N De Rancourt

Connected components of the general linear group of a real hereditarily indecomposable Banach space

Keywords: Mathematics Subject Classification (2020): Primary: 47B01; Secondary: 46B03, 54D05 Hereditarily indecomposable Banach spaces, connectedness of general linear groups, complex structures on real Banach spaces

We give a complete description of the structure of the connected components of the general linear group of a real hereditarily indecomposable Banach space, depending on the existence of complex structures on the space itself and on its hyperplanes. A side result is the fact that complex structures cannot exist simultaneously on such a space and on its hyperplanes.

, for the spaces p , 1 ď p ă 8, by Neubauer [12], for Cpr0, 1sq, L 1 pr0, 1sq, L 8 pr0, 1sq and 8 by Edelstein, Mityagin and Semenov [4],

Introduction

In this paper, unless otherwise specified, when speaking about a Banach space (or simply a space), we shall mean an infinite-dimensional Banach space, and by subspace of a Banach space, we shall always mean infinite-dimensional, closed subspace. By operator, we shall always mean bounded linear operator. The algebra of operators on a normed space X will be denoted by LpXq, and the general linear group of X, that is, the group of invertible elements of LpXq, will be denoted by GLpXq. The algebra LpXq, and all of its subsets, will always be endowed with the topology induced by the operator norm. 1 and for the spaces L p pr0, 1sq, 1 ă p ă 8, by McCarthy and Mityagin (the proof first appeared in the survey [START_REF] Mityagin | The homotopy structure of the linear group of a Banach space[END_REF] by Mityagin). On the other hand, it was proved by Douady [START_REF] Douady | Un espace de Banach dont le groupe linéaire n'est pas connexe[END_REF], that GLpX ˆY q, for X and Y being two distinct spaces among the p 's, 1 ď p ă 8, and c 0 , is not connected, both in the real and in the complex case. For more details and results on this topic, and more generally on the homotopy structure of general linear groups of infinite-dimensional Banach spaces, we refer to Mityagin's survey [START_REF] Mityagin | The homotopy structure of the linear group of a Banach space[END_REF].

A common point between all of the latter examples is that their spaces of operators are very rich. In this paper, we study the case of some spaces where on the contrary, the space of operators is very poor: hereditarily indecomposable Banach spaces. Definition 1.1. A Banach space is said to be hereditarily indecomposable (HI) if it contains no topological direct sum of two subspaces. HI spaces were introduced by Gowers and Maurey in [START_REF] Gowers | The unconditional basic sequence problem[END_REF], where they built the first example of such a space (this space, existing in a real and in a complex version, will be called Gowers-Maurey's space in this paper). It turns out that these spaces are very rigid. For example, in the complex case, the following result was proved by Gowers and Maurey in [START_REF] Gowers | The unconditional basic sequence problem[END_REF].

Theorem 1.2 (Gowers-Maurey). Every operator on a complex HI space X has the form λ Id X `S, where λ P C and S is a strictly singular operator.

Recall that an operator on X is said to be strictly singular if it induces no isomorphism between two subspaces of X. The set of strictly singular operators on X, that we will denote by SpXq in this paper, is a two-sided, closed ideal of LpXq. An analogue of Theorem 1.2 for real HI spaces, due to Ferenczi [START_REF] Ferenczi | Operators on subspaces of hereditarily indecomposable Banach spaces[END_REF], will be presented in Section 2 (Theorem 2.1).

A consequence of the smallness of the space of operators on an HI space is that similar methods as those used in the finite-dimensional case apply quite well to the study of connected components of GLpXq. This study was started by the author in [2], were the complex case was settled.

Theorem 1.3 (de Rancourt). If X is a complex HI space, then GLpXq is connected.

In the same paper, the author proved a partial result in the case of real HI spaces, that we present below. This result is stated in terms of homotopy between operators. If A Ď LpXq and S, T P A, we say that S and T are homotopic in A whenever there exists a continuous mapping r0, 1s Ñ A, t Þ Ñ T t , with T 0 " S and T 1 " T . Since GLpXq is open in LpXq, it follows that two operators in GLpXq are homotopic in GLpXq if and only if they are in the same connected component of GLpXq.

Also recall that an operator R P LpXq is a reflection of X if there exists H Ď X a hyperplane and x 0 P Xzt0u such that @x P H Rpxq " x and Rpx 0 q " ´x0 . All reflections of X are elements of GLpXq. Call an antireflection of X an operator of the form ´R, where R is a reflection of X. It is easy to see that any two reflections of the same space X are homotopic in GLpXq, and that the same holds for antireflections. The partial result about real HI spaces proved in [2] is the following: Theorem 1.4 (de Rancourt). Let X be a real HI space and let T P GLpXq. Then T is homotopic, in GLpXq, either to Id X , or to ´Id X , or to all reflections, or to all antireflections.

It immediately follows that GLpXq has at most four connected components. It is also remarked in [2] that this bound is attained in the case of the real Gowers-Maurey's space.

The goal of this paper is to give a complete description of the connected components of GLpXq when X is a real HI space. We will prove, in particular, that GLpXq necessarily has either 2 or 4 connected components, and that both can happen (see Theorem 2.5). It turns out that the structure of connected components of GLpXq is closely related to the existence of complex structures on X and on its hyperplanes, that has been widely studied by Ferenczi [START_REF] Ferenczi | Uniqueness of complex structure and real hereditarily indecomposable Banach spaces[END_REF] and Ferenczi-Galego [START_REF] Ferenczi | Even infinite-dimensional real Banach spaces[END_REF]. This paper is organized as follows. In Section 2, we review some properties of real HI spaces and of their complex structures, from papers [START_REF] Ferenczi | Operators on subspaces of hereditarily indecomposable Banach spaces[END_REF][START_REF] Ferenczi | Uniqueness of complex structure and real hereditarily indecomposable Banach spaces[END_REF] by Ferenczi and [7] by Ferenczi-Galego. At the end of the section, we state our main result, Theorem 2.5. The next two sections are devoted to its proof. In Section 3, we prove that whenever X is a real HI space, the identity of X can never be homotopic to a reflection in GLpXq (Theorem 3.1). In the course of the proof, some tools playing a similar role as the determinant in finite dimension will be introduced. In Section 4, we will finish the proof of Theorem 2.5 and, using the tools introduced in Section 3, we will give a complete description of the connected components of GLpXq depending on the existence of complex structures on X and on its hyperplanes (see Theorems 4.4, 4.5 and 4.6). It will also be proved that there cannot simultaneously exist complex structures on a real HI space and on its hyperplanes (Theorem 4.2).

Real HI spaces and complex structures

Theorem 1.2 is not always true for real HI spaces. However, the folloing analogue was proved by Ferenczi in [START_REF] Ferenczi | Operators on subspaces of hereditarily indecomposable Banach spaces[END_REF]: Theorem 2.1 (Ferenczi). Let X be a real HI space. Then the quotient algebra LpXq{SpXq is either isomorphic, as a unitary Banach algebra, to R, to C, or to the quaternion algebra H.

We will say that the space X has type R, type C, or type H depending on the case; the real Gowers-Maurey's space, for example, has type R. In the rest of this paper, we will denote by π X : LpXq Ñ LpXq{SpXq the canonical quotient map, and for λ P R, we will abusively identify the element π X pλ Id X q of LpXq{SpXq with the real number λ. Hence, R can be seen as a subalgebra of LpXq{SpXq. Of course, in the case of spaces of type C or H, we cannot identify the whole algebra LpXq{SpXq with C or H since this identification is in general not unique. Observe that if P P LpXq is a projection onto a finite-codimensional subspace Y , then the mapping LpY q Ñ LpXq, T Þ Ñ T ˝P induces an isomorphism from LpY q{SpY q onto LpXq{SpXq; in particular, a real HI space and its finite-codimensional subspaces have the same type.

If pX, } ¨}q is a real Banach space, we define a complex structure on X as a structure of complex vector space on X extending its structure of real vector space, together with a complex norm } ¨}C which, when seen as a real norm, is equivalent to the original norm } ¨}. Saying that a real Banach space X admits a complex structure is hence equivalent to say that X is isomorphic, as a real Banach space, to some complex Banach space; thus admitting a complex structure is an isomorphic property of real Banach spaces. We will identify two complex structures on X when the associated complex norms are equivalent. Hence, a complex structure on X is entirely determined by the action of the multiplication by i on the space X; this multiplication should be an operator I P LpXq such I 2 " ´Id X . Conversely, given such an operator I on a real Banach space X, we can define a complex structure on X by letting, for every x P X, ix :" Ipxq and }x} C :" sup aPC, |a|"1 }ax}. Thus, in the rest of this paper, we will identify the set of complex structures on X with the set of I P LpXq such that I 2 " ´Id X .

If X is a real Banach space, then X admits a complex structure if and only if its 2-codimensional subspaces do so. Indeed, if X admits a complex structure I, then given any hyperplane H Ď X, it is easy to see that the subspace HXIpHq of X has codimension 2 and is I-invariant; conversely, if a subspace Y Ď X of codimension 2 admits a complex structure, then X is isomorphic to Y ' C, so it admits a complex structure too. Thus, the problem of the existence of complex structures on finite-codimensional subspaces of X is settled once we know whether X and its hyperplanes admit complex structures. The following result of existence follows from results by Ferenczi and Galego [START_REF] Ferenczi | Even infinite-dimensional real Banach spaces[END_REF].

Theorem 2.2 (Ferenczi-Galego). Let X be a real Banach space and let u P LpXq{SpXq be such that u 2 " ´1. Then exactly one of the following conditions holds:

1. There exists U P LpXq with π X pU q " u such that U 2 " ´Id X ; 2. There exists U P LpXq admitting an invariant hyperplane H, with π X pU q " u, and such that pU ae H q 2 " ´Id H .

It follows that as soon as LpXq{SpXq contains a square root of ´1, then there exist a complex structure, either on X, or on its hyperplanes. (Of course, nothing prevents both to happen at the same time, since a complex structure on X and a complex structure on its hyperplanes could come from distinct elements of LpXq{SpXq; this is the case, for instance, for X " 2 .) In particular if X is a real HI space having type C or H, then either X, or its hyperplanes, admits a complex structure. Conversely, it is immediate that if either X, or its hyperplanes, admit a complex structure, then some element of LpXq{SpXq should have square ´1. Thus, real HI spaces of type R, and in particular the real Gowers-Maurey's space, admit no complex structures, neither their hyperplanes do.

The following notions were introduced by Ferenczi and Galego in the same paper [START_REF] Ferenczi | Even infinite-dimensional real Banach spaces[END_REF].

Definition 2.3.

1. The real Banach space X is said to be even if it admits a complex structure, and its hyperplanes do not.

2. The real Banach space X is said to be odd if it does not admit a complex structure, and its hyperplanes do.

This terminology comes from the finite-dimensional case: spaces of even dimension are even, and spaces of odd dimension are odd. It follows from the previous remarks that every real HI space X satisfies exactly one of the four following properties:

1. X has type R; 2. X is even; 3. X is odd; 4. both X and its hyperplanes admit a complex structure.

We have examples in the three first cases. As already said, the real Gowers-Maurey's space has type R. In [START_REF] Ferenczi | Uniqueness of complex structure and real hereditarily indecomposable Banach spaces[END_REF], Ferenczi built two examples of even real HI spaces, XpCq and XpHq, having respectively type C and H. Since hyperplanes of even spaces are obviously odd, and similarly, hyperplanes of odd spaces are even, it follows that hyperplanes of XpCq and of XpHq are odd HI spaces of respective types C and H. On the other hand, we will show in Section 4, as a byproduct of our work on general linear groups, that case 4. cannot possibly occur (see Theorem 4.2). This solves by the negative the following question by Ferenczi and Galego (still open in the general case) in the special case of HI spaces: Question 2.4 (Ferenczi-Galego). Does there exist a real Banach space X which is not isomorphic to its hyperplanes, such that both X and its hyperplanes admit a complex structure?

