F R Éd

Éric Bayart
email: frederic.bayart@math.univ-bpclermont.fr

Yanick Heurteaux
email: yanick.heurteaux@math.univ-bpclermont.fr

MULTIFRACTAL ANALYSIS OF THE DIVERGENCE OF FOURIER SERIES

A famous theorem of Carleson says that, given any function f ∈ L p (T), p ∈ (1, +∞), its Fourier series (Snf (x)) converges for almost every x ∈ T. Beside this property, the series may diverge at some point, without exceeding O(n 1/p ). We define the divergence index at x as the infimum of the positive real numbers β such that Snf (x) = O(n β ) and we are interested in the size of the exceptional sets E β , namely the sets of x ∈ T with divergence index equal to β. We show that quasi-all functions in L p (T) have a multifractal behavior with respect to this definition. Precisely, for quasi-all functions in L p (T), for all β ∈ [0, 1/p], E β has Hausdorff dimension equal to 1 -βp. We also investigate the same problem in C(T), replacing polynomial divergence by logarithmic divergence. In this context, the results that we get on the size of the exceptional sets are rather surprizing.

1. Introduction 1.1. Description of the results. The famous theorem of Carleson and Hunt asserts that, when f belongs to L p (T), 1 < p < +∞, where T = R/Z, the sequence of the partial sums of its Fourier series (S n f (x)) n≥0 converges for almost every x ∈ T. On the other hand, it can diverge at some point. This divergence cannot be too fast since, for any f ∈ L p (T) and any x ∈ T, |S n f (x)| ≤ C p n 1/p f p (see [START_REF] Zygmund | Trigonometric series[END_REF] for instance). In view of these results, a natural question arises. How big can be the sets F such that |S n f (x)| grows as fast as possible for every x ∈ F ? More generally, can we say something on the size of the sets such that |S n f (x)| behaves like (or as bad as) n β for some β ∈ (0, 1/p]?

To measure the size of subsets of T, we shall use the Hausdorff dimension. Let us recall the relevant definitions (we refer to [START_REF] Falconer | Fractal geometry: Mathematical foundations and applications[END_REF] and to [START_REF] Mattila | Geometry of Sets and Measures in Euclidian Spaces[END_REF] for more on this subject). If φ : R + → R + is a nondecreasing continuous function satisfying φ(0) = 0 (φ is called a dimension function or a jauge function), the φ-Hausdorff outer measure of a set E ⊂ R d is

H φ (E) = lim ε→0 inf r∈Rε(E) B∈r φ(|B|),
R ε (E) being the set of countable coverings of E with balls B of diameter |B| ≤ ε. When φ s (x) = x s , we write for short H s instead of H φs . The Hausdorff dimension of a set E is dim H (E) := sup{s > 0; H s (E) > 0} = inf{s > 0; H s (E) = 0}.

The first result studying the Hausdorff dimension of the divergence sets of Fourier series is due to J-M. Aubry [START_REF] Aubry | On the rate of pointwise divergence of Fourier and wavelet series in L p[END_REF].

Date: March 15, 2011.

Theorem 1.1. Let f ∈ L p (T), 1 < p < +∞. For β ≥ 0, define

E(β, f ) = x ∈ T; lim sup n→+∞ n -β |S n f (x)| > 0 .
Then dim H E(β, f ) ≤ 1βp. Conversely, given a set E such that dim H (E) < 1βp, there exists a function f ∈ L p (T) such that, for any x ∈ E, lim sup

n→+∞ n -β |S n f (x)| = +∞.
This result motivated us to introduce the notion of divergence index. For a given function f ∈ L p (T) and a given point x 0 ∈ T, we can define the real number β(x 0 ) as the infimum of the non negative real numbers β such that |S n f (x 0 )| = O(n β ). The real number β(x 0 ) will be called the divergence index of the Fourier series of f at point x 0 . Of course, for any function f ∈ L p (T) (1 < p < +∞) and any point x 0 ∈ T, 0 ≤ β(x 0 ) ≤ 1/p. Moreover, Carleson's theorem implies that β(x 0 ) = 0 almost surely and we would like to have precise estimates on the size of the level sets of the function β. These are defined as

E(β, f ) = {x ∈ T; β(x) = β} = x ∈ T; lim sup n→+∞ log |S n f (x)| log n = β .
We can ask for which values of β the sets E(β, f ) are non-empty. This set of values will be called the domain of definition of the spectrum of singularities of f . If β belongs to the domain of definition of the spectrum of singularities, it is also interesting to estimate the Hausdorff dimension of the sets E(β, f ). The function β → dim H (E(β, f )) will be called the spectrum of singularities of the function f (in terms of its Fourier series). By Aubry's result, dim H (E(β, f )) ≤ 1βp and, for any fixed β 0 ∈ [0, 1/p), for any ε > 0, one can find f ∈ L p (T) such that dim H

β 0 ≤β≤1/p E(β, f ) ≥ 1 -β 0 p -ε. Our first main result is that a typical function f ∈ L p (T) satisfies dim H (E(β, f )) = 1 -βp for any β ∈ [0, 1/p].
In particular, f has a multifractal behavior with respect to the summation of its Fourier series, meaning that the domain of definition of its spectrum of singularities contains an interval with non-empty interior.

Theorem 1.2. Let 1 < p < +∞. For quasi-all functions f ∈ L p (T), for any β ∈ [0, 1/p], dim H E(β, f ) = 1 -βp.
The terminology "quasi-all" used here is relative to the Baire category theorem. It means that this property is true for a residual set of functions in L p (T).

In a second part of the paper, we turn to the case of C(T), the set of continuous functions on T. In that space, the divergence of Fourier series is controlled by a logarithmic factor. More precisely, if (D n ) is the sequence of the Dirichlet kernels, we know that S n f ∞ ≤ D n 1 f ∞ , so that there exists some absolute constant C > 0 such that S n f ∞ ≤ C f ∞ log n for any f ∈ C(T) and any n > 1. As before, one can discuss the size of the sets such that |S n f (x)| behaves badly, namely like (log n) β , β ∈ [0, 1]. More precisely, mimicking the case of the L p spaces, we introduce, for any β ∈ [0, 1] and any f ∈ C(T), the following sets:

F(β, f ) = x ∈ T; lim sup n→+∞ (log n) -β |S n f (x)| > 0 F (β, f ) = x ∈ T; lim sup n→+∞ log |S n f (x)| log log n = β .
Theorem 1.1 indicates that, on L p (T), |S n f (x)| can grow as fast as possible (namely like n 1/p ) only on small sets: for every function f ∈ L p (T), dim H (E(1/p, f )) = 0. This property dramatically breaks down on C(T), as the following result indicates.

Theorem 1.3. For quasi-all functions f ∈ C(T), dim H F (1, f ) = 1.
Thus, for quasi-all functions f ∈ C(T), the partial sums (S n f (x)) n≥0 grow as fast as possible on big sets. We can also study the domain of the spectrum of singularities of f , namely the values of β such that F (β, f ) is non-empty. Like in the case of the space L p (T), this domain is for quasi-all functions of C(T) an interval with non-empty interior, so that a typical function f in C(T) has a multifractral behavior with respect to the summation of its Fourier series. However, the spectrum of singularities is constant! Theorem 1.4. For quasi-all functions f ∈ C(T), for any β ∈ [0, 1], F (β, f ) is non-empty and has Hausdorff dimension 1.

Theorem 1.4 indicates that the Hausdorff dimension is not precise enough to measure the size of the level sets F (β, f ). This leads us to introduce a notion of precised Hausdorff dimension, in order to distinguish more finely sets which have the same Hausdorff dimension. For s > 0 and t ∈ (0, 1], we consider

φ s,t (x) = x s exp (log 1/x) 1-t . Definition 1.5. Let E ⊂ R d . We say that E has precised Hausdorff dimension (α, β) if α is the Hausdorff dimension of E and • β = 0 if H φα,t (E) = 0 for every t ∈ (0, 1); • β = sup t ∈ (0, 1); H φα,t (E) > 0 otherwise. It is not difficult to check that φ s,t (x) ≤ φ s ′ ,t ′ (x) for small values of x iff s > s ′ or (s = s ′ and t ≥ t ′ ).
Thus the precised Hausdorff dimension is a refinement of the Hausdorff dimension. In particular it is a tool to classify sets that have the same Hausdorff dimension. The natural order for the precised dimension (s, t) is the lexicographical order which will be denoted by ≺. With respect to this order, we can say that the greater is the set, the greater is the precised dimension. Moreover, if (s, t) ≺ (s ′ , t ′ ) and (s, t) = (s ′ , t ′ ), then φ s ′ ,t ′ ≪ φ s,t . It follows that H φ s ′ ,t ′ (E) = 0 as soon as H φs,t (E) < ∞.

Our main theorem on C(T), which contains both Theorems 1.3 and 1.4, is the following:

Theorem 1.6. For quasi-all functions f ∈ C(T), for any β ∈ [0, 1], the precised Hausdorff dimension of

F (β, f ) is (1, 1 -β).
The paper is organized as follows. In the remaining part of this section, we introduce tools which will be needed during the rest of the paper. In Section 2, we prove Theorem 1.2 whereas in Section 3, we prove Theorem 1.6.

1.2.

A precised version of Fejér's theorem. Working on Fourier series, we will need results on approximation by trigonometric polynomials. Let k ∈ Z and e k : t → e 2πikt , so that, for any g ∈ L 1 (T) and any n ∈ N,

S n g : t → n k=-n
g, e k e k (t).

Let σ n g be the n-th Fejér sum of g,

σ n g : t → 1 n n-1 k=0 S k g(t).
σ n g is obtained by taking the convolution of g with the Fejér kernel

F n : t → 1 n sin(nπt) sin(πt) 2 .
If g belongs to C(T), (σ n g) n≥1 converges uniformly to g. For our purpose, we need to estimate how quick is the convergence. This is the content of the next lemma (part (1) rectifies a mistake in the proof of Lemma 12 in [START_REF] Aubry | On the rate of pointwise divergence of Fourier and wavelet series in L p[END_REF] and requires to replace θ ∞ /4 in Aubry's version by θ ∞ /2).

