Éphane St

Charpentier

S T Charpentier
email: stephane.charpentier@math.u-psud.fr

Éphane

COMPACT COMPOSITION OPERATORS ON HARDY-ORLICZ AND WEIGHTED BERGMAN-ORLICZ SPACES ON THE BALL

Keywords: 2000 Mathematics Subject Classification, Primary: 47B33 -Secondary: 32C22; 46E15 Carleson measure -Composition operator -Bergman-Orlicz space -Hardy-Orlicz space -angular derivative, 1

Using recent characterizations of the compactness of composition operators on Hardy-Orlicz and Bergman-Orlicz spaces on the ball ([2, 3]), we first show that a composition operator which is compact on every Hardy-Orlicz (or Bergman-Orlicz) space has to be compact on H ∞ . Then, although it is well-known that a map whose range is contained in some nice Korányi approach region induces a compact composition operator on H p (B N ) or on A p α (B N ), we prove that, for each Korányi region Γ, there exists a map φ : B N → Γ such that, C φ is not compact on H ψ (B N ), when ψ grows fast. Finally, we extend (and simplify the proof of) a result by K. Zhu for classical weighted Bergman spaces, by showing that, under reasonable conditions, a composition operator C φ is compact on the weighted Bergman-Orlicz space A ψ α (B N ), if and only if

In particular, we deduce that the compactness of composition operators on A ψ α (B N ) does not depend on α anymore when the Orlicz function ψ grows fast.

1. INTRODUCTION Let B N = z = (z 1 , . . .z N ) ∈ C N , ∑ N i=1 |z i | 2 <
1 denote the open unit ball of C N . Given a holomorphic map φ : B N → B N , the composition operator C φ of symbol φ is defined by C φ ( f ) = f • φ , for f holomorphic on B N . Composition operators have been extensively studied on common Banach spaces of analytic functions, in particular on the Hardy spaces H p (B N ) and on the Bergman spaces A p (B N ), 1 ≤ p < ∞. The continuity and compactness of these operators have been characterized in terms of Carleson measures ( [START_REF] Cowen | Composition Operators on Spaces of Analytic Functions[END_REF]). In dimension one, the boundedness of C φ for any φ : D → D is a consequence of the Littlewood subordination principle ( [START_REF] Shapiro | Composition operators and classical function theory[END_REF]). In C N , N > 1, it is well-known that there exists some map φ : B N → B N such that the associated composition operator is not bounded on H p (B N ). Whatever the dimension, it appears that both boundedness and compactness of C φ on H p (B N ) (resp. A p (B N )) are independent of p. On the other hand, every composition operator is obviously bounded on H ∞ and it is not difficult to check that C φ is compact on H ∞ if and only if φ ∞ < 1. Thus there is a "break" between H ∞ and H p (B N ) (resp. A p (B N )), for the compactness in dimension one, and even for the boundedness, when N > 1.

These observations first motivated P. Lefèvre, D. Li, H. Queffélec and L. Rodríguez-Piazza to study composition operators on Hardy-Orlicz spaces H ψ (D) (resp. Bergman-Orlicz spaces A ψ (D)) of the disc ( [START_REF] Èvre | Composition operators on Hardy-Orlicz spaces[END_REF][START_REF] Èvre | Some revisited results about composition operators on Hardy spaces[END_REF][START_REF] Èvre | Compact composition operators on H 2 (D) and Hardy-Orlicz spaces[END_REF][START_REF] Èvre | Compact composition operators on Bergman-Orlicz spaces[END_REF]), and then the author of [START_REF] Charpentier | Composition operators on weighted Bergman-Orlicz spaces on the ball[END_REF][START_REF] Charpentier | Composition operators on Hardy-Orlicz spaces on the ball[END_REF] to look at these questions in C N . These spaces both provide an intermediate scale of spaces between H ∞ and H p (B N ) (resp. A p (B N )) and generalize the latter. In particular, in [START_REF] Èvre | Some revisited results about composition operators on Hardy spaces[END_REF], the authors were interested in the question of whether there are some Hardy-Orlicz spaces on which the compactness of C φ is equivalent to that on H ∞ . In fact, they answer this question in the negative, by proving ([10, Theorem 4.1]) that, for every Hardy-Orlicz space H ψ (D), one can construct a surjective map φ : D → D which induces a compact composition operator C φ on H ψ (D). This result extends that obtained by B. MacCluer and J. Shapiro for H p (D) ( [START_REF] Maccluer | Angular derivatives and compact composition operators on the Hardy and Bergman spaces[END_REF]Example 3.12]). The same problem in the Bergman-Orlicz case has not yet been completely solved. In several variables, the situation is much more surprizing, as we show in [START_REF] Charpentier | Composition operators on weighted Bergman-Orlicz spaces on the ball[END_REF][START_REF] Charpentier | Composition operators on Hardy-Orlicz spaces on the ball[END_REF] that there exist some Hardy-Orlicz and Bergman-Orlicz spaces, "close" enough to H ∞ , on which every composition operator is bounded.

In this paper, we are mainly interested in the possibility to extend some known results about compactness of composition operators on classical Hardy or Bergman spaces, to the corresponding Orlicz spaces. We think that this study may outline some interesting phenomena and precise the link between the behavior of C φ and that of φ .

First of all, we come back to the "break" between H ∞ and H p , 1 ≤ p < ∞, for the compactness of C φ . There is no difference between being compact for C φ on one H p (B N ) and on every H p (B N ), while this property clearly depends on the Orlicz function ψ in H ψ (B N ). Therefore, we can wonder if the above question answered by [START_REF] Èvre | Some revisited results about composition operators on Hardy spaces[END_REF] was the good one; indeed, the study of C φ on Hardy-Orlicz spaces arises the following question: what can we say about a composition operator which is compact on every Hardy-Orlicz space? It turns out that such an operator has to be compact on H ∞ , which seems to us to be a positive result, because it confirms that Hardy-Orlicz spaces covers well the "gap" between every H p and H ∞ . This result also stands when we replace Hardy-Orlicz spaces by Bergman-Orlicz spaces.

Moreover, in Hardy or Bergman spaces, compactness (and boundedness) of composition operators is handled in terms of geometric conditions, emphasizing the importance of the manner in which the symbol φ approaches the boundary of B N . To be precise, if we denote by Γ (ζ , a) ⊂ B N , for ζ ∈ S N and a > 1, the Korányi approach region

Γ (ζ , a) = z ∈ B N , |1 -z, ζ | < a 2 1 -|z| 2 ,
it is known ( [START_REF] Maccluer | Compact composition operators on H p (B N )[END_REF]) that if φ takes the unit ball into a Korányi region Γ (ζ , a) with a small enough angular opening a, then C φ is compact on H p (B N ) and on A p α (B N ). When N = 1, the Korányi regions are just non-tangential approach regions. In this paper, we show that this result does not hold anymore for Hardy-Orlicz spaces on B N ; for Bergman-Orlicz spaces, we obtain such a result in dimension one only.

In [START_REF] Maccluer | Angular derivatives and compact composition operators on the Hardy and Bergman spaces[END_REF], the authors related the compactness of the composition operator C φ on H p (D) or A p α (D) to the existence of angular derivative for φ at the boundary. We say that the angular derivative of φ exists at a point ζ ∈ T if there exists ω ∈ T such that φ (z) -ω z -ζ has a finite limit as ζ tends non-tangentially to ζ through D. The Julia-Caratheodory Theorem then asserts that the non-existence of an angular derivative for φ at some ζ ∈ T is equivalent to

(1.1) lim z→ζ 1 -|z| 1 -|φ (z)| = 0.
Shapiro and Taylor [START_REF] Shapiro | Compact, nuclear and Hilbert-Schmidt composition operators on H 2[END_REF] pointed out that if C φ is to be compact on H p (D), then φ cannot have an angular derivative at even a single point in T, which may be written:

(1.2) lim |z|→1 1 -|z| 1 -|φ (z)| = 0.
In [START_REF] Maccluer | Angular derivatives and compact composition operators on the Hardy and Bergman spaces[END_REF], it is proven that (1.1) is not sufficient to the compactness of C φ on Hardy spaces of the unit disc in general, yet it is when φ is univalent. However, this condition is necessary and sufficient for C φ to be compact on every weighted Bergman spaces of the disc. The last main goal of this paper is to extend some of these results to Hardy-Orlicz and Bergman-Orlicz spaces of the unit ball.

In several variables, we can also define the angular derivative of φ : B N → B N at a point in the unit sphere S N and the Julia-Caratheodory Theorem also holds in B N [START_REF] Rudin | Function Theory in the Unit Ball of C n[END_REF]Theorem 8.5.6]). Here, as we already said, the situation is complicated by the fact that some composition operators are not bounded on Hardy or Bergman spaces, and the fact that even the boundedness of C φ on A p α (B N ) depends on α. In [START_REF] Zhu | Compact composition operators on Bergman spaces of the unit ball[END_REF], K. Zhu proves that C φ is compact on A p α (B N ) if and only if Condition (1.2) is satisfied, whenever C φ is bounded on some A p β (B N ), for some -1 < β < α. This assumption is somehow justified by the above observation and by [START_REF] Maccluer | Angular derivatives and compact composition operators on the Hardy and Bergman spaces[END_REF]Section 6], in which the authors show that, for any α > -1 and any 0 < p < ∞, there exists φ : B N → B N with no angular derivative at any point of S N , such that C φ is bounded on A p α (B N ) but not compact. There even exists such a map φ such that C φ is not bounded on A p α (B N ). In the present paper, we generalize Zhu's result to weighted Bergman-Orlicz spaces on the ball, by using recent characterizations of boundedness and compactness of composition operators on these spaces ( [START_REF] Charpentier | Composition operators on weighted Bergman-Orlicz spaces on the ball[END_REF]). We show that, if C φ is bounded on some

A ψ β (B N ), -1 < β < α, then it is compact on A ψ α (B N ) if and only if (1.3) lim |z|→1 ψ -1 1/ (1 -|φ (z)|) N(α) ψ -1 1/ (1 -|z|) N(α) = 0,
where N (α) = N + α + 1, under a mild and usual regularity condition on the Orlicz function ψ.

Our proof is quite simple, while that of K. Zhu uses a Schur test in H 2 (B N ) and the fact that the compactness of composition operators on H p (B N ) does not depend on p. Combining this result with the automatic boundedness of every composition operator on A ψ α (B N ) when ψ satisfies the ∆ 2 -Condition, we get that the compactness on such A ψ α (B N ) does not depend on α anymore. To be precise, C φ is compact on

A ψ α (B N ) if and only if lim |z|→1 ψ -1 (1/ (1 -|φ (z)|)) ψ -1 (1/ (1 -|z|)) = 0,
whenever ψ satisfies the ∆ 2 -Condition.