We are now ready to state our main theorem, which will be proved in the next two sections.

Theorem 2.5. Let X be a real HI space. Then exactly one of the three following condition holds.

1. X has type R, and GLpXq has exactly four connected components: one containing Id X , one containing ´Id X , one containing all reflections, and one containing all antireflections.

2. X is even, and GLpXq has exactly two connected components: one containing Id X and ´Id X , and one containing all reflections and all antireflections.

3. X is odd, and GLpXq has exactly two connected components: one containing Id X and all antireflections, and one containing ´Id X and all reflections.

In each of the latter cases, a characterization of operators belonging to each connected component will be given in Section 4, see Theorems 4.4, 4.5 and 4.6.

Reflections on real HI spaces

In this section, we prove the following theorem.

Theorem 3.1. Let X be a real HI space. Then Id X cannot be homotopic to a reflection in GLpXq.

In the proof of this theorem, we will need to use spectral theory, and hence to pass to the complexification of the real space X. This complexification, denoted by X C , is defined as the set of formal sums x `iy for x, y P X, endowed with the complex vector space structure defined by pa `ibqpx `iyq :" pax ´byq `ipay `bxq for a, b P R, and with the complex norm defined by }x `iy} :" sup θPR } cos θ ¨x `sin θ ¨y}; this makes it a complex Banach space. Given T P LpXq, we can define its complexification T C P LpX C q by T C px `iyq :" T pxq `iT pyq. When talking about the spectrum of the operator T , we will actually abusively talk about the spectrum of the operator T C ; the same convention will be adopted for all spectrum-related notions such as eigenvalues, multiplicities, etc. The spectrum of the operator T will be denoted by σpT q. It is easy to see that the spectrum of an operator T on a real Banach space is always conjugationinvariant, and that if λ is an eigenvalue of T , then so is λ, with the same multiplicity. It is also straightforward that the complexification of a Fredholm operator is itself Fredholm, with the same index. Lemma 3.2. Let X be a real HI space, and let T P LpXq. Let λ P σpT q X R be such that π X pT q ‰ λ. Then λ is an isolated eigenvalue of T with finite multiplicity.

Proof. Recall that the essential spectrum of T is the set of all µ P C such that T C ´µ Id X C is not semi-Fredholm. It is well known that every operator on an HI space is either strictly singular, or Fredholm with index 0 (see for example [2], Theorem 3.4). It follows that λ does not belong to the essential spectrum of T .

It was proved by Gowers and Maurey [START_REF] Gowers | The unconditional basic sequence problem[END_REF] that the spectrum of an operator on a real HI space is countable. It follows from [START_REF] Kato | Perturbation theory for linear operators[END_REF], Theorem 5.33, that every non-essential spectral value of T is an isolated eigenvalue with finite multiplicity. This is, in particular, the case for λ.

For I an open interval of R, denote by NS I pXq the set of all T P LpXq satisfying the two following conditions:

1. π X pT q R I;

2. σpT q X BI " ∅.

(Here, BI denotes the boundary of I in R.) The notation NS stands for non-singular. By Lemma 3.2, for T P NS I pXq, σpT q X I is made of finitely many isolated eigenvalues with finite multiplicities. We will denote by p I pT q the parity of the sum of the multiplicities of eigenvalues of T that are contained in I. Here, by a parity, we will mean an element of Z{2Z.

The next lemma, along with Lemma 3.5, is reminiscent of the proof of Proposition 8 in [START_REF] Ferenczi | Even infinite-dimensional real Banach spaces[END_REF].

Lemma 3.3. Let X be a real HI space and I be an open interval of R. Then the mapping p I : NS I pXq Ñ Z{2Z is locally constant.

To prove Lemma 3.3, we will need the following result of continuity of the spectrum. For a proof, see [START_REF] Kato | Perturbation theory for linear operators[END_REF], Chapter Four, Subsection 3.5. Proposition 3.4. Suppose T : Y Ñ Y is an operator on a complex space, and Γ is a rectifiable, simple closed curve in C such that Γ X σpT q " ∅. Denote by V the bounded connected component of CzΓ, and suppose that σpT q X V consists in finitely many eigenvalues of T with finite multiplicity. Then for every S P LpY q close enough to T , σpSqXV consists in finitely many eigenvalues of S and the sum of their multiplicities is equal to the sum of the multiplicities of the eigenvalues of T in V .

Proof of Lemma 3.3. Fix T P NS I pXq. Consider Γ a conjugation-invariant, rectifiable, simple closed curve in C, not intersecting σpT q, such that, denoting by V the bounded connected component of CzΓ, we have V X σpT q " I X σpT q, and I X p´}T C } ´1, }T C } 1q Ď V . This last condition implies that for S P NS I pXq close enough to T , we have σpSq X I Ď V .

Denote by n the sum of the multiplicities of the eigenvalues of T that are in I. By Proposition 3.4, for S P NS I pXq close enough to T , the sum of the multiplicities of the eigenvalues of S that are in V is equal to n. By invariance of the spectrum of S under conjugation, the sum of the multiplicities of the eigenvalues of S that are in V zI is even. Thus, for such an S, the sum of the multiplicities of the eigenvalues of S that are in I has the same parity as n, as wanted. Lemma 3.5. Let X be a real HI space having type C or H. Then p R : NS R pXq Ñ Z{2Z is constant.

Proof. Let S, T P NS R pXq; we show that p R pSq " p R pT q. We first claim that one of the segments rS, T s and rS, ´T s is contained in NS R pXq. If not, then both of their direct images under π X , that are respectively the segments rπ X pSq, π X pT qs and rπ X pSq, ´πX pT qs, intersect R, so there exist λ, µ P p0, 1q such that both u :" λπ X pSq `p1 ´λqπ X pT q and v :" µπ X pSq ´p1 ´µqπ X pT q are elements of R. Hence, p1´µqu`p1´λqv p1´µqλ`p1´λqµ " π X pSq P R, contradicting the assumption that S P NS R pXq.

Since p R pT q " p R p´T q, we can assume, without loss of generality, that the segment rS, T s is contained in NS R pXq; hence, the mapping p R is defined on this whole segment.

Since it is locally constant, and since locally constant mappings on connected sets are constant, we deduce that p R pSq " p R pT q.

As a consequence, if X is a real HI space having type C or H, we can define ParpXq P Z{2Z as the unique value taken by p R on NS R pXq (observe that in the case of a space of type R, this definition would make no sense since in this case, NS R pXq is empty). ParpXq will be called the parity of X. As it will turn out (see Proposition 4.3 below), this notion of parity coincindes with the one defined by Ferenczi and Galego in [START_REF] Ferenczi | Even infinite-dimensional real Banach spaces[END_REF] (Definition 2.3).

Observe that, for X a real HI space of type C or H, NS p´8,0q pXq and NS p0,`8q pXq form an open cover of GLpXq. Their intersection is precisely NS R pXq X GLpXq, and for every T P NS R pXq X GLpXq, we have p p´8,0q pT q `pp0,`8q pT q " p R pT q " ParpXq. So the following conditions correctly define a mapping Par : GLpXq Ñ Z{2Z:

• ParpT q :" p p´8,0q pT q for T P NS p´8,0q pXq;

• ParpT q :" ParpXq ´pp0,`8q pT q for T P NS p0,`8q pXq.

ParpT q will be called the parity of the operator T . Lemma 3.6. Let X be a real HI space having type C or H. Then Par : GLpXq Ñ Z{2Z is locally constant.

Proof. It is enough to show that it is locally constant on the two elements of the open cover tNS p´8,0q pXq, NS p0,`8q pXqu of GLpXq. This directly follows from the definition of ParpT q on each of these open sets and from Lemma 3.3.

Proof of Theorem 3.1. If the space X has type C or H, the result directly follows from Lemma 3.6, and for the fact that ParpId X q " 0 and ParpRq " 1 for every reflection R of X.

The case of spaces of type R has already been treated in [2], but we reproduce the proof here for completeness of this paper. If X has type R, then π X ae GLpXq takes values in Rzt0u. This implies that the image by π X of the connected component C of the identity in GLpXq is entirely contained in p0, `8q; in other words, C Ď NS p´8,0q pXq. Hence, by Lemma 3.3, p p´8,0q is constant on C; and its value is necessarily p p´8,0q pId X q " 0. Knowing that for a reflection R, we have p p´8,0q pRq " 1, this shows that no reflection belongs to C.

A complete description of the connected components of GLpXq

In this section, using tools developped in Section 3, we provide a complete description of the connected components of GLpXq, for X a real HI space. Our results will, in particular, imply Theorem 2.5. We start with some preliminary results.

• GL 0 pXq :" tT P GLpXq | π X pT q ă 0 and p p0,`8q pT q " 0u containing ´Id X ;

• GL 1 pXq :" tT P GLpXq | π X pT q ă 0 and p p0,`8q pT q " 1u, containing all antireflections.

Proof. The fact that GL 0 pXq, GL 0 pXq, GL 1 pXq, and GL 1 pXq contain respectively Id X , all reflections, ´Id X , and all antireflections, comes directly from the definitions. In particular, these four sets are nonempty. Now observe that every connected component of GLpXq is entirely contained in one of these four sets. Indeed, if C is such a connected component, then the continuity of π X shows that C is either entirely contained in GL 0 pXq Y GL 1 pXq, or in GL 0 pXq Y GL 1 pXq; and if, for instance, we are in the first case, then the local constancy of p p´8,0q that C is either contained in GL 0 pXq, or in GL 1 pXq. Since, by Theorem 1.4, GLpXq has at most four connected components, it follows that the sets GL 0 pXq, GL 0 pXq, GL 1 pXq, and GL 1 pXq are exactly the connected components of GLpXq.

Theorem 4.5. Let X be an even real HI space. Then GLpXq has exactly two connected components, listed below:

• GL 0 pXq :" tT P GLpXq | ParpT q " 0u, containing Id X and ´Id X ;

• GL 1 pXq :" tT P GLpXq | ParpT q " 1u, containing all reflections and all antireflections.

Proof. The facts that GL 0 pXq contains Id X and ´Id X , and that GL 1 pXq contains all reflections and all antireflections, are direct consequences of the definition of these sets and of Proposition 4.3. The local constancy of Par on GLpXq (see Lemma 3.6) shows that every connected component of GLpXq is entirely contained in one of these two sets. The combination of Lemma 4.1 and Theorem 1.4 shows that GLpXq has at most two connected components. Those are necessarily GL 0 pXq and GL 1 pXq.

Theorem 4.6. Let X be an odd real HI space. Then GLpXq has exactly two connected components, listed below:

• GL 0 pXq :" tT P GLpXq | ParpT q " 0u, containing Id X and all antireflections;

• GL 1 pXq :" tT P GLpXq | ParpT q " 1u, containing ´Id X and all reflections.

Proof. The proof is similar to this of Theorem 4.5.