Lemma 1.7. Let θ be a Lipschitz function on T, let n ∈ N and let x ∈ T. Suppose that θ ′ ∞ ≤ n and that θ(x) = 0. Then the two following inequalities hold:

|σ n θ(x)| ≤ 1 4 + 1 2 θ ∞ for any n ≥ 8 (1)
|σ n θ(x)| ≤ 4 + 1 4 θ ∞ for any n ≥ 4. (2)
Proof. We may assume that x = 0. Hence, σ n θ(0) = 1/2 -1/2 θ(y)F n (y) dy. Let us consider δ ∈ (0, 2] and n ≥ 4. On the one hand, for any y ∈ [0, 1/2),

0 ≤ F n (y) = sin 2 (nπy) n sin 2 (πy) ≤ 1 n(2y) 2 so that δ/n<|y|≤1/2 θ(y)F n (y) dy ≤ 1 2n θ ∞ +∞ δ/n dy y 2 = θ ∞ 2δ .
On the other hand, Using the convexity inequality sin n n+1 πy ≥ n n+1 sin(πy) and a change of variables, we see that (u n ) is non-increasing. To prove (1), we choose δ = 1 and we observe that u 8 = 0.2496... ≤ 1 4 . To prove (2), we choose δ = 2 and we observe that, since the maximum of

F n is F n (0) = n, |u n | ≤ 2n 2 2/n 0 ydy = 4.
1.3. The mass transference principle. The second main tool that we need in this paper is a method to produce sets with large Hausdorff dimension (Theorem 1.2) or with large precised Hausdorff dimension (Theorem 1.6). An efficient way to do this is to consider ubiquitous systems like this was done in [START_REF] Dodson | Patterson measure and Ubiquity[END_REF][START_REF] Jaffard | On lacunary wavelet series[END_REF]. This was later refined in [START_REF] Beresnevich | A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures[END_REF] to obtain a general mass transference principle, which we recall in the form that we need.

Theorem 1.8 (The mass transference principle). Let (x n ) n≥0 be a sequence of points in [0, 1] d and let (r n ) n≥0 be a sequence of positive real numbers decreasing to 0. Let also φ : R + → R + be a dimension function satisfying φ(s) ≫ s d when s goes to 0 and s -d φ is monotonic. Define

E = lim sup n B(x n , r n ) E φ = lim sup n B x n , φ -1 (r d n )
and suppose that almost every point of [0, 1] d (in the sense of the Lebesgue's measure) lies in E. Then, H φ (E φ ) = +∞.

We shall use it in the following situation.

Corollary 1.9. Let (q n ) be a sequence of integers and, for each n ∈ N, each k ≤ q n , let B k,n = B(x k,n , r n ) be a ball with center x k,n ∈ [0, 1] d and with radius r k,n such that lim n→+∞ max k (r k,n ) = 0. Let also φ : R + → R + be a dimension function satisfying φ(s) ≫ s d when s goes to 0 and s -d φ is monotonic. Define

B n = qn k=1 B k,n E = lim sup n B n B φ n = qn k=1 B(x k,n , φ -1 (r d k,n )) E φ = lim sup n B φ n .
Suppose that almost every point of [0, 1] d (in the sense of the Lebesgue's measure) lies in

E. Then, H φ (E φ ) = +∞.
Proof. Reordering the sequences (B k,n ) and (B φ k,n ) as (C j ) and (C φ j ), we can observe that lim sup

n B n = lim sup j C j = E lim sup n B φ n = lim sup j C φ j = E φ .
Thus the corollary follows from a direct application of Theorem 1.8.

Multifractal analysis of the divergence of the Fourier series of functions of L p (T)

In this section, we shall prove Theorem 1.2. Our method, which is inspired by [START_REF] Jaffard | On the Frisch-Parisi conjecture[END_REF], is divided into two parts. During the first one, we will construct a single function, which we call the saturating function, satisfying the conclusions of Theorem 1.2. During the second one, we will show how to derive a residual set from this single function.

2.1. The saturating function. Our intention is to construct a function g such that |S n g(x)| is big when x is close to a dyadic number. The following definition gives a precise meaning.

Definition 2.1. A real number x is α-approximable by dyadics, α ≥ 1, if there exist two sequence of integers (k n ), (j n ) such that x - k n 2 jn ≤ 1 2 αjn
and (j n ) goes to infinity. The dyadic exponent of x is the supremum of the set of real numbers α such that x is α-approximable by dyadics.

We denote by

D α = {x ∈ [0, 1]; x is α-approximable by dyadics} . It is easy to check that H β (D α ) = 0 for β > 1/α so that dim H (D α ) ≤ 1/α. On the other hand, it is well-known that dim H (D α ) ≥ 1 α .
Let us nevertheless show how this follows from Corollary 1.9. Indeed, D α can be described as a limsup set:

D α = lim sup j→+∞ 2 j -1 k=0 I α k,j
where the I k,j are the dyadic intervals

I k,j = k 2 j - 1 2 j , k 2 j + 1 2 j and I α k,j = k 2 j - 1 2 αj , k 2 j + 1 2 αj . Since 2 j -1 k=0 I k,j ⊃ [0, 1]
, Corollary 1.9 implies that H 1/α (D α ) = +∞. We are going to define g ∈ L p (T) such that the divergence index of g at x depends on the dyadic exponent of x. The greater the dyadic exponent will be, the greater the divergence index of g at x will be. To do this, we will classify the dyadic intervals following their center. Namely, each k/2 j can be uniquely written K/2 J with K / ∈ 2Z and 1 ≤ J ≤ j (such a center comes into play from the J-th generation). Let

I J = {K/2 J ; K / ∈ 2Z, 0 ≤ K ≤ 2 J -1} and I J,j = k 2 j ∈I J I k,j I ′ J,j = k 2 j ∈I J 2I k,j .
Here and elsewhere, when I is an interval and c is a positive real number, cI means the interval with the same center as I and with length c|I|. Observe that, when 1 ≤ J < j, the intervals 2I k,j , k 2 j ∈ I J don't overlap and the set I ′ J,j has measure 2 J-1 2 2-j . Observe also that when J is small with respect to j, the real numbers x in I J,j are well-approximated by dyadics K/2 J , since |x -K/2 J | ≤ 1/2 j . We first define a trigonometric polynomial with L p -norm 1 which is almost constant on each I J,j and which is big on I J,j when J is small. Lemma 2.2. Let j ≥ 1. There exists a trigonometric polynomial g j ∈ L p (T) with spectrum contained in [0, j2 j+1 ) such that • g j p ≤ 1;

• For any 1 ≤ J ≤ j and any x ∈ I J,j , we can find two integers n 1 and n 2 satisfying 0 ≤ n 1 < n 2 < j2 j+1 and such that

|S n 2 g j (x) -S n 1 g j (x)| ≥ 1 4j 2 -(J-j+1)/p .
Proof. We set for any 1 ≤ J ≤ j:

• χ J,j a continuous piecewise linear function equal to 1 on I J,j , equal to 0 outside I ′ J,j , and satisfying 0 ≤ χ J,j ≤ 1 and

χ ′ J,j ∞ ≤ 2 j ; • c J,j = 1 j 2 -(J-j+1)/p (c J,j is big when J is small); • g J,j = e (2J-1)2 j σ 2 j χ J,j .
It is straighforward to observe that the spectrum of g J,j is contained in [n J,j , m J,j ] with

n J,j = (2J -1)2 j -(2 j -1) m J,j = (2J -1)2 j + (2 j -1).
Thus, the spectra of the g J,j , 1 ≤ J ≤ j are disjoint. Moreover, g j,j p = 1 and for 1 ≤ J < j, g J,j p ≤ χ J,j p ≤ 2 (J-j+1)/p . We finally set

g j = j J=1 c J,j g J,j
and we claim that g j is the trigonometric polynomial we are looking for. First of all, the spectrum of g j is included in [n 1,j , m j,j ] which is contained in [0, j2 j+1 ). Moreover, the L p norm of g j is

g j p ≤ j J=1 1 j 2 -(J-j+1)/p g J,j p ≤ 1.
Pick now any x ∈ I J,j , 1 ≤ J ≤ j so that

|S m J,j g j (x) -S n J,j -1 g j (x)| = |c J,j g J,j (x)| = 1 j 2 -(J-j+1)/p |σ 2 j χ J,j (x)|.
Observing that χ J,j (x) = 1 and applying the first point of Lemma 1.7 to 1χ J,j , we find

|σ 2 j χ J,j (x)| ≥ 1 -|σ 2 j (1 -χ J,j (x))| ≥ 1 4 .
Thus,

|S m J,j g j (x) -S n J,j -1 g j (x)| ≥ 1 4j 2 -(J-j+1)/p
and the conclusion follows with n 2 = m J,j and n 1 = n J,j -1.

We are now ready to construct the saturating function. It is defined by

g = j≥1 1 j 2 e j2 j+1 g j .
Observe in particular that the functions e j2 j+1 g j have disjoint spectra (the spectrum of e j2 j+1 g j is contained in [j2 j+1 ; j2 j+2 ) ) and that g belongs to L p (T).

We then show that for any

x ∈ D α , α > 1, lim sup n→+∞ log |S n g(x)| log n ≥ 1 p 1 - 1 α .
Indeed, let x ∈ D α and let ε > 0 with αε > 1. We can find integers K and J with J as large as we want and K / ∈ 2Z such that

x - K 2 J ≤ 1 2 (α-ε/2)J . We set j = [(α -ε/2)J] the integer part of (α -ε/2)J and k such that k/2 j = K/2 J . Hence, x - k 2 j ≤ 1 2 (α-ε/2)J ≤ 1 2 j .
Using Lemma 2.2, we can find two integers n 1 and n 2 satisfying j2 j+1 ≤ n 1 < n 2 < j2 j+2 and such that

|S n 2 g(x) -S n 1 g(x)| = 1 j 2 |S n 2 (e j2 j+1 g j )(x) -S n 1 (e j2 j+1 g j )(x)| ≥ 1 4j 3 2 -(J-j+1)/p ≥ 1 4j 3 2 1 p j-j+1 α-ε/2 -1 ≥ C2 1 p (1-1 α-ε )j . It follows that we can find n ∈ {n 1 , n 2 } such that |S n g(x)| ≥ C 2 2 1 p (1-1 α-ε )j .
Combining the estimates on n and on |S n g(x)|, and since J (hence j, hence n) can be taken as large as we want, we get that lim sup

n→+∞ log |S n g(x)| log n ≥ 1 p 1 - 1 α -ε .
Since ε > 0 is arbitrary, we obtain in fact that for any x ∈ D α , lim sup

n→+∞ log |S n g(x)| log n ≥ 1 p 1 - 1 α .
At this point, it would be nice to get a lower bound for lim sup n→+∞ log |S n g(x)| log n for any x with dyadic exponent equal to α. Unfortunately, this does not seem easy and we will rather conclude by using an argument lying on Hausdorff measures. Indeed, define

D 1 α = x ∈ D α ; lim sup n→+∞ log |S n g(x)| log n = 1 p 1 - 1 α D 2 α = x ∈ D α ; lim sup n→+∞ log |S n g(x)| log n > 1 p 1 - 1 α .
We have already observed that

H 1/α (D 1 α ∪ D 2 α ) = H 1/α (D α ) = +∞. It suffices to prove that H 1/α (D 2 α ) = 0. Let (β n ) be a sequence of real numbers such that β n > 1 p 1 - 1 α
and lim

n→+∞ β n = 1 p 1 - 1 α .
Let us observe that

D 2 α ⊂ n≥0 E(β n , g).
Moreover, Theorem 1.1 implies that H 1/α (E(β n , g)) = 0 for all n. Hence,

H 1/α (D 2 α ) = 0 and H 1/α (D 1 α ) = +∞, which proves that dim H E 1 p 1 - 1 α , g ≥ 1 α .
By Theorem 1.1 again, this inequality is necessarily an equality. Finally, g satisfies the conclusions of Theorem 1.2, setting 1βp = 1/α.