We have to mention that Condition (1.3) is, in any case, necessary. Moreover, the authors of [START_REF] Èvre | Composition operators on Hardy-Orlicz spaces[END_REF] obtained such a result in dimension one, as announced in [START_REF] Li | Compact composition operators on Hardy-Orlicz and Bergman-Orlicz spaces[END_REF]. However, their proof uses the characterization of the compactness of composition operators in terms of the Nevanlinna counting function and is more complicated.

We organize our paper as follows: a first preliminary part is devoted to the definitions and the statements of the already known results we need. The main part contains the three most important results mentionned above.

Notation. Throughout this paper, we will denote by dσ N the normalized invariant measure on the unit sphere S N = ∂ B N , and by dv α = c α 1 -|z| 2 α dv, α > -1, the normalized weighted Lebesgue measure on the ball.

Given two points z, w ∈ C N , the euclidean inner product of z and w will be denoted by z, w , that is z, w = ∑ N i=1 z i w i ; the notation |•| will stand for the associated norm, as well as for the modulus of a complex number.

If α > -1 is a real number, we will denote by N (α) the quantity N + α + 1.

PRELIMINARIES

2.1. Hardy-Orlicz and Bergman-Orlicz spaces -Definitions. A strictly convex function ψ : R + → R + is called an Orlicz function if ψ (0) = 0, ψ is continuous at 0 and ψ (x)

x ----→ x→+∞ +∞.

If (Ω, P) is a probability space, the Orlicz space L ψ (Ω) associated to the Orlicz function ψ on (Ω, P) is the set of all (equivalence classes of) measurable functions f on Ω such that there exists some C > 0, such that Ω ψ | f | C dP is finite. L ψ (Ω) is a vector space, which can be normed with the so-called Luxemburg norm defined by

f ψ = inf C > 0, Ω ψ | f | C dP ≤ 1 .
It is well-known that L ψ (Ω) , • ψ is a Banach space (see [START_REF] Rao | Theory of Orlicz spaces[END_REF]).

Taking Ω = S N and dP = dσ N , the Hardy-Orlicz space H ψ (B N ) on B N is the Banach space of analytic functions f :

B N → C such that f H ψ := sup 0<r<1 f r ψ < ∞, where f r ∈ L ψ (S N ) is defined by f r (z) = f (rz). Every function f ∈ H ψ (B N ) admits a radial boundary limit f * such that f * ψ = sup 0<r<1 f r ψ < ∞ ([3, Section 1.3]).
For simplicity, we will sometimes denote by • ψ the norm on H ψ (B N ), emphasizing that H ψ (B N ) can be seen as a subspace of L ψ (S N ).

With Ω = B N and dP = dv α , α > -1, the weighted Bergman-Orlicz space A ψ α (B N ) is L ψ (B N )∩ H (B N )
, where H (B N ) stands for the vector space of analytic functions on the unit ball.

A ψ α (B N ) is a Banach space.
From the definitions, it is easy to verify that the following inclusions hold:

H ∞ ⊂ H ψ (B N ) ⊂ H p (B N ) and H ∞ ⊂ A ψ α (B N ) ⊂ A p α (B N ) for every Orlicz function ψ and any 1 ≤ p < ∞. Moreover, if ψ (x) = x p , for some 1 ≤ p < ∞ and for every x ≥ 0, then H ψ (B N ) = H p (B N ) and A ψ α (B N ) = A p α (B N ).

Four classes of Orlicz functions.

Let ψ be an Orlicz function. In order to distinguish the Orlicz spaces and to get a significant scale of intermediate spaces between L ∞ and L p (Ω), we define four classes of Orlicz functions.

-The two first conditions are regularity conditions: we say that ψ satisfies the ∇ 0 -condition if it satisfies one of the following two equivalent conditions:

(i) For any B > 1, there exists some constant

C B ≥ 1, such that ψ (Bx) ψ (x) ≤ ψ (C B By) ψ (y)
for any

x ≤ y large enough;

(ii) For any n > 0, there exists

C n > 0 such that ψ (Bx) n ψ (x) n ≤ ψ (C n y) ψ (By)
for any x ≤ y large enough.

Let us notice that (2) ⇒ (1) is obvious, while an easy induction allows to prove (1) ⇒ (2); the details are left to the reader.

If the constant C B can be chosen independently of B, then ψ satisfies the uniform ∇ 0 -Condition.

-The ∇ 2 -class consists of those Orlicz functions ψ such that there exist some β > 1 and some

x 0 > 0, such that ψ (β x) ≥ 2β ψ (x), for x ≥ x 0 .
-The third one is a condition of moderate growth: ψ satisfies the ∆ 2 -Condition if there exist x 0 > 0 and a constant K > 1, such that ψ (2x) ≤ Kψ (x) for any x ≥ x 0 .

-The fourth condition is a condition of fast growth: ψ satisfies the ∆ 2 -Condition if it satisfies one of the following equivalent conditions:

(i) There exist C > 0 and x 0 > 0, such that ψ (x) 2 ≤ ψ (Cx) for every x ≥ x 0 ;

(ii) There exist b > 1, C > 0 and x 0 > 0 such that ψ (x) b ≤ ψ (Cx), for every x ≥ x 0 ;

(iii) For every b > 1, there exist

C b > 0 and x 0,b > 0 such that ψ (x) b ≤ ψ (C b x), for every x ≥ x 0,b .
Finally, we mention that these conditions are not independent (see [START_REF] Èvre | Composition operators on Hardy-Orlicz spaces[END_REF]Proposition 4.7]):

Proposition 2.1. Let ψ be an Orlicz function.

(1) If ψ satisfies the uniform ∇ 0 -Condition, then it satisfies the ∇ 2 -Condition;

(2) If ψ satisfies the ∆ 2 -Condition, then it satisfies the uniform ∇ 0 -Condition.

For any 1 < p < ∞, every function x -→ x p is an Orlicz function which satisfies the uniform ∇ 0 -Condition, (so ∇ 2 and ∇ 0 -conditions too) and the ∆ 2 -Condition. At the opposite side, for any a > 0 and b ≥ 1, x -→ e ax b -1 belongs to the ∆ 2 -Class (and then to the uniform ∇ 0 -Class), yet not to the ∆ 2 -one. In addition, the Orlicz functions which can be written x → exp a (ln (x + 1)) b -1 for a > 0 and b ≥ 1, satisfy the ∇ 2 and ∇ 0 -Conditions, but do not belong to the ∆ 2 -Class.

For a complete study of Orlicz spaces, we refer to [START_REF] Śkii | Convex functions and Orlicz spaces[END_REF] and [START_REF] Rao | Theory of Orlicz spaces[END_REF]. We can also find precise and useful information in [START_REF] Èvre | Composition operators on Hardy-Orlicz spaces[END_REF], such as other classes of Orlicz functions and their links with each other.

S (ζ , h) = {z ∈ B N , |1 -z, ζ | < h} and S (ζ , h) = z ∈ B N , |1 -z, ζ | < h .
We say that a finite positive Borel measure µ on B N is a ψ-Carleson measure, ψ an Orlicz function, if

µ (S (ζ , h)) = O h→0 1 ψ (Aψ -1 (1/h N )) ,
uniformly in ζ ∈ S N and for some constant A > 0. µ is a vanishing ψ-Carleson measure if the above condition is satisfied for every A > 0 and with the big-Oh condition replaced by a little-oh condition.

A finite positive Borel measure

µ on B N is a (ψ, α)-Bergman-Carleson measure if µ (S (ζ , h)) = O h→0 1 ψ Aψ -1 1/h N(α) ,
uniformly in ζ ∈ S N and for some constant A > 0. µ is a vanishing (ψ, α)-Bergman-Carleson measure if the above condition is satisfied for every A > 0 and with the big-Oh condition replaced by a little-oh condition. When ψ satisfies the ∆ 2 -Condition, a (vanishing) ψ-Carleson measure (resp. (vanishing) (ψ, α)-Bergman-Carleson measure) is a (vanishing) Carleson measure (resp. (vanishing) Bergman-Carleson measure) (see [START_REF] Charpentier | Composition operators on weighted Bergman-Orlicz spaces on the ball[END_REF][START_REF] Charpentier | Composition operators on Hardy-Orlicz spaces on the ball[END_REF]Sections 3]).

For φ : B N → B N , we denote by µ φ the pull-back measure of σ N by the boundary limit φ * of φ , and by µ φ ,α that of dv α by φ . To be precise, for any [START_REF] Charpentier | Composition operators on Hardy-Orlicz spaces on the ball[END_REF]Section 3]). The main theorem is the following: Theorem 2.2. Let ψ be an Orlicz function which satisfies the ∇ 2 -Condition and let φ : B N → B N be holomorphic.

E ⊂ B N (resp. E ⊂ B N ), µ φ (E) = σ N (φ * ) -1 (E) and µ φ ,α (E) = v α φ -1 (E) .

Results for Hardy-Orlicz spaces (see

( Due to the non-separability of small Hardy-Orlicz spaces, [3, Theorem 3.2] is not a direct consequence of Carleson-type embedding theorems obtained in [3, Section 2]; however, if we follow the proofs of these embedding theorems directly for composition operators, by working on spheres of radius 0 < r < 1, then we get the following characterizations of both boundedness and compactness of composition operators: Theorem 2.3. Let ψ be an Orlicz function satisfying the ∇ 2 -Condition and let φ : B N → B N be holomorphic.

(1) If ψ satisfies the uniform ∇ 0 -Condition, then C φ is bounded on H ψ (B N ) if and only if there exists some A > 0 such that

(2.1) sup 0<r<1 µ φ r (S (ζ , h)) = O h→0 1 ψ (Aψ -1 (1/h N )) uniformly in ζ ∈ S N . (2) If ψ satisfies the ∇ 0 -Condition, then C φ is compact on H ψ (B N ) if and only if, for every A > 0, (2.2) sup 0<r<1 µ φ r (S (ζ , h)) = O h→0 1 ψ (Aψ -1 (1/h N )) uniformly in ζ ∈ S N .
In the further, we will see how these two characterizations are useful depending on the situations.

Results for Bergman-Orlicz spaces (see [2, Section 3]

). The main result is similar to that stated in the previous paragraph: Theorem 2.4. Let ψ be an Orlicz function, let α > -1 and let φ : B N → B N be holomorphic.

