF], [START_REF] Hale | Strong stabilization of neutral functional differential equations[END_REF]. Furthermore, when the closed-loop system has a delay term, we will show that it may be a neutral delay system or an advanced-type system. As a consequence, we will be able to derive stability and instability guarantees based on some well-known results concerning the latter two classes of systems.

Before presenting our main results, we will recall some preliminaries in the next section.

III. PRELIMINARIES

A. Neutral Delay Systems

Consider the neutral delay system of the form

α 1 y (a) + . . . + α a+1 y = β 1 y (a) τ + . . . + β a+1 y τ , (6)
for some a ∈ N and α i , β i ∈ R for all i ∈ {1, 2, . . . a + 1} with α 1 β 1 ̸ = 0. Results on well-posedness of system (6), i.e. existence and uniqueness of solutions, can be found in [20, Chapter 1], [START_REF] Hale | Introduction to functional differential equations[END_REF]Chapter 9].

System (6) admits the characteristic equation:

0 = α 1 z a + α 2 z a-1 + . . . + α a+1 -e -τ z β 1 z a + β 2 z a-1 + . . . + β a+1 . (7)
Lemma 1 (Neutral root chains [22, Item 2 Theorem 1]): If β 1 α 1 ̸ = 0, then equation [START_REF] Lafont | A model-free control strategy for an experimental greenhouse with an application to fault accommodation[END_REF] has an infinite number of roots given by the sequence:

z k = 1 τ log β 1 α 1 + i arg β 1 α 1 + 2kπ + g k ∀k ∈ Z, (8)
where g k = o (1); namely, for any c > 0, there exists

k c > 0 such that |g k | ≤ c for all k ∈ Z such that |k| ≥ k c .
Moreover, besides the sequence {z k }, equation ( 7) has a finite number of other roots. □

B. Advanced-Type Systems

Systems of advanced type are of the form

α 1 y (a) + . . . + α a+1 y = β 1 y (b) τ + . . . + β b+1 y τ , (9) for some a, b ∈ N, with b > a, α i ∈ R for all i ∈ {1, 2, . . . a + 1}, and β i ∈ R for all i ∈ {1, 2, . . . b + 1}, with α 1 β 1 ̸ = 0.
System (9) admits a characteristic equation of the form

0 = α 1 z a + α 2 z a-1 + . . . + α a+1 -e -τ z β 1 z b + β 2 z b-1 + . . . + β b+1 . ( 10
)
Lemma 2 (Unstable root chains [22, Item 3 Theorem 1]): If b > a and α 1 β 1 ̸ = 0, then the number of roots of [START_REF] Li | A Revisit to Model-Free Control[END_REF] with negative real part is finite. Additionally, [START_REF] Li | A Revisit to Model-Free Control[END_REF] has an infinite number of roots with arbitrarily large non-negative real parts. □

C. Neutral Delay Control Systems Subject to PD Control

We introduce the particular class of neutral delay control systems of the form

α 1 y (a) -y (a) τ + . . . + α a+1 (y -y τ ) = v, (11)
where v ∈ R is the control input, a ∈ N, α i ∈ R for i ∈ {1, 2, . . . a + 1}, and α 1 ̸ = 0.

System ( 11) is said to be (asymptotically, exponentially) stabilizable by PD control of order + 1) if there exist gains K d1 , K d2 , . . . , K d(b+2) ∈ R, with K d1 ̸ = 0, such that the control law

v := K d1 y (b+1) + K d2 y (b) + . . . + K d(b+2) y (12)
renders the origin of the resulting closed-loop system (asymptotically, exponentially) stable. We recall here that the closed-loop system is exponentially stable if there exist constants κ, σ > 0 such that for any initialization y([-τ, 0]), we have

|y(t)| ≤ sup s∈[-τ,0] |y(s)| κe -σt t ≥ 0. (13)
The following two facts are in order.

• When a > b + 1 or α 1 ̸ = K d1 , system [START_REF] Delaleau | A proof of stability of model-free control[END_REF] in closed loop with the PD controller ( 12) is of neutral type (6).

• When system [START_REF] Delaleau | A proof of stability of model-free control[END_REF] in closed loop with [START_REF] Liu | Model-Free Predictive Current Control for Three-Phase Power Converters With LCL Filter[END_REF] is such that

a = b + 1, α 1 = K d1 , ∃i ∈ {2, 3, ..., a + 1} : α i ̸ = K di , (14)
the closed-loop system is of advanced type [START_REF] Novel | A mathematical explanation via "intelligent" PID controllers of the strange ubiquity of PIDs[END_REF].

IV. MAIN RESULTS

We start showing that the closed loop of (1) using ( 4)-( 5) has the form of [START_REF] Delaleau | A proof of stability of model-free control[END_REF] subject to a PD control law in the form of [START_REF] Liu | Model-Free Predictive Current Control for Three-Phase Power Converters With LCL Filter[END_REF]. As a result, depending on the degree of the control system (1), its parameters, and the parameters (α, K) of the intelligent proportional controller, the closed-loop system is either a neutral delay system of the form [START_REF] Wang | Ultra-local model predictive control: A modelfree approach and its application on automated vehicle trajectory tracking[END_REF] or an advancedtype system of the form [START_REF] Novel | A mathematical explanation via "intelligent" PID controllers of the strange ubiquity of PIDs[END_REF]. In particular, we use Lemmas 1 and 2 to derive conditions, under which, the closed-loop system either fails to be exponentially stable or is unstable.

Remark 2: It is well known that if |β 1 /α 1 | < 1 in (6), then asymptotic stability is equivalent to exponential stability [START_REF] Kharitonov | Exponential estimates for neutral time-delay systems: An LMI approach[END_REF]. However, this is not true in general. For example, the origin of ẏ-ẏτ = -y is not exponentially stable but it is still asymptotically stable. Therefore, the fact that the intelligent proportional controller fails to guarantee stability in some scenarios does not mean that it cannot achieve asymptotic stability.

A. Equivalence between the iP Control of (1) and the PD Control of [START_REF] Delaleau | A proof of stability of model-free control[END_REF] We start introducing the following lemma.

Lemma 3: System (1) in closed-loop with the intelligent proportional controller in (4)-( 5) can be expressed as a neutral delay system [START_REF] Delaleau | A proof of stability of model-free control[END_REF], of the same order a, subject to a PD controller in [START_REF] Liu | Model-Free Predictive Current Control for Three-Phase Power Converters With LCL Filter[END_REF] of order (b + 1), whose gains K di , i ∈ {1, 2, . . . , b + 2}, satisfy

K d1 = - 1 α β 1 , K d(b+2) = - 1 α β b+1 K, and
K dj = - 1 α (β j-1 K + β j ) ∀j ∈ {2, . . . , b + 1}. (15)
□ Proof: Using (4), we obtain

β 1 u (b) + . . . + β b+1 u = β 1 u (b) τ + . . . + β b+1 u τ - 1 α β 1 Ky (b) + y (b+1) - 1 α β 2 Ky (b-1) + y (b) -. . . - 1 α β b+1 (Ky + ẏ) , (16)
which can be rewritten as

β 1 u (b) + . . . + β b+1 u = α 1 y (a) τ + . . . + α a+1 y τ - 1 α β 1 y (b+1) - 1 α β b+1 Ky - 1 α (β 1 K + β 2 ) y (b) -. . . - 1 α (β b K + β b+1 ) ẏ. ( 17
)
The closed-loop system is therefore given by

α 1 y (a) -y (a) τ + . . . + α a+1 (y -y τ ) = - 1 α β 1 y (b+1) - 1 α β b+1 Ky - 1 α (β 1 K + β 2 ) y (b) -. . . - 1 α (β b K + β b+1 ) ẏ, (18)
which is the structure of (11) subject to a PD controller [START_REF] Liu | Model-Free Predictive Current Control for Three-Phase Power Converters With LCL Filter[END_REF] of order (b + 1). Remark 3: From Lemma 3, we conclude that the strictly causal control system in (1) subject to the iP control in (4)-( 5) is under one of the following three forms:

• Advanced-type form (9): if there exist {K di } i∈{1,...,b+2} , such that ( 14)-( 15) hold.

• Undelayed form: if there exist {K di } i∈{1,...,b+2} , with a = b + 1, (15) holds, and α i = K di for all i ∈ {1, 2, ..., a + 1}.

• Neutral delayed form (6): when none of the conditions in the previous two items hold.

B. Instability and Lack of Exponential Stability

According to Lemma 3, the iP controller applied to (1) inherits the following limitations of the PD controller in [START_REF] Liu | Model-Free Predictive Current Control for Three-Phase Power Converters With LCL Filter[END_REF] applied to [START_REF] Delaleau | A proof of stability of model-free control[END_REF].