Remark 2.3. If α = 1, then β = 0 and the conclusion is a consequence of Carleson's Theorem.

The residual set.

To build the dense G δ -set, the idea is that any function whose Fourier coefficients are sufficiently close to those of the saturating function g on infinitely many intervals [j2 j+1 ; j2 j+2 ) will satisfy the conclusions of Theorem 1.2. Precisely, let (f j ) j≥1 be a dense sequence of polynomials in L p (T) with spectrum contained in [-j, j].

We define a sequence (h j ) j≥1 as follows:

h j = f j + 1 j
e j2 j+1 g j so that h jf j p goes to 0 and (h j ) j≥1 remains dense in L p (T). Observe also that the spectra of f j and h jf j don't overlap. Finally, let (r j ) j≥1 be a sequence of positive integers so small that, for any

f ∈ L p (T) with f L p ≤ r j , S n f ∞ ≤ 1 for any n ≤ j2 j+2 .
The dense G δ set we will consider is

A = l∈N j≥l B(h j , r j ).
Let f belong to A and let (j l ) l≥1 be an increasing sequence of integers such that f belongs to B(h j l , r j l ) for any l. Then, for any α > 1, we define J l = [j l /α] + 1 (which is smaller than j l if l is large enough) and

E = lim sup l→+∞ I J l ,j l .
For any x ∈ E one can find j = j l as large as we want, the corresponding J = J l and 1 ≤ k ≤ 2 j -1 such that x belongs to I k,j with k/2 j ∈ I J .

Observe that f = f j + 1 j e j2 j+1 g j + (fh j ). By Lemma 2.2, we can find two integer n 1 and n 2 satisfying j2 j+1 ≤ n 1 < n 2 < j2 j+2 and such that

|S n 2 (e j2 j+1 g j )(x) -S n 1 (e j2 j+1 g j )(x)| ≥ 1 4j 2 -(J-j+1)/p .
Using the definition of the r j , we obtain

|S n 2 f (x) -S n 1 f (x)| ≥ 1 4j 2 2 -(J-j+1)/p -|S n 2 (f -h j )(x)| -|S n 1 (f -h j )(x)| ≥ 1 4j 2 2 -(J-j+1)/p -2 so that |S n 2 f (x)| ≥ C j 2 2 -(J-j+1)/p or |S n 1 f (x)| ≥ C j 2 2 -(J-j+1)/p . Observing that      max(log n 2 , log n 1 ) = j log 2 + O(log j) log j -2 2 -(J-j+1)/p = 1 p 1 -1 α j log 2 + O(log j)
we find in particular that, for any x ∈ E,

lim sup n→+∞ log |S n f (x)| log n ≥ 1 p 1 - 1 α .
On the other hand, let us write

I J l ,j l = 1≤K<2 J l , K / ∈2Z K 2 J l - 1 2 j l , K 2 J l + 1 2 j l
and remark that for any l, since J l ≥ j l /α,

1≤K<2 J l , K / ∈2Z K 2 J l - 1 2 j l /α , K 2 J l + 1 2 j l /α ⊃ [0, 1].
Hence, we can apply Corollary 1.9 to get H 1/α (E) = +∞. We now conclude exactly as in Section 2.1 to get H 1/α (E 1 ) = +∞, with

E 1 = x ∈ E; lim sup n→+∞ log |S n f (x)| log n = 1 p 1 - 1 α . Finally, dim H E 1 p 1 - 1 α , f ≥ 1 α
and f satisfies the conclusions of Theorem 1.2, setting 1βp = 1/α.

Remark 2.4. During the construction , we didn't use that the spectra of the functions e j2 j+1 g j are disjoint, because we considered each one separately. We could also define h j by h j = f j + 1 j e j+1 g j .

Remark 2.5. The above construction can be carried on L 1 (T). Namely, for quasi-all f ∈ L 1 (T), we obtain for any β ∈

[0, 1], dim H (E (β, f )) ≥ 1 -β.
However, we cannot go further because Carleson's Theorem dramatically breaks down in L 1 (T) and we do not have Theorem 1.1 at our disposal in this context. The study of what happens exactly on L 1 (T) is a very exciting open question.

Multifractal analysis of the divergence of the Fourier series of functions of C(T)

We turn to the proof of Theorem 1.6. We follow a strategy close to that of Section 2. First of all, we will un upper bound for the precised Hausdorff dimension of the sets F(β, f ) (hence, of the sets F (β, f )) for any f ∈ C(T) and any β ∈ (0, 1). Second, we will build polynomials with small L ∞ -norms and such that their Fourier series have big partial sums on big intervals. These polynomials will be the blocks of our final construction. Working on C(T) adds several difficulties which will be explained when we will encounter them.

3.1. The sets F(β, f ) cannot be too big. We shall prove the following lemma (recall that φ s,t (x) = x s exp (log 1/x) 1-t ).

Lemma 3.1. Let β ∈ (0, 1) and f ∈ C(T). Then, for any γ > 1β,

H φ 1,γ F(β, f ) = 0.
In particular, the precised Hausdorff dimension of F(β, f ) cannot exceed (1, 1β).

Proof. A key point in Aubry's proof of Theorem 1.1 is the Carleson-Hunt theorem which asserts that, for any g ∈ L p (T), 1 < p < +∞,

S * g p ≤ C p g p where S * g(x) = sup n≥0 |S n g(x)|.
On C(T), a weak inequality (also due to Hunt) remains valid (see [START_REF] Arias De Reyna | Pointwise convergence of Fourier series[END_REF]Theorem 12.5]): there are two absolute constants A, B > 0 such that, for every f ∈ C(T) and every y > 0,

λ {x ∈ T ; S * f (x) > y} ≤ Ae -By/ f ∞ .
Here, λ denotes the Lebesgue measure on T. So, let β ∈ (0, 1) and f ∈ C(T). We may assume f ∞ ≤ 1. For any M > 0, we introduce

F(β, f, M ) = x ∈ T; lim sup n→+∞ (log n) -β |S n f (x)| > M .
Since F(β, f ) = M >0 F(β, f, M ), we just need to prove that H φ 1,γ F(β, f, M ) = 0 for every M > 0. From now on, we fix some M > 0. We pick any x ∈ F(β, f, M ) and n x large enough such that

|S nx f (x)| ≥ M (log n x ) β .
Such an inequality remains true in an interval around x whose size is not so small. Precisely, because n x can be assumed to be large and since the L 1 -norm of the Dirichlet kernel D n behaves like 4 π 2 log n, we may assume that

S nx f ∞ ≤ (log n x ) f ∞ ≤ log n x . By Bernstein's inequality, (S nx f ) ′ ∞ ≤ n x log n x . Let I x = x - M 2n x (log n x ) 1-β , x + M 2n x (log n x ) 1-β .
For any y ∈ I x , we get

|S nx f (y)| ≥ M 2 (log n x ) β . ( 3
) (I x ) x∈F (β,f,M ) is a covering of F(β, f, M ).
We can extract a Vitali's covering, namely a countable family of disjoint intervals

I i , i ∈ N, of length M n i (log n i ) 1-β such that F(β, f, M ) ⊂ i 5I i . Let us finally set, for any q ≥ 1, U q = i; 2 q+1 ≥ M (log n i ) β 2 > 2 q
. Without loss of generality, we may assume the n i so large that q U q = N. By applying Hunt's theorem,

λ ({x; S * f (x) > 2 q }) ≤ Ae -B2 q .
Now, by (3), the set {x; S * f (x) > 2 q } contains the disjoint intervals I i , for i ∈ U q . Thus,

i∈Uq |I i | ≤ Ae -B2 q .
Moreover, for any i ∈ U q , it is not hard to check that

|I i | ≥ Ce -D2 q/β
for some positive constants C, D which do not depend on q. Picking any α such that 1β < α < γ, we get

i∈Uq φ 1,α (5|I i |) = i∈Uq 5|I i | exp (log(1/5|I i |)) 1-α ≤ 5   i∈Uq |I i |   exp D2 q/β -log 5C 1-α ≤ 5Ae -B2 q +D ′ 2 q(1-α)/β .
Since 1α < β, this shows that there exists

C 0 < +∞ such that i∈N φ 1,α (5|I i |) = q∈N i∈Uq φ 1,α (5|I i |) ≤ C 0 .
Remember that i 5I i is a covering of F(β, f, M ) and that the I i can be chosen as small as we want. We can then conclude that

H φ 1,α (F(β, f, M )) ≤ C 0 . In particular, H φ 1,γ F(β, f, M ) = 0, since φ 1,α ≫ φ 1,γ .
Remark 3.2. The functions φ 1,γ , for γ > 1β, are not optimal in the statement of the previous lemma. We can replace them by any function φ(x) = x exp (log 1/x) β ε(x) with ε(x) goes to 0 as x goes to 0.

3.2.