The two last results show that the parity of an operator plays a similar role as the sign of its determinant in finite-dimension. More precisely, if X is a finite-dimensional normed space, then the mappings p I , for I an interval of R, can be defined on the whole LpXq, and ParpXq and ParpT q for T P GLpXq can thus be correctly defined. We have that for all T P GLpXq, e iπ ParpT q is exactly the sign of the determinant of T . Moreover, Proposition 4.3 and Theorems 4.5 and 4.6 are still valid in this case, replacing even and odd by even-dimensional and odd-dimensional, respectively. Some results of this paper have been independently proved by Maurey [START_REF] Maurey | Théorie spectrale et opérateurs sur un espace HI réel[END_REF], in particular Theorems 3.1 and 4.2. In this text, he also proves some results about the existence of quaternionic structures on a real HI space and on its finite-codimensional subspaces, using quite different methods.

Acknowledgments. The author would like to thank Bernard Maurey for very interesting questions and discussions on this topic; research that led to this paper was initiated by one of his questions. The author would also like to thank Valentin Ferenczi for advice on this topic, and the anonymous referee for useful comments and corrections.

The author was supported by FWF Grant P29999.

Proposition 4.3. Let X be a real HI space having type C or H. Then X is even iff ParpXq " 0, and is odd iff ParpXq " 1.

Proof. If X is even, then Id X and ´Id X are homotopic in GLpXq by Lemma 4.1. By Lemma 3.6, it follows that ParpId X q " Parp´Id X q. We know that ParpId X q " p p´8,0q pId X q " 0, and that Parp´Id X q " ParpXq ´pp0,`8q p´Id X q " ParpXq, so it follows that ParpXq " 0.

A similar argument shows that if X is odd, then ParpXq " 1. Since by Theorem 4.2, every real HI space of type C or H is either even, or odd, we deduce that the two implications we just proved are actually equivalences.

We now describe the structure of the connected components of GLpXq, for X a real HI space, in the three cases defined by Theorem 4.2, in the three theorems below. Their combination, along with Theorem 4.2, implies Theorem 2.5. Theorem 4.4. Let X be a real HI space having type R. Then GLpXq has exactly four connected components, listed below:

• GL 0 pXq :" tT P GLpXq | π X pT q ą 0 and p p´8,0q pT q " 0u, containing Id X ;

• GL 1 pXq :" tT P GLpXq | π X pT q ą 0 and p p´8,0q pT q " 1u, containing all reflections;


Philippe Lasaygues

Régine Guillermin

Jean-Pierre Lefebvre

Distorted Born diffraction tomography applied to inversing the ultrasonic field scattered by a non-circular infinite elastic tube

Keywords: Distorted Born Diffraction Tomography PACS numbers: 43.20.Fn Ultrasonic tomography, Born approximation, distorted Born iterative method, diffracted field, quantitative imaging, non-circular elastic tube

come

INTRODUCTION

The results of studies on the ultrasonic characterization and imaging of elastic tubes have many potential technical applications in fields such as medicine, geophysics, underwater acoustics and non-destructive testing (NDT). However, ultrasonic wave propagation is greatly perturbed by the difference in the acoustic impedance (acoustic impedance contrast) between the scatterer and the surrounding medium (soft tissues, water or coupling gel), which results in considerable parasite events such as the refraction, attenuation and scattering of the waves.

Many authors have dealt in the past with the ultrasonic characterization and imaging of elastic tubes. Their main aim has usually been to assess the thickness of the tube (as part of its geometrical conformation) and to calculate the speed of sound crossing the shell. To study the shape of elastic tubes, a sinogram (giving the diffracted angle vs. time) can be obtained from experimental or numerical data recorded in the plane perpendicular to their generator using radial or cross-section methods of measurement and imaging processes. In the present approach, this problem is viewed in terms of the scattering of ultrasonic waves by a solid cylindrical cavity. Ultimately, the aim is to be able to solve inverse scattering problems in the case of tubes, but for this to be possible, it is necessary to solve the corresponding forward problem. Several methods have been applied so far to the forward problem, including integral equation methods, hybrid FEM methods, and the geometrical theory of diffraction. The latter method provides asymptotic approximations for diffracted fields, which are valid at large distances from the diffracting body.

Actually all these approaches involve the "classical" problem of minimization of the differences between modeling data and measurements. Several strategies can be used for this purpose, which make it possible to model the forward problem and the inverse problem simply, efficiently and accurately. Based on the first-order Born approximation [START_REF] Born | Principles of Optics[END_REF], ultrasonic tomography [START_REF] Roberts | Reflection mode diffraction tomography[END_REF] is one of these strategies [START_REF] Kak | Computerized tomography with X-Rays, emission, and ultrasound sources[END_REF], which is known to be a potentially valuable method of imaging objects with a similar acoustical impedance to that of the surrounding homogeneous medium (in soft tissue characterization, [START_REF] Hiller | Ultrasound computerized tomography using transmission and reflected mode: application to medical diagnosis[END_REF] and underwater acoustics [START_REF] Howe | Recent advances in acoustical tomography[END_REF]). But difficulties arise when it is proposed to obtain quantitative tomograms using acoustical parameters such as the velocity or the attenuation of the wave, or, tomograms of more highly contrasted media (in the case of hard tissues [6 , industrial process tomography [START_REF] Eberhard | Ultrasonic tomography for nondestructive evaluation[END_REF][START_REF] Lasaygues | NDE Application of ultrasonic reflexion tomography[END_REF], etc.).

Finding solutions here involves either using iterative schemes [START_REF] Lasaygues | Progress toward in vitro quantitative imaging of human femur using compound quantitative ultrasonic tomography[END_REF] and/or performing extensive studies on the limitation of the first-order Born approximation [START_REF] Slaney | Limitations of imaging with first-order diffraction tomography[END_REF].

In this paper, we describe the theoretical foundations of first-order Born tomography, recall the limits of this method when dealing with highly contrasted scatterers (although the local fluctuations of the acoustical characteristics of the cross-section will be weak in such cases), and present a high-order Born tomographic method named distorted Born diffraction tomography [START_REF] Lu | Image reconstruction with acoustic measurement using distorted Born iteration method Ultrasonic Imaging[END_REF][START_REF] Chew | Waves and fields in Inhomogeneous Media[END_REF]. Lastly, the performances and limitations of both tomographic approaches are compared in the basis of experimental data.

The theoretical development of the first-order Born tomographic method is asymptotic, as described using the Green's function in the case of a homogeneous medium. The unknown object function, which has been extended to the case of weakly contrasted and weakly heterogeneous media, is linearly related to the measured field via a Fourier transform, and the inverse problem is related to the filtered back-projections algorithm [START_REF] Lefebvre | La tomographie ultrasonore en réflexion[END_REF]. In the case of more highly contrasted media, this strategy was adapted experimentally to take into account physical phenomena such as the wave refraction, where the problem can be reduced to the study of a fluid-like cavity buried in an elastic cylinder surrounded by water. Since this iterative experimental method, which is known as compound ultrasonic tomography, gives excellent results, the only limitations here are the heavy data processing required and the complex acoustical signals resulting from the multiple physical effects involved.

Nonlinear inversion methods with higher-order levels of approximation have therefore been investigated, including the distorted Born iterative method [START_REF] Haddadin | Multiple frequency distorted Born iterative method for tomographic imaging[END_REF][START_REF] Haddadin | Imaging strongly scattering media using a multiple frequency distorted Born iterative method[END_REF], which is generally applied to solving electromagnetic [START_REF] Tijhuis | Multiple-frequency distortedwave Born approach to 2D inverse profiling[END_REF][START_REF] Tijhuis | Theoretical and Computational Aspects of 2-D Inverse Profiling[END_REF][START_REF] Belkebir | Special section: Testing inversion algorithms against experimental data[END_REF] and optical [START_REF] Ye | Modified distorted Born iterative method with an approximate Fréchet derivative for optical diffusion tomography[END_REF] inverse scattering problems. However, very few ultrasonic experimental reconstructions are available in the literature [START_REF] Guillermin | Imaging an object buried in the sediment bottom of a deep sea by linearized inversion of synthetic and experimental scattered acoustic wavefields[END_REF].

The ultrasonic distorted Born diffraction tomography approach based on this iterative method makes it possible to obtain quantitative images. Our theoretical development is algebraic and the iterations are performed numerically by solving the forward and inverse problems at every iteration after calculating an appropriate Green's function; the previous iteration serves in each case to define the surrounding medium with a variable background. Contrary to what occurs with the original approach, distorted tomography requires only a single series of (experimental or numerical) data.

II.

CONFIGURATION AND STATEMENT OF THE PROBLEM

Physical considerations

Since the acoustic impedance of elastic tubes is more highly contrasted than that of the surrounding water (the scatterer is immersed in a water tank), the ultrasonic propagation is perturbed here by the refraction, attenuation and scattering of the waves [START_REF] Anderson | The application of ray tracing towards a correction for refracting effects in computed tomography with diffracting sources[END_REF]. The crosssections of the tube were taken to be isotropic. This scatterer supports the propagation of more complex waves, such as those occurring in elastic volumes, but in this study, it will be modeled like "fluid", since only compressional P-waves and P-P diffraction scattering are taken into account.

Since the frequency range used in the cross-section imaging is [0.15 -1] MHz, the wavelengths occurring in the tube -typically 2 to 20 mm at compression wave velocities of between 2000 and 3000 m/s -are much larger than the macroscopic porosity and the micro-structural scale of the shell ( nm). Our tubes will therefore be assumed to be weakly heterogeneous (i.e., to consist of a homogenized equivalent medium) and the ultrasonic waves in the shell will be less strongly disturbed. The latter assumption is necessary to be able to introduce a linearized propagating theory and to use an asymptotic approximation. On the other hand, the wavelength ([1.5, 10] mm) measured in water

( s m c o / 1500 
) was similar to the diameter ( 10 mm) of the object and the vibration mode was resonance.

2.

Geometrical considerations

The canonical 2-D geometry we have adopted here is depicted in Figure 1 ) is assumed to be constant everywhere in o  . It will be assumed that an a priori value is used for  1 ( r  ).

The aim of this study is to determine the radii and c 1 ( r  ). However, we assume that the radii are restricted to a limited set of values.

III. ULTRASONIC DIFFRACTED TOMOGRAPHY

The aim of ultrasonic diffraction tomography is to reconstruct the spatial distribution of some geometrical and physical parameters of an object from scattered ultrasonic measurements.

These measurements are carried out on variably densely spaced sets of transmitter and receiver positions and the frequencies of an interrogating wave. We are then faced with a forward scattering problem, i.e., predicting the pressure field when the scattering medium and the incident field are assumed to be known, as well as with the inverse scattering problem, i.e., retrieving the parameters of the medium from the measured or simulated incident and scattered fields. This inverse problem is non-linear and ill posed [START_REF] Bui | Introduction aux problèmes inverses en mécanique des matériaux[END_REF][START_REF] Wirgin | Some quasi-analytic and numerical methods for acoustical imaging of complex media[END_REF], and there is generally no single solution. It is necessary to find a way of eliminating solutions, which do not correspond to reality.

Basic ultrasonic diffraction tomography principles have by now been clearly established in the case of weakly varying media such as low-contrast structures, i.e. almost homogeneous media [START_REF] Hiller | System Analysis of Ultrasound Reflection Mode Computerized Tomography[END_REF].A constant reference medium can therefore be chosen (i.e., approximations will be made with a constant background). The scattering problem can be linearized by using the first-order Born approximation, and if the Green's function of the unperturbed problem (the If the contrast between the media increases, however, the first-order Born approximation will no longer be valid and we will have to resort to other strategies. The first strategy adopted in this case [START_REF] Ouedraogo | Contrast and velocity ultrasonic tomography of long bones[END_REF][START_REF] Lasaygues | Progress toward in vitro quantitative imaging of human femur using compound quantitative ultrasonic tomography[END_REF] consists in iteratively correcting the measured or simulated data, depending on the reflection and refraction behavior of the waves propagating through the water/object interface. This strategy, which is known as compound tomography, makes it possible to use the Born approximation, by, "virtually" modifying the Green's function of the scatterer at each iteration, i.e. in each experiment. The disadvantage of this procedure is that as many experiments as iterative steps have to be carried out.