• The PD controller in [START_REF] Liu | Model-Free Predictive Current Control for Three-Phase Power Converters With LCL Filter[END_REF] cannot guarantee exponential stability if a > b + 1. • When a = b + 1, the resulting closed-loop system can be of advanced type, and thus unstable. • Even when the closed-loop system is not of advanced type, when the coefficient of y (a) τ is, in norm, larger than one, then the closed-loop system is unstable.

The aforementioned facts are applied to assess the instability of (1) in closed loop with (4)- [START_REF] Choi | Model-free control of automotive engine and brake for Stop-and-Go scenarios[END_REF].

Theorem 1: Consider system (1), with a > b, in closed loop with the intelligent proportional controller in (4)- [START_REF] Choi | Model-free control of automotive engine and brake for Stop-and-Go scenarios[END_REF]. Then, for any delay τ > 0, the following properties hold:

• If a > (b + 1), then, for any α ∈ R * and K ∈ R, the origin of the closed-loop system is not exponentially

• If a = (b + 1) and α = -β 1 /α 1 , then for any K ∈ R such that α a+1 α ̸ = -β b+1 K or α i α ̸ = -(β i-1 K + β i )
for some i ∈ {2, 3, ..., b + 1}, the origin of the closedloop system is unstable.

• If a = (b + 1) and α ̸ = -β 1 /α 1 , then, for any K ∈ R, the origin of the closed-loop system is unstable if

α 1 α 1 + β 1 /α > 1, (19)
and not exponentially stable if

α 1 α 1 + β 1 /α = 1. (20)
□ Proof: From Lemma 3, we know that the closed-loop of (1) using ( 4)-( 5) is governed by the equation

α 1 y (a) + α 2 y (a-1) + . . . + α a-b-1 y (b+2) + α a-b + 1 α β 1 y (b+1) + α a-b+1 + (β 1 K + β 2 ) α y (b) + . . . + α a + (β b K + β b+1 ) α ẏ + α a+1 + β b+1 K α y -α 1 y (a) τ -. . . -α a+1 y τ = 0. (21)
The characteristic equation of this system is

z a + α 2 α 1 z a-1 + . . . + α a-b-1 α 1 z b+2 + (α a-b + β 1 /α) α 1 z b+1 + 1 α 1 α a-b+1 + (β 1 K + β 2 ) α z b + . . . + 1 α 1 α a + (β b K + β b+1 ) α z + (α a+1 + K (β b+1 /α)) α 1 -e -τ z z a + α 2 α 1 z a-1 + . . . + α a+1 α 1 = 0. ( 22
)
If a > (b + 1), using Lemma 1, we conclude that the system admits infinitely many characteristic roots of the form

z k = i 2kπ τ + o(1) k = 0, ±1, ±2, . . . (23)
In other words, there exists a sequence of characteristic roots that tends towards the imaginary axis. The origin is therefore not exponentially stable.

We consider now the case where a = (b + 1). Hence, the characteristic equation of the system is given by

(α 1 + β 1 /α) α 1 z a + 1 α 1 α 2 + (β 1 K + β 2 ) α z a-1 + . . . + 1 α 1 α a + (β b K + β b+1 ) α z + (α a+1 + K (β b+1 /α)) α 1 -e -τ z z a + α 2 α 1 z a-1 + . . . + α a+1 α 1 = 0. ( 24
)
If we have α = -β 1 /α 1 , then the previous equation becomes

1 α 1 α 2 + (β 1 K + β 2 ) α z a-1 + . . . + 1 α 1 α a + (β b K + β b+1 ) α z + 1 α 1 α a+1 + K β b+1 α -e -τ z z a + α 2 α 1 z a-1 + . . . + α a+1 α 1 = 0. (25)
If, in addition, 25) is the characteristic equation of an advancedtype system. As a result, using Lemma 2, we conclude that it possesses infinitely many roots with arbitrarily large real parts, which leads to instability of the closed-loop system.

α a+1 α ̸ = -β b+1 K or there exists i ∈ {2, 3, ..., b + 1} such that α i α ̸ = -(β i-1 K + β i ), then equation (
If α ̸ = -β 1 /α 1 , the characteristic equation of the closedloop system is given by [START_REF] Chen | Delay-dependent stability criterion for neutral time-delay systems[END_REF]. Hence, if we let

θ := arg α 1 α 1 + β 1 /α (26)
and apply Lemma 1, we conclude that the system admits infinitely many characteristic roots of the form

z k = 1 τ log α 1 α 1 + β 1 /α + i (θ + 2kπ) + o(1), (27)
for k = 0, ±1, ±2, . . .. In other words, there exists an infinite number of characteristic roots, for which, the real parts converge to

1 τ log α 1 α 1 + β 1 /α . (28)
The origin of the closed-loop system is therefore unstable if [START_REF] Hale | Strong stabilization of neutral functional differential equations[END_REF] holds, and fails to be exponentially stable if [START_REF] Fridman | Introduction to Time-Delay Systems : Analysis and Control[END_REF] is verified.

Remark 4: Note that the non-exponential and instability properties in Theorem 1 hold for any delay τ > 0, although the closed-loop system is exponentially stable when τ = 0. This property is known as the small time-delay effect [START_REF] Hale | Effects of small delays on stability and control[END_REF], [START_REF] Hale | Strong stabilization of neutral functional differential equations[END_REF]. This property is due to the transcendental term e -τ z in the characteristic equation of the system, which generates infinitely many characteristic roots as τ varies by an infinitesimal amount from zero.

C. A Stability Result

Inspired by [START_REF] Chen | Delay-dependent stability criterion for neutral time-delay systems[END_REF], we propose sufficient conditions to guarantee exponential stability of the closed-loop system using the iP controller.

Theorem 2: Consider system (1), with a = b+1, in closed loop with the intelligent proportional controller in (4)-( 5) such that α ̸ = -β 1 /α 1 , and

α 1 α 1 + β 1 /α < 1. (29)
Then, the origin is exponentially stable if

s( Â) < 0, Â := 0 a-1 I a-1 A a+1 A a . . . A 2 , ( 30
) τ |(α a+1 ... α 2 )| + |α 1 | < |ᾱ 1 |, (31)
where s( Â) is the spectral of Â, and

0 > (A a+1 A a ... A 2 ) (|α 1 | + τ |(α a+1 α a ... α 2 )|) + |ᾱ 1 |µ( Â), (32)
where µ( Â) is the logarithmic norm of Â, and for all i ∈ {1, 2, ..., b + 2}, we have A i := (-ᾱ i + α i )/ᾱ 1 , ᾱi := α i -K di , and K di is given in [START_REF] Fliess | Closed-loop parametric identification for continuous-time linear systems via new algebraic techniques[END_REF]. □ Remark 5: Note that it is not difficult to adapt the statement of Theorem 2 when using the general intelligent PD controller defined recursively by

u = u τ - 1 α K 1 y + K 2 ẏ + . . . + K ν y (ν-1) + y (ν) , (33)
which corresponds to the ultra-local form

y (ν) (t) = αu(t) + F (t) ∀t ≥ 0, ν ∈ N. (34)
Note that the resulting closed-loop system is of the form ᾱ1 y (ν+b) + . . . + ᾱν y = α 1 y (a) τ + . . . + α a+1 y τ , (35) for v := b + ν + 1 and ᾱi ∈ R for all i ∈ {1, 2, . . . ν + b + 1}.

Here we distinguish between two cases:

• When the smallest index i ∈ {1, 2, ..., b + ν + 1} such that ᾱi ̸ = 0 guarantees ν + b -i + 1 ≤ a, then the tools in Lemmas 1 and 2 used to study the iP controller can be used. Indeed, the resulting closed-loop system must have one of the forms listed in Remark 3.

• When ν +b-i+1 > a, the resulting closed-loop system has none of the forms listed in Remark 3. It is rather a delayed differential equation, which, according to [START_REF] Krall | On the real parts of zeros of exponential polynomials[END_REF], can only have a finite number of unstable characteristic roots. The stability analysis for this particular case will be considered in future work.

V. EXPERIMENTAL RESULTS

In this section, we illustrate some of the obtained theoretical results on an experimental benchmark, which consists of the electronic throttle valve depicted in Figure 2. This is a butterfly valve used for flow-control applications. The valve's commercial reference is 03L128063. The output of the valve is the opening angle θ, which is regulated by imposing an input voltage u on a 12V DC motor. The motor is controlled using the SHIELD-MD10 board. The input u is applied by generating a PWM signal from an Arduino Mega 2560 via the Arduino IDE. The output θ is measured using Fig. 2: Electronic Throttle Valve the magnetic angle sensor KMA221. We refer to [START_REF] Witrant | A data-driven control methodology applied to throttle valves[END_REF] for more details on the experimental test bench.