The basic construction. When we try to build explicitely a continuous function whose Fourier series diverges at some point, say 0, a natural way is to consider polynomials P with small L ∞ norm, and satisfying nevertheless that |S n P (0)| is big for some large value of n. The easiest examples are

P N (x) = e N (x) N j=1 sin(2πjx) j ,
since the sequence (

P N ∞ ) N ≥1 is bounded whereas |S N (P )(0)| ∼ 1 2 log N . Moreover, this example is in some sense optimal since S N f ∞ ≤ C(log N ) f ∞ for any f ∈ C(T).
In our context, we need to find a polynomial P which satisfies a similar property not only at one point, but on a set which is rather big since at the end we want to construct sets of divergence with Hausdorff dimension 1. This does not seem to be the case for P N , the reason being that |(S N P ) ′ (0)| behaves like N , which is much bigger than S N P (0).

To tackle this problem, we start from a construction of Kahane and Katznelson in [START_REF] Kahane | Sur les ensembles de divergence des séries trigonométriques[END_REF] (see also [START_REF] Katznelson | An Introduction to Harmonic Analysis[END_REF]) which they use to prove that every subset of T of Lebesgue measure 0 is a set of divergence for C(T). Since we want to control both the size of the sets E and the index n such that S n P (x) becomes larger than some given real number for any x ∈ E, the forthcoming lemma needs very careful estimations. Lemma 3.3. Let β ∈ (0, 1), δ ∈ (0, 1) and K ≥ 2. Then there exist an integer k ≥ K, an integer n as large as we want and a trigonometric polynomial P with spectrum contained in [0, 2n -1] such that

• |P (x)| ≤ 1 for any x ∈ T; • log |S n P (x)| ≥ (1 -δ)β log log n for any x ∈ I β k ,
where

I β k = k-1 j=0 j k - 1 2k exp (log k) β ; j k + 1 2k exp (log k) β .
Proof. Let us first describe the idea of the proof. We shall construct a trigonometric polynomial Q with spectrum in [1, n -1] and with the following properties: |ℑm Q| is small and |Q| is large on a set E. We then set P = e n × ℑm Q, so that P ∞ is small. On the other hand, writing

Q = n-1 k=1 a k e k , 2iℑm Q = -n-1 k=1 a k e -k + n-1 k=1 a k e k , so that |S n (P )| = 1 2 n-1 k=1 a k e n-k = 1 2 n-1 k=1 a k e k = 1 2 |Q|
is large on E. The construction of Q will be done by taking log f , the logarithm of an holomorphic function defined on a neighbourhood of the closed unit disk D (which allows to control the imaginary part of log f while the modulus of it can be large), and by taking a Fejér sum of log f . We now proceed with the details. The proof uses holomorphic functions and it is better to see T as the boundary of the unit disk D. To avoid cumbersome notations, the letter C will denote throughout the proof a positive and absolute constant, whose value may change from line to line. Let k ≥ K whose value will be fixed later. We set:

ε = 1 k exp (log k) β z j = e 2πij k , j = 0, . . . , k -1 f (z) = 1 k k-1 j=0 1 + ε 1 + ε -z j z .
f is holomorphic in a neighbourhood of D. We claim that f satisfies the following four properties.

(P1): ∀z ∈ D, ℜef (z) ≥ Cε; (P2): ∀z ∈ I β k , |f (z)| ≥ ℜef (z) ≥ C exp (log k) β ; (P3): ∀z ∈ T, |f (z)| ≤ C exp (log k) β ; (P4): ∀z ∈ T, f ′ (z) f (z) ≤ C ε 3
. Indeed, for any z ∈ D and any j ∈ {0, . . . , k -1},

ℜe 1 + ε 1 + ε -z j z = 1 + ε |1 + ε -z j z| 2 ℜe 1 + ε -z j z ≥ 1 + ε (2 + ε) 2 × ε ≥ Cε, (4)
which proves (P1). To prove (P2), we may assume that z = e 2πiθ with θ ∈ -ε 2 ; ε 2 . Then

ℜe 1 + ε 1 + ε -z 0 z = 1 + ε |1 + ε -z| 2 ℜe 1 + ε -z ≥ C ε .
If we combine this with (4), we get

ℜef (z) ≥ C kε + k -1 k Cε ≥ C kε = C exp (log k) β .
which gives (P2).

Conversely, we want to control sup z∈T |f (z)|. Pick any z = e 2πiθ ∈ T. By symmetry, we may and shall assume that |θ| ≤ 1 2k . Then we get

1 + ε 1 + ε -z 0 z ≤ C ε .
Now, for any j ∈ {1, . . . , k/4}, we can write

|1 + ε -z j z| ≥ |ℑm(z j z)| ≥ sin 2πj k -2πθ ≥ 2 π × 2π j k -θ ≥ 4 k j - 1 2 .
Taking the sum,

k/4 j=1 1 + ε 1 + ε -z j z ≤ k(1 + ε) 4 k/4 j=1 1 j -1/2 ≤ Ck log k.
In the same way, we obtain

k-1 j=3k/4 1 + ε 1 + ε -z j z ≤ Ck log k whereas |1 + ε -z j z| ≥ C for any j ∈ [k/4, 3k/4], so that 3k/4 j=k/4 1 + ε 1 + ε -z j z ≤ Ck.
Putting this together, we get

|f (z)| = 1 k k-1 j=0 1 + ε 1 + ε -z j z ≤ C 1 kε + log k + 1 ≤ C exp (log k) β .
Finally, it remains to prove (P4). We observe that

f ′ (z) = 1 k k-1 j=0 (1 + ε)z j (1 + ε -z j z) 2 .
We do not try to get a very precise estimate for |f ′ (z)| (this is not useful for us). We just observe that |1 + εz j z| 2 ≥ ε 2 for any j ∈ {0, . . . , k -1} and any z ∈ T, so that

|f ′ (z)| ≤ C ε 2 .
If we combine this with (P1), we get (P4).

We are almost ready to construct P . The next step is to take h(z) = log(f (z)), which defines a holomorphic function in a neighbourhood of D by (P1). Moreover, |ℑm(h(z))| ≤ π/2 for any z ∈ D and h(0) = 0. Now, we look at the function h on the boundary of the unit disk D, that is we introduce the function g(x) = h(e 2iπx ) defined on the circle T = R/Z. Properties (P2), (P3) and (P4) can be rewritten as

∀x ∈ I β k , |g(x)| ≥ (log k) β -C ∀x ∈ T, |g(x)| ≤ (log k) β + C ∀x ∈ T, |g ′ (x)| ≤ Ck 3 exp 3(log k) β .
Let now n be the smallest integer such that Ck 3 exp 3(log k) β ≤ n. We also have g ′ ∞ ≤ n and we can apply the second part of Lemma 1.7 to the function θ(t) = g(t)-g(x)

when x ∈ I β k . Recall that θ ∞ ≤ 2(log k) β + C. We get |σ n θ(x)| ≤ (log k) β 2 + C
and we can conclude that

|σ n g(x)| ≥ |g(x)| -|σ n θ(x)| ≥ (log k) β 2 -C.
We finally set

P = 2 π e n σ n (ℑmg) = 2 π e n ℑm(σ n g).
It is straightforward to check that P ∞ ≤ 1 (recall that σ n is a contraction on C(T)), and that the spectrum of σ n g is contained in [1, n -1] (ĝ(0) = 0 since h(0) = 0). Now, the simple algebraic trick exposed at the beginning of the proof shows that

|S n P (x)| = 1 π σ n g(x) ,
so that, for any x ∈ I β k ,

|S n P (x)| ≥ 1 2π (log k) β -C.
This leads to log

|S n P (x)| ≥ β log log k -C.
On the other hand,

log log n ≤ log 3 log k + 3(log k) β + log C ≤ log log k + C. Finally, log log |S n P (x)| log log n ≥ β log log k -C log log k + C ≥ (1 -δ)β,
provided k has been chosen large enough. Moreover, n can be chosen as large as we want since n → +∞ when k → +∞.

Remark 3.4. The fact that we have to compare log log n and log |S n | helps us for the previous proof. Even if n and k do not have the same order of growth, this is not apparent when we apply the iterated logarithm.

Remark 3.5. During the construction, the integers k and n can't be chosen independently : they satisfy n -1 ≤ Ck 3 exp 3(log k) β ≤ n where C is an absolute constant. If we want to construct a polynomial P satisfying the conclusion of Lemma 3.3 with a large value of n, we need also to choose a large value of k.

The conclusion.

We are now going to prove the full statement of Theorem 1.6. At this point, the situation is less favourable than in the L p -case. There, the basic construction done at each step j did not depend on the index of divergence that we would like to get. We had the same function g j which worked for all indices of divergence, and it was the dyadic exponent of x which decided how large was |g j (x)|. The construction done in Lemma 3.3 is less efficient, because the polynomial P does depend on the expected divergence index β (the index β is a parameter of the definition of f above). We have to overcome this new difficulty and the solution will be to introduce redundancy in the construction of the G δ -set.

As usual, we start from a sequence (f j ) j≥1 of polynomials which is dense in C(T). For convenience, we assume that f j ∞ ≤ j for any j and that the spectrum of f j is contained in [-j, j]. Furthermore, we fix four sequences (α l ), (β l ), (δ l ) and (ε l ) with values in (0, 1) and such that:

• (β l ) is dense in (0, 1) and l → β l is one to one; • l ε l ≤ 1; • (δ l ) and (α l ) go to zero.

• δ l < 1/3. Let now j ≥ 1. By induction on l = 1, . . . , j, we build sequences (P j,l ), (n j,l ) and (k j,l ) satisfying the conclusions of Lemma 3.3 with β = β l , δ = δ l and K = j (to ensure that lim j→+∞ k j,l = +∞) and we will decide how large should be n j,l during the construction. According to Remark 3.5, these constraints on n j,l will determine the values of the k j,l . We then set

g j := f j + α j j l=1
ε l e n j,l P j,l so that g jf j ∞ ≤ α j j l=1 ε l P j,l ∞ ≤ α j . In particular, the sequence (g j ) remains dense in C(T). Recall that the spectrum of f j is included in [-j, j] and observe that the spectrum of e n j,l P j,l lies in [n j,l , 3n j,l -1]. If we suppose that n j,1 = j + 1 and n j,l+1 ≥ 3n j,l , we can conclude that the spectra of f j , e n j,1 P j,1 , • • • , e n j,j P j,j are disjoint. Let now x belongs to I β l k j,l for some l ≤ j. Then

S 2n j,l g j (x) ≥ α j ε l S n j,l P j,l (x) -α j l-1 m=1 ε m P j,m ∞ -j ≥ α j ε l S n j,l P j,l (x) -α j -j.
Because we can choose n j,l as large as we want in the process, we may always assume that the choice that we have done ensures that S 2n j,l g j (x) ≥ α j ε l 2 S n j,l P j,l (x) .