Our second strategy involves the algebraic inversion of the scattered field, based on the distorted Born iterative method, using iterative numerical steps and just one experiment.

A.

First-order Born tomography

In the case of our configuration, we take o  to denote the known part (the background) identified by the density  0 and the velocity c 0 , and 1  to denote the unknown part (the perturbation) identified by the (variable

) density   r  1 
and the (variable

) velocity   r c  1 .
In the case of weak scattering (  

r  1    r c  1   0 c 0 )
, the temporal equation that best describes the acoustic propagation/diffusion processes occurring in the medium (including the boundary and Sommerfeld conditions) results from the Pekeris equation and is given by: Equation 1:

a p a a a p S p k k p p k grad . log grad ) ( 0 1 2 1 2 0 2 0                        
where a p is the acoustic pressure, 0 k and ) r ( 1 k are the complex wave numbers.

Upon introducing

) r ( ) r ( - - ) r ( ) r ( 1 2 0 2 1          k k f , the contrast function, 2 / 1 0 1 with ,                a p p and p S q  
, we obtain : Equation 2:

q p f p k p ) r ( 2 0     
If the Green's function of the non-perturbed problem is known, we can determine the total pressure field in  o using the Lippmann-Schwinger integral equation:

Equation 3: ' r ) ' r ( ) ' r ( ) ' r , r ( ) r ( ) r ( 0        d p f G p p RO i   
To determine the pressure in the heterogeneous region (RO), the Lippmann-Schwinger equation must be solve but the problem is a non-linear. The intuitive solution consists in linearizing the problem using the Born approximation. We assume that the diffracted field is negligible in comparison with the incident field in RO, which is valid in the case of weakly heterogeneous media: Equation 4:

0 ) r ( ) r ( ) r ( ) r ( 1 2 0 2 1             k k f
Upon introducing f 0 = f under the Born approximation, we obtain the following forward problem:

Equation 5: ' r ) ' r ( ) ' r ( ) ' r , r ( ) r ( ) r ( ) r ( 0 0         d p f G p p p i RO i s    
Let us assume that the incident field is a plane wave (in the far field of the transducer): Equation 6:

) r . n exp( ) r , r ( E 0 E     ik A p i 
where E n  is the incident normal vector.

Within the framework of the Born approximation, the scattered field measured by a receiver at point R r  is:

Equation 7: r ) r ( ) r ( ) r r ( ) r ( 0 R 0 R       d p f , G p i RO s   with Equation 8:     r . n exp 4 r exp r 2 4 ) r , r ( R 0 R 0 R 0 R 0       ik ik k i G     
where R n  is the observation normal vector.

The far field Born approximation of the scattered field is given by: Equation 9:

    r r n exp ) r ( r n exp 4 r exp r 2 4 ) r ( 0 0 0 0 0          d ik A f ik ik k i p RO s . . E R R R R             Equation 10:     r r . ) n n ( exp ) r ( 4 r exp r 2 4 ) r ( E R 0 0 R 0 R 0 R         d ik f ik k i A p RO s       
If we perform measurements on a circle surrounding the object, we obtain:

Equation 11:   r r . ) n n ( exp ) r ( ) r ( E R 0 0 R       d ik f B p RO s    
where B is known and depends only the frequency when the radius of the circle is given. This equation is the 2-D Fourier transform of the contrast function f 0 . The inverse problem, i.e. the restitution of f 0 , is an inverse Fourier transform. The spectral aperture of the frequency space depends on

) n n ( E R 0    k
and is therefore not complete. This aperture is then experimentally limited at several points inside a limited bandwidth/range/field depending on the frequency range of the transducers. Furthermore, the coverage of the Fourier plane of the object by the data obtained is only partial and it is therefore necessary to introduce interpolations between the projections to be able to use a classical 2-D Fourier transform. Our algorithms are based on a special angular and linear scanning process performed on the radial measurements of the spatial Fourier transform [START_REF] Lefebvre | Born ultrasonic reflexion tomography: some limits and improvements[END_REF]. It therefore suffices to cross the Cartesian coordinates with the polar coordinates in the Fourier space to be able to carry out the reconstruction without having to interpolate the object. This is done using the algorithm of reconstruction by summation of filtered back-projections [START_REF] Kak | Reconstruction from Projections: Application in Computerized Tomography[END_REF][START_REF] Devaney | A filtered backprojection algorithm for diffraction tomography[END_REF].

B. Limits in the case of a high-impedance contrast scatterer

In the ultrasonic characterization of high-impedance contrast scatterers, it is not possible to use the first-order Born approximation with a constant background. Bodies of this kind are quite heterogeneous, and their acoustical characteristics are quite different from those of the surrounding soft medium.

If we assimilate the present tube imaging problem to the identification of a water-like cavity present in an elastic cylinder, and adopt the previous acoustical assumptions about the weak heterogeneity of the shell and the propagation, the first-order Born approximation with a variable background can be used if the background is defined as the fixed, solid part without any hollow (the water) and the perturbation, i.e. the object to be reconstructed, is defined as the cavity. But there will be a bias in the thickness measurements and shape imaging. In fact, "classical" reconstruction tools based on the first-order Born approximation assume the occurrence of straight ray propagation, whereas a large deviation of the incident beam occurs at the interface between the water and the tube, due to the difference between the acoustic properties of the two media. Due to the refraction effects, diffraction occurs on the irregularly shaped scatterer. Our approach was designed first to cancel out the refraction effects by using a specific experimental set-up in order to impose straight ray propagation inside the shell during high-frequency (> 500 kHz) tomographic measurements [START_REF] Lasaygues | Progress toward in vitro quantitative imaging of human femur using compound quantitative ultrasonic tomography[END_REF][START_REF] Ouedraogo | Transmission ultrasonic tomography of human bone: Limit and applications[END_REF]. This procedure requires exactly determining the position and the geometrical shape of the scatterer. Our initial attempts on these lines have been improved using signal and image processing methods [START_REF] Lasaygues | Deconvolution and wavelet analysis on ultrasonic reflexion tomography[END_REF], but this strategy is still too complicated to be suitable for experimental purposes.

The solution proposed in this paper consists in iteratively calculating the Green's function of the inhomogeneous problem beginning with the first-order Born approximation previously defined. The first perturbation operator is the same as with a homogeneous background and all the perturbation parameters are defined in terms of the characteristics of the medium. The first results obtained were similar to those obtained in the case of a constant background, with various effects at the water/shell interface. In the next step, the inverse problem is solved using a procedure to solve the forward problem whereby the contrast function f of the scatterer has been determined at the previous step.

C. Distorted Born diffraction tomography

The distorted Born diffraction tomography is based on the distorted Born iterative (DBI) method. This approach involves performing iterative minimization on the discrepancy between actual measurements and the integral representation of the field obtained by solving a forward problem with an estimated contrast function.

The integral expression is discretized using the moment method. The inhomogeneous region (RO) is replaced by an array of elementary square cells Ω pq (Figure 2) which are small enough for ) r (  k and ) r (  p to be presumably constant therein. This leads to a discretized form of Equation 3: 12. ) ( pq X p  is determined using the collocation method.

Equation 12: ' r ) ' r , r ( ) X ( ) X ( ) r ( ) r ( 1 1        d G p f p p pq pq n p n q pq i       
Writing Equation 12 for all points

  X r   n α ,..., 2 , 1 ,  
, we obtain a system of 2 n linear equations in terms of 2 n unknowns:

Equation 13: ) X ( ' r ) ' r , X ( ) X ( ) X ( 1 1 pq n p n q pq q p i p d G f p pq                          
This can also can be expressed as the linear matrix equation:

P F P i ) (  
where

               nn i i i X p X p P   . 11 ,                nn X p X p P   . 11
the incident and total pressure fields and

               nn X f X f F   .
11 the contrast function.

In the inverse problem, the incident field is known, the total field is known at some M measurements points ,

j r  M j ,..., 2 , 1 
; and we have to determine the unknown contrast function F in (RO). Via Equation 12, the inverse problem can be expressed as a non-linear system of M equations in terms of the

... 1 , ' r ) ' r , r ( ) X ( ) X ( ) r ( j 1 1 j              
This can also can be written in the form:

F P Q P s ) (  , with               M s s s r p r p P   . 1
To solve this equation, we use the distorted Born iterative method (DBI), which consists in solving the problem using successive linear estimates and updating the total field and the Green's function at each iteration. The initial guess is provided by the first-order Born solution 0 F . Then, if the l-order solution l f is assumed to be known, the (l+1)-order solution verifies [START_REF] Chew | Waves and fields in Inhomogeneous Media[END_REF][START_REF] Haddadin | Ultrasound inverse scattering for tomographic imaging self-focusing array[END_REF]:

Equation 15:     d ) , r ( G ) X ( p ) X ( f ) X ( f ) r ( p ) r ( p n p n q pq j l pq l pq l pq l j l s j s          1 1 1 ' r ' r -        
Where l G is the inhomogeneous Green's function of the object l f immersed in  0. l p and p l s are the total and scattered pressure fields for this scatterer l f illuminated by a source point placed at E (Figure 1). These quantities are calculated using the integral representation [START_REF] Chew | Waves and fields in Inhomogeneous Media[END_REF] and by solving a forward problem for the known scatterer l f . The DBI procedure can be written in condensed form as follows: Equation 16:

                  ] [ ) ( )] ( [ ) ( 1 1 0 0 l l l l l s s i l l i s i F F P Q P P P F P F P Q P P P
The iterative procedure is continued until

l l l F F F  1
is smaller than  (which is taken to be % 5 ).

It is worth noting that l

Q is generally a non-square and ill-conditioned matrix. A mean-square solution of the inverse problem is obtained by performing a Singular Values Decomposition (SVD) associated with a Tikhonov regularization procedure [START_REF] Hansen | Regularization tools: a Matlab package for analysis and solution of ill-posed problems[END_REF].

IV. WATER TANK MEASUREMENTS

The experimental setup used here was designed for performing diffraction measurements. The acoustic device involved is able to move with various degrees of freedom and used to analyze samples in all directions. The positions of the target and the transducers can be adjusted. In particular, the operator can impose precise rotations and translations on the transmitter and the receiver.

Ultrasounds were generated using a pulse receiver device and piezo-composite wide-band transducers. The nominal frequency of the transducers was 0.25 MHz and the usable bandwidth of the transducers was approximately 135 to 375 kHz.

The object was placed in the center of the measuring system. Ultrasonic measurements were performed in water at room temperature. The sector scanned was 360° with both transducers, and the angular increment was "5-degree" (72 x 72 signals). Transmitted and received ultrasound Radio-Frequency (RF) signals were digitized (8 bits, 20 MHz) using an oscilloscope, stored (4000 samples) on a computer via a General Purpose Interface Bus for off-line analysis.

A. Simulations

We first tested the efficiency of the distorted Born iterative algorithm on simulated values generated using a boundary integral equation (BIE) method [ 32 ]. The transmitter was rotated in"5-degree" steps and the receiver, in 10-degree steps. Five frequencies were used for the simulations: 150 kHz, 180 kHz, 250kHz, 300 kHz and 350 kHz. Results are shown in Figure 7. In this simulation, we assumed the densities  1 and  o to be identical.