Given the angular reference θ r in Figure 3, we suppose that, locally, the behavior of the tracking error y := θ -θ r is governed by equation (1). We implement the iP controller in (4)-( 5) with τ := 0.05s, which is imposed by the sampling time of the test bench

The output derivative ẏ at t k is approximated using the Euler backward method, i.e., ẏk = (y k -y k-1 )/τ . As a result, the discrete-time controller u k at t k is given by

u k := u k-1 + 1 ατ (-(Kτ + 1)y k + y k-1 ) . ( 36
)
The initialisation of the controller is u([-τ, 0]) = 0.

In Figure 3, we plot the valve response for α := 1/(0.01τ ) = 2000 and K := 2α = 4000, which shows oscillations of θ around θ r . Hence, exponential stability is not achieved. If the error y := θ -θ r is governed by (1) with a = b + 1, then this lack of exponential stability necessarily means that the sufficient conditions for exponential stability that we derived in Theorem 2 are not satisfied. To verify this fact, we propose to identify such a model, using the System-Identification Matlab Toolbox [START_REF] Ljung | System identification toolbox[END_REF]. The system is identified in closed-loop operation, to focus on the closedloop frequency range. For a = 2 and b = 1, we obtained the model ÿ + 32.16 ẏ + 1875y = 65.82 u -85.89u,

whose output, using the input signal generating the response in Figure 3, matches the response in Figure 3 at the precision of 64.9%; see Figure 4. One can check that the sufficient conditions for exponential stability in Theorem 2 are not satisfied by (37) subject to (4)-( 5), mainly because the gain α is too large. The intuition we gain from Theorem 2 is that it is more likely to achieve exponential stability when α is small than when it is large, as long as the system can be described by (1) with a = b + 1. At the same time, we need to make sure that K/α is not too large. Following this intuition, we select α = 2.5 and K = 5. The corresponding valve response is shown in Figure 5, where exponential convergence of θ towards θ r is observed.

VI. CONCLUSION

We analyzed the effect of the iP controller on the stability of linear control systems. Inspired by the literature on neutral delay and advanced-type systems, we derived sufficient conditions making the closed-loop system either unstable or not exponentially stable. Some other conditions are derived to guarantee exponential stability. In particular, we confirm, via theory and experiment, that the iP controller in (4)- [START_REF] Choi | Model-free control of automotive engine and brake for Stop-and-Go scenarios[END_REF] does not yield to a closed-loop dynamics of the form ẏ = -Ky. In future work, we would like to study the effect of approximating ẏ on the stability of the closed-loop system. Furthermore, we would like to consider the more general classes of intelligent PD and intelligent PID controllers.

Fig. 1 :

 1 Fig.1: The iP control algorithm applied to system (1).

Fig. 3 :Fig. 4 :

 34 Fig. 3: Valve response for α = 2000 and K = 4000.

REFERENCES [ 1 Fig. 5 :

 15 Fig. 5: Valve response for α = 2.5 and K = 5.


Silvia Cerruto

Daniela Ferrarello
email: daniela.ferrarello@unict.it

Mathem-Ethics for teenagers: An experiment about "loci"

Keywords: Mathematics education, ethics, personal change, teaching for robust understanding

The aim of this paper is to show an experiment of giving to mathematics' topics (on geometric loci) ethical meanings, so that students can own the mathematical topic, seen as connected with their own lives, and can use mathematical reasoning to reflect on their own states of mind and feelings. The theoretical frameworks to design the activity are the Teaching for Robust Understanding Framework and the Mathem-Ethics methodology. Investigated students are teenagers. Analyses of students' productions are briefly described at the end of the paper.

Introduction

Too many students exhibit alienation from mathematics [START_REF] Nardi | Is mathematics T.I.R.E.D.? A profile of quiet disaffection in the secondary mathematics classroom[END_REF], too often reduced to a ready set of rules to memorize and blindly apply [START_REF] Ofsted | Mathematics: Made to measure[END_REF][START_REF] Foster | Resisting reductionism in mathematics pedagogy[END_REF]. Mathematics is seen as far from students' lives and at most useful for technical reasons. "Discussions about the connections between mathematics and democracy amongst the general populace have not been explicitly well rehearsed, other than to either assume that mathematics has nothing to do with anything political, being neutral in form and practice, so it has nothing to do with democracy, which is something political; or that it is implicitly democratic." (Swanson & Appelbaum, 2012, p 1). But nowadays research in mathematics education and teachers' effort aim more and more towards the development of skills useful for students seen as future citizens in democratic societies [START_REF] Jablonka | Mathematical Literacy[END_REF][START_REF] Skovsmose | Mathematical literacy and globalisation[END_REF], [START_REF] Aguilar | On the links between mathematics education and democracy: A literature review[END_REF]. And teachers, and mathematics' teachers, have to take into account their ethical obligations [START_REF] Ernest | The ethical obligations of the mathematics teacher[END_REF], and rethink mathematics' teaching in view of a social dimension. But still, if mathematics can be useful for a future role of citizens, students do not think of mathematics as connected to their own present lives. Mathematical interventions aiming to a life change have been designed by one of the authors in prison, with adults inmates [START_REF] Ferrarello | Vietato non toccare [Don't not touch][END_REF], [START_REF] Battaglia | Non si può dividere per zero! Fare Matem-Etica in carcere [It's not possible to divide by zero! Doing Mathem-Ethics in prison[END_REF], [START_REF] Ferrarello | Mathem-Ethics in prison: how mathematics can enhance social skills[END_REF], starting to use a methodology called Mathem-Ethics. Starting from one of the topics described by Ferrarello et al. (2021), a mathem-ethical path was experimented in a regular class of teenagers in the south of Italy, for the master degree thesis of one of the authors. The mathematical content is on loci (conics, to be precise), and it is used to make students think of their states of mind. We used a wordplay because of the double meaning of the Italian word "luogo", used both for "place" and for "locus" (in mathematical sense). In this paper we will use the Latin word "locus", both in a mathematical sense (a conic) and in a metaphorical sense (a state of mind). From the "little prince worlds", used to explore students feelings on their "undesired places", we talked especially about ellipse and hyperbola. All the conics were constructed by means of mathematical machines [START_REF] Bartolini Bussi | Macchine matematiche: dalla storia alla scuola[END_REF]. The machines were introduced in a virtual version, on GeoGebra, and presented by means of GeoGebra classrooms. Paper worksheets were used too, to collect students' thoughts and feelings.

The activity was based on the Teaching for Robust Understanding (TRU) framework together with the Mathem-Ethic methodology. Results are analysed by considering the following research question: "Is it possible to reach a robust understanding of a rich content, with an ownership of the topic, by means of a mathem-ethical path and to make teenager students reflect on their own lives by using mathematics?"

Theoretical framework

We chose to design the activity and review the results by using the Teaching for Robust Understanding (TRU) framework proposed by [START_REF] Schoenfeld | An Introduction to the Teaching for Robust Understanding (TRU) Framework[END_REF] and the Mathem-Ethics methodology [START_REF] Ferrarello | Mathem-Ethics in prison: how mathematics can enhance social skills[END_REF]. The TRU framework consists of five dimensions (D1, D2, D3, D4 ad D5) for powerful classrooms, described in Table 1.

Table 1: The five dimensions of the TRU model

The framework identifies these five dimensions, useful for a truly effective teaching/learning context and fostering a deep understanding, achieving a robust teaching/learning. Briefly, "if the content is rich, the students get to engage, they get powerful ideas, they build on each other's ideas, they can build positive identities with the teacher adjusting the level of instruction so that it is right for the students to engage productively" (Schoenfeld, video in https://truframework.org/). A peculiarity of the TRU framework is that it does not tell the teacher how to teach. There are many ways to reach a robust understanding. For instance, a laboratorial methodology [START_REF] Anichini | Matematica 2003. La matematica per il cittadino[END_REF], together with a mathematical discussion [START_REF] Bandura | Self-efficacy: The exercise of control[END_REF], could be a good way to engage all the students (D3) and make them think, letting them the possibility to be active thinkers (D1) and produce their own ideas (D4), without any fear of being judged (D5), even if the content is rich and not trivial (D2). Our project was focused especially on Dimension 4 (Agency, Ownership, and Identity) and Dimension 3 (Equitable Access to Content), with the aid of Mathem-Ethics, as a methodology. The Mathem-Ethics is a methodology to teach/learn mathematics, by giving precise ethical meanings to mathematical topics. The muse of the Mathem-Ethics is Polya's "How to solve it" [START_REF] Polya | How to Solve It[END_REF], where several methodologies to solve mathematical problems are analysed. Reading "How to solve it" with a wider look, we think that mathematics could help to solve life problems. The idea is to link a mathematical entity (a definition, a theorem, a procedure, …) to a personal attitude (a state of mind, a feeling, an action in life, …), with the final target to face personal problems with a similar method used to solve the connected mathematical problem. It is possible to design mathem-ethical activities, aiming both to mathematical and social skills. Giving to mathematics meanings related to life can help students to see mathematics alive and be engaged with it, but also better understand topics with a less formal but deeper approach. Solving mathematical problems can give students a self-efficacy [START_REF] Bandura | Self-efficacy: The exercise of control[END_REF]: the attitude to solve mathematical problems connected with life problems get students used to think that it is possible to solve life problems and can be useful for supporting students in their lives [START_REF] Ferrarello | Mathem-Ethics in prison: how mathematics can enhance social skills[END_REF]. The method was successfully experimented by one of the authors in a prison context [START_REF] Ferrarello | Vietato non toccare [Don't not touch][END_REF], [START_REF] Battaglia | Non si può dividere per zero! Fare Matem-Etica in carcere [It's not possible to divide by zero! Doing Mathem-Ethics in prison[END_REF], [START_REF] Ferrarello | Mathem-Ethics in prison: how mathematics can enhance social skills[END_REF], and here, for the first time, out of prison, with teenager students.