(

) If ψ satisfies the uniform ∇ 0 -Condition, then C φ is bounded from A ψ α (B N ) into itself if and only if µ φ is a (ψ, α)-Bergman-Carleson measure. (2) If ψ satisfies the ∇ 0 -Condition, then C φ is compact from A ψ α (B N ) into itself if and only if µ φ is a vanishing (ψ, α)-Bergman-Carleson measure. (3) If ψ satisfies the ∆ 2 -Condition, then C φ is bounded (resp. compact) from A ψ α (B N ) into itself if 1
(3) C φ is compact on H ∞ (B N ); (4) φ ∞ < 1. Proof. It is well-known that C φ is compact on H ∞ (B N ) if and only if φ ∞ < 1. Using the fact that a composition operator is compact on H ψ (B N ) (resp. A ψ α (B N )) if and only if for every bounded sequence ( f n ) n ⊂ H ψ (B N ), f n ψ ≤ 1, (resp. ( f n ) n ⊂ A ψ α (B N ), f n A ψ α ≤ 1)
which tends to 0 uniformly on every compact subset of B N , then

f n • φ ψ ---→ n→∞ 0 (resp. f n • φ A ψ α ---→ n→∞ 0), it is not difficult to show that if φ ∞ < 1, then C φ is compact on H ψ (B N ) (resp. A ψ α (B N )
.) It remains to prove (1)⇒( 4) and ( 2)⇒(4). We first deal with the proof of (1)⇒( 4). We will use the necessary part of Theorem 2.3. Let us assume that φ induces a compact composition operator on every Hardy-Orlicz space. According to (2.2), this means that sup 0<r<1 sup

ζ ∈S N µ φ r (S (ζ , h)) = o h→0 1 ψ (Aψ -1 (1/h N )) ,
for every A > 0 and every Orlicz function ψ, which in turn implies

(3.1) sup 0<r<1 sup ζ ∈S N µ φ r (S (ζ , h)) ≤ 1 ψ (Aψ -1 (1/h N )) ,
for every A > 0, for every Orlicz function ψ and for h sufficiently small. We intend to show that sup 0<r<1 sup

ζ ∈S N µ φ r (S (ζ , h)) = 0,
for all 0 < h ≤ h 0 , h 0 ∈ (0, 1). By contradiction, we suppose that sup 0<r<1 sup

ζ ∈S N µ φ r (S (ζ , h)) = 0 for every h > 0, since h -→ sup 0<r<1 sup ζ ∈S N µ φ r (S (ζ , h))
is an increasing function on (0, 1). A straightforward computation shows that inequality (3.1) is satisfied for every A > 0, for every Orlicz function ψ and for h small enough, if and only if we have, by putting x = 1/h,

(3.2) ψ -1 x N ψ -1 1/ sup 0<r<1 sup ζ ∈S N µ φ r (S (ζ , 1/x)) ≤ 1 A ,
for every A > 0, for every Orlicz function ψ and for x large enough. The following lemma ensures that this cannot occur: ν (g (x)) ≥ δ > 0, for every x large enough.

We assume for a while that this lemma has been proven, and we finish the proof of Theorem 3.1. With the notations of the lemma, we put

f (x) = x N and g (x) = 1/ sup 0<r<1 sup ζ ∈S N µ φ r (S (ζ , 1/x)) .
It is clear that lim x→+∞ f (x) = +∞; since C φ is supposed to be compact on every H ψ (B N ), it is in particular bounded on H p (B N ) ([3, Corollary 3.5]), then we have g (x) ----→ x→+∞ +∞ (Theorem 2.2, 3). Now, the above lemma provides a constant δ > 0 and a continuous increasing concave function ν, tending to infinity at infinity, such that

ν x N /ν    1 sup ξ ∈S N µ φ S f (ξ , 1/x)    ≥ δ > 0,
for every x large enough. It is not difficult to check that ν can be constructed such that ψ = ν -1 is an Orlicz function, i.e. such that

x ν (x) ----→ x→+∞ +∞ (
that is what we are doing in the proof of the lemma below). Therefore, we get a contradiction with Condition (3.2), so we must have

sup 0<r<1 sup ξ ∈S N µ φ r S f (ξ , h) = 0,
for every h > 0 small enough. It follows that there exists some 0 < r 0 < 1 such that

(3.3) sup 0<r<1 µ φ r (C (r 0 , 1)) = 0,
where C (r 0 , 1) = {z ∈ B N , r 0 < |z| < 1}. We intend to show that φ -1 (C (r 0 , 1)) = / 0, which should give the result. Let 0 < r < 1 and let us look at the set φ -1 r (C (r 0 , 1))

∩ S N = {ζ ∈ S N , φ r (ζ ) ∈ C (r 0 , 1)} . Condition (3.3) implies σ N φ -1 r (C (r 0 , 1)) ∩ S N = 0. Since φ r is continuous on B N , φ -1
r (C (r 0 , 1)) ∩ S N must be an open subset of S N and then must be empty. So we have proven that, for any r ∈ (0, 1),

{ζ ∈ S N , φ r (ζ ) ∈ C (r 0 , 1)} = φ -1 (C (r 0 , 1)) ∩ rS N = / 0, where rS N = z ∈ B N , |z| = r , hence φ -1 (C (r 0 , 1)) = 0<r<1 φ -1 (C (r 0 , 1)) ∩ rS N = / 0.
The proof in the Bergman-Orlicz context is much easier. Proceeding as above and using the necessary part of the second point of Theorem 2.4, we get that Condition µ φ (C (r 0 , 1)) = 0 must hold, for some 0 < r 0 < 1. By continuity of the map φ on B N , φ -1 (C (r 0 , 1)) cannot be but empty.

To finish the proof, we have to do that of Lemma 3.2:

Proof of Lemma 3.2. The proof will be constructive. Let f and g be given as in the statement of the lemma. We are going to build by induction a sequence (a n ) n which will be of interest in the construction of the desired function ν. We put a 0 = 0, a 1 = 1, and we deduce a n+2 from a n and a n+1 in the following way: we define

b n+2 = sup {g (x) , f (x) ≤ a n+1 } and a n+2 = max {b n+2 , a n+1 + (a n+1 -a n )} .
We observe that:

(1) If f (x) ≤ a n+1 , then g (x) ≤ a n+2 ; (2) a n+2 -a n+1 ≥ a n+1 -a n ≥ 1.
We now construct the concave function ν as a continuous affine one, whose derivative is equal to

ε n = 1 √ n (a n+1 -a n )
on the interval (a n , a n+1 ), and with ν (0) = 0. Of course ν is increasing and then maps [0, +∞[ into itself. Since ε n is decreasing, because of 2) above, ν is concave. In order to check that ν tends to infinity at infinity, we compute ν (a n ):

ν (a n+1 ) = ν (a n ) + ε n (a n+1 -a n ) = ν (a n ) + 1 √ n . Therefore ν (a n+1 ) = ∑ n+1 k=1 1 √ k which shows that lim x→+∞ ν (x) = +∞, since a n → +∞.
We now check that ν

• f (x) ν • g (x)
is bounded below by some constant δ > 0, when x is big enough.

Let x ∈ [0, +∞[, and let n be an integer such that

a n ≤ f (x) ≤ a n+1 ; we have ν ( f (x)) ≥ ν (a n ).
Using the first property of the sequence (a n ) n above, we get ν (g (x)) ≤ ν (a n+2 ). This yields, for

n ≥ 1, ν ( f (x)) ν (g (x)) ≥ ν (a n ) ν (a n+2 ) = ∑ n k=1 1 √ k ∑ n+2 k=1 1 √ k ≥ δ > 0,
hence the result.

Korányi regions and compactness of C φ on Hardy-Orlicz and Bergman-Orlicz spaces.

For ζ ∈ S N and a > 1, we recall that the Korányi approach region Γ (ζ , a) of angular opening a is defined by

Γ (ζ , a) = z ∈ B N , |1 -z, ζ | < a 2 1 -|z| 2 .
[5, Theorem 6.4] and the third part of Theorem 2.2 yields the following result: ( Indeed, the first point is nothing but the continuity of every composition operator on the disc, and the second one is contained in [5, Proposition 3.25] which says that, whenever φ (D) is contained in some nontangential approach region in D, then C φ is Hilbert-Schmidt in H 2 (D), hence compact on every H p (D), 1 ≤ p < ∞.

) If φ (B N ) ⊂ Γ (ζ , b N ), then C φ is bounded on H ψ (B N ); (2) If φ (B N ) ⊂ Γ (ζ , b), for some ζ ∈ S N and for some 1 < b < b N , then C φ is compact on H ψ (B N ). 1
(2) Following the proof of [4, Theorem 3.3], it is not difficult to show that the boundedness or compactness of C φ on H ψ (B N ) implies that on A ψ α (B N ), for any α > -1, as soon as the Orlicz function ψ satisfies the ∆ 2 -Condition. Thus, the first two points of the previous theorem also holds for Bergman-Orlicz spaces.

The following result shows that Theorem 3.3 does not hold as soon as the Orlicz function grows fast.

Theorem 3.5. Let ψ be an Orlicz function satisfying the ∆ 2 -Condition. Then, for every ζ ∈ S N and every b > 1, there exists a holomorphic self-map φ taking

B N into Γ (ζ , b), such that C φ is not compact on H ψ (B N ).
Remark 3.6. Observe that there is no assumption on N.

Proof of Theorem 3.5. The proof will use the necessary part of the second point of Theorem 2.2. First of all, we recall that ∆ 2 -Condition implies ∇ 2 -Condition (see Paragraph 2.2). We denote by e 1 the vector (1, 0 . . .0) in C N . It is clearly enough to prove the theorem for ζ = e 1 . For any b > 1, we set

(3.4) β = 2 cos -1 (1/b) π in (0, 1)
. We need a lemma whose proof is included in that of [START_REF] Cowen | Composition Operators on Spaces of Analytic Functions[END_REF]Theorem 6.4]: Proof. Without going into details, we briefly give the ideas of the proof. It uses the deep Alexandrov's result which gives the existence of non-constant inner functions in B N ( [START_REF] Alexandrov | Existence of inner functions in the unit ball[END_REF]). Therefore, we consider a function φ which can be written

φ = κ • ϕ, 0 ′ ,
where 0 ′ is the (n -1)-tuple (0, . . ., 0), ϕ is an inner function with ϕ (0) = 0, and where κ is a biholomorphic map from D onto the non-tangential approach region Γ (1, b) in the disc, defined by

Γ (1, b) = z ∈ D, |1 -z| < b 2 1 -|z| 2 .
One can show that the lower-estimate (3.7) holds for this map φ , using the fact that inner functions ϕ are measure preserving maps of S N into T (see [15, p. 405]) in the following sense:

σ N (ϕ * ) -1 (E) = σ 1 (E) ,
for any Borel set E in T.

Let φ be as in the statement of the theorem. According to the necessary part of the second point of Theorem 2.2, the previous lemma ensures that, if we show that for any Orlicz function ψ satisfying the ∆ 2 -Condition, for any β ∈ (0, 1), there exists some A > 0 such that

(3.6) 1 ψ (Aψ -1 (1/h N )) ≤ h 1/β ,
for every h small enough, then C φ would not be compact on H ψ (B N ). Now, putting y = ψ -1 1/h N , an easy computation implies that (3.6) is equivalent to ψ (y) The previous remark leads us to say some words about weighted Bergman-Orlicz spaces in dimension one. Indeed, we can adapt the proof of Lemma 3.7 to get the following result. Lemma 3.9. Let α > -1, let b > 1 and let β be defined by (3.4). There exists a holomorphic map

φ : D → D, φ (D) ⊂ Γ (1, b), such that (3.7) v α φ -1 (S (1, h)) ≥ Ch (2+α)/β ,
for some constant C > 0 depending only on α, φ and b.