Taking the logarithm, we find log S 2n j,l g j (x) ≥ log S n j,l P j,l (x) + log ε l + log α jlog 2

≥ (1δ l )β l log log(n j,l ) + log ε l + log α jlog 2

≥ (1 -2δ l )β l log log(2n j,l )

provided again that we have chosen n j,l very large. We then fix r j > 0 so small that, for any f ∈ B(g j , r j ) (the balls are related to the norm • ∞ ), for any l ≤ j, S 2n j,l f -S 2n j,l g j ∞ ≤ 1/2.

Observe that for every real number t ≥ 1, we have log(t -1/2) ≥ log(t)log 2. For any x ∈ I β l k j,l with l ≤ j, we get log S 2n j,l f (x) ≥ log S 2n j,l g j (x)log 2 ≥ (1 -2δ l )β l log log(2n j,l )log 2 ≥ (1 -3δ l )β l log log(2n j,l ) if n j,l are chosen sufficiently large such that δ l β l log log(2n j,l ) ≥ log 2. We finally set A = p∈N j≥p B(g j , r j ), and we claim that A is the dense G δ set we are looking for. Indeed, let f belong to A and let (j p ) be an increasing sequence of integers such that for every p ≥ 0, f ∈ B(g jp , r jp ). We consider β ∈ (0, 1) and choose p 0 such that

β 1 , • • • , β jp 0 ∩ (0, β) = ∅.
Such a p 0 exists since the sequence (β l ) l≥1 is dense in (0, 1). For every p ≥ p 0 , let l p be chosen in {1, • • • , j p } such that ββ lp = inf{ββ l ; l ≤ j p and β > β l }.

Since the sequence (β l ) is dense in (0, 1), β lp < β for p ≥ p 0 and β lp → β. Moreover, since l → β l is one to one, it is clear that l p is non decreasing and goes to +∞.

Observe that, for p ≥ p 0 , I β k jp ,lp ⊂ I It follows that φ(x) ≤ x exp (log(1/2x)) β ≤ φ 1,1-β (x) and H φ 1,1-β (F ) = +∞. We now conclude exactly as in the L p -case, using Lemma 3.1 to replace Aubry's result. Namely, we set and we observe that Lemma 3.1 guarantees that H φ 1,1-β (F 2 ) = 0. Thus, H φ 1,1-β (F 1 ) = +∞ and the precised Hausdorff dimension of F (β, f ), which contains F 1 , is at least (1, 1β). By Lemma 3.1, it is exactly (1, 1β).

Remark 3.6. It is amazing that, with our method, it is easier to prove Theorem 1.6 and to deduce Theorem 1.4 from it than to prove Theorem 1.4 directly. Indeed, to ensure that the sets F(β, f ) are big, we need to know that the sets F (β ′ , f ) are small for β ′ > β. This cannot be done if we stay within the notion of Hausdorff dimension.

Remark 3.7. The method developed above allows us to construct a "concrete function" that satisfies the conclusion of Theorem 1.6. More precisely, it suffices to consider

g = +∞ j=1 1 j 2 j l=1
ε l e n j,l P j,l

2 y

 2 dy := u n .

  β lp k jp ,lp , so that, for any x ∈ I β k jp ,lp , settingN p = 2n jp,lp , log |S Np f (x)| ≥ (1 -3δ lp )β lp log log(N p ).In particular, setting F = lim sup p I β k jp ,lp , we get that lim supn→+∞ log |S n f (x)| log log n ≥ βfor any x ∈ F . Now, we can apply Corollary 1.9 with a jauge function φ satisfyingφ -1 (y) = y exp -(log(1/2y)) β to obtain H φ (F ) = ∞.Observe that if y = φ(x), then y = x exp (log(1/2y)) β and log x ≤ log y.

F 1 =

 1 x ∈ F ; lim sup n→+∞ log |S n f (x)| log log n = β F 2 = x ∈ F ; lim sup n→+∞ log |S n f (x)| log log n > β

with the constraint 3n j,j < n j+1,1 to ensure that the blocks j l=1 ε l e n j,l P j,l have disjoint spectra. Such a function is some kind of saturating function in the continuous case.


Ji Ming
email: j.ming@qub.ac.uk

Timothy J Hazen

James R Glass

Combining Missing-Feature Theory, Speech Enhancement, and Speaker-Dependent / -Independent Modeling for Speech Separation

Keywords: speech separation, speech enhancement, speaker modeling, speech recognition, missing-feature theory, posterior union model

This paper considers the separation and recognition of overlapped speech sentences assuming single-channel observation. A system based on a combination of several different techniques is proposed. The system uses a missing-feature approach for improving crosstalk/noise robustness, a Wiener filter for speech enhancement, hidden Markov models for speech reconstruction, and speaker-dependent/-independent modeling for speaker and speech recognition. We develop the system on the Speech Separation Challenge database, involving a task of separating and recognizing two mixing sentences without assuming advanced knowledge about the identity of the speakers nor about the signal-to-noise ratio. The paper is an extended version of a previous conference paper submitted for the challenge.

Introduction

There are currently two major approaches to speech enhancement. One approach assumes the availability of single-channel data (i.e., the speech and noise are available only in a single mixed form), and the other assumes the availability of multi-channel data (i.e., the speech and noise are available in different combination forms, from a set of two or more spatially-distributed transducers). Current single-channel techniques include optimal filtering, for example, spectral subtraction, Wiener filtering, Kalman filtering, or subspace decomposition (see, for example, [START_REF] Boll | Suppression of acoustic noise using spectral subtraction[END_REF] When statistics of the noise are not available, they are predicted using previous data without significant speech content (e.g., [START_REF] Martin | Noise power spectral density estimation based on optimal smoothing and minimum statistics[END_REF][START_REF] Cohen | Noise spectrum estimation in adverse environments: improved minima controlled recursive averaging[END_REF].

These algorithms work for stationary or slowly-varying noise, but less so for speech-like or heavily nonstationary noise. This is because of the weak predictability of fast-varying noises.

In some applications (e.g., meeting-room or car environments), it is possible to place several microphones to simultaneously record speech and background sounds. Based on the multi-channel data, assuming mutual independence between the sources, it is possible to separate the individual source signals without having to assume prior information. The approach, so called blind source separation, has been studied in speech enhancement as a means of removing the requirement for prior information about the noise (see, for example, [START_REF] Cichocki | Advances in blind source separation[END_REF]. The multi-channel approach is not the focus of this paper.

In this paper, we study the problem of separating and recognizing overlapped speech sentences assuming single-channel data. In this research, the background noise is crosstalk speech. This problem is challenging not only because the noise is nonstationary, but also because the noise has characteristics of speech signals. It could be more difficult to separate this type of noise from the targeted speech than for other non-speech noises. In the paper, we describe an approach that combines several techniques as a possible solution. We develop the approach on the Speech Separation Challenge database [START_REF] Cooke | The 2006 speech separation challenge[END_REF], involving a task of separating and recognizing two overlapped sentences spoken by the same or two different speakers, assuming the availability of only single-channel data. Our proposed system includes speaker dependent and independent modeling for speaker and speech recognition, missing-feature processing for crosstalk and noise robustness, Wiener filtering for speech enhancement, and hidden Markov models (HMMs) for speech reconstruction.

The remainder of the paper is organized as follows. Section 2 provides an overview of the proposed system for the speech separation challenge, for separating and recognizing two mixing sentences given single-channel data. Section 3 presents the details of the algorithms used to implement the system. Speech separation experiments are described in Section 4, followed by a summary in Section 5.

Overview of Proposed System

Fig. 1 illustrates the structure of the proposed system. The input speech waveform is divided into short-time frames, denoted by w t . Each w t is a mixed signal of target and masker, of unknown speaker identities and an unknown targetto-masker ratio. For convenience, we note the sentence with a higher energy ratio as the primary sentence, and the sentence with a lower energy ratio as the secondary sentence. Here, the energy ratio between the two sentences can be defined in the same way as conventional global signal-to-noise ratio, by treating one sentence as speech and the other as noise. Our system separates the two sentences in five steps, operating in sequence.

In Step 1, the system aims to identify the primary sentence by exploiting its higher energy ratio and hence potentially higher recognition accuracy. In the recognition, the lower energy secondary sentence is treated as noise. A speaker-dependent (SD) recognition system is used to select and recognize the primary sentence from the mixed signal. The SD system contains a set of acoustic HMMs for each speaker, with each HMM modeling an appropriate lexical unit for the speaker. For the speech separation challenge task, wholeword HMMs are used. Each HMM is a subband union model [START_REF] Ming | A posterior union model with applications to robust speech and speaker recognition[END_REF], which uses a missing-feature approach to improve the robustness to the crosstalk noise. It is assumed that the HMM sequence for the primary sentence, matching both the word sequence and speaker characteristics, is likely to produce maximum probability due to the higher energy ratio (and hence smaller distortion) of the primary sentence, and due to the speaker discrimination and improved noise robustness of the HMMs. Thus, the SD recognizer producing the highest probability HMM sequence is selected, with the HMM sequence defining a most-likely primary sentence. This approach is also applied to the situations in which the two mixed sentences have similar energy ratios. As observed in our experiments, when the global signal-to-noise ratio between the two sentences is 0 dB, the system chooses between the two sentences randomly, depending on which sentence produces a higher global probability.

In Step 2, the primary sentence recognized in Step 1 is reconstructed using an algorithm exploiting the most-likely state sequence of the sentence. The reconstructed speech process consists of short-time spectral estimates X<1> t , and the corresponding waveform estimate x<1> t . Here, we use the superscript < 1 > to note the variables associated with the primary sentence. Likewise, later we the superscript < 2 > to note the variables associated with the secondary sentence. The short-time spectral estimates X<1> t will be passed to

Step 3 for enhancing the secondary sentence, described below.

In Step 3, a Wiener filter is used to enhance the signal of the secondary sentence by filtering out the primary sentence from the mixed signal. The short-time spectral estimates for the primary sentence, produced in Step 2, are used in the operation. The operation takes the short-time spectra of the mixed signal W t as input, and generates enhanced short-time spectra Ŵ <2> t for the secondary sentence.