Description of the project and methods

The class, 11-th grade at "Galilei-Campailla" high-school in Modica (Sicily), consisted of 24 students, the main part well disposed toward mathematics. But the aim was to propose a content that is rich, not only from a mathematical but also from an educational point of view. The researcher (one of the authors) presented the project to the class teacher. It is important, in fact, to have a close collaboration between researchers and teachers; many researchers have explored this cooperation, promoting interventions carried out directly by researchers and teachers, together [START_REF] Wiliam | On the purpose of mathematics education research: making productive contributions to policy and practice[END_REF]. The whole project was structured in two meetings, for a total of 3 hours. It is possible to design short-term interventions in mathematics education that can alleviate the typical problems of classroom practice: in this way teachers can benefit from the observed methodology without disrupting their curricular activity [START_REF] Stylianides | Seeking research-grounded solutions to problems of practice: classroom-based interventions in mathematics education[END_REF]. The activities involved the use of technological devices and the dynamic geometry software GeoGebra. GeoGebra was used to construct virtual version of mathematical machines, useful to build geometric loci (conics). Students were divided into groups of 4, in order to implement a collaborative learning, making students able to learn from each other, formulate conjectures and solve problems together. The researcher presented the problems ("what does this virtual machine do?") and guided the discussion, interacting with students, observing them working, helping if necessary. After a worksheet on the "Little Prince" story, and constructions of already known loci (circumference and parabola), students were guided to construct ellipses and hyperbolas, both accompanied by astronomic applications (Kepler laws for ellipse and Fly-past effect for hyperbola). The two meetings were audio-recorded. Analysed data came from the transcription of these recording, from paper worksheets produced by students and from digital worksheets on GeoGebra classrooms. Textual data were analysed from a qualitative point of view by using the Discourse Analysis: "Analysis is based on the details of speech […] or writing that are arguably deemed relevant in the context and that are relevant to the arguments the analysis is attempting to make." (Gee, 2005, p. 117). The first author selected those phrases among the data related to the important issues of the research question; the second author organized the data, guided by the themes (the 5 dimensions of the TRU and the Mathem-Ethic), but both authors paid attention to any other possible result seeming to emerge from the data.

The mathematics' loci

As mentioned, we dealt with conics, as mathematics' loci, all constructed by means of virtual mathematical machines. In this section, we only enter into the details of the mathematical construction of an ellipse and a hyperbola by using the so-called director circle. We start with the construction of the ellipse (Figure 1): we set a circle c, called director circle, with centre in a fixed point A and radius r. We also set a point B free to move on the boundary of c and a fixed point C inside c. We construct the line a, that is the axis of the (not explicitly constructed) line segment BC. Finally, we construct the point L, intersection point between r and a. The geometric locus described by L, whenever the point B moves on c is the ellipse whose foci are A and C. In fact, since a is the axis of BC, 𝐿𝐵 ̅̅̅̅ = 𝐿𝐶 ̅̅̅̅ . So, the sum 𝐴𝐿 ̅̅̅̅ + 𝐿𝐶 ̅̅̅̅ = 𝐴𝐿 ̅̅̅̅ + 𝐿𝐵 ̅̅̅̅ = 𝑟. And r, that is the radius of c, is a fixed distance. We can visualize the ellipse in GeoGebra also by moving the point B and considering all the positions taken by the line a (all with different colours in Figure 1). The ellipse is get by an envelope of tangent lines. This is useful to visualize the ellipse as a "closed" locus, being the white part, the only not covered by coloured lines, limited in space. (GeoGebra applet: https://www.geogebra.org/m/c5a3ykvg).

Figure 1: Ellipse with director circle on GeoGebra

To construct a hyperbola by means of the director circle, we can use the previous construction, with just two differences: point C has to be fixed outside c, and point L is the intersection of axis a with the line R that is the line containing segment r (Figure 2). The geometric locus described by L, whenever the point B moves on c is the hyperbola whose foci are A and C. In fact, since a is the axis of BC, 𝐿𝐵 ̅̅̅̅ = 𝐿𝐶 ̅̅̅̅ . So, the difference |𝐴𝐿 ̅̅̅̅ -𝐿𝐶| ̅̅̅̅̅ = |𝐴𝐿 ̅̅̅̅ -𝐿𝐵| ̅̅̅̅̅ = 𝑟. And r, that is the radius of c, is a fixed distance. We can visualize the hyberbola in GeoGebra also by moving the point B and considering all the positions taken by the line a (all with different colours in Figure 2). The hyperbola is get by an envelope of tangent lines. This is useful to visualize the hyperbola as an "open" locus, being the white part, the only not covered by coloured lines, not limited in space (GeoGebra applet: https://www.geogebra.org/m/myagrzqv). The Mathem-Ethics' loci

The Mathem-Ethic meaning of the Mathematics' content have two steps: "Loci", aimed at the construction of the ellipse, as a "closed" locus; and "Changes", aimed at the passage from ellipse to hyperbola, by dragging a point outside the director circle. The step "Loci" is the consideration that an ellipse is a finite geometric locus, in the Riemann's sense, i.e. it has infinite points, but it is limited in space. The ethical meaning is related to the feeling to dwell loci that we do not want to belong to, and to feel confined into these loci, without the possibility to escape, such as the characters of the "Little prince" story. Students were invited to think about those loci and to express their feelings. The step "Changes" is the consideration that an escape from those closed loci that we do not like is possible, such as the transformation of an ellipse into a hyperbola, when a point (C in figure 2), is outside the director circle. But it is not always easy. It requires an effort: the dragging of C. Because if you want to change the locus, you have to change conditions to be in the locus (no more the sum of the distance from two fixed points, as in the ellipse, but the difference, as in the hyperbola): you should leave your comfort zone, the closed locus, to reach the new open locus.

Brief discussion and conclusion

Here we briefly analyze the results of the effectiveness of the TRU framework and the Mathem-Ethics methodology, in the school context in which the experimentation took place. The research question, "Is it possible to reach a robust understanding of a rich content, with an ownership of the topic, by means of a mathem-ethical path and to make teenager students all reflect on their own lives by using mathematics?", links back to the dimensions of the TRU (see Table 1), and we analyze Mathem-Ethics at the end of section. Students showed to be engaged and to have reached an agency, ownership and identity (D4 of the TRU) thanks to the reflection of their own lives:

Student: I take with myself a little life lesson, more than a mathematics' one, […] it has stayed with me the way we linked mathematics with things we live everyday […] We were more involved with respect to a normal lecture, because we had fun and we learned at the same time.

Since the first activity (the "Little Prince" worksheets) students opened their own hearts:

Student: I would like to succeed in doing what I want, without depriving myself of things I like, just for fear of the others' thinking. Student:

I am stuck in fear of the judgment of the people surrounding me […] I wish to be myself with everyone, because I often wear a mask in order to please the others.

and many other comments identifying the locus they feel confined in as the fear of the others' judgements. They also claimed to own the mathematical topics:

Student: I liked the practical application of the studied concepts, because it will not forget anything [...]; the memory, indeed, needs images to be able to work actively. I also clarified some doubts I had on this topic.

Since the first geometric locus, the circumference, was introduced, all the students (D3 of TRU) were involved and showed the availability to help each other: Good! You can help who is left behind. This collaboration among students was possible also because they knew that they would not be evaluated: a formative assessment (D5 of TRU) let students be free to help each other and free to make mistakes, because every mistake is not judged, but rather used to reach a better comprehension. At the beginning, in fact, some students seemed to be confused about the functioning of the virtual machine to construct loci, for instance parabola, as in the dialogue involving student F we report:

Reseacher: What does this machine realize? Student:

A parabola! It is evident Reseacher:

And why is it a parabola? Student F: I see that it is a parabola, but I'm not able to understand the reason why this conic comes out.