For the seek of completeness, we prefer to give some details of the proof of this lemma, in order to point out the slightly difference with that of [START_REF] Cowen | Composition Operators on Spaces of Analytic Functions[END_REF]Theorem 6.4,[START_REF] Charpentier | Composition operators on Hardy-Orlicz spaces on the ball[END_REF]].

Proof. We consider a biholomorphic map κ from D onto

Γ (1, b) = z ∈ D, |1 -z| < b 2 1 -|z| 2 ,
for some b > 1. As it is explained in the proof of [START_REF] Cowen | Composition Operators on Spaces of Analytic Functions[END_REF]Theorem 6.4,[START_REF] Charpentier | Composition operators on Hardy-Orlicz spaces on the ball[END_REF]], if β ∈ (0, 1) is defined by (3.4), then the function g (z) := 1 -κ (z) (1z) β is continuous and non-zero in D∩V , where V is a closed disc (with non-empty interior) centered at 1, and κ -1 (S (1, h)) ⊂ D ∩ V , for h > 0 sufficiently small. Then, for such h, we follow the computation at the end of the proof of [START_REF] Cowen | Composition Operators on Spaces of Analytic Functions[END_REF]Theorem 6.4,[START_REF] Charpentier | Composition operators on Hardy-Orlicz spaces on the ball[END_REF]

] to get κ -1 (S (1, h)) ⊃ S 1, Ch 1/β ,
for some constant C > 0, depending only on κ and b. Therefore,

v α κ -1 (S (1, h)) ≥ Ch (2+α)/β ,
where C > 0 depends on α, κ and b. The proof of the previous proposition does not work directly when N > 1, because we do not know if there exists a non-constant inner function which is measure-preserving from B N to D in the following sense:

v α φ -1 (E) = A α (E) ,
for any E ⊂ D, where A α is the weighted area measure in D.

3.3.

Another characterization of the compactness of C φ on weighted Bergman-Orlicz spaces.

The following result generalizes that obtained in [START_REF] Zhu | Compact composition operators on Bergman spaces of the unit ball[END_REF] for classical Bergman spaces:

Theorem 3.11. We assume that α > -1. Let φ : B N → B N be holomorphic and let ψ be an Orlicz function which satisfies the ∇ 0 -Condition. We assume that C φ is bounded from A

ψ β (B N ) into itself for some -1 < β < α. Then C φ is compact from A ψ α (B N ) into itself if and only if (3.8) lim |z|→1 ψ -1 1/ (1 -|φ (z)|) N(α) ψ -1 1/ (1 -|z|) N(α) = 0.
Proof. The proof of the necessary part is the same as that of [START_REF] Èvre | Composition operators on Hardy-Orlicz spaces[END_REF]Theorem 5.7]. We deal with the proof of the sufficiency of (3.8). Without loss of generality, we assume that φ (0) = 0. According to the second point of Theorem 2.4, by the convexity of the Orlicz function ψ, it is sufficient to show that for every B > 0, there exists h 0 ∈ (0, 1), such that

(3.9) ψ -1 1/µ φ ,α (S (ξ , h)) ≥ Bψ -1 1/h N(α) ,
uniformly in ξ ∈ S N , and for any 0 < h < h 0 . Let α and β be as in the statement of the theorem. We have

µ φ ,α (S (ξ , h)) = φ -1 (S(ξ ,h)) 1 -|z| 2 α dv (z) ≤ 2 α-β sup z∈φ -1 (S(ξ ,h)) (1 -|z|) α-β φ -1 (S(ξ ,h)) 1 -|z| 2 β dv (z) = 2 α-β sup z∈φ -1 (S(ξ ,h)) (1 -|z|) α-β µ φ ,β (S (ξ , h)) ≤ 2 α-β sup z∈φ -1 (S(ξ ,h)) (1 -|z|) α-β 1 ψ C β ψ -1 1/h N(β ) , (3.10)
where the last inequality stands for some constant C β ≥ 1 and for h small enough, since C φ is supposed to be bounded on A ψ β (B N ). Now, since αβ > 0, the hypothesis (3.8) is equivalent to the fact that, for any A > 0,

(1 -|z|) α-β ≤ 1 ψ Aψ -1 1/ (1 -|φ (z)|) N(α) α-β N(α)
, whenever |z| is close enough to 1. Moreover, let us observe that if z ∈ φ -1 (S (ξ , h)), then

1 -|z| ≤ 1 -|φ (z)| ≤ |1 -φ (z) , ξ | < h
so that, for any A > 0, sup z∈φ -1 (S(ξ ,h))

(1 -|z|) α-β ≤ 1

ψ Aψ -1 1/h N(α) α-β N(α)
, for any h > 0 small enough, using the fact that ψ is a non-decreasing function and that αβ > 0. Thus, it follows from (3.10) that µ φ ,α (S (ξ , h)) ≤ 2 α-β 1

ψ Aψ -1 1/h N(α) α-β N(α) 1 ψ C β ψ -1 1/h N(β ) ,
for any A > 0 and h small enough. Using (3.9), the last inequality ensures that C φ will be compact on A ψ α (B N ) if, for any B > 0, there exists a constant A > 0 such that (3.11) ψ -1 ψ Aψ -1 1/h N(α) α-β N(α) .ψ C β ψ -1 1/h N(β ) ≥ Bψ -1 1/h N(α) , for h small emough. Putting x = ψ -1 1/h N(α) , (3.11) is equivalent to α) .ψ C β ψ -1 (ψ (x)) N(β )/N(α) , which is in turn satisfied, using the convexity of ψ and C β ≥ 1, if ψ (Bx) N(α) ≤ ψ (Ax) α-β .ψ (x) N(β ) , for x large enough. Let us notice that this last inequality is equivalent to ψ (Bx) N(β )/(α-β ) ψ (x) N(β )/(α-β ) ≤ ψ (Ax) ψ (Bx) , for x large enough, which is nothing but the ∇ 0 -Condition (see Paragraph 2.2).

ψ (Bx) ≤ ψ (Ax) α-β N(
Remark 3.12. We mention that the proof of the necessary part of the previous theorem does not use the boundedness of C φ on some "smaller" weighted Bergman-Orlicz space. Also, it is not necessary to assume that ψ satisfies the ∇ 0 -Condition.

Since every composition operator is bounded on every A ψ α (B N ) as soon as ψ satisfies the ∆ 2 -Conditon, we have the following corollary: = 0, since ψ satisfies the ∆ 2 -Condition. Indeed, it is easy to deduce from the definition of the ∆ 2 -Condition that, if a > 1, then ψ -1 (x a ) ≤ Cψ -1 (x) for some constant C > 0 and for x large enough.

This corollary highlights an important difference with the classical weighted Bergman case: when ψ satisfies the ∆ 2 -Condition, the compactness (as well as the boundedness, Theorem 2.4) of composition operators on A ψ α (B N ) does not depend on α > -1.

Yet, this independency does not stand for α = -1, i.e. for Hardy-Orlicz spaces; indeed, it was shown ([7, Theorem 5.8]) that there exists some Orlicz function ψ which satisfies the ∆ 2 -Condition (to be precise, ψ (x) = e x 2 -1) such that there exists a holomorphic self-map of D inducing a compact operator on A ψ α (D), but not compact on H ψ (D). Nevertheless, the same proof as that of the necessary part of Theorem 3.11 for Hardy-Orlicz spaces yields: ψ -1 (1/ (1 -|z|)) = 0.

2. 3 .

 3 Background results. All the results of the present paper are based on characterizations of the boundedness and compactness of composition operators on Hardy-Orlicz and Bergman-Orlicz spaces ([2, 3]). As I already said, these characterizations essentially depend on the manner in which the Orlicz function grows. The characterizations of the boundedness and compactness of C φ involve adapted Carleson measures, and then geometric notions. For ζ ∈ S N and 0 < h < 1, let us denote by S (ζ , h) and S (ζ , h) the non-isotropic "balls", respectively in B N and B N , defined by

Theorem 3 . 3 .

 33 Let ψ be an Orlicz function satisfying the ∆ 2 ∩∇ 2 -Condition. Let also φ : B N → B N be holomorphic. We assume that N > 1 and we fix b N = (cos (π/ (2N))) -1 .

)

  Both of the above results are sharp in the following sense: given c > b N , there exists φ with φ (B N ) ⊂ Γ (ζ , c), for some ζ ∈ S N , and C φ not bounded on H ψ (B N ); there also exists some φ with φ (B N ) ⊂ Γ (ζ , b N ), for some ζ ∈ S N , and C φ not compact on H ψ (B N ).Remark 3.4. (1) If N = 1, the two first point of the previous theorem are true if we put b 1 = +∞ and Γ (ζ , +∞) = D.

Lemma 3 . 7 .

 37 Let b > 1 and let β be defined by(3.4). There exists a holomorphic map φ : B N → B N , with φ (B N ) ⊂ Γ (e 1 , b), such that(3.5) σ N φ -1 (S (e 1 , h)) ≥ Ch 1/β , for some constant C > 0 depending only on φ and b.

Now, it isProposition 3 . 10 .

 310 sufficient to argue as in in the end of the proof of Theorem 3.5 to get: Let α > -1, let b > 1 and let ψ be an Orlicz function satisfying the ∆ 2 -Condition. There exists a holomorphic map φ : D → D, with range contained in some nontangential approach region Γ (ζ , b), ζ ∈ T, such that the induced composition operator C φ is not compact on A ψ α (D).

Corollary 3 . 13 .ψ -1 1 /

 3131 Let α > -1, let ψ be an Orlicz function satisfying the ∆ 2 -Condition and let φ :B N → B N be holomorphic. Then C φ is compact on A ψ α (B N ) if and only if lim |z|→1 ψ -1 (1/ (1 -|φ (z)|)) ψ -1 (1/ (1 -|z|)) = 0.Proof. It is sufficient to remark that we have lim|z|→1 (1 -|φ (z)|) N(α) ψ -1 1/ (1 -|z|) N(α) = 0 ⇐⇒ lim |z|→1 ψ -1 (1/ (1 -|φ (z)|)) ψ -1 (1/ (1 -|z|))

Proposition 3 . 14 .

 314 Let φ : B N → B N be holomorphic and let ψ be an Orlicz function. If C φ is compact on H ψ (B N ), then lim |z|→1 ψ -1 (1/ (1 -|φ (z)|))

  1) If ψ satisfies the uniform ∇ 0 -Condition, then C φ is bounded from H ψ (B N ) into itself if and only if µ φ is a ψ-Carleson measure. (2) If ψ satisfies the ∇ 0 -Condition, then C φ is compact from H ψ (B N ) into itself if and only if µ φ is a vanishing ψ-Carleson measure. (3) If ψ satisfies the ∆ 2 -Condition, then C φ is bounded (resp. compact) from H ψ (B N ) into itself if and only if µ φ is a Carleson measure (resp. a vanishing Carleson measure). (4) If ψ satisfies the ∆ 2 -Condition, then C φ is bounded on H ψ (B N ). first two points are contained in [3, Theorem 3.2]; according to [5, Theorem 3.35], the third point means that, if ψ satisfies the ∆ 2 -Condition, then C φ is bounded (resp. compact) on H ψ (B N ) if and only if it is on H p (B N ) (see [3, Corollary 3.4]). The last point is [3, Theorem 3.7].