In Step 4, speech recognition is performed on the enhanced signal for the secondary sentence. A speaker-independent (SI) system is used for the recognition, which consists of an acoustic-linguistic HMM trained using data from all the speakers. The SI system is again a subband union model, for improving robustness to the residual noise in the enhanced signal. The use of an SI system in place of the SD system is found to be important for the recognitionfor greater robustness to the distorted speaker characteristics in the enhanced signal, caused by the Wiener filtering operation.

In Step 5, the secondary sentence recognized in Step 4 is reconstructed, using an algorithm similar to that for reconstructing the primary sentence.

Again, the reconstructed speech process consists of short-time spectral estimates X<2> t and the corresponding waveform estimate x<2> t . By this process, the system produces speech recognition results for both the primary and secondary sentences. The system therefore implements a "complete" separation process: taking the mixed speech waveform as input, and producing separated target and masker waveforms as output, along with the speech recognition results for both mixing sentences. In the following section we describe each component of the system in more detail.

More Details of Proposed System

Subband union model for recognition

The subband posterior union model [START_REF] Ming | A posterior union model with applications to robust speech and speaker recognition[END_REF]) is used to build both the SD and SI recognition components. As shown in Fig. 1, they have input y t and y <2> t , respectively, both representing a short-time feature vector consisting of subband features. The union model is a missingfeature approach, aiming to focus the recognition on uncorrupted subbands thereby improving the robustness to crosstalk interference and/or noise. Let y = {y(1), y(2), ..., y(B)} be a frame feature vector, consisting of B independent subbands y(b), subject to crosstalk and/or noise corruption. The union model is used to select the clean or usable subbands for recognition. Without assuming prior information about the corruption, the reliable subbands may be defined as the subbands that maximize the probability of the state for frame y. Denote by ŷ an estimate for the reliable subbands, which is a subset of y, then ŷ = arg max z⊂y p(s|z) [START_REF] Boll | Suppression of acoustic noise using spectral subtraction[END_REF] where p(s|z) is the probability of state s given subband feature set z. Using Bayes' Rules p(s|z) can be expressed as

p(s|z) = p(z|s)p(s) s p(z|s )p(s ) (2)
where p(z|s) is the state-conditioned probability of z, p(s) is a state prior, and the summation in the denominator is over all possible states for frame z. For clean-data trained HMMs, clean data are most likely to produce maximum probabilities for the correct states. Therefore, it is possible to find the clean or reliable subbands by selecting the subbands that maximize the probability of a potential state, as implemented in Eq. ( 1).

Searching for the optimal set of reliable subbands to maximize the state probability can be computationally expensive, of a complexity O(2 B ), for a system using a large number of subbands B. This problem can be relieved by replacing the probability p(ŷ|s), for the sought optimal set ŷ, with the probability of the union of all subsets in y of the same size as ŷ. Assuming that ŷ contains Q subbands, the union probability can be expressed as [START_REF] Ming | Robust speech recognition using probabilistic union models[END_REF] p(

z⊂y Q B z|s) ∝ z⊂y Q B p(z|s) ( 3 )
where y Q B is the collection of all subsets of Q subbands chosen from the full B subbands in y, and the proportionality is due to ignoring the joint probabilities between the different subsets. Since Eq.(3) contains marginal probabilities of all possible feature subsets, it contains the marginal probability of the optimal feature subset that can be assumed to dominate the sum because of the best data-model match. Therefore, Eq.( 3) can be used in place of p(ŷ|s) for maximum-probability based recognition. Note that the union probability is not a function of the identity of ŷ but only a function of the size of ŷ.

Therefore, substituting Eq. (3) into Eq. ( 2) for p(z|s), we reduce the problem of finding the optimal set of reliable subbands to finding the optimal number of reliable subbands, but not the exact set, resulting in a lower complexity O(B). This can be expressed as

Q = arg max Q p(s|Q, y) ( 4 )

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where, by definition,

p(s|Q, y) = z⊂y Q B p(z|s)p(s) s z⊂y Q B p(z|s )p(s ) (5)
As noted by us [START_REF] Jancovic | Combination of multiple feature streams for robust speech recognition[END_REF] and independently by [START_REF] Chan | Efficient computation of the frame-based extended union model and its application in speech recognition against partial temporal corruptions[END_REF], assuming independence between the subbands, an efficient, recursive algorithm exists for calculating the union probability Eq. ( 3) for Q from 1 to B.

The above model, named posterior union model, can be incorporated into an HMM by replacing the conventional state-emission probability with the state probability optimized for the number of reliable subbands, i.e., max Q p(s|Q, y) [START_REF] Ming | A posterior union model with applications to robust speech and speaker recognition[END_REF]. Operating on a frame-by-frame basis, the optimal subband selection offers robustness to nonstationary corruption.

HMM-based speech reconstruction

An algorithm is developed for reconstructing the short-time spectral sequences X<1> t by an inverse FFT, using the short-time phase P t from the mixed input signal w t .

Wiener filtering for speech enhancement

Given the estimate X<1> t for the primary sentence, we can obtain an estimate

Ŵ <2> t
for the secondary sentence by removing X<1> t from the mixed input W t , assuming all three quantities in the same short-time FFT magnitude format.

The enhanced signal Ŵ <2> t is then used as the input for the SI component for recognizing the secondary sentence. In the system, a Wiener filter is used for the enhancement:

Ŵ <2> t (f ) = H t (f )W t (f ).
The short-time filter function has a simple form:

H t (f ) = P Ŵ <2> t (f ) P Wt (f ) (8)
where P Wt (f ) is a smoothed periodogram of the mixed input signal w t , and

P Ŵ <2> t (f
) is a smoothed periodogram of the secondary sentence estimated using the following spectral subtraction

P Ŵ <2> t (f ) = P Wt (f ) -[g X<1> t (f )] 2 (9)
where [ X<1> t (f )] 2 is the codeword-based periodogram for the primary sentence treated as noise, and g is a gain factor for matching the gain of the codeword to the gain of the primary sentence in the mixed observation W t (f ).

In the system, g is decided on a sentence-by-sentence basis, by minimizing

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the sentence-level mean square error between X<1> t (f ) and W t (f ) over all periodogram bins and frames:

g = arg min g T t=1 f [W t (f ) -g X<1> t (f )] 2 (10)
Solving Eq. ( 10) results in

g = T t=1 f W t (f ) X<1> t (f ) T t=1 f [ X<1> t (f )] 2 (11)
It is assumed that P Ŵ <2> t (f ) = αP Wt (f ) if the subtraction in Eq. ( 9) results in a negative value, where α defines the maximum attenuation. An α = 0.3 is used in the system.

Experimental Results

Database and acoustic-linguistic modeling

The above system has been tested on the Speech Separation Challenge database [START_REF] Cooke | The 2006 speech separation challenge[END_REF], containing a two-talker speech recognition task.

The database consists of 34 speakers (16 female, 18 male). The sentences by each speaker have a command-like form, all of an identical grammatical structure: <command:4> <color:4> <preposition:4> <letter:25> <digit:10> <adverb:4>, where the number in the brackets indicates the number of choices at each point. Of the six words forming a sentence, the color, letter and number are defined as the keywords for recognition. For each speaker, 500 sentences are available for training. For testing, pairs of sentences, one being treated as "target" and the other being treated as "masker", are mixed at different target-to-masker ratios (TMRs) to form the test sentences. The database provides test data at 7 different TMRs: 6, 3, 0, -3, -6, -9 dB and clean, where

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"clean" corresponds to the test data without masker speech. Each test TMR condition contains 600 test sentences, of which, one third are masked by the same talker, one third are masked by talkers of the same gender, and the remaining are masked by talkers of different genders. By definition of the database, of the two mixing sentences forming a test case, one will contain the word "white". This is the target sentence. The recognition task is to identify the letter and number in the target sentence.

In our experiments, we use a 13-subband, 39-stream feature vector to represent each frame. This frame vector is derived from the output of a 27-channel melwarped filter bank detailed as follows. The speech signal, sampled at 25 kHz, is divided into frames of 20 ms at a frame period of 10 ms. Each frame is analyzed by a 512-point FFT, followed by a 27-channel mel-warped filter bank producing 27 log-scale energies. The 27 log filter-bank energies are then passed to a high-pass filter H(z) = 1z -1 for decorrelation [START_REF] Nadeu | On the decorrelation of the filter-bank energies in speech recognition[END_REF], resulting in 26 decorrelated log filter-bank energies (DLFBE).

The final frame feature vector, i.e., y t and y <2> t , is formed by grouping the 26 DLFBE uniformly into 13 subbands, with the addition of the first-order and second-order derivatives for each subband, resulting in a 13-subband, 39stream frame feature vector for modeling by the SD and SI union models for recognition. The 257 short-time FFT magnitudes derived from the FFT are used to form the codewords, associated with the states/mixture components of the SD/SI model, for speech reconstruction.

Each word is modeled by a 14-state left-to-right HMM without state skipping, with one mixture per state in the SD model (trained using speaker-specific data) and 32 mixtures per state in the SI model (trained using all speaker's data). Each mixture component is a Gaussian density with a diagonal covariance matrix. Both the SD and SI recognizers adopt a word bigram language model applied to the Viterbi algorithm for finding the most-likely state sequence given a test signal. The language models reflect the grammatical con-straint defined above for identifying the primary/secondary sentences; the SI recognizer is additionally subjected to a no-repetition constraint in identifying the secondary sentence, i.e., the keywords that have been recognized for the primary sentence are not assumed to occur again in the secondary sentence. This is indicated in Step 4, Fig. 1, as an additional input containing the disallowed primary keywords into the SI component. To cope with the condition that there may be only one sentence/speaker in the test signal, a silence state, trained using data without speech and allowed to have an unlimited number of self loops, is included in the SI model to absorb the signal from the Wiener filter with the only sentence being removed from the input signal.

In the following we describe two separation experiments. The first shows the system for recognizing the target sentence containing the specified keyword.

The second shows the system for recognizing and reconstructing both mixing sentences.

Recognizing target sentence

Of the two mixing sentences, the target sentence contains keyword "white".

The task is to recognize the remaining keywords, letter and number, in this target sentence. We achieve this by simultaneously identifying the target sentence and recognizing the target keywords using the system described in Fig. 1.

We run the recognition with two system configurations. In the first configuration, the language model for the SD recognition component forces the word "white" while the language model for the SI recognition component disallows the word "white". This produces two recognized sentences, with respective probability scores p SD(w) (for the primary sentence from the SD component with word "white"), and p SI(no w) (for the secondary sentence from the SI component without word white). In the second configuration, the language models for the SD and SI components are swaped, i.e., SD disallowing word "white"

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while SI forcing word "white". This produces two new recognized sentences, with respective probability scores p SD(no w) (for the primary sentence without word "white"), and p SI(w) (for the secondary sentence with word "white").