[…] I got it, but anyway I still don't understand Student F is not afraid to share his difficulty. The proposed content is not trivial, it is challenging (D2 of TRU). After some notes from a student and the researcher, the following dialogue took place:

Student F:

And what is the focus here? The vertex upside in the rhombus, right? Reseacher:

Exact. Very good! Student F:

Now, finally, it is clear! Similar discussions took place for the construction of ellipse. Students did not listen to a ready lesson, but rather they constructed their own knowledge, starting from the exploration of the virtual machine and showing to be real thinkers (D1 of TRU):

Student:

This non-traditional method has made everything easier [...] I find it useful […] because we are the ones who directly experience and work.

More, as we can read from their worksheets, they reached the definition of hyperbola by themselves, through the dragging of point C (Figure 2). And here the Mathem-Ethic part comes out. If in the first meeting students expressed their thoughts about the feeling to be confined in a closed locus (as the others' judgement), in the second meeting they dragged the point outside the circle and realized that a change is possible, and they could deal with the difficulty, free themselves in an open locus:

Student: I'm taking home a little life lesson […] thanks to this meeting I am more willing to change "locus" in my life Student:

I think that a moment of crisis allows you to question ourselves and to have a broader conception of ourselves.

Of course, a change can be hard to reach, as in the fly-past effect, and it requires a meeting with an important situation (as a big planet) or a huge use of fuel, to change the orbit:

Student: From this experience I understood that change costs and that when you want to change, you must be willing to travel new and unknown paths [...]. This method is fruitful because it allows for reflection and deepening of the examined objects [...]. I think this experience is positive and a source of enrichment of mathematical knowledge which, before participating, I did not think it was connected so much to our lives

Two more aspects arose from data, not directly connected to the research question. Firstly, students claimed to have clarified doubts on topics already known (which was not a target for us), suggesting that the Mathem-Ethic methodology could be used to strengthen mathematical concepts. Secondly, they put in evidence the importance of a collaborative way of learning.

Figure 2 :

 2 Figure 2: Hyperbola with director circle on GeoGebra

Figure 3 :

 3 Figure 3: Mathem-ethic correspondences in the mathem-ethic path Students reflected about the possibility of changing conditions to change their undesired loci.

  used GeoGebra, how can I do? Reseacher: You can work together trying to solve the problem […] Student: I got it! Can I help my classmates? Reseacher:

Acknowledgement

This research was supported by the research project "Programma Ricerca di Ateneo UNICT 2020-22 linea 2", Engineering solutions for sustainable development of agricultural buildings and land.


Philippe Lasaygues

Compound quantitative ultrasonic tomography of long bones using wavelets analysis

Keywords: Ultrasonic tomography, bone imaging, wavelets analysis 1

Compound Quantitative Ultrasonic Tomography (CQUT) is used to long bones imaging. We showed that an iterative tool might be used to provide, from reflection tomography, qualitative images of the shape of the object, and to provide, from transmission tomography, quantitative images of the velocity map. Both tomographies are based on ultrasonic propagation in bones, particularly perturbed by this high-contrasted heterogeneous medium. Reflected and transmitted signal are composed of several packages of waves. We propose a novel algorithm based on the wavelet analysis tool adapted to ultrasonic signals that allow the transmitted signals to be cleaned and filtered and the useful information to be separated from the unwanted noise.

INTRODUCTION

Quantitative ultrasonic tomography is used to long bones imaging. In previous works, we showed that an iterative tool might be used to provide, from reflection tomography, qualitative images of the shape of the object, and to provide, from transmission tomography, quantitative images of the velocity map. Both tomographies are based on ultrasonic propagation in bones, particularly perturbed by this high-contrasted heterogeneous medium. Reflected and transmitted signal are composed of several packages of waves, which had followed various pathways within the cortical shell of long bones. Signal and image processing have a large important part in the complete process.

Transmission tomography is based on the knowledge of the Time-Of-Flight (TOF) measured on the extracted ultrasonic signal and related to the useful velocities of the ultrasonic waves throughout the shell. To optimise our TOF detection processing, we propose a novel signal processing based on the wavelet analysis adapted to ultrasonic signals that allow the transmitted signals to be cleaned and filtered and the useful information to be separated from the unwanted noise.

COMPOUND ULTRASONIC TOMOGRAPHY

Ultrasonic Tomography (UT) is based on a linearization of the inverse acoustic scattering problem. It allows perturbations (theoretically small) of a reference medium to be visualized.

For long bones, the acoustic impedance of the cortical is highly contrasted compared to the surrounding soft tissues (or water), and so, perturbs ultrasonic propagation (refraction, attenuation and diffraction). Therefore, bone imaging is a non-linear inverse problem with no single solutions. However, for some hypotheses (weak heterogeneous medium, straight lines waves propagation) and a priori knowledge of the analyzed bones, we showed that one solution is possible [START_REF] Ouedraogo | Contrast and velocity ultrasonic tomography of long bones[END_REF].

Our algorithm combines two tomographies, Ultrasonic Reflection Tomography (URT) and Ultrasonic Transmission Tomography (UTT). URT [START_REF] Lasaygues | Cancellous and cortical bone imaging by reflected tomography[END_REF] provides the realistic shape of the bodies and UTT is especially adapted to mapping local velocities inside the area defined. On the final image, the mean geometrical and acoustical parameters may be measured and used to improve the dimensioning and the quantification, ensuring, the complete iterative Compound UT in which the behaviour and the convergence are excellent.

Long human bones are effectively irregular hollow tubes (Figure 1) and should support the propagation of more complex waves similar to elastic volume waves.

Close to the external water/bone interface, the wave is deviated from its initial position because of the refraction throughout the shell. The corrections of this refraction are based on the knowledge of the dimensions and on a velocity ground level (Snell-Descartes laws).

WAVELET ANALYSIS OF TRANSMITTED SIGNALS

For transmission tomography, projections must be constructed from determination of the TOF between both transducers. The experimental ultrasonic acquisition protocol enabled us, according to the compensation procedure, to digitize correctly the transmitted signals [START_REF] Ouedraogo | Multistep compensation technique for ultrasound tomography of bone[END_REF], composed of several packages of waves which had followed various pathways within the object (cortex and/or medullary and/or cortex) [START_REF] Ouedraogo | Caractérisation géométrique et Acoustique des os long par tomographie ultrasonore[END_REF].

Various algorithms of existing T.O.F. measurements are based on various definitions of transit times of pulse, such as zero-crossing or thresholding. Our algorithm was based on detecting the arrival time of the signal corresponding to the wave propagating at, or close to, the longitudinal velocity within the shell ("fluid" modeling).

The level of the detection accepted for each signal was fixed by a subjective threshold. Then, the number of bad T.O.Fs detected increased because these thresholds are very sensitive to the signal-to-noise-ratio (SNR), and are not correctly adapted to all the values (usually 15 6 or 20 6 ) that we had to automatically treat. Indeed, the choice of the lower limit of the threshold was fixed for each view angle in relation to the noise. If we extracted the useful signal from this noise and if we standardized all the packages with a criterion which did not perturb their temporal position, the errors decrease. Our idea was to exploit wavelet analysis, which allowed simultaneous joint studies in the time and frequency domains.

Two aspects were considered in particular: the significant spectral bandwidth of the emitted signal; that is, the transmitted signal without the object (Figure 2) and the axial resolution of the transducer, brought back to the support of this emitted signal. Then we defined a Region-Of-Interest (R.O.I.) consisting of this time and frequency bandwidth for the wavelet analysis, filtering and synthesis (sum of the wavelet coefficients). In the medullary hollow zone, the processing of the signals is generally easy and the T.O.F. are calculated with a margin for error lower than 2% (Figure 3). In the solid cortex (Figure 4 & 5), signals are much more disturbed.

For example, on the Figure 5, three lots of waves were distinguished with different amplitudes and the first-arriving signal was the lowest and the arbitrary first-time could be 115 µs. The upper signal arrived in water after the T.O.F. (123.35 µs) and so, did not correspond to a direct propagation inside the shell. This signal was analyzed and the R.O.I. was defined. Figures 4-d display the curves after filtering. The SNR was higher than before processing, and the signal was filtered for low and high frequencies. The resolution and the quantification of the first two packages had improved.

Hence, the T.O.F. detections increased and the false values decreased. The same threshold can be used for all filtering signals for all view angles. The advantage of our algorithm was that the noise reduction, the thresholding and the T.O.F. detection could be made simultaneously on the image of the wavelet coefficients.