	The

  and only if µ φ is a Bergman-Carleson measure (resp. a vanishing Bergman-Carleson measure). (4) If ψ satisfies the ∆ 2 -Condition, then C φ is bounded on A According to [5, Theorem 3.37], the third point means that, if ψ satisfies the ∆ 2 -Condition, then C φ is bounded (resp. compact) on A ψ α (B N ), if and only if C φ is bounded (resp. compact) on A p α (B N ). Let φ : B N → B N be a holomorphic map. The following assertions are equivalent: (1) C φ is compact on H ψ (B N ), for every Orlicz function ψ;(2) For some α > -1, C φ is compact on A

	3. MAIN RESULTS
	3.1. Compactness of C φ on every Hardy-Orlicz or Bergman-Orlicz spaces. The following
	theorem completes both Theorem 2.2 and Theorem 2.4.
	Theorem 3.1.

ψ α (B N ).

ψ α (B N ), for every Orlicz function ψ;

  Remark 3.8. When N = 1, the proof of Lemma 3.7 can be simplified: first, because the existence of a non-constant inner function in the unit disc is trivial, and then because it clearly suffices to take ϕ (z) = z, what just turns the proof of Lemma 3.7 into considering a biholomorphic map κ from D onto an non-tangential approach region.

	1 Nβ ≤ ψ (Cy) for some constant C > 0. We conclude the proof by noticing that this latter condition is trivial if 0 < 1 Nβ ≤ 1 (which al-
	lows to recover the compactness part of the third point of Theorem 3.3), while it is nothing but
	∆ 2 -Condition if	1 Nβ	> 1 (see Paragraph 2.2).


Christophe Debain
email: christophe.debain@cemagref.fr

Florent Malartre

Pierre Delmas

Roland Chapuis

Thierry Humbert

Michel Berducat

Smart Range Finder and Camera Fusion -Application to Real-Time Dense Digital Elevation Map Estimation

This paper is about environment perception for navigation system in outdoor applications. Unlike other approaches that try to detect an obstacle of binary state, we consider here a Digital Elevation Map (DEM). This map has to be built in regards to the guidance system's needs. These needs depend on the vehicle capabilities, its dynamics constraints, its speed etc... Starting with the navigation system's needs, our goal is to estimate a precise and dense DEM. Our approach is based on a SLAM algorithm combining a one map rangefinder and a camera. Thus we can estimate both the displacement between two laser scan and have a good density of the reconstructed DEM. The approach has been validated both using simulated realistic data and in real conditions with our robot in outdoor environement. The estimation of the DEM has been done successfully in realtime (approx. 40ms per loop).

I. INTRODUCTION

The design and the conception of mobile robots in highly unstructured outdoor environments as agricultural applications is still an open issue. Many teams in the world are working on this subject addressing different research domains. [START_REF] Kelly | Minimum throughput adaptive perception for high speed mobility[END_REF] introduces the difficulties encountered when autonomy is given to a vehicle which has to move in real contexts. Lots of scientific problems have to be considered in this huge subject as perception of the environment [START_REF] Chanier | Map fusion based on a multi-map slam framework[END_REF], control in difficult situation like sliding terrain [START_REF] Cariou | Adaptive control of four-wheel-steering off-road mobile robots: Application to path tracking and heading control in presence of sliding[END_REF] with stability constraints [START_REF] Bouton | A rollover indicator based on a tire stiffness backstepping observer: Application to an all-terrain vehicle[END_REF], obstacle avoidance [START_REF] Khatib | Real-time obstacle avoidance for manipulators and mobile robots[END_REF]... All these subjects are essential and have to be considered in an elegant manner in order to be efficient and reliable. Since 2004, our team decided to address these problems from a new point of view. Since the beginning of the robotic domain, the most part of the research teams have considered the robotic problems from the sensors point of view. For instance, usually when we want to localise a robot, we use a sensor like a GPS (global localization) or a camera (local localization) and a fusion is achieved. Then, powerful algorithms need to be developed to extract features and the result is the input of the control algorithm in order to automatically drive vehicles.

In our case, we consider the autonomy of the robot and we try to model what is useful for this autonomy. For automatic guidance it means that we should try to extract data in order to optimize a criterion that takes into account the required precision, the needed integrity, safety ... For instance, in our work on the SLAG concept (Simultaneous Localization And Guidance), we show that the perception system can stop to work if the guidance conditions are fulfilled [START_REF] Tessier | A cognitive perception system for autonomous vehicles[END_REF]. The main idea in this concept is to call a resource (sensor or algorithm) when it is necessary evaluating the ratio between gain and cost of each resource. If all the needs of the robot are fulfilled, no resource are called. This concept has been use for the first time in [START_REF] Tessier | Fusion of active detections for outdoor vehicle guidance[END_REF] to automatically guide a vehicle (see figure 1) and no "obstacle notion" was considered [START_REF] Delmas | Environment characterization and path optimization to ensure the integrity of a mobile robot[END_REF], the automatic adaptation to different context and the reliability of the demonstration were particularly appreciated.

In our current works we want to apply this concept to insure the physical integrity of the robot. More precisely the goal to reach is to ensure the vehicule has the capability to cross the ground in front it, in a given corridor. Il this application, the system requires the ground geometry in a dynamic zone (depending on its speed, and the wished navigation corridor). The purpose of this paper is to obtain the Digital Elevation Map (DEM) in real-time with a good accuracy. In the proposed approach this area is obtained using both a camera and a laser scanner coupled in a SLAM process. The guidance system will therefore searches for the most optimal path in regards to the criteria of stability and speed.

Obtaining a DEM is fairly easy if we use dense data sensors (such as a Velodyne, for example). If we use more conventional sensors (such as camera and classical rangefinders) obtaining the DEM is much more difficult.

Using a single camera, although, seems attractive in terms of implementation and cost issues. Nevertheless major problem occur: density of the DEM, stability problems, drift and scale ratio in usual SLAM technics.

To stabilize the solutions, the approach suggested here combines two sensors: a camera and a lidar in a single SLAM process. To reach this goal, the wished analysis area is backprojected in the image frame with a flat-ground assumption. The region of interest obtained is divided into a large number of points (usually a few hundreds). The one map laser scanner provides a dense set of 3D points in front of the vehicle (at approximately a 5m distance). These points are tracked in the image in a SLAM process in order to obtain iteratively a dense DEM. Depending on the needs of the the guidance system, points can be added and tracked between each laser scan to fill the lack of data.

The paper is organized as follows: first, we present classical approaches in section II. In section III, our approach is presented and the main parts of the system is exposed. We will show the obtained results with our approach in section IV. Finally, conclusions and future works will be addressed in section V.

II. RELATED WORKS A. Obstacle detection

Usually, obstacles detection is based on detecting and classifying them with binary state: safe, unsafe. In [START_REF] Manduchi | Obstacle detection and terrain classification for autonomous off-road navigation[END_REF], authors describe how to segment stereo pictures and decide what is an obstacle and what is not. This segmentation does not rely on the vehicle's dynamical constraints and so may be unusable depending on vehicle state (a small stone may not be an obstacle at low speed, but can be at high speed). The same comments can be done for [START_REF] Michels | High speed obstacle avoidance using monocular vision and reinforcement learning[END_REF] where authors interpret pictures from monocular camera to find obstacles after a learning process. All those methods rely on data interpretation and/or 2D 1/2 representation to perceive potential obstacle. On the other hand, for the vehicle safety and if we consider optimal path generation, a 3D representation of the environment is more suitable for the navigation system. Therefore, a DEM is considered to be able to find best path according to both dynamics and integrity constraints of the vehicle.

B. Digital Elevation Map

In order to obtain the optimal path, we need a 3 dimensionnal representation of the environment to satisfy the guidance system. It is also important to know the uncertainty on this 3D representation to choose the safest path. For example, the guidance system could make a mistake if it is detecting a flat ground with a big uncertaincy that can affect the physical integrity of the vehicle. The localization of the mobile robot on this map must also be considered for the control system and to chose an optimal path. In this paper, we will consider the world as static. Therefor, moving object are not considered here.

C. Vision-only Digital Elevation Map

Many researchers work on the DEM estimation problem using stereo-vision, optical flow, etc (for example see [START_REF] Bonin-Font | Visual navigation for mobile robots: A survey[END_REF]). In outdoor conditions, dense stereo approaches are not well suited because of the baseline required to obtaine good results. Optical flow approaches are rather restricted to obstacle movements rather than ground reconstruction. Nowadays a popular method is the Simultaneous Localization And Mapping approach (SLAM) which consists in finding the pose of the vehicle and building a map in the same process [START_REF] Lemaire | Vision-based SLAM: stereo and monocular approaches[END_REF][START_REF] Diebel | Simultaneous localization and mapping with active stereo vision[END_REF][START_REF] Civera | 1-point ransac for ekf-based structure from motion[END_REF]. If we consider vision-only SLAM, the main problem is that visual features used for the process are usually extracted from remarkable points (Harris points [START_REF] Harris | A combined corner and edge detection[END_REF] or SIFT [START_REF] Lowe | Object recognition from local scaleinvariant features[END_REF]). Therefore the elevation map could have a lack of information big enough to make the guidance system fail between these remarkable points. In fact it is really necessary to control the amount of data to perceive from the environment in order to fulfill the needs of the guidance system. On the other side, the control system must adapt itself if our perception system can't provide enough accurate data (due to a lack of computation time or insufficient sensor accuracy).

A solution coming up is to use stereo pair camera to build the elevation map in the SLAM process. The problem remains that the accuracy of far points is dependant of the stereo baseline. The accuracy is also influenced by the good calibration of the cameras.

D. Lidar-only Digital Elevation Map

Besides the high cost 3D laser scanner solutions a few works use one map scanners. Indeed, having only one scan does not give the whole map in front of the robot. Nevertheless, it is still possible to build a DEM using a one map rangefinder. For example, consider the vehicle is equiped with a Lidar inclined to look at the ground. We make the assumption that this mobile robot is perfectly localised (Position and Orientation). Consequently we are able to build the DEM as long as the vehicle moves. Obviously, the granularity of the reconstruction depends on the vehicle speed. The space between two acquisitions can be modulated if we control the speed. But the faster we move, the more accurate map we need. The consequence is that if we want to go fast, we cannot because we have a large unobserved space between two rays proportional to the vehicle speed (fig. 2).

E. Our approach

In conclusion, using a rangefinder alone is not sufficient (even a 3D rangefinder) because the quantity of information is not sufficient and not controllable. But a camera can provide high density data on demand. On the other hand, a rangefinders is usually used on mobile robot for safety (detecting pedestrians for example). This paper proposes a method to combine the accurate data from the rangefinder with the dense and controllable data from a camera in a SLAM process.