The contrast function f therefore depended only on the compressional wave velocity.

(a) (g -h) 19 iterations at 250 kHz; (i -j) 13 iterations at 300 kHz; (k -l) 12 iterations at 350 kHz The initial frequency was chosen so that the first-order Born solution does not include any artifacts. Previous studies [START_REF] Haddadin | Ultrasound inverse scattering for tomographic imaging self-focusing array[END_REF] have shown that if the phase shift resulting from the presence of the scatterer is greater than , the model will include large artifacts. We therefore chose a frequency such as kHz 150 2 c 0 1

(b) (c) (d) (e) (f) (g) (h) (i) ( j ) ( k ) ( l )
1 0    c c d c f
(where d is the largest dimension of the scatterer). The initial solution is given by the first-order Born estimate. At each frequency, the initial solution was the final result of the iterations performed at the previous frequency. The numerical simulations were found to be satisfactory on the qualitative level (as regards the shape, dimensions, and location ). From the quantitative point of view (velocity measurements), the algorithm was found to overestimate the velocity, which depends strongly on the regularization parameter used for the mean-square inversion procedure. A study is now in progress with a view to optimizing this parameter [START_REF] Hansen | Numerical tools for analysis and solution of Fredholm integral equations of the first kind[END_REF][START_REF] Hansen | Analysis of discrete ill-posed problems by mean of the L-curve[END_REF].

B. Experimental

We then compared the performances of the first-order Born tomography and the distorted Born diffraction tomography with a set of experimental data. With the transducers used, which had a nominal frequency of 250 kHz, the frequency range was limited to the [135,375] kHz bandwidth. The image resolution was therefore limited by frequency range available. Figure 8 and Figure 9 give the results obtained using the classical back-projection algorithm.

In this algorithm based on the first-order Born approximation, all the available frequencies between 135 kHz and 375 kHz were used. Because of the large impedance contrast and since the wavelength was similar to the size of the scatterer, the image is bound to include considerable artifacts and the resolution is not satisfactory. In addition, the image obtained here does not give any quantitative information about the speed of sound. The first-order Born tomography was based on a single-scattering theory and there will be a bias in the assessment of the shell thickness. It only gives a qualitative picture of the "perturbation", i.e. of the external boundaries.

Figure 10 gives the results obtained using the distorted Born diffraction tomography. First it is worth noting that in our experimental data, differences in the density were observed between the target and the surrounding medium that are not taken into account in the inversion scheme. It can be seen here that the resolution and the quality of the modeled contrast improved gradually, as with the inversion of the numerical data (Figure 7). The final result of the iteration process was satisfactory, even as far as the value of the contrast was concerned. As in the case of the model, the geometry was fairly accurate, whereas the velocity was less accurately estimated, with a relative error of about 5 %. As in the case of the simulations, this was mainly due to the regularization parameter selected.

VI.

CONCLUSION

This paper deals with the two-dimensional imaging of a non-circular elastic tube using distorted Born iterative tomography. The scope of the ultrasonic Born tomography was thus extended from low impedance contrast media (the classical domain) to higher impedance contrast domains, using a correction scheme with the following features: the wave propagation and the associated Green's function of the medium are determined at the previous step. The ultrasonic propagation is greatly perturbed by the difference in the acoustic impedance between the cross-section of the scatterer and the surrounding homogeneous reference material, which generates large parasite processes. The strategy used to solve the problem was based on the distorted Born iterative method, where the iterations were performed numerically, based on a single experimental measurement. This strategy did not require any a priori information to give a reasonable contrast, but this information was more necessary for the iteration procedure. The examples given in this paper involve a noncanonical homogeneous shape, and the results were based on numerical simulations and experiments. The results obtained here are most promising because the geometrical and physical parameters of the scatterer were accurately reconstructed.

Various way of improving this method will now be investigated. One of the most important aspects of the inversion scheme is the choice of regularization procedure. In this paper, the Tikhonov regularization method was used, and work is in progress on how to choose the regularization parameter. This classic regularization method yields promising good results, but one of its drawbacks is that the whole image is smoothed with this method, and the edges are not accurately reconstructed. To solve this problem and improve the quality of the images (especially with highly contrasted targets), other "edge-preserving" regularizing methods will have to be employed [START_REF] Charbonnier | Deterministic edgepreserving regularization in computed imaging[END_REF][START_REF] Lobel | A new regularization scheme for inverse scattering[END_REF]

  . Let us take an infinite space o  in a linear homogeneous uncompressible fluid (a water-like substance, for example), named the background. Let us also take an elastic cylindrical cavity with an arbitrary cross-section 1  with generators parallel to the z-axis, immersed in the surrounding medium o  and illuminated by a point source at the point E. We define the position vector r  in the xy-plane as follows:  o ( r  ) denotes the density of the surrounding fluid-like medium and the hollow area, c o ( r  ) denotes the velocity of the propagating wave,  1 ( r  ) denotes the density of the cavity and c 1 ( r  ) denotes the velocity of the propagating longitudinal wave.

Figure 1 :

 1 Figure 1: Non-circular cylindrical elastic cavity scattering: geometrical and acoustical configuration

  background o  ) is known, the forward problem can be solved with the Lippmann-Schwinger integral equation and the far field solution of the equation can be calculated. The inverse problem is then solved by performing an asymptotic development and the "classical" tomographic algorithm will yield the perturbation with respect to the reference problem. This leads to a linear relation between the object function (or contrast function) and the scattered field, particularly in the far-field (2-D or 3-D Fourier transform), which makes it possible in principle to reconstruct the object function in almost real time based on a sufficiently large set of scattering data.

Figure 2 :

 2 Figure 2: Discretization of the object In the forward problem, the contrast function and the incident wave are known. The first step consists in finding the pressure field ) ( pq X p  in (RO). The scattered pressure field can then be calculated elsewhere via Equation12. ) ( pq X p  is determined using the collocation method.

Figure 3 :Figure 4 :Figure 5 :

 345 Figure 3: Water tank and experimental configuration

Figure 6 :

 6 Figure 6: NEUKADUR ProtoCast 113 TM resin tube dimensions (left),

Figure 7 :

 7 Figure 7: Ultrasonic distorted Born iterative tomography of the non-circular elastic tube. Simulated data, left: tomograms 40 x 40 pixels, right: x-pixel profiles drawn at y = 0 mm; (ab) initial solution at 150 kHz; (c -d) 17 iterations at 150 kHz; (e -f) 11 iterations at 180 kHz;

Figure 8 Figure 8 :Figure 9 :Figure 10 :

 88910 Figure8shows the first-order Born reflection tomography of the scatterer. No corrections or signal processing have been carried out on these images. The sinogram corresponds to the modulus of the Hilbert transform of the initial RF-signals. Only one transducer was used, and it was placed in 72 positions around the object with a 5-degree step. Figure9shows the first-


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

Loïc Le Marrec

tic Cy lin dri cal Tube

an In fi nite Elas

IN TRO DUC TION

The development of ultrasonic characterization and imaging methods for use on elastic cylindrical tubes is of great importance in several medical and technical fields, where these methods are required for bone imaging and nondestructive testing of steel tubes, for example. In the case of elastic tubes showing high impedance contrast with the surrounding medium (soft tissues in medical applications, water or coupling gel in non de struc tive testing), ultrasonic propagation is greatly perturbed, which gives rise to imaging problems.

Many au thors have dealt with ul tra sonic char ac ter iza tion and im ag ing of elas tic tubes. The aim of most pre vi ous stud ies has been to as sess the thick ness of the tube (as part of its geomet ri cal con for ma tion) and to cal cu late the speed of sound (an acous ti cal pa ram e ter) of the compressional wave cross ing the struc ture. To study the shape of elas tic tubes, mea surements can be made in a plane par al lel to the gen er a tor axis (in the z-di rec tion), us ing an 'ax ial or trans verse mea sure ment' pro ce dure, or in a per pen dic u lar plane, us ing a 'ra dial or cross-sec tional mea sure ment and im ag ing' pro ce dure. Our group is par tic u larly in ter ested in the cross-sec tional im ag ing method, which we have ap proached as a prob lem where ul tra -sonic waves are scat tered by a solid cy lin dri cal cav ity. The ul ti mate aim is to find a means of solv ing in verse scat ter ing prob lems in the case of tubes. One of the pre req ui sites to to achieve this aim is to solve the cor re spond ing for ward prob lems. Sev eral meth ods have been ap plied to solv ing for ward prob lems, in clud ing in te gral equa tion meth ods, hy brid fi nite-el ements meth ods and the geo met ri cal the ory of dif frac tion. The lat ter method pro vides as ymptotic ap prox i ma tions for dif fracted fields, which are valid at high fre quen cies and at large dis tances from the dif fract ing body. Sev eral im ag ing strat e gies are pos si ble in this con text but to ob tain real time im ag ing, sim ple, ef fi cient and ac cu rate nu mer i cal meth ods of mod eling the for ward and in verse prob lems are re quired.

Our Ul tra sonic Re flec tion To mog ra phy (URT) method, based on the in verse Born approx i ma tion, is one of these strat e gies. [START_REF] Lasaygues | High res o lu tion low fre quency ul tra sonic to mog ra phy[END_REF] This method has proved to be ac cu rate and gives prom is ing re sults in terms of the shape of the tubes tested (qual i ta tive to mog ra phy). In the case of ma te ri als show ing high acous ti cal im ped ance con trast with the sur round ing me dium, how ever, it is not pos si ble to re con struct the in ner cav ity or to de ter mine the acous ti cal param e ters (such as the ve loc ity and/or the at ten u a tion of the sound waves) with out cor rect ing for wave prop a ga tion through the wa ter/shell in ter face. The main rea son for this prob lem is that the Born ap prox i ma tion solv ing the for ward prob lem is valid for ob jects show ing a weak im ped ance con trast with the sur round ing me dium (high fre quency ap prox i ma tion). Mod el ing an elas tic tube with a model where the acous tic prop er ties are too sim i lar to those of the sur round ing me dium re sults in tomo grams con tain ing geo met ri cal ab er ra tions as far as the di men sions of the shell are con cerned.

In this pa per, a new model based on a ca non i cal ap prox i ma tion is pre sented for solv ing the for ward prob lem. The mod el ing pro ce dure used here is based on the In ter cept ing Ca non i cal Body Ap prox i ma tion (ICBA) method de vel oped by A. Wirgin and his group. [START_REF] Scotti | Shape re con struc tion us ing dif fracted waves and ca non i cal so lu tions[END_REF][START_REF] Wirgin | Some quasi-an a lytic and nu mer i cal meth ods for acous ti cal im ag ing of com plex me dia[END_REF][START_REF] Marrec | Ef fi cient shape re con struc tion of non-cir cu lar tubes us ing broad band acous tic mea sure ments[END_REF] In con trast to straight ray the ory, no lim ited con fig u ra tion of the scat tered field is re quired. The ICBA method con sists in re plac ing the real scat terer at each ob ser va tion point by an ob ject hav ing sim ple geo met ri cal char ac ter is tics and known acous ti cal prop er ties, and the so lu tion of the for ward prob lem has to be ca non i cal (i.e., it has to be pre vi ously de ter mined an a lyt i cally). For ex am ple, in the case of a noncircular elas tic tube, a cir cu lar tube hav ing the same in ner and outer ra dii as the lo cal ra dii of the real ob ject in the mea sure ment di rec tion would provide a suit able model.