CQUT OF FEMUR

The human sample was a femoral specimen of approximately 32 ± 5 mm for the external diameter and 16 ± 2 mm for the internal diameter. The a priori mean-velocities were 3400 m/s in bone and 1478 m/s in water. In previous studies, the femur was compared to a circular tube, but this approximation (Figure 6-left) was not suitable enough. The fluid in the internal shape was reconstructed with a correct velocity value (≈ 1500 m/s) and the dimension of this cavity was 15 -17 mm. The greatest external shape (30 -34 mm) was exact and the mean-velocity estimated in the cortical shell was 3150 m/s.

5.

Figure 1 .

 1 Figure 1. Supposed pathways of the wave propagation throughout cortical long bone for CQUT

Figure 2 .

 2 Figure 2. Transmitted signal in water

Figure 3 .Figure 4 .Figure 5 .

 345 Figure 3. Wavelets analysis of a transmitted signal through the medullary hollow zone (Zone 1)

  Figure 6-right displays the final image obtained by CQUT.

Figure 6 .

 6 Figure 6. Human femur CQUT, without (left) and with(right) signal processing


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

Wavelet based deconvolution method in ultrasonic tomography

Keywords: Ultrasonic tomography, hard tissues, deconvolution, wavelet transform

This paper deals with the quantitative and qualitative ultrasonic imaging of media contrasting strongly with the background (bones, for example). The wave propagation in these structures, which involves on several phenomena, generates ultrasound signals forming packages with different time and frequency signatures. To correlate the complex physical processes involved with the various packets forming the signals, it is advisable to separate the useful information (about the various packages, for example) from the parasite information (electronic noise). After testing several possible methods (filtering, deconvolution) with variable levels of success, it was proposed here to increase the resolution of the ultrasound signals, using a method of processing which was optimized by simultaneously analyzing the time and frequency parameters. For this purpose, algorithms based on the wavelet decomposition of the signals were used and suitable transmitted signals correlated with the parameters of the experimental device and the incident wave, were selected. The mathematical properties of the transmitted signal could be adapted themselves well to this time and frequency based approach. This method, which we have called the "wavelet based deconvolution method", makes it possible to determine the transfer function of the scatterer, which we attempted to reconstruct.

A. Introduction

Medical imaging is the main field in which ultrasonic computed tomography (UCT) is applied. UCT, which involves the linearisation of the inverse problem of acoustic wave propagation, can be used to detect small perturbations occurring against a reference background. In the case of soft tissues, the mean medium (the background) serves as a homogeneous reference medium, and the waves cross an almost homogeneous and isotropic medium with similar acoustical properties to that of water. When the contrasts are too strong (in the case of bone structures for example), linear theories no longer reasonably apply, and the presence of some physical phenomena associated with the wave propagation makes it necessary to change the methods used for the acquisition of the ultrasonic signals. The more the frequency of the wave increases, the more its propagation will obviously be perturbed. To overcome these problems, the use of low ultrasound frequencies (< 3 MHz) provides an effective possible alternative [START_REF] Lasaygues | High Resolution Low Frequency Ultrasonic Tomography[END_REF]: when the frequency of the transmitted wave decreases, certain zones turn out to be more homogeneous and the attenuation is also reduced. However, if the depth of the field increases, the resolution of the signals and hence that of the reconstructed images is bound to decrease. Even with low frequencies, the complexity of the wave propagation process generates extremely complex acoustic signals consisting of several packets with different signatures, which it is often difficult to analyze (in terms of the wavepaths, volume, guided or surface waves, and attenuation) and to interpret.

To improve the quality of the signals used either to reconstruct a tomographic image or to characterize the physical processes involved, or both, we focused on the signal processing. In the particular case where ultrasonic imaging is applied to structures, which contrast strongly with the background, optimizing the signal processing used seemed to be a useful approach because tissue characterization requires quantified acoustical information extracted from signals. Several approaches to this problem have been used so far, such as filtering, spectrum analysis and a method involving the deconvolution of the signals using a characteristic transfer function of the experimental device [START_REF] Lasaygues | Déconvolution en tomographie ultrasonore basse fréquence[END_REF]. The latter operation improves the information in the high and low frequency ranges. The alternative method presented in the present study, which is based on a multi-scale decomposition procedure of the signals, makes it possible to process all the information available in terms of frequency and time. But this analysis alone does not suffice to optimize the signal processing because the acoustic signature of the transmitted wave, i.e. the incident wave reaching the scatterer, also has to match some or all of the wavelets' mathematical properties. The main aim of this study is to show the usefulness of this method based on the optimized shaping of the electronically transmitted signal associated with the incident wave propagating through the media, in comparison with the conventional signal processing approach.

B. Acoustical statements

The frequencies used here (1 MHz) where performing UCT on strongly contrasted materials are therefore lower than those formerly used in practice (for clinical and other purposes, > 3 MHz). This increases the wavelength of the transmitted wave (1.5 mm in water) and decreases the quantity of energy lost during the propagation of the wave. It also increases the depth of penetration of the wave into the medium. The scatterer analyzed will be then taken to consist of a quasi-homogeneous medium. But this reduces the accuracy of the separating power and finally

hal-00440745, version 1 -11 Dec 2009

Author manuscript, published in "Wavelet based deconvolution method in ultrasonic tomography, Austria (2007)" the resolution decreases. It is then more difficult on the tomogram to identify details such as boundaries (see Fig. 1). The propagation of the waves will result in signals with specific characteristics, which will depend on the pathways taken and on the mode of data acquisition (reflection, transmission or diffraction) used. The following examples (see Fig. 2) show how difficult it is to analyse the acoustical signatures. In this example, working in the transmission mode (but similar complexity occurs in the reflection and diffraction modes), a different signature and different packets can be detected, depending on whether the wave is propagating through a solid region (2), a medium solid region (1) comparable to a multi-layer (solid/fluid/solid) material, or the region near the border contrasting with the background (3).

C. Material and methods

The test bench

The test bench we have developed consists in a main symmetrical arm carrying two secondary transversal arms. Each arm supports an aligned transducer, which can be moved linearly or turned. )

)( ( ) )( ( ) ( t h e h t h x t s M T M ⊗ ⊗ = ⊗ = , ( 1
)
where e(t) is the electric signal conveyed to the transmitter, h T (t) is the pulse response of the two transducers (which is assumed to be known and to be identical), and h M is the pulse response of the scatterer.

When the input x(t) cannot be reasonably approximated by a Dirac function, the function h M (t) has to be deduced from the output measurement s(t) and then we must deal with deconvolution.

C.1. Wavelet based deconvolution

Since e(t) is generally comparable to a unit impulse function (i.e. a Dirac delta function), the input will be equal to the transducer's response ) (

) ( t h t x T ≈
. Let us now generate a wave corresponding to a transmitted signal x(t) having suitable properties for specific processing method, such as wavelet analysis.

If x(t) is a wavelet denoted ( )

t J ϕ
which is centered on the (dilation) scale denoted J (J ∈ Z), we have:

( ) ( ) t h t t s M J ⊗ ϕ = ) ( , (2)
Time-scale decomposition of the signal s(t) on a dyadic grid [START_REF] Lasaygues | Application de l'analyse en ondelettes en tomographie ultrasonore[END_REF][START_REF] Lasaygues | Progress toward in vitro quantitative imaging of human femur using Compound Quantitative Ultrasonic Tomography[END_REF] makes it possible to calculate the coefficients X j , ( ) ( ) ( ) ( )

t h t s(t) t X j, J M j, j 0 0 φ ⊗ ϕ ⊗ = φ ⊗ = (3)
where ( )

t j 0 ,
φ is a wavelet centered on the scale j ( Z j∈ ). The properties of the wavelet decomposition (an orthogonal decomposition in this case) are such that the coefficients, Xj nullify everywhere except for j = J, which makes it possible to directly determine the response of the scatterer without encountering any further filtering effects.

( ) ( )

     ≠ = = δ ⊗ = J j if J j (t) if h t ) (h t X M M j 0 , ( 4
)
If it is possible to process the transfer function of the experimental device in such a way that it is identical to a wavelet function, this algorithm directly yields the response of the scatterer.

D. Results, simulation and experiments

Let us take the case of a multi-layer plane interface (water/Plexiglas/water) with thickness D and compressional wave velocity c M = 2700 m/s. The case of the interface at a depth of 2 mm is interesting because the wavelength in the water at 1 MHz (equal to 1.5 mm for a compression wave velocity of 1500 m/s) is close to the depth, and the signals have therefore merged, which affects the resolution (see Fig. 4.). Based on the time at which the maximum coefficient occurred in the 1st group and the time at which the maximum coefficient was reached in the 2nd group, we obtained the following thickness values (Table 1). When the echoes are fairly well separated, it is also possible to discriminate between the -input and outputinterfaces, and the error was about 8%. With the waveletbased deconvolution method, the error was reduced to 0.099 %. At a thickness of 2 mm, the discrimination based on the pulse echo was more difficult (the error was about 62%), whereas using the wavelet-based deconvolution method, the error at the interface was only about 5.5%.