Then a decision is made to choose either the first or second configuration result as output dependent on which of the joint probabilities, p SD(w) p SI(no w)

or p SD(no w) p SI(w) , is greater. Table 1 presents the recognition results by the system.

Recognizing and reconstructing both mixing sentences

The following describes further experiments of using the proposed system to recognize and reconstruct both mixing sentences from the mixed signal.

The task is slightly different from the above task in that we do not aim a specific sentence; instead, we aim to recognize the keywords for each of the mixing sentences. In the experiments, we run the system only once for each mixed test signal, using the more "general" language models described in Section 4.1 without aiming a specific sentence. Also, we consider all three keywords, color/letter/number, in the recognition instead of two keywords in the above experiments. As described in Section 2, the proposed system is capable of producing both speech recognition results and reconstructed speech waveforms simultaneously for both mixing sentences.

For each test signal, the system produces two recognized sentences O 1 , O 2 . We compare these two recognized sentences with the transcripts of the two input sentences I 1 , I 2 . There are two possible matches between them:

(1) O 1 -I 1 , O 2 -I 2 , and (2) O 1 -I 2 , O 2 -I 1 .
The match producing a higher overall word accuracy rate is used to summarize the final results. Table 2 shows the word accuracy rates for color/letter/number in the target and masker sentences, respectively, produced by the system.

As indicated in Fig. 1, the proposed system uses the union model, Wiener filtering, and speaker-independent modeling for improved separation and recognition performance. To understand the contribution of each of these components, we rerun a set of experiments from using a basic system without these components, to using a refined system with these components added one after another, till the final system. Table 3 shows the improvement on the word accuracy rates for the target and masker sentences, averaged over all the talkers.

With reference to Fig. 1, the basic system uses the same speaker-dependent models (in Step 1) for recognizing both the primary and secondary sentences; it uses the full set of subband features for recognition and has no Wiener filter.

Therefore, it separates the two sentences by just disallowing repetition of the primary keywords in the secondary sentence. The use of union model for the speaker-dependent models allows the selection of optimal set of subbands for recognition, which reduces the crosstalk noise and offers improved recognition accuracy throughout all noisy conditions. The addition of Wiener filtering improves recognition accuracy for the sentences with low signal-to-noise ratios.

For example, it increase the accuracy rate for the target sentences at TMR = -9 dB, from 26.7% to 31.8%, and the accuracy rate for the masker sentences at TMR = 6 dB, from 38.2% to 42.1%. The improvement, however, is smaller for the sentences with high signal-to-noise ratios. This is because the filtering operation tends to alter the characteristics of the speaker while removing the crosstalk noise, thereby causing a mismatch between the speaker-dependent model and the filtered signal for recognition. Replacing the speaker-dependent model with a speaker-independent model may help reduce the mismatch and thereby gain more benefit from the noise reduction. This is evident in Table 3, which shows that the use of a union-based speaker-independent model improves the recognition performance for the filtered signals.

Finally, Fig. 2 shows an example of the reconstructed signals for the target and masker sentences, generated by the codeword-based algorithms described

  ; Lim & Oppenheim, 1979; McAulay & Malpass, 1980; Ephraim & Trees, 1995; Gannot, Burshtein & Weinstein, 1998; Jensen & Heusdens, 2007). Other existing single-channel techniques include optimal estimation, for example, minimum mean-square error or maximum a posteriori estimators (see, for example, Ephraim & Malah, 1984; Ephraim, 1992; Sameti et al., 1998; Lotter & Vary, 2005; Hendriks & Martin, 2007). Most techniques produce estimates of short-time speech spectra by filtering the noise components, based on knowledge about the statistics (e.g., power spectra, variances, or signal-to-noise ratio) of the noise and speech.

Fig. 1 .

 1 Fig. 1. Schematic diagram of the proposed system for speech separation.

Fig. 2 .

 2 Fig. 2. Separation and reconstruction of sentence t20-lwwd7n-m6-lrwe8a, TMR = 0 dB, DG. From top: mixed signal, reconstructed target sentence, reconstructed masker sentence.

  , the subband SD model produces an estimate for the primary speaker/sentence, which can be represented by m for the speaker and ŝt , t = 1, 2, ..., T , for the most-likely state sequence of the primary sentence spoken by the speaker. The m and ŝt can be used to retrieve a clean codeword sequence A m,ŝt , t = 1, 2, ..., T , for reconstructing the spectra and waveform of the primary sentence, thereby separating the sentence from the mixed signal. A m,ŝt represent the estimate of the short-time log FFT magnitude of the primary sentence. The corresponding waveform estimate, x<1> inverse FFT, assuming that the short-time phase can be approximated by the phase of the mixed frame, P t[START_REF] Lim | Enhancement and bandwidth compression of noisy speech[END_REF].The above method, modified slightly, can be applied within the SI recognition component for reconstructing the signal of the secondary sentence based on the SI recognition result. The difference is that in the SI model a codeword is estimated for each mixture component within each state, thereby obtaining a good resolution for reconstructing the speaker individualities. Denote by A k,i nent within the state: kt = arg max k,Q p(ŝ t , k|Q, y <2> ). Therefore a codeword sequence A kt,ŝt , t = 1, 2, ..., T , addressed jointly by the most-likely state sequence ŝt and most-likely mixture-component sequence kt , can be retrieved as an estimate for the short-time log FFT magnitudes of the secondary sentence:

	Let ln X<1>
	ln X<2>
	s ,k	z⊂(y <2> ) Q B z⊂(y <2> ) Q p(z|s, k)p(k|s)p(s) B p(z|s , k )p(k |s )p(s )	(7)
	where p(z|s, k) is the probability of feature set z on state s and mixture compo-
	nent k, p(k|s) is the mixture weight in state s, and p(s) is a prior probability of
	state s. Given the most-likely state ŝt for frame y <2> t	, the most-likely mixture
	component can be obtained by choosing the maximum-probability compo-

t , X<2> t , and waveforms x<1> t , x<2> t , of the primary and secondary sentences based on the recognition results from the SD and SI components (Step 2 and 5). In training the SD/SI subband HMMs, a prototype spectrum -suitable for speech reconstruction -is estimated for each HMM state or mixture component using the training data frames assigned to the state or mixture component. In the system, the average log FFT magnitude, taken over all the training frames within the state or mixture component, is used as the prototype spectrum (codeword). Consider the SD recognition component. Denote by A m,i the codeword for speaker m in state i. Given a mixed test sentence w t , t = 1, 2, ..., T t = t , can be obtained from X<1> t by an the codeword for mixture component k in state i. The maximization described in Section 3.1, for estimating the reliable subbands, can be moved inside the state and applied over the individual mixture components, to obtain a mostlikely mixture component for each given frame for reconstruction. Let y <2>

denote an input frame consisting of subband features for the SI HMM system.

The maximized state probability, used as the state-emission probability within the system, is defined as

max Q p(s|Q) = k max Q p(s,

k|Q, y <2> ) ( 6 ) where p(s, k|Q, y <2> ) is the union-based probability of state s and mixture component k given y <2> , defined similarly to Eq. (5) as p(s, k|Q, y <2> ) = t = A kt,ŝt . The corresponding waveform estimate x<2> t can be obtained from X<2>

Table 1

 1 Word accuracy rates (%) for recognizing the letter/number keywords in the target sentences containing keyword "white", for different target-to-masker ratios (TMRs), and for two mixing sentences from the same talker (ST), same gender (SG), and different gender (DG).

	TMR (dB)	ST	SG	DG Average
	clean				95.17
	6	73.08 85.75 86.75	81.42
	3	61.54 80.45 82.25	74.08
	0	52.49 65.36 72.75	63.08
	-3	46.15 56.42 62.75	54.75
	-6	38.24 41.89 49.25	43.00
	-9	32.81 31.56 38.00	34.17

Table 2

 2 Simultaneous recognition of both mixing sentences, showing the respective word accuracy rates for the target and masker sentences for the color/letter/number

	keywords.
	TMR			Target				Masker
	(dB)	ST	SG	DG Average	ST	SG	DG Average
	clean				96.94
	6	79.94 89.01 90.00	86.00	48.72 55.87 55.50	53.11
	3	70.44 83.79 87.17	80.00	55.35 67.04 69.83	63.67
	0	60.03 74.67 80.50	71.22	57.92 74.86 80.83	70.61
	-3	54.45 66.67 69.00	62.94	66.67 83.99 88.33	79.06
	-6	48.72 54.38 58.50	53.67	75.57 89.76 94.00	85.94
	-9	43.59 43.95 48.17	45.22	85.82 93.29 95.67	91.33

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in Section 3.2. More examples of the reconstructed signals in a WAV format can be found in (Ming, Hazen & Glass, 2007).

Summary

This paper described a system for the separation and recognition of two overlapped sentences, given only single-channel data. The system was built upon a combination of several different techniques, aiming to exploit simultaneously the speaker, energy ratio, language-model constraint, training data and acoustic model information, enhanced by the missing-feature theory for ignoring mismatches, to identify and separate the two mixing sentences. The system was tested on the two-talker database from the Speech Separation Challenge, and showed useful improvements. Some of the techniques used in the system were applied earlier to speaker verification [START_REF] Ming | A comparative study of methods for handheld speaker verification in realistic noisy conditions[END_REF].


J Ludo

MD Peter Vanopdenbosch

MD Jan W Dedeken

MD PhD Stephan Apa Casselman

MD. Vlaminck

Dr Ludo J Vanopdenbosch
email: 3250452359ludo.vanopdenbosch@azbrugge.be

Dr Tony Van Havenbergh

Kris De Smet

Thomas Somers

Dr Dimitri Hemelsoet

Prof Koen

Paemeleire

Peter Soors

MRI with intrathecal gadolinium to detect a CSF leak: a prospective open label cohort study

Keywords: intracranial hypotension, cerebrospinal fluid, magnetic resonance imaging

Objectives: The techniques currently used to detect a CSF leak are indium radio nucleotide scan or CT scan with intrathecal iodinated contrast agent. They have a low spatial and temporal resolution and are unpleasant for the patient. This open label prospective observational cohort study was designed to investigate the feasibility, success ratio, complications and therapeutic consequences of MRI with gadolinium administered by lumbar puncture to detect a CSF leak.