A. Introduction

In this section, we present our method to efficiently build a DEM flexible enough to fit the guidance needs. An Extended Kalman Filter will be used. Our method consists in combining the 3D data provided by the lidar to the corresponding points in the image in a SLAM process. We propose to observe the rangefinder points in the camera in order to have both an accurate estimation of the 3D environment (for the DEM) and an observation of this environment in the camera (for the localization). The camera informations are used to extend our world perception thanks to the high density of information it gives. Figure 3 shows the system.

Here the difficulties are due to the projection of the 3D points on the camera being non-remarkable since we wish the guidance to control the data density and not the environment. The technics used in the state of the art to track points in pictures flow are not usable here because authors try to matches points extracted from the environment as being remarkable points. Therfore, we have developed algorithms able to track common points in picture giving a patch and a region of tracking. Usually the points we track come from a laser scan: their 3D position is therefore known with a low uncertainty (assuming calibration stage has been done previoulsy between camera and laser scanner, see III-E). Since most of the points come from a laser scan, the SLAM stability is very high and it is possible to reconstruct points out of the laser scan to fill the DEM in wished positions.

Notice we only use a subset of the points given by the laser scanner (usually 30). We will see later that all the scanner points will be used to compose the dense MAP (see section III-F).

B. Problem formulation

First, the state vector of our system is composed of 6 parameters of the vehicle pose X v = (X,Y, Z, α, β , γ) T plus 3 parameters per points of our local map X p = (x p , y p , z p ) T , p ∈ [1, n] which give the state vector [START_REF] Bonin-Font | Visual navigation for mobile robots: A survey[END_REF].

X = (X T v , X T 1 , • • • , X T n ) T = (X,Y, Z, α, β , γ, x 1 , y 1 , z 1 , . . . , x n , y n , z n ) T (1)
n represents the number of points necessary to fullfil the guidance system's needs depending on its dynamical constraints (speed, vehicle caracteristic ...). The calculated points are initialized into the focus area provided by the application. The 3D points initilization will be done as follow:

• m < n points will be initialized with the informations coming from the Lidar. We can found the 3D coordinates of the rangefinder's 2D points in the camera's frame using the transition matrix P L-C . Determination of the P L-C matrix will be done in section III-E. This matrix is used to transfer the points from the lidar frame to the camera frame. The uncertainty associated is given by the rangefinders manufacturer (a few centimeters). Those points will be refered as lidar points.

• l = nm points will be initialized in the world frame to fill the focus area of the guidance system. Their initialization will be made assuming they are on the ground and with a certain uncertainty (typically 0.5m). Those points will be refered as camera points. Thanks to the rangefinder, the initial state of the EKF is very close to the real state and its covariance is low for lidar points. These conditions will guaranty a good convegence of the filter.

The EKF needs to have access to the equation of the state evolution and the state observation. In our case, the observation is given by the projection of the state points in the camera's frame (u, v). To update the state vector we also need the jacobian matrix of those equations.

These equations and their respective jacobian matrices are given in [START_REF] Malartre | Digital elevation map estimation by vision-lidar fusion[END_REF]. The next step will be to update the state vector X for each time t.

C. Global method v.s Point to Point method

Our objective is to estimate the whole state vector X for each time t. We assume we are able to track n t ≤ n points X i of the state vector in the next image using a patch (recorded in the previous image or when we added this point in the state vector) in a small traking area (deduced, as usual, from the covariance of the vehicle state estimation). This gives us observation sets (u i , v i ) T with (i ∈ [1, n t ]) to feed the SLAM process.

However the EKF can be processed in two different ways: (1) we can feed the filter with all the data we got in the current image and (2) we can provide only one observation at a time and repeat the process for all the data. The first method (described in Algorithm 1 and referenced as GM) have some drawbacks. In this case, the size of the tracking region is constant and used for every points. That means that we cannot focus our attention taking account succesfull matches and the signal noise ratio won't be optimal. Thus the probability for a false positive match is increased. We also have to deal with the inversion of a 2n t × 2n t matrix where n t is the number of points from the state vector sucessfully matched. This inversion is very time consuming.

On the other hand algorithm (2) is described in Algorithm 2 and refered as PTPM. In thois case, we see that the size of the tracking region is given by the current observation covariance. The first point wil make the process converge and the area of interest for the nexts points will be drastically reduced in size. Thus, we have a size of tracking region which is focused after Compute the observation jacobian.

4:

Compute the observation covariance.

5:

Track the point in the image flow.

6:

Update the EKF with the innovation. 7: end for each iteration of the EKF (but we have nt iterations for a given image). The benefit are both in computation time and rejection of false positive matchings. Furthermore, unlike the first method we only have to compute n t 2 × 2 matrices (the computational times are approximately divided by n t ).

Those two methods are currently implanted on our software architecture. This paper will also compare results for the two methods.

D. Unsynchronized data

We have now all the equations to implement the EKF for our Vision-Rangefinder system.

However we also need to fill our state vector with data's from the rangefinder. For this, we have to project data from the rangefinder frame in the camera frame and acquire patches on the image where the laser impact occures (to be observed several times). The problem is that the data from the rangefinder are not synchronized with the camera's pictures (see fig. 2)

Thanks to the previous work in our lab. [START_REF] Tessier | A real-time, multi-sensor architecture for fusion of delayed observations: Application to vehicle localisation[END_REF] we can recover an optimal state vector at the desired date. In this case we use a spline interpolation of the 6 parameters of the localization to retrieve the camera position and orientation at the rangefinder date.

Once the position of this camera is found, the transition matrix

P (C [L-date] -C [C-date] )
has to be computed. This matrix projects points of camera frame at rangefinder's date to the nearest (in acquisition time) valid camera frame. A valid camera frame is a frame where a picture has been acquired.

The equation to project rangefinder points X L to a valid camera frame X c is given by eq.( 2).

X c = P (C [L-date] -C [C-date] ) P L-C X L (2)
At this moment, the points can be added to the state vector with an initial covariance. Finaly we have described how to add points in the state vector. Together with the EKF equations and the main algorithm to be processed we now have all informations and steps to perform our SLAM process.

E. Calibration of a Camera-Rangefinder set

In section III we saw that a transition matrix is needed in order to use the rangefinder points in the camera. This transition matrix is simply a 4 × 4 matrix composed of 3 rotations and 3 translations. We have used software already developped in our laboratory. The principle is described in [START_REF] Pujas | Etalonnage d'un systme de fusion camra-tlmtre pour la fusion multi-sensorielle[END_REF] even if we envizage to use more recent methods (such as [START_REF] Zhang | Extrinsic calibration of a camera and laser range finder[END_REF]) in near future. To achieve the extrinsic calibration we use an orthogonal pattern (Figure 5).

The vehicle moves forward to the pattern and a dataset composed of camera picture and rangefinder scan is acquired. The camera dataset is used to compute the position and orientation of the camera in the pattern mark (P P-C ). Then the rangefinder dataset and the vehicle trajectory are used to compute the position and orientation of the pattern in the rangefinders frame (P L-P ). The calibration matrix is then simply deduced using eq. ( 3).

P L-C = P L-P P P-C

(3)

F. Dense Digital Elevation Map

The presented SLAM is able to compute a large list of 3D points in real-time (about 150). But this is usually insufficent for the guidance process which needs a map of about one thousand points to evolve at a raisonnable speed (2 m/s). Thus we have to gather a more dense DEM. This task is easily obtained thanks to the rangefinder. Even if all the rangefinders points have not been used in the SLAM process, they can be used to fill the DEM because of their good accuracy.

To do this, we simply have to put the rangefinder slices in the estimated position frame thanks to equation 4. Then, we have to use all our 3D points (from rangefinders and from SLAM) to fill the DEM in accordance to the guidance process needs (density, observation range and length). For points in the DEM not been observed by the rangefinder, we still have enough computation time to include regular camera points in the SLAM process and estimate them.

X DEM = P (C [L-date] ) P L-C X L (4)
Therefore the DEM can be estimated in a local frame at the date wanted by the guidance process. In the following section we will present the results obtained with our approach.

IV. RESULTS

In this section, we present the results of the DEM estimation based on our camera and rangefinder fusion. The comparison will be made between the two algorythms presented in this paper (GM and PTPM, see section III-C). First, in order to quantify the reconstruction errors, we will use a simulator we have developped. Notice we have made two movies 1 showing (1) the real-time behaviour of our method (algorithm PTPM) implemented in real-time on our robot (see figure 1 and figure 6) and (2) the behaviour using our simulator (real-time implementation) where a similar vehicle evolves in a simulated environment. The vehicle is equiped with a pinhole camera (resolution of 800x600) and a rangefinder inclined to look at the ground (impact at about 3 meters on flat ground). The frequency of our camera is set to 25 Hz and the frequency of the rangefinder is set to 12.5 Hz. Our vehicle is moving at about 1.5 m/s. The simulated vehicle is equipped with the same sensors as the real one. We drove the vehicle (both in simulation and real case) manually and, in the simulated case, the reconstructed DEM is compared to the ground truth provided by the simulator.

1 http://www.lasmea.univ-bpclermont.fr/ ftp/pub/chapuis/ICARCV2010

A. DEM estimation

In this case, we compare the reconstructed points coming from the state vector to the ground truth given by the simulator that provides an elevation grid of the environment in front of the vehicle (5x5 meters large). After each iteration, point's height of the state vector are compared with the elevation grid. We have acquired a data set which will be used for both algorithms. Fig. 7 shows the comparison procedure whereas fig. 8 reports the reconstruction error in height of the state points for the whole experimentation (100m long: see movie 1 ). Fig. 9 shows the global world map reconstructed at the end.

The difference with the real simulated world comes from the drift of localization (the world in our simulator is very flat but SLAM is very sensitive to this: a much more complicated world would give better drift).

Concerning the real exprimentation (see movie), the calibration matrix P L-C from the rangefinder to the camera frame has been extracted as mentionned in this paper. The experimentation shows the vehicle moving on an environment over a 100 meters long trajectory (Fig. 10). The DEM reconstruction is shown in the Digital Elevation Map window in the video. This window represents a top view of the 3D points in the local frame. Points shown in green levels are points near the ground (ie. ±10cm). Points in red or blue levels are points respectively higher and lower than the ground.

B. SLAM localization evaluation

Fig. 10 represents a top-view of the followed trajectory with all trajectories computed by our methods (in the simulation case). The error of localization is reported in fig. 11. Similar results are obtained in real situation (see [START_REF] Malartre | Digital elevation map estimation by vision-lidar fusion[END_REF]).

We can see that we are able to follow that kind of trajectory with our camera and rangefinder fusion with a very small drift.

All those results in this paper have been processed in realtime with a C++ software [START_REF] Tessier | A real-time, multi-sensor architecture for fusion of delayed observations: Application to vehicle localisation[END_REF]. The computation time in regards to the number of points is represented in Fig. 12. As expected, the PTPM method is performed faster than the GM method. For exemple, computation time for the whole EKF process includind tracking is tipicaly 40ms for 130 points with the PTPM method (State vector of size 396). This software is independant of the data used (simulation or real experimentation).