It is not easy to test this method experimentally because approaching the numerical transmitted signal x(t) in the form of wavelets makes it necessary to process an initial function, which has to be filtered by the transducer to obtain a specific wavelet. Several approaches were previously tested with various levels of success, such as those based on direct deconvolution and simulated annealing, and especially manual determination, which consists in iteratively producing the signal by correcting the quadratic error at every iteration.

The experimental sample used here was a noncircular homogeneous isotropic tube made of artificial resin (NEUKADUR ProtoCast 113 TM ). Its maximum internal and external diameters, which were measured with a caliper, were 6 and 12 mm, respectively. The density of the resin was ρ 1 ≈ 1150 kg /m 3 , and the mean velocity of the compressional wave was c 1 ≈ 2400 ± 50 m/s. After performing wavelet-based decomposition on the all cross-sectional signals received, the definition of the outer and inner boundaries of the cylinder was obtained with a much better resolution.

E. Aknowledgements

The author is grateful for the technical assistance and manpower provided by Louise Matet and Laurent Marsac at the Ecole Supérieure des Ingénieurs de Luminy (ESIL) in Marseille and by Karine Degrendele at the Ecole Centrale de Marseille.

F. Literature

Fig. 1

 1 Fig.1 Effects of the frequency on the resolution of ultrasonic computed tomography performed on adult human thighbone (left) 500 kHz, (right) 2.25 MHz.

(Fig. 2 .

 2 Fig.2. Ultrasonic signals depending on the bone region involved: (1) multi-layer solid/fluid/solid (2) solid (3) limit contrasting with the backgroundIn the case of objects showing a strong contrast with the background, a large proportion of the incident energy is scattered (or simply reflected) and the transmitted proportion is refracted (deviated) after modal conversion at the interface between the water/scatterer/central cavity, at various angles. The diffraction will not be isotropic. The various wave packets constituting the signals depend decisively on the pathway taken and on the nature of the waves (compressional or shear waves, surface or guided waves). The idea of using a specific signal processing method is to extract as much information as possible about these various packets by analyzing the time and frequency signatures of the signals. It is therefore necessary to decompose these signals so as to identify

Fig. 3 .

 3 Fig.3. Ultrasonic control of a high-contrast scatterer The electro-acoustic device and the transducers therefore serve as a continuous, linear stationary causal filter. The input and output signals are connected by convolution:

Fig. 4 .

 4 Fig.4. Simulated ultrasonic signals transmitted in plane interfaces at various depths D (left) 5 mm, (right) 2 mm ; (height), pulse echo (bottom), wavelet at J = -4 , 512 samples, frequency rate 20 MHz, velocity of compression waves = 2700 m/s

Fig. 5 .

 5 Fig.5. Dyadic grid corresponding to the scale J of non-zero coefficients (height) pulse echo (bottom), wavelet J = -4 (left) 5 mm, (right) 2 mm, 512 samples, frequency rate 20 MHz, velocity of compression waves = 2700 m/s

Fig. 6 .

 6 Fig.6. Ultrasonic computed tomography of a non-circular elastic tube. (height) pulse echo (bottom), wavelet J = -4 (left) sinograms, RF-signals, 180 projections through 360°, 1024 samples; (right) tomogram with 300 x 300 pixels. The surrounding fluid-like medium (and that present in the hollow area) was water at a temperature of 18 ° (ρ o = 1000 kg /m 3 , c o = 1480 m/s). The transmitter and the receiver were placed 17.5 cm to the right of the center of the bench. The sinogram drawn up contained 180 projections distributed over 360° with an angular increment of 2°. The signals contained 1024 distributed samples occurring every 50 ns (frequency rate of 20 MHz). The nominal frequency of the transducer was 1 MHz.

Table 1

 1 Real thickness (mm) Calculated thickness (mm) (error %)

		Pulse echo	Wavelet
	5	5.4 (8 %)	4.995 (0.099 %)
	2	3.24 (62 %)	1,89 (5.5 %)

Paper #1052 Presented at the International Congress on Ultrasonics, Vienna, April 9 -13, 2007, Session R05: Biomedical Ultrasound


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

Emilie Franceschini

Eric Debieu

Loïc Brancheriau
email: brancheriau@cirad.fr

Non-destructive diagnosis of the integrity of green wood using ultrasonic computed tomography

Keywords: Ultrasonic Computed Tomography, Wood Imaging, simulation and experiments

Ultrasonic computed tomography (UCT) was used to assess the integrity of green wood. Woods are heterogeneous air-coupled, orthorhombic materials. Because of the difference in acoustic impedance between the material and the surrounding medium (water or coupling gel), the ultrasonic wave propagation is greatly perturbed by physical processes such as the refraction, attenuation and scattering of the waves. UCT belongs to the inverse scattering class of problems and the aim of this study was therefore to present a strategy for simply, efficiently and accurately developing a reconstruction algorithm. UCT is based on several assumptions, such as the presence of a low contrast medium (the biological medium), a large frequency range (a broadband pulse), and dense and complete sets of projections. If these conditions are ideally fulfilled, we can reconstruct images of the impedance of the medium. This technique involves an algorithm based on first-order Born approximation methods. To date we have tested the qualitative aspects of this imaging technique and part of the quantitative aspects by performing numerical simulations and on our tomographic testing ground. The latter is composed of a base supporting a rotating and translating mechanical structure that can hold several transducers, and a numerical system and computing and recording the projections.

When performing non-destructive assessments on wood, we would have liked to obtain quantitative images related to acoustic parameter of wood (impedance, speed of sound, attenuation). However, since the problem is non-linear, a low-frequency method involving signal and image processing was used.

A. Introduction

Acoustical properties such as the ultrasonic attenuation or velocity have been reported to reflect the strength of wood [START_REF] Bucur | Acoustic properties of wood in tridimensional representation of slowness surfaces[END_REF] and to predict in vivo decay processes [START_REF] Dunlop | Testing of poles by using acoustic pulse method[END_REF]. However, the ultrasound methods currently available for assessing green wood status are limited to performing measurements in the transmission mode through a section. These methods therefore give average values recorded along the ultrasound pathway through the section. They are not suitable, however, for assessing the spatially heterogeneous acoustical properties of wood. The use of Ultrasonic Computed Tomography (UCT) in this context would constitute a significant step toward more accurate wood quality assessments. Although UCT imaging has been found to provide a valuable tool for imaging soft biological materials [START_REF] Mensah | Enhanced Diffraction Tomography[END_REF], little attention has been paid so far to the use of this method for imaging harder media such as green wood. L. Brancheriau et al. have published the preliminary results of an acoustic wave propagation study on green wood [START_REF] Dikrallah | Experimental analysis of acoustic anisotropy of green wood by using guided waves[END_REF]. The interactions occurring between ultrasound and wood, especially in the case of green wood, make the imaging process particularly intricate. For instance, classical reconstruction algorithms are based on the assumption that straight ray propagation is involved, whereas the incident beam is liable to be refracted at the interface between the soft background (coupling gel, water) and the wood, depending on the incident angle, due to the great difference between the acoustic properties of the two media. Diffraction, which is closely related to refraction, can also result from the irregular shape of the samples. The anisotropy of the ultrasound propagation processes occurring through green wood can also sometimes require the use of sophisticated reconstruction tools.

Despite the many difficulties encountered when attempting to perform quantitative ultrasonic tomography on wood, the UCT imaging of wood provides an attractive non-invasive tool with which it is possible to perform quantitative 3-D reconstruction of the geometry of trees and determine acoustical properties such as the ultrasound velocity and attenuation, depending on the type of measurements (the time of flight or amplitude) performed. In previous studies by G. Nicolotti [START_REF] Nicolotti | Application and comparison of three tomographic techniques for detection of decay in trees[END_REF] and V. Bucur [START_REF] Bucur | Acoustics of wood[END_REF], projection data were obtained in the transmission mode through a wood specimen, which could be inversed using classical tomographic reconstruction algorithms [START_REF] Kak | Principles of computerized tomographic imaging[END_REF]. Previously published preliminary results showed that the tomographic reconstruction of an academic test object and green wood was feasible, using ultrasonic transmission tomography techniques. The authors therefore hypothesized that the reconstruction of the outer boundary of the samples and the inner boundaries of the various regions of decay would provide information about the strength of the wood.

However, these studies were seriously limited by several factors. First, the boundaries of the object can't be easily reconstructed from the data set acquired in the transmission mode, but from data set acquired in the reflection mode, which gives an improved tomography of the shape of the scatterer. Furthermore, the mesh in transmission-UCT is very often too large, and the cavities of diameter lower than 30 mm are generally imperfectly identified. Secondly, inversing the data set acquired in the transmission mode to recover the sound velocity image of a cross-section requires the signals to be unambiguously identified, and the time-of-flight to be measured accurately from a signal with a low signal-to-noise ratio. The complex interactions between the incident ultrasonic wave field and wood give rise to a complex transmitted field with several output signals, which somehow makes it difficult to identify the signals.