Here we deal with the case of an elas tic cy lin dri cal tube, which is in fi nite in the z-di rec tion and is im mersed in a ho mo ge neous fluid, namely wa ter. Note that the real ob ject is not nec essary cen tered or cir cu lar, as re quired in the case of the model used for ICBA. Its cav ity is taken to be weakly het er o ge neous in com par i son with the wave length. In fact, we con sider weak the lo cal fluc tu a tions in the acous tic char ac ter is tics of the cross-sec tion of the scat terer. This is an it er a tive tomographic ap proach re quir ing pre lim i nary a pri ori in for ma tion. We dis tin guish be tween high-level a pri ori knowl edge about pa ram e ters such as the den sity, veloc ity of the waves prop a gat ing in the shell, and thick ness of the 'medullar' ca nal, and low-level a pri ori in for ma tion about pa ram e ters such as the outer ra dius or acous tic pa ram eters of the sur round ing me dium. We will there fore fo cus on de ter min ing the thick ness of the tube by study ing the res o lu tion of the im ages. Other phys i cal pa ram e ters such as the den sities and ve loc i ties are set in this work at their ex act val ues. A sinogram (dif fracted an gle vs. time) based on ex per i men tal mea sure ments and nu mer i cal sim u la tions is drawn up. The cross-sec tional im age of the tube is first de ter mined us ing URT and a fil tered back-pro jections al go rithm. [START_REF] Lefebvre | A linearized in verse prob lem: acous tic im ped ance to mog ra phy of bi o log i cal me dia[END_REF] The de ter mined ex ter nal bound ary of the scat terer is then used as the low-level a pri ori in for ma tion. The sec ond ap proach (that based on ICBA) was de vel oped in or der to op ti mize the in verse prob lem. The ac cu racy of the bound aries and, hence, the thickness is thus im proved. The par tic u lar case of a tube hav ing a sim i lar thick ness to the wavelength of the wave in the wa ter will be stud ied.

The clas si cal ul tra sonic to mog ra phy method is de scribed in sec tion 3. The ICBA ap proach to the for ward and in verse prob lem is pre sented in sec tion 3 and re sults ob tained on nu mer ical and ex per i men tal scat ter ers are given. The re sults pre sented in sec tion 4 are of con sid erable im por tance in the field of ul tra sonic im ag ing.

CON FIG U RA TION AND STATE MENT OF PROB LEM

Phys i cal con sid er ations

The elas tic and acous tic prop er ties of the ma te rial in the cross-sec tion of the tube are taken to be iso tro pic. This elas tic scat terer me di ates the prop a ga tion of com plex elas tic vol ume waves, namely compressional P-waves and shear ver ti cal SV-waves (shear hor i zon tal SH-waves are not taken into ac count here). Since the acous tic im ped ance of the tube contrasts strongly with that of the sur round ing wa ter (the scat terer is im mersed in a wa ter tank), the ul tra sonic prop a ga tion is greatly per turbed by re frac tion, at ten u a tion and scat ter ing of the waves.

6

The fre quency band width used here in the cross-sec tion im ag ing was 0.25-1MHz. The wave lengths oc cur ring in the tube -typ i cally 2 -16 mm at com pres sion waves ve loc i ties rang ing be tween 2,000 and 4,000 m/s -were there fore much greater than the mac ro scopic po ros ity and the microstructural scale of the shell (» 1 mm). The tube was there fore taken to be weakly het er o ge neous (small per tur ba tions of the prop a ga tion) and to con sist of a ho moge nized equiv a lent me dium. It was as sumed that the ul tra sonic wave in the shell was not disturbed by the microstructure. The lat ter as sump tion is nec es sary to be able to in tro duce a linearized prop a gat ing the ory and to adopt an as ymp totic ap prox i ma tion. On the other hand, the wave length (1.5, 6 mm) in wa ter (c 0 » 1,500 m/s) is smaller than the di am e ter (» 10 ± 2 mm) of the ob ject. The ka prod uct, where k = w/c is the wave num ber and a is the mean-radius of the tube, ranges be tween 6and 25 and the con fig u ra tion is there fore nonres o nant.

Geo met ri cal con sid er ations

In this sec tion, it is pro posed to study the ho mo ge neous noncircular solid tube, the phys ical prop er ties of which are known. The ca non i cal 2-D ge om e try con sid ered here is shown in fig ure 1. The sur round ing me dium con sists of a lin ear, ho mo ge neous, uncompressible fluid (a wa ter-like fluid, for ex am ple). The noncircular elas tic cy lin dri cal cav ity has gen er a tors placed par al lel to the z-axis and is im mersed in the sur round ing me dium . We de fine in the xy-plane: r o the den sity and c o the ve loc ity of the prop a gat ing wave in the sur round ing medium and the hol low area, r 1 the den sity of the cav ity and c 1p and c 1t the ve loc ity of the prop agat ing lon gi tu di nal and transversal waves. We take r 1 (q m ) and r 2 (q m ), to de note the ra dius of the ex ter nal and in ter nal bound aries, re spec tively, of the cav ity at the ob ser va tion an gle q m . It will be as sumed that high-level a priori knowl edge is used for r 1 and c 1 . The ex ter nal radius r 1 , and in ter nal ra dius r 2 are un known and the aim of this study is to de ter mine these ra dii. How ever, we as sume that these ra dii are in cluded in a lim ited set of val ues (cor re spond ing to the low level a pri ori in for ma tion).

UL TRA SONIC SCAT TER ING PROB LEM

Here, we deal with the for ward scat ter ing prob lem, i.e., pre dict ing the scat tered field when the scat ter ing me dium and in ci dent field are as sumed to be known, as well as with the in verse scat ter ing prob lem, i.e., re triev ing the bound aries of the me dium from the mea sured (or simu lated) in ci dent and scat tered fields, all the ma te rial prop er ties be ing known. First, we look at the for ward prob lem in or der to ob tain the state equa tion, which is es sen tial to solv ing the in verse prob lem.

Since in verse prob lems are nonlin ear and ill posed, one of the con se quences is that there is no sin gle so lu tion. In most ul tra sonic scat ter ing prob lems, ap prox i mate mod els (such as the first-or der Born ap prox i ma tion or ca non i cal ap prox i ma tion) [START_REF] Wirgin | Some quasi-an a lytic and nu mer i cal meth ods for acous ti cal im ag ing of com plex me dia[END_REF] are used and it is gen er ally nec es sary to find a means of elim i nat ing so lu tions, which do not cor re spond to the real param e ters (Delamare et al. [START_REF] Delamare | Ghosts in Born im ages of a layer probed by acous tic waves[END_REF] have pro posed a tech nique based on Born ap prox i ma tion). To detect the bound aries of a high-con trast cy lin dri cal cav ity, we es tab lish that:

1. The scat ter ing prob lem can be linearized us ing the first-or der Born ap prox i ma tion method. If the Green func tion of the un per turbed prob lem (the sur round ing me dium W 0 ) is known, we can solve the di rect prob lem with the Lippmann-Schwinger in te gral equa tion and the far field so lu tion of the equa tion can be cal cu lated. The in verse prob lem can then be solved us ing a tomographic al go rithm to ob tain a per tur ba tion map with re spect to the ref erence prob lem.

2. A ra tio nal method of im prov ing the res o lu tion con sists in com bin ing ca non i cal ap prox ima tion with the anal y sis of a func tional (cost-func tion) mea sur ing the gap be tween the scattered and pre dicted fields. The func tional cho sen for this pur pose is gen er ally the least-mean square es ti mate ob tained as fol lows: ( ; ; , ) t w r F the es ti mated scat tered field. These fields are func tion of the pul sa tion w, the ob ser va tion point r x m = (r m , q m ) and in ci dent an gle q i , this lat ter pa ram e ter be ing ex pressed through the dif fracted an gle f = q Iq m . The esti mated field de pends on the un knowns (ra dii) it is pro posed to re con struct. We de note these un knowns us ing gen eral term, the vec tor t m .

As a ca non i cal ap prox i ma tion of the in ter ac tions be tween the wave and the body, the Inter cept ing Ca non i cal Body Ap prox i ma tion (ICBA) has proved to be an ef fi cient for ward solver both in and be yond the res o nant do main in the case of full and hol low elas tic cyl inders. We will see be low how an ICBA ex trap o la tion of the nonres o nance dif frac tion can be used to image the shape of the cavity.

Born modeling and reflection tomography

The goal of ul tra sonic to mog ra phy is to re con struct the spa tial dis tri bu tion of some of the geo met ri cal pa ram e ters of an ob ject from scat tered ul tra sonic mea sure ments. These measure ments are car ried out at vari ably-dense sets of trans mit ter and re ceiver po si tions and inter ro gat ing wave fre quen cies.

The in verse scat ter ing prob lem is solved us ing a Born ap prox i ma tion, which gives a partic u larly sim ple and at trac tive lin ear re la tion be tween the ob ject func tion and the scat tered field, par tic u larly in the far-field (2D or 3D Fou rier trans form). This makes it pos si ble, in prin ci ple, to re con struct the ob ject func tion prac ti cally in real time when a suf fi ciently large set of scat ter ing data is avail able.

As ymp totic con sid er ations

The ba sic prin ci ples of URT have by now been clearly es tab lished in the case of weaklyvary ing me dia such as low-con trast struc tures, i.e., al most ho mo ge neous me dia. [START_REF] Hiller | Sys tem anal y sis of ul tra sound re flec tion mode com put er ized to mog ra phy[END_REF] The ref erence me dium cho sen can there fore be a con stant me dium (form ing a con stant back ground), yield ing an In verse Born Ap prox i ma tion (IBA). In the case of our con fig u ra tion, we take W 0 to de note the known back ground ex pressed in terms of the den sity r 0 and the ve loc ity c 0 , and W 1 to de note the un known per tur ba tion (scat terer) ex pressed in terms of the ho mo ge neous den sity r 1 and the ve loc ity c 1 .

Un der weak scat ter ing con di tions (r 1 c 1 » r 0 c 0 ), the fol low ing tem po ral equa tion based on the Pekeris equa tion de scribes the acous tic prop a ga tion pro cesses oc cur ring in the me dium (in clud ing the bound ary and Sommerfeld con di tions):

where p x t m ( , )
r is the acous tic pres sure.

a x c x c c x = - ( ( ) ) / ( ) 1 2 0 2 1 2 r
r is the qua dratic fluc tu a tion of ve loc ity and x( )

r x = log( ( ) / ) Z x Z 1 0
r is the log a rith mic fluc tu a tion of acous tic im ped ance, Z x 1 ( )

r = r 1 1 ( ) ( ) r r x c x .
If the Green func tion of the nonper turbed prob lem is given, we can find the to tal pres sure field in W o us ing the Lippmann-Schwinger in te gral equa tion:

where p x t s m ( , ) de notes the scat tered field in W o given by: UL TRA SONIC RE FLEC TION TO MOG RA PHY OF ELASTIC TUBE 5 Un der backscat ter ing con di tions ( r r n n = -0 ) and within the first-or der Born ap prox i ma tion (sin gle-scat ter ing), the as ymp totic so lu tion of Eq.( 3) is:

) ( ). , ( 2
1 ) ( ). , ( ) , ( ) ( ) , ( . . 1 2 2 2 0 2 2 2 0 m m m m m m m x d a gr t x p d a gr x d a gr t x p d a gr t t x p c x t x p t c r r r r r r r r r r r a x ¶ ¶ a ¶ ¶ - + - = ú û ù ê ë é D + - (2)
where h d ( ) 1 ¥
is the first-or der far-field re sponse of the me dium: where x 3 is the r n 0 axis. In this case, the pa ram e ter a is elim i nated. In the space-fre quency do main, the back scat ter ing trans fer func tion is given by: where is the Fou rier trans form of the log a rith mic fluc tu a tion of the acous tic im ped ance and r r K kn = -2 0 is the wave vec tor. If one takes only the am pli tude of the ech oes in ac count, one ob tains a n r K d ( ) ( , ) $( )

1 0 ¥ = r r w
, where $( ) r K r is the spa tial Fou rier trans form of the re flec tivity of the me dium r( r x). The pur pose of URT is to ob tain re flec tivity im ages from back scat ter ing mea sure ments. Ro tat ing the emit ter-re ceiver trans ducer around the 'ob ject' and emit ting broad band pulses at each po si tion leads to the same sit u a tion as in X-ray to mog ra phy; one ob tains slice-byslice spec tral cov er age of the ob ject spec trum $( ) r K r . Any tomographic re con struc tion al gorithm can there fore be used to per form the re con struc tion. We chose an al go rithm of sum mation of the backprojections of fil tered pro jec tions.