Methods: Patients were selected with either confirmed liquorrhoea, recurrent bacterial meningitis or symptoms and MRI findings of spontaneous intracranial hypotension. High resolution T1 weighted MRI with fat suppression of the spinal column at 1 hour and of the brain at 6 and 24 hours post injection of 0.5 ml of gadolinium were performed.

Results: 27 patients were included. The clinically suspected CSF leak was found in 6 of 8 patients with liquorrhoea, 3 of 5 patients with recurrent bacterial meningitis and 9 of 14 patients with spontaneous intracranial hypotension. The procedure was easy to perform and generally well tolerated. One patient developed streptococcal meningitis in the hours following the procedure but recovered completely with antibiotic treatment. 17 of 18 patients in whom a dural defect was found underwent surgery. All patients became symptom free after closure of the dural leak.

Conclusions: Spinal cord and brain MRI after intrathecal gadolinium injection is an easy to perform and accurate technique for detection of a dural defect with excellent anatomical detail.

Background

The diagnosis of a cerebrospinal fluid (CSF) leak can be suspected in patients with orthostatic headache or a sweet tasting clear nasal drip. Headache attributed to low CSF pressure is characterised by a dull headache that worsens within 15 minutes of sitting or standing and that can be associated with tinnitus, hyperacusis, neck stiffness, photophobia and nausea.

Nasal CSF drip can be confirmed with high specificity and sensitivity by beta-trace protein detection. Spinal CSF leaks can occur after lumbar puncture, complicating spinal surgery or epidural infiltrations. Cranial CSF leaks can be a complication of head trauma, surgery to the ear and nose or neurosurgical interventions. Idiopathic intracranial hypotension or spontaneous intracranial hypotension was recently reported to be caused by spinal epidural leaks. Cranial MRI with intravenous gadolinium can show meningeal thickening and contrast enhancement, downward displacement of the cerebellar tonsils, subdural collections, engorgement of veins and pituitary hyperaemia. The treatment of a CSF leak usually consists of pain medication and prolonged bed rest. A blood patch is successful in up to 75% of patients. (2) With a blood patch 20-50 ml autologous blood is injected in the lumbar epidural space. The mode of action is not known. Refractory spinal and cranial CSF leaks require surgery. The techniques currently used to detect and localise a dural defect with CSF leak are indium radio nucleotide scan or CT scan with intrathecal iodinated contrast agent. They have significant limitations in spatial and temporal resolution and important side effects with meningeal irritation, large volumes to be injected, imaging artefacts and irradiation dosage.

Intermittent leaks can be missed with both techniques. (2) MRI with intrathecal gadolinium was reported to be safe and a promising technique to detect CSF leaks. [START_REF] Zheng | Intrathecal gadolinium-enhanced MR myelography and cisternography: a pilot study in human patients[END_REF][START_REF] Tali | Intrathecal gadolinium (gadopentetate dimeglumine)-enhanced MR cisternography used to determine potential communication between the cerebrospinal fluid pathways and intracranial arachnoid cysts[END_REF][START_REF] Aydin | MRI cisternography with gadoliniumcontaining contrast medium: its role, advantages and limitations in the investigation of rhinorrhoea[END_REF] Intravenous administration of gadolinium is rarely complicated with nephrogenic systemic fibrosis. There is no information whether this could be an issue with intrathecal administration. The intravenous dosage usually is 20 ml, while only 0.5 ml is injected intrathecally.

Methods

We included patients 18 y or older with either nasal liquorrhea confirmed by beta-trace protein, recurrent bacterial meningitis or symptoms and MRI findings of spontaneous intracranial hypotension. When a spinal leak was suspected, a lumbar epidural blood patch was performed and if needed repeated up to 3 times. A lumbar puncture was performed at lumbar level L3-4 with a 22 Gauge atraumatic needle and 0.5 ml of gadolinium (Magnevist, Schering, Germany) was injected. The patient was positioned in Trendelenburg position for 30 minutes. From February 2005 till January 2006 MRI scanning was performed with a 1.5

Tesla Philips Achieva (Best, The Netherlands) and from then onwards with a 3 Tesla Philips Xseries Quasar Dual Achieva (Best, The Netherlands) . The full spine was scanned at 1 hour post injection with high resolution T1 with fat suppression with coronal and sagittal slices and axial slices through regions of interest. Thin T1 with fat suppression coronal slices of the head including all paranasal sinuses, the middle ear and mastoid bone were taken at 6 and 24 hours post injection. All scans were performed and reported by the same neuroradiologist (JWC). The local ethics committee reviewed and approved this study design. Written informed consent was obtained from all patients.

Results

Between February 2005 and January 2009, 27 patients were included: 8 with confirmed nasal CSF drip, 5 with recurrent bacterial meningitis and 14 with spontaneous intracranial hypotension (table 1). The average age was 45 (18-73), 18 patients were female, 9 male. In 6 out of 8 patients with confirmed nasal CSF drip a dural leak was detected. This leak was through the ethmoid bone in all, mainly at the olfactory recessus and lamina cribrosa. A CSF leak was found in 3 out of 5 patients with recurrent meningitis. Two of these leaks were through the ethmoid bone in close association with the olfactory bulb. One patient had a leak through the roof of the petrous bone into the middle ear. In 9 of 14 patients with spontaneous intracranial hypotension we found a CSF leak. All 9 were spinal epidural leaks mainly at the cervicodorsal junction (illustration 1). The intrathecal injection of 0.5 ml of Gadolinium by lumbar puncture was well tolerated. No new headaches occurred. We did not observe drowsiness, behavioural disturbances, attention deficit or epileptic seizures. One patient with liquorrhea developed meningitis 12 hours after intrathecal gadolinium injection presenting with rapidly worsening headache, fever and obtundation. CSF obtained within two hours of the first symptoms contained 29600 white blood cells /mm3, with a 93% neutrophilic formula. CSF lactate was 14.2 meq/l and glucose 0 mg/dl. Treatment with 2 g ceftriaxon and 1 g vancomycin intravenously was started promptly. Streptococcus Pneumoniae was cultured in CSF and the patient was treated for 14 days with IV ceftriaxon 2 g bid with full recovery. Documented nasal CSF leaks were closed by the ENT specialist with endoscopic nasal surgery using fibrin glue (Tissucol, Baxter Healthcare, Deerfield, IL, USA).

The spinal epidural CSF leaks but one were closed by the neurosurgeon. One spinal epidural leak resolved spontaneously. The neurosurgeon observed dural tears associated with minor osseous protrusions or a herniated thoracic disc. In some cases an artificial dural patch was applied. All patients who underwent surgery had a complete resolution of symptoms.

Discussion

This prospective observational cohort was set up to study the feasibility, success ratio, complications and therapeutic consequences of MRI with intrathecal gadolinium. We did not make a direct comparison with iodinated contrast CT myelography or cisternography with radiolabeled indium since it was found unethical to subject patients to three sequential intrathecal contrast injections. Positive predictive values and sensitivity can therefore not be calculated. MRI with intrathecal gadolinium proved to be easy to perform. The contrast ratio and detail of the scans was excellent with superior spatial and temporal resolution. The gadolinium remains in the cranial cavity over 24 hours permitting detection of intermittent leaks. We found a leak in 18 of 27 carefully selected patients (67%). The intrathecal injection of gadolinium was well tolerated. No behavioural disturbances, seizures or new headaches occurred. It must be noted that the majority of the patients had orthostatic headaches at presentation, worsening of headache was not reported. One patient developed bacterial meningitis in the hours after the injection, he was treated with antibiotics and made a full recovery. The cultured Streptococcus Pneumoniae suggests a nasal entry through the documented ethmoidal CSF leak rather than a complication of the intrathecal injection with gadolinium. 17 of 18 patients in whom a dural defect was found successfully underwent surgical closure of the defect and became symptom free. All patients with spontaneous intracranial hypotension in which a leak was found had the same MRI finding of spinal epidural collections (illustration 1). The gadolinium enhanced CSF leak through a dural defect was usually seen at the cervicodorsal junction in association with a small herniation of an intervertebral disc. Recent publications have reported similar findings. (2, 6, 7, 8, 9) In a minority of patients we could not detect a CSF leak which is also in concordance with other observations. (10) The fact that a spinal epidural leak being the cause of spontaneous intracranial hypotension was only reported in the last years makes us believe that MRI is superior to CT with intrathecal iodinated contrast or radiolabeled indium cisternography.

Illustration legend

Competing interests and funding

There are no competing interests. This study was not sponsored by industry, government or institutional funding. 2008;71:1902-1906

illustration 1 :

 1 spinal epidural CSF leak. Axial (A,B) and sagittal (C) 4 mm thick T1-weighted images of the thoracal spinal column with fat suppression following intra-thecal gadolinium injection. (A): A disc-osteophyte protrusion is causing a disruption of the anterior wall of the dural sac and gadolium is leaking through a defect in the dura mater (arrow) into the anterior epidural spaces on the left and right side.(B): At an adjacent level one can see another disc herniation, but the signal intensity of the intradural gadolinium is much higher than the gadolinium-CSF intensity in the surrounding epidural collection (arrow), indicating that the entry of gadolinium in the epidural space is not situated at this level. (C): On the sagittal image a dubbel fluid space can be seen anterior to the medulla: intrathecal gadolinium is separated from the gadolinium in the epidural collection by the dura mater (arrowheads).

9

  Watanabe A, Horikoshi T, Uchida M, Koizumi H, Yagashita T, Kinouchi H. Diagnostic value of spinal MR imaging in spontaneous intracranial hypotension syndrome. AJNR Am J Neuroradiol 2009;30:147-151 10 Leep Hunderfund AN, Mokri B. Orthostatic headache without CSF leak, Neurology

Acknowledgements My colleagues Dr. Olivier Deryck, Dr. Geert Vanhooren, Dr. Michel Van Zandijcke and Dr. Kristof Verhoeven for useful comments and correcting the manuscript. R. Chahal for language assistance. Patients were referred by: Dr. Patrick Staels, Department of Neurosurgery, Dr. Stephan Vlaminck and Dr. Rudi Kuhweide, Department of Nose-earthroat, Algemeen Ziekenhuis Sint Jan Brugge -Oostende, Brugge; Dr. Jean Delbecq, Department of Neurology, Regionaal Ziekenhuis Sint Maria, Halle; Dr. Mieke De Weweire,