The main objectives of our project were first to take full advantage of UCT in the diffraction and transmission modes by developing a technique for the cross-sectional imaging of wood, and secondly, to refine the signal processing technique by developing a suitable signal filtering process based on wavelet decomposition to improve the assessments based on time of flight measurements.

The latter point was addressed in a previous paper [START_REF] Lasaygues | Wavelet based deconvolution method in ultrasonic tomography[END_REF]. The present paper focuses on the qualitative (non parametric) image obtained using reflection-UCT to detect decay in a green wood. Reflection-UCT, which is a particular case of the general diffraction-UCT approach, consists in forming an image of the difference in contrast between the different layers of the wood (i.e. the shape imaging) from the projection data resulting from backscattering.

By performing a numerical simulation, we will study the effects of several patterns of decay and several configurations of "artificial" wood on the reflection-UCT data obtained. The experimental test performed on a green wood sample using a tomographic scanner showed the advantages and the limitations of the classical algorithms.

B. Assumptions adopted in UCT

Reflection-UCT can be used to detect (theoretically small) perturbations in the medium studied. In the case of soft, low-contrast media such as biological tissues, weak scattering approximations are generally used, such as the approximations of Born or Rytov [START_REF] Greenleaf | Introduction to Computer Ultrasound Tomography[END_REF].

Contrary to what occurs with soft media, the acoustical contrast between wood and the surrounding reference medium (coupling gel or water) is so great that the problem is intrinsically non-linear in terms of the unknown physical and geometrical parameters, and there generally exists no single solution.

One solution therefore consists in finding the homogenized or "equivalent" material parameters of a spatially bounded heterogeneous medium, and adapting the acquisition and reconstruction software initially developed for used with soft media.

Trees consist of long circular cylinders surrounded by an irregular thick bark. These cylinders are generally assimilated to orthotropic materials with a principal axis of symmetry oriented along the vertical axis [START_REF] Goodman | Orthotropic Elastic Properties of Wood[END_REF]. In this study, the transversal cross-sections of the wood to be imaged were taken to be isotropic and were therefore assumed to mediate the propagation of more complex waves, similar to volume waves Since the typical compression bulk wave velocity recorded in the transverse direction was 1400 -1600 m/s, the wavelength at the frequency used here (~ 0.5 MHz) was of the order of 2.8 to 3.2 millimeters, which is much larger than the usual structural heterogeneities (~ 10 micrometers). Wood can therefore be said to be a homogeneous medium on the scale of the wavelength and the Born approximation can be used.

The wavelength in water (~ 3 mm) was much shorter than the diameter of the object (approximately 100 -200 mm). Ray theory was applied and straight lines inside the object were taken to approximate the wave propagation paths. Since only signal contributions associated with compression waves were of interest here, a weakly inhomogeneous fluid can be used to model wood.

C. Experimental set-up

The experimental setup was designed to perform all UCT-measurements (in the reflection, diffraction and/or transmission modes). In included an acoustic device, which could be moved with various degrees of freedom and used to analyze the samples in all directions. The position of the target and the transducers could be adjusted. In particular, the operator could prescribe the exact positions of the transmitter and the receiver. During all the measurements, the wave velocity in the water tank was 1480 m/s. Ultrasounds were generated using a Panametric® pulse/receiver and Imasonic® piezo-composite wide-band transducers with a nominal frequency of 0.5 MHz. The object under investigation was placed in the centre of the measuring system. The sector scanned was typically 72° / 360°, (i.e. 72 view angles), and the angular increment used was 5°. The typical backscattered signals were composed of 4096 samples regularly spaced at intervals of 50 nsec (20 MHz). In the reflection mode, the projections corresponded to the backscattering echoes and to the complete Radio-Frequency signal. The reconstruction algorithm was the algorithm of the UCT based on the summation and the spreading of the filtered projections. The size of the images was 300 x 300 pixels.

D. Results

Reflection-UCT was applied to a numerically simulated test-target and to a green wood sample.

The numerically simulated test-target was a homogeneous and isotropic circular wood cylinder with a diameter of 30 mm (see Fig. 1-a). The mean velocity of the compressional wave was 1500 m/s in water, and 1600 m/s in the wood. The sample contained two inclusions corresponding to decay (c 1 = 700 m/s and c 2 = 1300 m/s).

The spruce wood sample (picea abies) was green, fresh, streamlined, surrounded by bark, and measured 125 mm in diameter (see Fig. 2-a). The mean velocity of 1400 m/s measured at the center of a piece of wood with plane parallel faces from the same specimen, was used as the required a priori value. A mixture of glue and sawdust was placed in a 30 mm-hole drilled in the center of the sample. This image was geometrically accurate as far as the actual dimension of the object was concerned.

D.1. Simulation test

The result was acceptable, since the Born approximation was used and the background consisting of water with an a priori wave velocity equal to 1500 m/s. The perturbation in this case was introduced by the two holes simulating decay. The most highly contrasted area therefore naturally focused on the inner holes rather than on the external boundaries of the target. In fact, in this example, the bark was not simulated, and the difference in velocity between the water and the wood was very insignificant (100 m/s). It was not in fact possible for the algorithm to discriminate between the two domains. The reference background used for the inversion procedure was finally the set composed of the water and the main wood. Since the difference in velocity between the wood and the holes increased, they could easily be identified in this example.

D.2. Wood specimen

Fig. 2 shows the reconstructed reflection-UCT of a spruce wood sample (picea abies). The difficulties encountered were mainly due to the strong attenuation of the waves occurring during their propagation. Signals were therefore amplified but not saturated.

Besides the attenuation of the waves during the propagation, it is worth mentioning that the effects of the bark on the signals were of great importance. Large amounts of air are probably imprisoned between the healthy tree and the bark. The acoustic impedances are very different and this affects the wave propagation by brutal interrupting. Given the irregularity of the surface texture, it is likely that only a small part of the incident energy will penetrates into the wood, and a large proportion will be directly backscattered without penetrating into the healthy wood. This idea is in line with the strong echoes, which, if we assume that they are not attenuated between the healthy wood and the bark, can be taken to reflect the thickness of the bark.

The diameter of the wooden sample was correctly assessed (125 mm), and the cavity was positioned well in the center, but it was impossible to measure its diameter exactly. Here again, the problem is due to the low contrast in the sound speed between the region of interest and the rest of the sample. In addition, it was necessary to assume (as we did at the beginning) that we were dealing with isotropic conditions.

E. Conclusion

Here we have presented wood ultrasonic imaging method based on the use of Ultrasonic Computed Tomography. Due to the acoustical properties of wood (the attenuation and impedance contrast), working with wood is more complex than dealing with soft tissues, which made it necessary to adapt the linear inversion procedure usually applied to the data, using a linear approximation of the direct problem, the Born approximation. It was necessary to study the non-linear inverse problem. But in our case, wood was assumed to be equivalent to an internally weakly contrasted object (the local fluctuations in the acoustic characteristics of a cross-section are weak) immersed in a homogeneous reference medium (water). An ultrasonic tomography procedure was then applied. The results of the reconstructed geometry are similar to the real values.

This research could now be continued on two possible lines. The first approach is based on the Born iterative method, where the iterations are calculated numerically from experimental data [START_REF] Lasaygues | Distorted Born Diffraction Tomography: limits and applications to inverse the ultrasonic field scattered by an non-circular infinite elastic tube[END_REF]. The second possible approach is based on modeling the forward problem using the Intercepting Canonical Body Approximation method, which reduces the cost of the calculations. The latter method consists in taking the wood covered with bark and the decay it contains to be a non-circular multi-layer cylinder, in which every layer (bark, wood, decay) is homogeneous. Contrary to approaches based on straight ray theory, this method requires no limited approximations and no a priori information. The scattered field is completely integrated [START_REF] Franceschini | An optimization method for quantitative impedance tomography[END_REF].

F. Literature

Fig. 1 -

 1 Fig.1-b gives the ultrasonic sinogram and fig.1-c, the reflection-UCT of the numerically simulated test-target.

Fig. 1 .

 1 Fig.1. Numerically simulated test-target consisting of a homogeneous and isotropic circular wood cylinder 30 mm in diameter. (a) Sample; (b) ultrasonic reflection sinogram; (c) tomogram

Fig. 2 .

 2 Fig.2. Reflection-UCT of spruce wood sample (picea abies) (a) geometry (b) ultrasonic reflection sinogram; (c) tomogram; (d-e) x and y pixel profiles drawn at x = y = 0 mm