( )

2 0 3 2 2 ' 3 2 3 0 ) 1 ( ) ( 4 , t c x R m d x d x x c t x h = ¥ þ ý ü î í ì = ò r r r x ¶ ¶ (7) ( ) ) ( 2 , 2 0 ) 1 ( K k n H d r r x w - = ¥ (8) ò - = 3 3 . ) ( ) ( ˆR x K i x d e x K r r r r r x x ò ¢ þ ý ü î í ì - + ¢ - - = ÷ ÷ ø ö ç ç è ae ¢ - - 3

To mog ra phy of a high-im ped ance con trast scat terer

In the ul tra sonic char ac ter iza tion of high con trast bod ies, it is not pos si ble to use the IBA with a 'con stant' back ground. For ex am ple, hard bi o log i cal tis sues (such as bones) are quite het er o ge neous and their acous ti cal char ac ter is tics are very dif fer ent from those of the surround ing soft tis sues (or wa ter in vi tro). The prob lem aris ing with the tube-thick ness im aging is how to iden tify a wa ter-like cav ity within an elas tic cyl in der. In this case, an IBA with a 'vari able back ground' can be used. The back ground can be de fined in terms of the fol low ing two parts: on the one hand, a solid part with out hol low/wa ter, and on the other hand, the pertur ba tion, i.e., the ob ject to be re con structed, namely, the cav ity. Based on the acous ti cal assump tions adopted, the shell is weakly het er o ge neous (in com par i son with the wave length) and there fore gives off weak ech oes. IBA can there fore be used to ob tain a good ap prox i mation for the in verse scat ter ing prob lem. The al go rithm of tomographic re con struc tion by sum ma tion of the fil tered backpro jec tions can be used with some sig nal pro cess ing re finements. [START_REF] Lasaygues | As sess ing the cor ti cal thick ness of long bone shafts in chil dren us ing two-di men sional ul trasonic dif frac tion to mog ra phy[END_REF] Due to the high con trast im ped ance, ar ti facts are gen er ated in the im age by the scatter ing pro cess (the Born ap prox i ma tion is a sin gle-scat ter ing the ory). How ever, the main re sult will be pre served: the ba sis un der ly ing the 'per tur ba tion' will be de tected, i.e., a qual ita tive im age of the cav ity will be ob tained.

Wide-band imaging using canonical approximation

An a lyt i cal con sid er ations

In this sec tion, we deal with the ul tra sonic im ag ing of 2-D elas tic me dia based on multifrequency, multiview syn thet i cally-and/or ex per i men tally-gen er ated data. The goal is to de ter mine a geo met ri cal pa ram e ter of the non-cir cu lar elas tic tube cross-sec tion from mea surements of the scat tered pres sure field re sult ing from in ter ac tions be tween the scat terer and the known in ci dent wave. We fo cus on the high-fre quency do main, where the ob ject is larger than the wave length of the ul tra sonic wave (ka > 1).

In ( , ) r and de notes the in ci dent field expressed by: where w is the pul sa tion and r K is the wave vec tor in the sur round ing me dium W o . The acous tic fields are gov erned by the Helmholtz equa tion and the scat tered field sat isfies the usual con di tions (Sommerfeld ra di a tion con di tions, con ti nu ity of pres sure and normal ve loc i ties at the in ter face).

The in verse prob lem is ap proached us ing the 'In ter cept ing Ca non i cal Body Ap prox i mation' (ICBA) method de vel oped by A. Wirgin et al [START_REF] Wirgin | Some quasi-an a lytic and nu mer i cal meth ods for acous ti cal im ag ing of com plex me dia[END_REF] to solve the in verse scat ter ing prob lem in the res o nance do main. Know ing the po si tion of the re ceiver in re la tion to the emit ted ul trasound source, it is as sumed that the scat ter ing pro cess at ob ser va tion point r x m is iden ti cal to that which would have been pro duced by an acous ti cally iden ti cal cir cu lar cyl in der with an 'ap par ent' ra dius equal to the ra dius of the ob ject in the mea sure ment di rec tion sub jected to the same con di tions of insonification. The main ad van tage here is that this pro ce dure can be used even in cases where there is a high im ped ance con trast.

The es ti mated field P e is the field scat tered from a cen tered cir cu lar cyl in der hav ing the same den sity r 1 and the same ve loc ity c 1 as the ob ject and ra dius r e 1 equal to the lo cal ra dius of the ob ject in the mea sure ment di rec tion q m . Us ing the par tial wave ex pan sion method, we ob tain the fol low ing ex pres sion for r x m = (r m , q m ) Î W :

where k f c [START_REF] Lasaygues | High res o lu tion low fre quency ul tra sonic to mog ra phy[END_REF] is the first-kind Hankel func tion of the or der n, e n is the Neumann fac tor (e n = 2 for n > 1), and the scat ter ing co ef fi cient B n is ex actly eval u ated us ing the Rayleigh-Fou rier method.

0 0 2 = p / , H n ( )

11

To es ti mate the un known r r m e m = 1 ( ) q , we seek to min i mize the cost func tion, Since a broad band in ci dent pulse is used, a set of back scat ter ing mea sure ments at L consec u tive fre quen cies (cor re spond ing to 99% of the pulse spec trum) is avail able for a given q m . If these multifre quency data are treated in de pend ently, the num ber of ad mis si ble so lutions will in crease. In the same way, search ing for the min ima of each of the cost func tion cor re spond ing to the var i ous fre quen cies and then ap ply ing a postpro cess ing al go rithm to de ter mine the unique so lu tion re quires as many minimizations as fre quen cies. This is also the case with meth ods based on an it er a tive fre quency al go rithm. Whereas the method presented here is based on the minimization of a sin gle cost func tion, the 'mean cost func tion' is de fined by

Nu mer i cal sim u la tion

To eval u ate the va lid ity of the method, we first per formed im ag ing on the ba sis of syn thesized data. The mea sure ments are ob tained by solv ing the acous tic wave prop a ga tion problem in 2-D nu mer i cally. The method used for this pur pose was based on mixed fi nite el e ments for the space discretization and on a cen tered 2nd or der fi nite dif fer ence scheme for the time discretization. The ob ject is a noncir cu lar cyl in der, the cross-sec tion of which is shown in fig ure 5. The acous ti cal prop er ties used in the sim u la tions are r 0 = 1,000 kg/m 3 , c 0 = 1,500 m/s, r 1 = 1,800 kg/m 3 and c 1 = 4,000 m/s. Note that in this sim u la tion, all the ma te ri als are fluid-like and, in par tic u lar, the ob ject is not elas tic.

P x f P m

( , ) r is mea sured at 16 points equally dis trib uted on a cen tered mea sure ment ring with ra dius r m = 12 mm (near field). The wave form of the in ci dent pulse is a nor mal ized second de riv a tive of a Gaussi an func tion. Two prob ing pulses are sim u lated: the 'low fre quency

( ) ( ) ( ) ( ) ( ), cos , ) ( , 0 0 ) 1 ( å = - = N n m i m n n n m n m e n r k H i f B f A f x P q q e q r (11) 4 10 10 . . - - = < å b t s N N N n n ( ) ( ) 2 , ; , ) , ; ( f x P f x P f x F m m e m P m m r r r t t - = (12) å = = L l l m m m m f x F L x F 1 ) , ; ( 1 )
; ( r r ( t t (13) insonification' (LFI) pulse with a cen tral fre quency f c = 250 kHz and the 'high fre quency insonification' (HFI) with f c = 1 MHz. Since the mean ra dius of the ob ject is 6 mm, the first insonification cor re sponds to an ex ci ta tion in the res o nance re gion (ka » 6) while the sec ond insonification is be yond the res o nance re gion (ka » 25). Since ICBA is a mono chro matic forward solver, the time-do main mea sure ments are con verted into fre quency do main data using a Fast Fou rier Trans form al go rithm.

Fre quency be hav ior

To an a lyze the role of the fre quency in the in ver sion al go rithm, we first com pute the sin gle fre quency cost func tion (Eq.( 12)) at dif fer ent fre quen cies. The pa ram e ter of in ter est is the lo cal ra dius r 1 4 ( / ) p = 5.4 mm and the data are the scat tered field mea sured at q m = p/4 (i.e., backpro jec tion, q i = p + q m ).

Fig ure 2a gives the cost func tion for three dif fer ent fre quen cies, f 1 = 250 kHz, f 2 = 500 kHz and f 3 = 1 MHz. Note that each of the cost func tions has a min i mum (not al ways global) value, which is near the ex act so lu tion. How ever, other es ti mates (e.g., r e 1 = 2.5 mm) also cor re spond to the min ima of all the cost func tions. Care ful postpro cess ing is there fore required when us ing this al go rithm with a small num ber of fre quen cies.

Upon plot ting the lo cal cost func tion vs. all the fre quen cies of HFI (Fig. 2b), it was observed that at a given fre quency, a pe ri odic dis tri bu tion of the min ima oc curs and the pe riod cor re sponds to the half wave length. The at trac tion do main of each so lu tion can be said to be one half wave length wide and to be cen tered on each ad mis si ble so lu tion. More over, a line of lo cal min ima can be ob served at which the es ti mated pa ram e ter is ap prox i mately equal to the so lu tion and var ies weakly with the fre quency. This sug gests that the re con struc tion will never be per fect but that an ap prox i mate so lu tion can be ob tained with a res o lu tion smaller than the wave length.

Fig ure 3 gives the mean cost func tion Eq.( 13) cor re spond ing to the re con struc tion of the lo cal ra dius with HFI. The sum in Eq.( 13) is based on three fre quency do mains D j=1,2,3 centered on the cen tral fre quency and cover 33%, 66% and 99%, re spec tively, of the avail able fre quency range. The vari a tions in the cost func tion de crease and the min i mum cor re sponding to the op ti mum so lu tion be comes more ob vi ous (i.e., eas ier to es ti mate) as the fre quency range in creases. Av er ag ing over the fre quency has a con struc tive ef fect on the min i mum, which is fre quency in vari ant, and a de struc tive ef fect on the oth ers. More over, the fact that the at trac tion do main of the global min i mum con serves the same width in all fre quency do - mains means that this width de pends more strongly on the cen tral fre quency than on the frequency range.