V. CONCLUSIONS AND FUTURE WORKS

We have presented an efficient method to recover a dense elevation map whose density could be easily controllable. Our system can satisfy the guidance process goal by controlling the amount of data reconstructed. This need must be taken in account if we want to achieve the automatic guidance whatever the evolution speed is. In this paper, we proposed a SLAM based system combining a laser scanner and a camera. The obtained system is able to provide a DEM of hundreds points to fill the guidance needs with good stability. Two methods have been presented, the PTPM method showed slightly better results than GM method with a smaller computation time. The dense Digital Elevation Map is recovered thanks to the rangefinder and the 6 parameter localization provided by our system.

Those solutions has been implanted in real time on a common laptop configuration and has been demonstrated on both realistic simulated configuration and on our real robot at approximatively 2m/s speed. Depending on the application needs, it would be straighforward to work at high speeds (5 to 10 m/s). A future step will be to check unobserved space of the map and to fill them with camera's point estimated with our SLAM (since we still have enough computation time). Finally, our next step will be to combine this intelligent environment perception with an attentional control system. This will allow the vehicle to move at an optimal speed taking into account both its dynamic and its perceptive capabilities.

Fig. 1 .

 1 Fig. 1. The vehicle AROCO

Fig. 2 .

 2 Fig. 2. Unobservability of the rangefinders

Fig. 3 .

 3 Fig. 3. Camera and rangefinder sensors fusion

Algorithm 1

 1 SLAM global method: GM 1: Prediction of the state evolution. 2: Compute the observation jacobian. 3: Compute the observation covariance. 4: Tracking of state points observed in the image flow. 5: Compute the observation jacobian in regards to the points succesfully tracked. 6: Update the EKF with the innovation.

Algorithm 2 SLAM

 2 Point to Point method: PTPM 1: Prediction of the state evolution. Require: n t = Number of tracked points 2: for i = 1 to n t do 3:

Fig. 4 .

 4 Fig. 4. Interpolating the Rangefinder position and orientation in the 'nearest' camera frame.

Fig. 5 .

 5 Fig. 5. The pattern used to calibrate the camera-rangefinder system

Fig. 6 .

 6 Fig. 6. Internal view of the experimentation. Our vehicle AROCO in a real outdoor environment. On the left : the camera picture with projected state points. On the right : the DEM.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Comparison procedure of the reconstruction error

Fig. 9 .Fig. 10 .

 910 Fig. 9. The world built by our system.

Fig. 11 .

 11 Fig. 11. Trajectory error between slam estimation and the ground truth.

Fig. 12 .

 12 Fig. 12. Computation time of the whole process. Global method and Point per Point method.


Vahid Khanagha
email: vahid.khanagha@inria.fr

Khalid Daoudi

Oriol Pont

Hussein Yahia

IMPROVING TEXT-INDEPENDENT PHONETIC SEGMENTATION BASED ON THE MICROCANONICAL MULTISCALE FORMALISM

Keywords: phonetic segmentation, non-linear speech processing, multiscale signal processing, complex signals and systems

In an earlier work, we proposed a novel phonetic segmentation method based on speech analysis under the Microcanonical Multiscale Formalism (MMF). The latter relies on the computation of local geometrical parameters, singularity exponents (SE). We showed that SE convey valuable information about the local dynamics of speech that can readily and simply used to detect phoneme boundaries. By performing error analysis of our original algorithm, in this paper we propose a 2-steps technique which better exploits SE to improve the segmentation accuracy. In the first step, we detect the boundaries of the original signal and of a low-pass filtred version, and we consider the union of all detected boundaries as candidates. In the second step, we use a hypothesis test over the local SE distribution of the original signal to select the final boundaries. We carry out a detailed evaluation and comparison over the full training set of the TIMIT database which could be useful to other researchers for comparison purposes. The results show that the new algorithm not only outperforms the original one, but also is significantly much more accurate than state-of-the-art ones.

INTRODUCTION

In an earlier work [START_REF] Khanagha | A novel text-independent phonetic segmentation algorithm based on the microcanonical multiscale formalism[END_REF], we proposed a radically new approach for textindependent (TI) phonetic segmentation based on the Microcanonical Multiscale Formalism (MMF). Using methods from the statistical physics field, MMF provides accurate analysis of the nonlinear dynamics of complex signals. It relies on the estimation of local geometrical parameters, the singularity exponents (SE), which quantify the degree of predictability at each point of the signal. When correctly defined and estimated, these exponents can provide valuable information about the local dynamics of complex signals and has been successfully used in many applications ranging from signal representation to inference and prediction [START_REF] Yahia | Motion analysis in oceanographic satellite images using multiscale methods and the energy cascade[END_REF][START_REF] Pont | Empirical evidences of a common multifractal signature in economic, biological and physical systems[END_REF][START_REF] Pont | Description, modeling and forecasting of data with optimal wavelets[END_REF]. We showed that MMF is valid for speech signal and that a simple analysis of SE yields a fast and relatively accurate TI phonetic segmentation algorithm. Experiments on TIMIT showed that, while our approach is simple and conceptually novel, it still achieves better segmentation accuracy than state-of-the-art methods. Moreover our algorithm is almost parameter/threshold free, which is a major advantage w.r.t. traditional algorithms and a desirable property for text and language independent applications.

By performing error analysis of our original MMF-based algorithm, in this paper we propose a technique which better shows the strength of SE and further improves the segmentation accuracy. This technique is a 2-step procedure. In the first step, we detect the phoneme boundary candidates on the given signal and on a low-pass filtered version using our MMF-based algorithm. We then consider the set of all detected boundaries (those of the signal and those of its filtered version) as candidate boundaries. In the second step, we use dynamic windowing and Log-Likelihood Ratio Test (LLRT) to decide which are the correct boundaries. This 2-step structure is similar to that of traditional segmentation methods where there is a boundary pre-selection followed by statistical tests to make the final decision [START_REF] Almpanidis | Robust detection of phone boundaries using model selection criteria with few observations[END_REF].

Another objective of this paper is to address a common difficulty in comparing TI segmentation methods which is the diversity of evaluation datasets and also incoherencies in performance measures. Indeed, while the literature is rich in phonetic segmentation methods, it is relatively poor in material for performance comparison. Most of the papers report either on undefined subsets of known databases or on personal/unaccessible databases. Moreover, it is often difficult to analyze the reported accuracy scores because of the diversity of measures used. This makes it difficult to make fair comparisons between different segmentation algorithms. To the best of our knowledge, the only TI and unsupervised segmentation algorithms that report on known and accessible database are [START_REF] Esposito | Text independent methods for speech segmentation[END_REF] and [START_REF] Dusan | On the relation between maximum spectral transition positions and phone boundaries[END_REF]. The latter reports on the full training set of TIMIT and has the advantage of providing different scores with different sizes of tolerance windows. [START_REF] Qiao | Unsupervised optimal phoneme segmentation: Objectives, algorithm and comparisons[END_REF] and [START_REF] Lin | Phone boundary detection using sample-based acoustic parameters[END_REF] also report on the same dataset but the algorithms they propose are not fully unsupervised. The former assumes prior knowledge of the the number of phonemes in the utterance, while the latter uses prior knowledge of all manual transcriptions to train a neural network. We will thus report our results on the full training set of TIMIT and compare them to [START_REF] Dusan | On the relation between maximum spectral transition positions and phone boundaries[END_REF]. Using different sizes of tolereance windows, we provide detection, insertion and oversegmentation rates and also 2 different global performance measures, F1 and R-value. By doing so, we attempt to provide results which are easy to interpret and compare with.

The paper is structured as follows. In Section 2 we briefly introduce MMF and our previous work on its application to phonetic segmentation. In Section 3 we present our new MMF based segmentation algorithm. The experimental results are presented in Section 4 and we draw our conclusion in Section 5.

OVERVIEW ON PREVIOUS WORK

In [START_REF] Khanagha | A novel text-independent phonetic segmentation algorithm based on the microcanonical multiscale formalism[END_REF] the very first steps in applying MMF for catching non-linear dynamics of speech signal were taken. In this section, we briefly introduce MMF and our previous work on its application to the phonetic segmentation of speech signal.

Microcanonical Multiscale Formalism

MMF is based on the computation of the local scaling exponents of a given signal, whose distribution is the key quantity defining intermittent dynamics of the signal. These exponents are a useful tool for the study of geometrical properties of signals, and have been used in a wide variety of applications ranging from signal compression to inference and prediction [START_REF] Pont | Empirical evidences of a common multifractal signature in economic, biological and physical systems[END_REF][START_REF] Pont | Description, modeling and forecasting of data with optimal wavelets[END_REF]. These exponents are associated to the evaluation of a local power-law scaling behavior at each point in the signal domain. The validity of MMF for a given signal s(t) relies on the existence of such relationship for at least one scale-dependent functional Γr, for each time instance t and for small scales r:

Γr (s(t)) = α(t) r h(t) + o " r h(t) " r → 0 (1)
where h(t) is the so-called singularity exponent (SE). Turiel et al. [START_REF] Turiel | Microcanonical multifractal formalism: a geometrical approach to multifractal systems. part 1: singularity analysis[END_REF] proposed a method for accurate estimation of SE by choosing Γr to be the gradient-modulus measure:

Γr (s(t)) := 1 Λ(Br) Z Br (t) dτ |s ′ (τ )| (2)
It is shown that, the exponent associated to the corresponding power law characterizes the information content and the dynamical transitions of the signal in terms of the scale [START_REF] Frisch | Turbulence: The legacy of A.N. Kolmogorov[END_REF][START_REF] Turiel | The multi-fractal structure of contrast changes in natural images: from sharp edges to textures[END_REF]. Practical implementation to avoid noise and discretization artifacts consists in using a continuous wavelet transform t) . We use the Lorentzian wavelet because it provides an accurate estimation for small exponents which are the most informative ones [START_REF] Turiel | Multifractal measures: definition, description, synthesis and analysis. a detailed study[END_REF].

TΨ [|s ′ |] (r, t) ∝ r h(
In fact, for a given point, the smaller the value of SE is, the higher predictability is in the neighborhood of this point [START_REF] Turiel | Microcanonical multifractal formalism: a geometrical approach to multifractal systems. part 1: singularity analysis[END_REF]. It has been established that the critical transitions of the system occur at these points, and this fact has been successfully used in many applications [START_REF] Yahia | Motion analysis in oceanographic satellite images using multiscale methods and the energy cascade[END_REF][START_REF] Pont | Description, modeling and forecasting of data with optimal wavelets[END_REF][START_REF] Pont | On optimal wavelet bases for the realization of microcanonical cascade processes[END_REF].

Application of MMF to the phonetic segmentation

The validity of MMF and the availability of precise estimates of SE for speech signal, was proved by an extensive evaluation on TIMIT database. We then showed how SE convey valuable information about dynamical transitions of the speech signal. Indeed, since different phonemes should have different geometrical and statistical properties, we expected the corresponding SE to have different behavior inside the boundaries of each phoneme. This was verified by evaluating the time evolution of the distribution of SE. This led us to the development of an automatic segmentation algorithm, by exploiting the easiest interpretation of the changes in distributions, which is the change in averages. In other words, we expect that different phonemes have different averages of exponents compared to their neighboring phonemes. We proposed to use the primitive of the SEs function over time as an estimator of the instantaneous average:

ACC(t) = Z t t 0 dτ h(τ ) (3)
The resulting functional, with a detrending to enhance the presentation, is plotted in Figure 1a. As expected, this new functional revealed the changes in distribution in a more precise way. Indeed, inside each phoneme the functional ACC is almost linear. Moreover, there is a clear change in the slope at the phoneme boundaries. Extensive observations over different sentences confirm this behavior, and thus the strength of the proposed functional, Eq. ( 3).

In order to develop an automatic segmentation algorithm, a very simple solution was employed to fit a piecewise linear curve to ACC and identify the breaking points. To do so, we performed a left-toright search to find the hypothesized boundaries as the points where the mean squared error of the linear fit is below a certain threshold. Finally we reject the nodes located in the silence as the points where the average power in a 30ms window is less than -30dB.

IMPROVING THE MMF-BASED SEGMENTATION

In this section we present a technique which better exploits the strength of singularity exponents in order to improve the accuracy of the phonetic segmentation algorithm described above. This technique is motivated by some observations about the behavior of the latter algorithm at some particular phoneme transitions. This leads us to propose a two-steps algorithm where we first pre-select candidate boundaries and then use statistical hypothesis test to make the final decision.

By performing error analysis of the MMF-based algorithm, we first observed that some of the misssed boundaries correspond to transitions between fricatives/stops and vowels. We also observed that transitions between speech and low energy segments (such as pauses and epenthetic silence) display strong and easy to detect changes in the slopes of ACC. Indeed, the singularity exponents of low energy segments have high positive values, while they are mostly negative in active speech segments. Motivated by these observations and the fact that fricatives/stops are essentially high-band signals, we propose to compute ACC on a low-pass filtered version of the utterance. By doing so fricatives/stops-vowels transitions will be converted into silence-speech transitions which are much easier to detect as shown in Figure 1b. It is known that most of the spectral energy of fricatives is located above 2000Hz and, for most stops, the active frequency bands start at 1800Hz. We thus choose the cutoff frequency of the low-pass filter as 1800Hz.

The second and most important observation we made is related to the statistical distribution of singularity exponents. We observed that some missed boundaries correspond to neighboring phonemes which have a quite distinctive difference in their SE distribution. However, the change in their averages is not strong enough to be translated as a change in the slope of ACC, and thus it is not captured by the simple curve-fitting procedure. It is then natural to think about including a statistical hypothesis test over SE distributions in our segmentation algorithm in order to detect such boundaries.

Motivated by these observations, we develop new segmentation algorithm which consists in 2 steps. First, we use our original MMFbased algorithm to detect the boundaries of the original and filtered signal. We gather all the detected boundaries and consider them as candidate boundaries. In the second step, we make the final decision by performing a dynamic windowing over these candidates followed by Log Likelihood Ratio Test (LLRT) over SE distributions of the original signal. We use a Gaussian hypothesis because our purpose is to detect changes in mean and variance. More precisely, for each candidate c i we consider the large window Z = [ci-1, ci+1] and the two smaller windows X = [ci-1, ci] and Y = [ci, ci+1]. We then compute LLR statistic to decide between the two hypothesis:

• H0 : SE of Z are generated by a single Gaussian.

• H1 : SE of Z are generated by two Gaussians on X and Y .

If H1 is significantly likelier than H0, we select ci as a boundary. Otherwise, ci is removed from the candidates list. We emphasize here that SE of the filtered signal are used only in the first step. The final decision is made upon the information conveyed by SE of the original signal. We also emphasize that this new algorithm is still simple and efficient as the original one.

EXPERIMENTAL RESULTS

Our evaluation is carried out on the full Train set of the TIMIT database which contains 4620 sentences uttered by 462 speakers. This set contains a wide variabilitiy of speakers and is balanced for dialectical coverage which is desirable for the evaluation of TI segmentation methods.

Performance measures

The segmentation quality can be evaluated and analyzed using three "partial" scores: the Hit Rate (HR) which is the rate of correctly detected boundaries; the False Alarm Rate (F A) which is the rate of erroneously detected boundaries and the Over Segmentation Rate (OS). These three scores are defined as:

HR = NH NR , OS = NT -NR NR , F A = NT -NH NT (4)
where, NT is the total number of detected boundaries, NH is the number of correctly detected boundaries and NR is the total number of boundaries in the reference transcription. In order to to assess the overall quality of a segmentation method, a global measure which simultaneously takes these scores in to account is required. A well known measure is the F1-value:

F1 = 2 × P CR × HR P CR + HR (5)
where P CR = 1 -F A is the precession rate. Another global measure, called the R-value, which is supposed to be more accurate than F1 has been recently proposed in [START_REF] Okko | An improved speech segmentation quality measure: the R-value[END_REF]. This measure makes more emphasize on over-segmentation by arguing that better hit rates might be achieved by simply adding random boundaries without any algorithmic improvement. This measure evaluates how close one is to the ideal segmentation R = 1:

r1 = p (1 -HR) 2 + OS 2 , r2 = HR -OS -1 √ 2 (6)
R = 1 - |r1| + |r2| 2 (7)

Results

Using different sizes of tolerance windows, we provide comparison of segmentation results for 3 methods. In the first one, we give the results reported in [START_REF] Dusan | On the relation between maximum spectral transition positions and phone boundaries[END_REF]. We mention here that [START_REF] Dusan | On the relation between maximum spectral transition positions and phone boundaries[END_REF] report scores with 0ms, 10ms and 20ms tolerance windows. However, their approach is frame-based with a 10ms frame step size and they convert each manual boundary to the closest frame position. Thus, 5ms has to be added to their window size in order to make a fair comparison with our sample-based approach which has the finest possible resolution. In the second one, we provide the results using our original MMF-based algorithm [START_REF] Khanagha | A novel text-independent phonetic segmentation algorithm based on the microcanonical multiscale formalism[END_REF] summarized in section 2.2, we call it MMF-ACC. In the third one, we present the results obtained using our new algorithm described in section 3, we call it MMF-LLRT. Table 1 presents HR, FA and OS for the 3 methods. The first observation is that MMF-LLRT outperforms MMF-ACC for the 3 scores and all tolerance windows. In particular, a significant improvement is made in FA and OS. This shows that, as expected, some of the insertions introduced by the curve fitting procedure has been corrected by the LLRT. The second observation is that MMF-LLRT yields considerably much higher accuracy than [START_REF] Dusan | On the relation between maximum spectral transition positions and phone boundaries[END_REF]. In particular, the smaller tolerance window is, the higher relative improvement is. This shows that MMF-LLRT is better suited for high precision detection of phoneme boundaries. To this regard, we can mention another interesting comparison with [START_REF] Lin | Phone boundary detection using sample-based acoustic parameters[END_REF] which is also a sample-based segmentation method as ours. In [START_REF] Lin | Phone boundary detection using sample-based acoustic parameters[END_REF], it is reported that 43.5% of their 86.8% detection output is located within the first bin of the cumulative histogram of distances from true boundaries. This corresponds to 43.5% × 86.8% = 37.75% hit rate with 7.5ms tolerance.

With MMF-LLRT we obtain 44% hit rate which is significantly more accurate. More importantly, the algorithm in [START_REF] Lin | Phone boundary detection using sample-based acoustic parameters[END_REF] is supervised (all manual transcriptions are used to train a neural network) while ours is fully unsupervised. Table 2 presents the performance of each of the 3 methods when evaluated using the global measures F1 and R. The same observations we made above still hold for the global performance evaluation. Indeed, MMF-LLRT still outperforms MMF-ACC for both F1 and R. Moreover, about 6% (resp. 10%) improvement in R-value and 4% (resp. 10%) in F1-value is achieved for 25ms of tolerance (resp. 5ms and 15ms). This is a significant gain in accuracy that shows the strength of singularity exponents in revealing the transitions fronts between phonemes.

Finally we emphasize an important feature of our algorithm which is its insensitivity to the threshold of the linear curve fitting. Figure 2 displays the R-value for different thresholds. we used a subset of 30 randomly selected sentences to compute these values. One can see that with about 400% change in the value of threshold, that variance of changes in R-value is less than 0.5%. Thus, we can fairly consider that our algorithm is threshold-free. This is a major advantage as most of the TI methods require accurate threshold tuning.

Fig. 2: The sensitivity of the MMF-LLRT to threshold.

CONCLUSIONS

By performing error analysis of our original MMF-based phonetic segmentation algorithm, we presented a new technique which confirms the strength of singularity exponents in detecting phoneme boundaries, and which further improves the segmentation accuracy. We provided a detailed evaluation and comparison using the full training set of TIMIT that could be useful to other researchers for comparison purposes. The results show that the new algorithm not only outperforms the original one, but also is significantly much more accurate than state-of-the-art ones. We are still at the beginning of exploration of speech analysis from the MMF perspective.

The encouraging results we obtained so far suggest that the MMF has indeed big potential in speech processing and should be further investigated. This will be the purpose of future communications.

Fig. 1 :

 1 Fig. 1: (a) A speech signal from TIMIT and its ACC functional. (b) TOP: Two examples of ACC functional for the original signal where the change in the slopes are not clear, BOTTOM: The ACC functional for the low-passed filtered signal. The changes in slopes are clearer. Phoneme boundaries are marked with vertical red lines.

Table 1 :

 1 The comparative table of segmentation results ("-" means not available). The scores are reported as percentages.

	tolerance	score	Dusan et al [7]	MMF-ACC
		HR	22.8	31.7
	5ms	FA	79.7	70.2
		OS	12.8	6.4
		HR	-	52.8
	10ms	FA	-	50.4
		OS	-	6.4
		HR	59.2	65.5
	15ms	FA	47.5	38.4
		OS	12.8	6.4
		HR	-	72.4
	20ms	FA	-	31.94
		OS	-	6.42

Table 2 :

 2 The comparative table of global performance measures.

	tolerance	score	Dusan et al [7]	MMF-ACC	MMF-LLRT
	5ms	R-value F1-value	0.29 0.21	0.39 0.31	0.41 0.32
	10ms	R-value F1-value	--	0.57 0.51	0.60 0.53
	15ms	R-value F1-value	0.60 0.55	0.68 0.63	0.70 0.65
	20ms	R-value F1-value	--	0.74 0.70	0.76 0.72
	25ms	R-value F1-value	0.73 0.71	0.77 0.74	0.79 0.75
	30ms	R-value F1-value	--	0.79 0.76	0.81 0.77