In fig ure 4, we com pare the re con struc tion ob tained us ing the full-fre quency spec trum (do main D 3 ) with HFI and LFI. Note that the num ber of lo cal min ima in creases with the central fre quency f c . In both cases, the deep est min i mum is very close to the so lu tion and will be re ferred to from now on as the 'op ti mal es ti mate.' The ab so lute er rors in each fre quency domain (0.019 mm with LFI and 0.048 mm with HFI) are far smaller than the wave length in both cases. The res o lu tion seems to be in de pend ent of the fre quency do main, al though this pa ram e ter af fects the width of the at trac tion do main.

Bound ary eval u a tion

Hav ing es ti mated the lo cal ra dii, it is now pro posed to de fine the to tal bound ary as a function r(q) of po lar an gle q. Con sid er ing a smooth bound ary, we suppose the fol low ing form of this func tion, This can be in ter preted as a Fou rier de com po si tion at a fi nite or der of the bound ary. Note that in this case the un knowns are the co ef fi cients j s and y s . Each lo cal mean cost func tion is min i mized on D 3 tak ing 15 ini tial guesses equally dis trib uted over 3 mm and 10 mm us ing a quasiNew to nian al go rithm (NAG For tran sub rou tine E04JYF). The op ti mum lo cal es timates are in tro duced into the left side of Eq.( 14) and thus de fine a lin ear sys tem, which is q y q j q (14) solved in or der to find j s and y s . In this ex am ple, we use S = 4 (the ad di tional terms are not sig nif i cant). Both LFI and HFI yield sat is fac tory re con struc tions (Fig. 5).

EX PER I MEN TAL RE SULTS

Water-tank measurements

The ex per i men tal setup used here was de signed for per form ing dif frac tion mea sure ments. The acous tic de vice could be moved with var i ous de grees of free dom to an a lyze sam ples in all di rec tions. The po si tions of the tar get and trans duc ers could be ad justed. In par tic u lar, the op er a tor could pre scribe ex act ro ta tions and trans la tions on the trans mit ter and re ceiver. Dur ing all these mea sure ments, the wave ve loc ity in the wa ter tank was 1,480 m/s.

Ultrasounds were gen er ated us ing a Panametric ® pulse/re ceiver and Imasonic ® piezo-compos ite wideband trans duc ers. The nom i nal fre quency of the trans duc ers was 1 MHz and the us able band width of the trans duc ers was ap prox i mately 0.5-1.5 MHz. The ax ial res o lu tion was 3 mm in wa ter (with a pulse of 4 ms) and the lat eral and az i muth res o lu tions were both 2 mm. Ech oes were there fore oc cur ring as the re sult of in ter ac tions with the ul tra sonic beam at a vol ume of ap prox i mately 12 mm 3 . The ob ject was placed in the cen ter of the bench. Ul tra sonic mea sure ments were performed in wa ter at room tem per a ture. The sec tor scanned was typ i cally 180°/360°, i.e., there were 180 view an gles with an an gu lar in cre ment of 2°. In our ap pli ca tion, the dif frac tion mode was re duced to the re flec tion mode with a sin gle trans ducer. In this case, the dif fraction an gle cor re sponded to f = q i + q m = 180° (p/2 rad), and the pro jec tions cor re sponded to the back scat ter ing ech oes. Trans mit ted and re ceived ul tra sound radio fre quen cy (rf) sig nals (1,024 sam ples in all) were dig i tized (8 bits, 20 MHz) us ing a nu mer i cal os cil lo scope and stored on a com puter via a Gen eral Pur pose In ter face Bus for off-line anal y sis. The whole sys tem ar chi tec ture was based on two per sonal com put ers, one of which was de voted to the au to matic pi lot ing of the data ac qui si tion pro cess and the other to sig nal process ing and im age re con struc tion. The im age size was 512 ´ 512 pix els.

Circular elastic tubes

When test ing our meth ods, we man aged to re con struct ac a demic test-tar gets suc cess fully. Al though the in for ma tion in side the tubes is also im por tant and in ter est ing, it has been impos si ble so far to ob tain this in for ma tion re li ably at the ex per i men tal level. How ever, the den si ties and ve loc i ties can be es ti mated us ing sim ple parallelepiped-cut sam ples gen er ated in ul tra sonic ex per i ments. To check the re sults of these ex per i ments, we com pared the ra dii of the tubes ob tained from URT with those ob tained from ICBA.

The test tar gets used here were ho mo ge neous, iso tro pic cir cu lar Plexi glas tubes. The density was r 1 = 1,200 kg/m 3 and the mean ve loc ity of the compressional wave was c 1 = 2,700 m/s. The ex ter nal ra dius (r 1 ) of the tubes was 6 mm and the in ner cav ity ra dius (r 2 ) was 4 mm. Note that the real ob ject was not cen tered here, con trary to what is as sumed to be the case in the ca non i cal ap prox i ma tion.

Fig ure 6 gives the re sults of the sec ond set of ex per i ments. The sinogram was ob tained on the tube af ter per form ing an gu lar scan ning in the same con fig u ra tion as the pre vi ous measure ments.

Since the ax ial res o lu tion of the trans mit ted sig nal is 3 mm (with a 4-ms pulse), the boundary ech oes on the shell (shell thick ness: about 2 mm) are con fused and the re sult ing tomogram does not clearly show the in ner cav ity. The ob ject is over sized and the ex ter nal bound ary is there fore not cor rectly as sessed. On the tomogram (Fig. 6b), it can be noted that the ra dius was given as 12 mm in stead of 6 mm. In ad di tion, the wave length of the 1 MHz wave is 1.5 mm in wa ter, which is also sim i lar to the wall thick ness. Because of the low imped ance con trast, the res o lu tion lim its of first-or der Born to mog ra phy were reached. It is there fore strictly im pos si ble to de ter mine or de fine the ex ter nal bound ary and the thick ness of the tube from the tomogram (Fig. 6b).

The value of us ing a ca non i cal ap prox i ma tion rather than a Born ap prox i ma tion in ad dition to an an a lyt i cal ap proach can be clearly seen from fig ure 7. With the ICBA method, the di men sions of the tube were fairly ac cu rately de ter mined (<7%). The ab so lute er ror in the thick ness of the tube was one tenth of the wave length in the sur round ing me dium. This re sult FIG. 5 Lo cal (as ter isk) and bound ary (dot) re con struc tion, LFI (a) and HFI (b). The op ti mal lo cal es ti ma tion is as ter isk and cir cle and the ac tual sec tion is solid plot. The mean er ror on the bound ary is 0.02 mm for LFI and 0.17 mm for HFI. is most en cour ag ing, es pe cially con sid er ing the de vi a tion of the real ob ject from the cen ter of the mea sure ment frame, which in tro duces a con sid er able dif fer ence be tween the real scattered field and the model (cen tered tube). Thanks to the an a lytic model used for the in ver sion pro ce dure, the whole sec tion was re con structed in only a few min utes. The ef fi ciency and robust ness of this method have been es tab lished in the case of sev eral nu mer i cal ex am ples by Le Marrec et al. [START_REF] Marrec | Multi-fre quency Quan ti ta tive im ag ing of high con trast ob jects: canon i cal ap prox i ma tion[END_REF] The mean cost func tion of the back-scat ter ing data was used here as a function of the lo cal ra dius, while mean val ues were used for the ve loc i ties and the den sity. The ad mis si ble so lu tion for the lo cal ra dius there fore seems to be in de pend ent of small er rors in the acous ti cal pa ram e ters. This makes sec tion re con struc tion pos si ble with out any am bi guity, even if the as sump tions about the acous ti cal pa ram e ters are wrong. This does not mean of course that back scat ter ing re con struc tion makes it pos si ble to es ti mate acous ti cal pa ram e ters.

Mea sur ing pa ram e ters other than the backscat ter ing, such as the dif frac tion or trans mission, as so ci ated with an ad ap ta tion of the ICBA method gives better shape re con struc tion and rea son ably ac cu rate acous ti cal pa ram e ters.

CON CLU SION

This pa per deals with the two-di men sional im age re con struc tion of tubes us ing ul tra sonic Born to mog ra phy and a new ca non i cal ap prox i ma tion. The lat ter im prove ment makes it possi ble to ex tend the scope of to mog ra phy from lower im ped ance con trast me dia (clas si cal appli ca tions) to higher im ped ance con trast sit u a tions, even when the ul tra sonic prop a ga tion is greatly per turbed by the dif fer ence in acous tic im ped ance be tween the scat terer and the sur - round ing me dium. The strat egy used here to solve this prob lem was based on com par i sons be tween the ex per i men tal dif fracted field and the ca non i cal so lu tion ap proached as a forward prob lem. The al go rithm adopted, us ing the an a lyt i cal so lu tion to the lo cal for ward prob lem and an it er a tive pro cess to re cover the un knowns, is fast enough to yield real-time in for ma tion about the shape. A sin gle fre quency does not suf fice to de ter mine the cost function and the in ver sion is im proved by us ing all the fre quen cies pres ent in the broad band of the trans duc ers. The method pre sented here is ro bust and is not per turbed by the ex per i mental mea sure ments or small er rors in the ma te rial prop er ties. The re sults are most prom is ing, and, in par tic u lar, this method gives an im age with an er ror that is lower than the wave length. Var i ous ways of im prov ing this method will be in ves ti gated in the future. At pres ent, the al go rithm is be ing im proved as re gards the de tec tion of the global min ima and re con struction tests are be ing car ried out on ob jects with noncir cu lar ge om e try. This method needs to be op ti mized to be able to es ti mate acous ti cal and phys i cal pa ram e ters, such as the ve loc i ties and the den sity, more ac cu rately. For this pur pose, sev eral so lu tions are be ing in ves ti gated, such as the pos si bil ity of in tro duc ing a larger num ber of scat ter ing mea sure ment an gles in or der to sta bi lize the in verse prob lem.

ACKNOWLEGMENTS

This work bene fited from the very fruit ful help of T. Scotti and J.-P. Lefebvre from the Lab o ra tory of Me chan ics and Acous tics (LMA), UPR CNRS 7051, Mar seille, France.

FIG. 1

 1 FIG. 1 Noncircular cy lin dri cal elas tic cav ity: geo met ri cal and acous ti cal con fig u ra tion.

  One can cal cu late the far field so lu tion of this equa tion in the di rec tion r of an ide ally pulsed plane-wave insonification, with the in ci dence r

12

 12

FIG. 2

 2 FIG. 2 Re con struc tion of r 1 (p/4). (a) am pli tude of cost func tion at three fre quen cies, f 1 (solid), f 2 (dash) and f 3 (dot). (b) sin gle fre quency cost func tion as a func tion of fre quency and es ti ma tion pa ram e ter (HFI).

FIG. 3

 3 FIG.3Mean cost func tion for r 1 (p/4) in three fre quency do mains: D 1 (dot) » 33%, D 2 (dash) » 66% and D 3 (solid) » 99% of avail able fre quency range.

FIG. 4

 4 FIG.4 Mean cost func tion for on D 3 » 99% of avail able fre quency range, LFI (dash) and HFI (solid).

FIG. 6

 6 FIG. 6 Ul tra sonic to mog ra phy of a cir cu lar plexi glas tube (ka » 25). (a) sinogram, 180 backprojections through 360°, 1024 sam ples. (b) tomogram 512 ´ 512 pix els. (c, d) x and y pixel pro files drawn at x = y = 0 mm.

FIG. 7

 7 FIG. 7 In ver sion by ICBA of a cir cu lar plexi glas tube (ka » 25). 180 backprojections through 360°, 1024 sam ples F e = 20 MHz, first step c 1 = 2,500 m/s, r 1 = 1,200 kg/m 3 . (a) Cost func tion r ext at q m = p/2. (b) Cost func tion r int at q m = p/2. (c) geo met ri cal re con struc tion.

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